ExplAIner: A Declarative Query Language for Explaining Classification Models

# ExplAIner: A Declarative Query Language for Explaining Classification Models
Source: [https://arxiv.org/html/2607.06407](https://arxiv.org/html/2607.06407)
,Pablo BarcelóPontificia Universidad Católica de ChileSantiagoChile,Diego BustamantePontificia Universidad Católica de ChileSantiagoChile,Jose CaraballPontificia Universidad Católica de ChileSantiagoChile,María Alejandra SchildPontificia Universidad Católica de ChileSantiagoChileandBernardo SubercaseauxCarnegie Mellon UniversityPittsburghPennsylvaniaUSA

###### Abstract\.

The XAI community has studied a wide range of queries and scores for explaining predictions of ML models\. From a data management perspective, this proliferation of explanation notions calls for declarative query languages in which such notions can be specified, combined, and analyzed uniformly\. In this paper, we develop such a framework for Boolean models\. We first revisitFOIL, an interpretability query language for black\-box models, and show that it has two fundamental limitations: it cannot express central optimality\-based explanation queries, and its evaluation problem over decision trees is hard for every level of the polynomial hierarchy\. We then introduceExplAIner, a query language based onFOILwith an extended vocabulary and a layered structure\. We show thatExplAInercan express a broad family of explanation notions, including abductive, contrastive, feature\-based, and distance\-based queries\. We also prove that the evaluation problem for each query inExplAInerbelongs to the Boolean hierarchy over every class of Boolean models for which some basic predicates can be evaluated in polynomial time\. In particular, that property holds for deterministic and decomposable Boolean circuits\. Finally, we introduceOpt\-FOIL, an optimization\-oriented fragment ofExplAInerfor computing explanations that are minimal with respect to strict partial orders, and prove that its evaluation problem is inFPNP\\mathrm\{FP\}^\{\\mathrm\{NP\}\}under the same tractability assumptions\. These complexity results have a direct algorithmic consequence: a fixed ExplAIner query can be evaluated with a fixed number of calls to a SAT solver, while a notion of explanation specified inOpt\-FOILcan be computed with a polynomial number of such calls\. This is particularly relevant in formal XAI, where SAT solvers have been successfully used to compute explanations for several classes of ML models\.

## 1\.Introduction

#### Explainability as a query\-language problem\.

The increasing use of machine learning \(ML\) models in decision\-making systems has created a pressing need for principled methods to understand the predictions produced by such models\. This need has led to a large body of work in explainable AI \(XAI\)\(Gunning and Aha,[2019](https://arxiv.org/html/2607.06407#bib.bib3); Guidottiet al\.,[2019](https://arxiv.org/html/2607.06407#bib.bib76); Arrietaet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib74); Molnar,[2022](https://arxiv.org/html/2607.06407#bib.bib56)\), and in particular to a variety of queries, scores, and explanation notions aimed at identifying why a model classifies a given input in a particular way\(Marques\-Silva,[2023](https://arxiv.org/html/2607.06407#bib.bib63),[2024](https://arxiv.org/html/2607.06407#bib.bib317); Darwiche,[2023](https://arxiv.org/html/2607.06407#bib.bib62)\)\. Examples include abductive explanations, contrastive explanations, counterfactual\-style queries, and feature\-necessity or feature\-relevance notions\(Ignatievet al\.,[2019](https://arxiv.org/html/2607.06407#bib.bib18); Darwiche and Hirth,[2020](https://arxiv.org/html/2607.06407#bib.bib66); Ribeiroet al\.,[2018](https://arxiv.org/html/2607.06407#bib.bib52); Huanget al\.,[2023](https://arxiv.org/html/2607.06407#bib.bib64)\)\.

From a data management perspective, this proliferation of explanation notions suggests a natural question: rather than designing a separate algorithm or formalism for each explanation task, can we develop a declarative language in which users specify what explanation they are looking for? This is in line with a long tradition in databases: complex computational tasks are exposed through query languages with well\-defined syntax and semantics, while the study of their expressive power and evaluation complexity provides a principled understanding of what can be asked and how hard it is to answer\(Abiteboulet al\.,[1995](https://arxiv.org/html/2607.06407#bib.bib114); Kanellakis,[1990](https://arxiv.org/html/2607.06407#bib.bib78); Vardi,[1982](https://arxiv.org/html/2607.06407#bib.bib240); Papadimitriou and Yannakakis,[1999](https://arxiv.org/html/2607.06407#bib.bib253)\)\. In this view, a model becomes an object over which one poses queries, and explanation notions become fixed queries evaluated over that object\(Arenaset al\.,[2021a](https://arxiv.org/html/2607.06407#bib.bib10); Arenas,[2024](https://arxiv.org/html/2607.06407#bib.bib1)\)\.

This perspective has several advantages\. First, it provides a uniform framework for comparing and combining explanation notions\. This is important because there is no single explanation concept that is best suited for all users, models, or applications; in many cases, the most informative explanation is obtained by combining several criteria\(Doshi\-Velez and Kim,[2017](https://arxiv.org/html/2607.06407#bib.bib53); Marques\-Silva and Ignatiev,[2023](https://arxiv.org/html/2607.06407#bib.bib72)\)\. Second, it makes it possible to study explainability through standard database\-theoretic lenses, such as expressiveness and evaluation complexity\(Abiteboulet al\.,[1995](https://arxiv.org/html/2607.06407#bib.bib114); Vardi,[1982](https://arxiv.org/html/2607.06407#bib.bib240); Libkin,[2004](https://arxiv.org/html/2607.06407#bib.bib8)\)\. Third, it opens the door to the development of general optimization techniques for the operators of a query language for explainability\. Such techniques can reduce the evaluation time of several explainability queries simultaneously, rather than treating each query in isolation\.

A central issue in such a framework is how to measure the complexity of query evaluation\. Since an explanation notion is intended to be specified by a fixed formula of the language, the appropriate measure is*data complexity*: the query is fixed, while the input consists of the model representation and the instance to be explained\(Vardi,[1982](https://arxiv.org/html/2607.06407#bib.bib240)\)\. From this perspective, polynomial\-time data complexity is desirable, but it is not the only meaningful tractability target\. Many explanation tasks are inherently computationally demanding\(Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41); Wäldchenet al\.,[2021](https://arxiv.org/html/2607.06407#bib.bib13)\), and therefore a useful explainability language should also allow controlled forms of non\-polynomial complexity\. In particular, data complexity inPNP\\mathrm\{P\}^\{\\mathrm\{NP\}\}remains a reasonable and useful target: it corresponds to computation with a polynomial number of calls to an NP oracle, and it is compatible with the use of SAT solvers as evaluation engines\. This complexity level is especially appropriate in our setting because the inputs are model representations, not database instances in the traditional sense\. In contrast with large relational databases, the tree\-based models commonly considered in formal explainability are often of moderate size, and SAT\-based methods have been successfully used to compute explanations for such models\(Ignatiev and Silva,[2021](https://arxiv.org/html/2607.06407#bib.bib46); Izza and Marques\-Silva,[2021](https://arxiv.org/html/2607.06407#bib.bib48); Yuet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib47)\)\. Thus, in this paper we regard polynomial time andPNP\\mathrm\{P\}^\{\\mathrm\{NP\}\}as desirable data\-complexity bounds for an explainability query language\.

#### Toward declarative languages for model interpretability\.

A first step in this direction was taken by Arenas et al\.\(Arenaset al\.,[2021a](https://arxiv.org/html/2607.06407#bib.bib10)\), who introduced FOIL, a first\-order interpretability logic for querying ML models\. FOIL is model\-agnostic: it treats a model as a black box and provides access to its positive instances together with the natural subsumption relation over partial instances\. This simple design makes FOIL an appealing foundational language, and it is expressive enough to capture several basic explanation notions\.

However, model\-agnosticism also has limitations\. If the language is too weak, it cannot express explanation concepts that are central in practice\. If it is too unconstrained, its evaluation problem may become too complex to support query evaluation in the sense expected from a database\-oriented framework\. Thus, the challenge is to design a language that balances two requirements\. On the one hand, it should be expressive enough to capture a broad family of explanation queries, including optimality\-based notions such as minimum or maximum explanations\. On the other hand, it should have well\-behaved evaluation and computation problems over relevant classes of Boolean models\.

In this paper, we address this challenge by developing a declarative framework for explaining Boolean models\. Our setting is not tied to decision trees\. Instead, we consider models abstractly as Boolean functions, while also studying concrete representation classes such as deterministic and decomposable Boolean circuits and decision trees\. This allows us to separate the logical specification of explanation queries from the representation\-dependent complexity of evaluating them\. Such a separation is particularly natural from a database perspective, where query specification and query evaluation over different representation classes are treated as distinct but connected problems\.

#### The limitations of FOIL

We begin by revisiting FOIL from the perspective of query\-language design\. We show that, despite its foundational role, FOIL does not satisfy the requirements above\. First, FOIL lacks the expressive power needed to capture some natural optimality\-based explanation notions\. In particular, we prove that minimum abductive explanations cannot be expressed in FOIL, even when the underlying model is restricted to be a decision tree\. This shows that the limitation is not caused by the use of complex model classes, but by the expressive resources of the language itself\. Second, FOIL has high evaluation complexity\. We prove that, for every level of the polynomial hierarchy, there is a fixed FOIL formula whose evaluation problem over decision trees is hard for that level\. Thus, even on a class of models traditionally regarded as interpretable, unrestricted FOIL does not provide the kind of controlled data complexity that one would expect from a practical declarative language for explanations\. This complements earlier complexity\-theoretic approaches to model interpretability, which study the difficulty of answering explanation queries over different model classes\(Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41); Wäldchenet al\.,[2021](https://arxiv.org/html/2607.06407#bib.bib13)\)\.

#### ExplAIner: a tractable logic for explanation queries\.

Motivated by these limitations, we introduceExplAIner, a first\-order logic designed to express explanation queries over Boolean models while retaining controlled evaluation complexity\. The language extends the basic FOIL vocabulary with a relation that compares partial instances according to the number of defined features\. This addition is essential for expressing optimality conditions based on cardinality, such as minimum abductive explanations\(Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41); Darwiche and Hirth,[2020](https://arxiv.org/html/2607.06407#bib.bib66)\)and maximum contrastive explanations\(Ignatievet al\.,[2019](https://arxiv.org/html/2607.06407#bib.bib18); Huanget al\.,[2023](https://arxiv.org/html/2607.06407#bib.bib64)\)\.

ExplAIneris organized in layers\. Its atomic layer captures structural properties of partial instances; its quantified layer allows formulas to refer to the behavior of the model by combining formulas from the atomic layer with the predicates𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}, which express whether all completions of a partial instance are classified positively or negatively; and its topmost layer permits Boolean combinations of explanation properties\. This organization is designed to provide enough expressive power for explanation tasks while keeping evaluation under control\.

We show thatExplAInercan express the explanation notions studied in this paper, including weak abductive explanations, abductive explanations, and minimum abductive explanations\(Ignatievet al\.,[2019](https://arxiv.org/html/2607.06407#bib.bib18); Darwiche and Hirth,[2020](https://arxiv.org/html/2607.06407#bib.bib66); Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41); Audemardet al\.,[2022a](https://arxiv.org/html/2607.06407#bib.bib314)\); weak contrastive explanations, contrastive explanations, and maximum contrastive explanations\(Ignatievet al\.,[2019](https://arxiv.org/html/2607.06407#bib.bib18); Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41); Darwiche,[2023](https://arxiv.org/html/2607.06407#bib.bib62)\); minimum change required and maximum change allowed\(Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41)\); and necessary features and relevant features\(Izzaet al\.,[2021](https://arxiv.org/html/2607.06407#bib.bib14); Huanget al\.,[2023](https://arxiv.org/html/2607.06407#bib.bib64)\)\. At the same time, we prove that the evaluation problem forExplAInerbelongs to the Boolean hierarchy over every class of models for which the basic𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}checks are tractable\. This includes decision trees and also richer representation classes with suitable tractability properties, such as deterministic and decomposable Boolean circuits\(Darwiche and Marquis,[2011](https://arxiv.org/html/2607.06407#bib.bib24); Arenaset al\.,[2021b](https://arxiv.org/html/2607.06407#bib.bib9)\)\.

#### Opt\-FOIL: computing explanations\.

Evaluation is only one part of the problem\. A language for explainability should also support the computation of explanations\. For this reason, we introduceOpt\-FOIL, an optimization\-oriented fragment built from the quantified layer ofExplAInertogether with a minimization operator over definable strict partial orders\.

Opt\-FOILcaptures explanation tasks in which one seeks an object satisfying a logical specification and minimal with respect to a user\-defined preference order\. This includes standard subset\-minimal explanations, cardinality\-minimum explanations, and distance\-based notions such as minimum change required\. By changing the order, the same formalism can also express maximality\-based notions, such as maximum contrastive explanations and maximum change allowed\.

Our main computational result forOpt\-FOILis that its computation problem belongs toFPNP\\mathrm\{FP\}^\{\\mathrm\{NP\}\}over every class of models for which𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}can be evaluated in polynomial time, whereFP\\mathrm\{FP\}is the class of functions that can be computed in polynomial time\. In the terminology above, this means that computing explanations specified inOpt\-FOILhas controlled data complexity: the formula is fixed, and the cost is measured as a function of the model representation and the input instance\. This placesOpt\-FOILwithin the complexity regime identified above as suitable for declarative explainability languages, while allowing the language to capture optimization\-based explanation tasks that are unlikely to admit polynomial\-time algorithms in full generality\.

#### Technical contributions\.

The following are the technical contributions of the paper\.

- •We prove two limitations of FOIL over decision trees\. On the expressiveness side, we show that no FOIL formula can define the minimum abductive explanation query\. On the complexity side, we prove that for every levelΣkP\\Sigma\_\{k\}^\{P\}of the polynomial hierarchy, there is a fixed FOIL formula whose evaluation problem isΣkP\\Sigma\_\{k\}^\{P\}\-hard\.
- •We introduceExplAIner, a logic based onFOILwith an extended vocabulary and a layered structure: the atomic layer, the quantified layer, and the fullExplAInerlayer\. The vocabulary ofExplAInerconsists of the predicates⊆\\subseteq,⪯\\preceq,𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}, and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}, where⊆\\subseteqis the subsumption relation on partial instances and⪯\\preceqcompares partial instances by their number of defined features\. We show that both⊆\\subseteqand⪯\\preceqare necessary by proving that neither relation is first\-order definable from the other over decision trees\.
- •We show thatExplAIneris expressive enough to encode the explanation queries considered in the paper, including weak abductive explanations, subset\-minimal abductive explanations, cardinality\-minimum abductive explanations, weak and maximal contrastive explanations, minimum change required, maximum change allowed, necessary features, and relevant features\. Each of these notions is expressed by a fixed query inExplAIner\.
- •We establish complexity bounds for the three layers ofExplAIner\. First, we show that the evaluation problem for queries in the atomic layer can be solved in polynomial time over every class of Boolean models\. Second, for every class of Boolean models over which the predicates𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}can be decided in polynomial time, we show that the evaluation problem for queries in the quantified layer is inNP\{\\rm NP\}\. Moreover, we show that there exists a query in the quantified layer whose evaluation problem isNP\{\\rm NP\}\-complete over the class of decision trees\. Third, for every class of Boolean models over which𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}can be decided in polynomial time, we show that the evaluation problem forExplAInerqueries is in the Boolean hierarchy; equivalently, such queries can be evaluated with a fixed number of calls to anNP\{\\rm NP\}oracle\. Furthermore, we show that for every levelBHk\{\\rm BH\}\_\{k\}of the Boolean hierarchy, there exists anExplAInerquery whose evaluation problem isBHk\{\\rm BH\}\_\{k\}\-hard over the class of decision trees\. Importantly, the assumption that the predicates𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}can be decided in polynomial time is not specific to decision trees\. This condition also holds for more general classes of Boolean models, including deterministic and decomposable Boolean circuits\. Thus, the upper bounds above apply beyond tree\-based representations and cover circuit classes that are central in knowledge compilation\(Darwiche and Marquis,[2011](https://arxiv.org/html/2607.06407#bib.bib24)\)\.
- •We defineOpt\-FOILas an optimization\-oriented fragment ofExplAIner, based on the quantified layer together with a minimization operator over strict partial orders\. For every class of Boolean models over which the predicates𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}can be decided in polynomial time, we show that the computation problem forOpt\-FOILqueries is inFPNP\\mathrm\{FP\}^\{\\mathrm\{NP\}\}\. Hence, explanations specified inOpt\-FOILcan be computed with a polynomial number of calls to anNP\{\\rm NP\}oracle\.
- •As a result of independent interest, we use Presburger arithmetic to show that the problem of verifying whether a sentence over the atomic layer ofExplAIneris valid is decidable\. This result is needed to provide an effective syntax forOpt\-FOIL, sinceOpt\-FOILrequires verifying that a sentence in the atomic layer defines a strict partial order\.
- •Finally, we show that, under standard complexity\-theoretic assumptions,Opt\-FOILis strictly contained inExplAIner, andExplAIneris strictly contained inFOILover the extended vocabulary consisting of the predicates⊆\\subseteq,⪯\\preceq,𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}, and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}\.

#### Organization of the paper\.

The remainder of the paper is organized as follows\. Section[2](https://arxiv.org/html/2607.06407#S2)introduces the basic notions used throughout the paper, including Boolean models, partial instances, model representations, and the explanation queries studied in our framework\. Section[3](https://arxiv.org/html/2607.06407#S3)revisitsFOILand establishes its limitations in terms of expressiveness and evaluation complexity\. Section[4](https://arxiv.org/html/2607.06407#S4)introducesExplAInerand proves its main expressiveness and evaluation results\. Section[5](https://arxiv.org/html/2607.06407#S5)presentsOpt\-FOIL, our optimization\-oriented language for computing explanations, together with its computational guarantees\. Section[6](https://arxiv.org/html/2607.06407#S6)presents concluding remarks and directions for future work\. Finally, the appendix contains supplementary material, including technical proofs that are deferred for readability\.

## 2\.Background

We begin by introducing the main components of our framework, followed by a review of various explainability queries that will be addressed in the subsequent sections\.

### 2\.1\.Models and instances

We use an abstract notion of a model of dimensionnn, and define it as a Boolean functionℳ:\{0,1\}n→\{0,1\}\\mathcal\{M\}:\\\{0,1\\\}^\{n\}\\to\\\{0,1\\\}\.111We focus on Boolean models, as is common in formal XAI research\(Wäldchenet al\.,[2021](https://arxiv.org/html/2607.06407#bib.bib13); Audemardet al\.,[2022b](https://arxiv.org/html/2607.06407#bib.bib280); Cabodiet al\.,[2024](https://arxiv.org/html/2607.06407#bib.bib61)\)\.We writed​i​m​\(ℳ\)\{\\textit\{d\}im\}\(\\mathcal\{M\}\)for the dimension of a modelℳ\\mathcal\{M\}\. Apartial instanceof dimensionnnis a tuple𝐞∈\{0,1,⊥\}n\\mathbf\{e\}\\in\\\{0,1,\\bot\\\}^\{n\}, where⊥\\botis used to represent undefined features\. We define𝐞⊥=\{i∈\{1,…,n\}∣𝐞​\[i\]=⊥\}\\mathbf\{e\}\_\{\\bot\}=\\\{i\\in\\\{1,\\dots,n\\\}\\mid\\mathbf\{e\}\[i\]=\\bot\\\}\. Aninstanceof dimensionnnis a tuple𝐞∈\{0,1\}n\\mathbf\{e\}\\in\\\{0,1\\\}^\{n\}, that is, a partial instance without undefined features\. For every instance𝐞\\mathbf\{e\}of dimensiond​i​m​\(ℳ\)\{\\textit\{d\}im\}\(\\mathcal\{M\}\), we writeℳ​\(𝐞\)\\mathcal\{M\}\(\\mathbf\{e\}\)for the value assigned byℳ\\mathcal\{M\}to𝐞\\mathbf\{e\}\.

Given two partial instances𝐞\\mathbf\{e\},𝐞′\\mathbf\{e\}^\{\\prime\}of dimensionnn, we write𝐞⊆𝐞′\\mathbf\{e\}\\subseteq\\mathbf\{e\}^\{\\prime\}and say that𝐞\\mathbf\{e\}issubsumedby𝐞′\\mathbf\{e\}^\{\\prime\}if and only if, for everyi∈\{1,…,n\}i\\in\\\{1,\\ldots,n\\\}, whenever𝐞​\[i\]≠⊥\\mathbf\{e\}\[i\]\\neq\\bot, we have𝐞​\[i\]=𝐞′​\[i\]\\mathbf\{e\}\[i\]=\\mathbf\{e\}^\{\\prime\}\[i\]\. In other words,𝐞′\\mathbf\{e\}^\{\\prime\}can be obtained from𝐞\\mathbf\{e\}by replacing some occurrences of⊥\\botby Boolean values\. For example,\(1,⊥\)\(1,\\bot\)is subsumed by\(1,0\)\(1,0\), but it is not subsumed by\(0,0\)\(0,0\)\. A partial instance𝐞\\mathbf\{e\}can be seen as a compact representation of the set of instances𝐞′\\mathbf\{e\}^\{\\prime\}such that𝐞\\mathbf\{e\}is subsumed by𝐞′\\mathbf\{e\}^\{\\prime\}\. Such instances𝐞′\\mathbf\{e\}^\{\\prime\}are called thecompletionsof𝐞\\mathbf\{e\}and the set of all of them is denoted bycomp​\(𝐞\)\\textit\{comp\}\(\\mathbf\{e\}\)\.

For eachnn, partial instances of dimensionnnare partitioned inton\+1n\+1levels: for eachi∈\{0,…,n\}i\\in\\\{0,\\ldots,n\\\}, leveliiconsists of all partial instances with exactlyiidefined features\. Given partial instances𝐞\\mathbf\{e\},𝐞′\\mathbf\{e\}^\{\\prime\}, we write𝐞⪯𝐞′\\mathbf\{e\}\\preceq\\mathbf\{e\}^\{\\prime\}and say that𝐞\\mathbf\{e\}is on a*less or equal level*than𝐞′\\mathbf\{e\}^\{\\prime\}if and only if\|𝐞⊥\|≥\|𝐞⊥′\|\|\\mathbf\{e\}\_\{\\bot\}\|\\geq\|\\mathbf\{e\}^\{\\prime\}\_\{\\bot\}\|\. In other words,𝐞′\\mathbf\{e\}^\{\\prime\}has at least as many defined features as𝐞\\mathbf\{e\}\.

We will also use the symbols⊂\\subsetand≺\\precfor the corresponding strict relations\.

In several proofs, we write𝐞⋅𝐞′\\mathbf\{e\}\\cdot\\mathbf\{e\}^\{\\prime\}for the concatenation of both instances\. Moreover, for a values∈\{0,1,⊥\}s\\in\\\{0,1,\\bot\\\}, we denote by\{s\}n\\\{s\\\}^\{n\}the partial instance of dimensionnnwhose entries are all equal toss\.

We next introduce several classes of Boolean functions that will be used throughout the paper\.

#### Boolean Circuits\.

A*Boolean circuit*of dimensionnnis a directed acyclic graph over a set of variables\{x1,…,xn\}\\\{x\_\{1\},\.\.\.,x\_\{n\}\\\}such that:

1. \(i\)Every node without incoming edges is either avariable gateor aconstant gate\. A variable gate is labeled with a variable, and a constant gate is labeled with either0or11;
2. \(ii\)Every node with incoming edges is alogic gate, and is labeled with a symbol∧\\land,∨\\loror¬\\lnot\. If it is labeled with the symbol¬\\lnot, then it has exactly one incoming edge;
3. \(iii\)Exactly one node does not have any outgoing edges, and this node is called theoutput gate\.

Given a Boolean circuitCCand an instance𝐞\\mathbf\{e\}of dimensionnn, the valueC​\(𝐞\)C\(\\mathbf\{e\}\)is defined as the value of the output gate ofCCwhen we evaluateCCon input𝐞\\mathbf\{e\}\. Note that we are identifying inputs of the circuit as instances of the Boolean model\.

Several restrictions of Boolean circuits with good computational properties have been studied\.

#### Negation Normal Form\.

A*negation normal form*\(𝖭𝖭𝖥\\mathsf\{NNF\}\) circuit of dimensionnnis a Boolean circuit of dimensionnnsuch that the incoming edge of every negation gate comes from a variable gate\.

#### Determinism and decomposability\.

LetXXbe a set of variables, letCCbe a circuit overXX, and letggbe a gate ofCC\. We defineCgC\_\{g\}to be the Boolean circuit overXXinduced by the set of gatesg′g^\{\\prime\}ofCCfor which there exists a directed path fromg′g^\{\\prime\}togginCC\. Note thatggis the output gate ofCgC\_\{g\}\. An∨\\lor\-gateggofCCis said to be*deterministic*if, for every pairg1g\_\{1\},g2g\_\{2\}of distinct input gates ofgg, there is no instance𝐞\\mathbf\{e\}such thatCg1​\(𝐞\)=Cg2​\(𝐞\)=1C\_\{g\_\{1\}\}\(\\mathbf\{e\}\)=C\_\{g\_\{2\}\}\(\\mathbf\{e\}\)=1\. The circuitCCis called*deterministic*if every∨\\lor\-gate ofCCis deterministic\. For every gateggofCC, define𝗏𝖺𝗋​\(g\)\\mathsf\{var\}\(g\)as the set of variablesx∈Xx\\in Xsuch that there exists a variable gate labeled byxxinCgC\_\{g\}\. An∧\\land\-gateggofCCis said to be*decomposable*if for every pairg1g\_\{1\},g2g\_\{2\}of distinct input gates ofgg, we have𝗏𝖺𝗋​\(g1\)∩𝗏𝖺𝗋​\(g2\)=∅\\mathsf\{var\}\(g\_\{1\}\)\\cap\\mathsf\{var\}\(g\_\{2\}\)=\\emptyset\. The circuitCCis called*decomposable*if every∧\\land\-gate ofCCis decomposable\.

x2x\_\{2\}x3x\_\{3\}¬\\negx4x\_\{4\}∧\\land∨\\lor∧\\landx1x\_\{1\}Figure 1\.A𝖽​\-​𝖣𝖭𝖭𝖥\\mathsf\{d\}\\text\{\-\}\\mathsf\{DNNF\}circuit of dimension 4\.Example of a four\-variable d\-DNNF circuit\.The figure shows a four\-variable circuit\. The output is an and gate with two inputs: variable x1 and an or gate\. The or gate has two inputs\. One input is variable x2\. The other is an and gate whose three inputs are not x2, x3, and x4\.

#### Deterministic Decomposable Negation Normal Form\.

A*deterministic decomposable negation normal form*\(𝖽​\-​𝖣𝖭𝖭𝖥\\mathsf\{d\}\\text\{\-\}\\mathsf\{DNNF\}\) circuit of dimensionnnis an𝖭𝖭𝖥\\mathsf\{NNF\}circuit of dimensionnnthat is both deterministic and decomposable\. An example is shown in[Figure 1](https://arxiv.org/html/2607.06407#S2.F1)\.

#### Binary Decision Diagram\.

A*binary decision diagram*\(𝖡𝖣𝖣\\mathsf\{BDD\}\) of dimensionnnis a directed acyclic graph with a unique root, and whose nodes and edges are labeled as follows: \(i\) every leaf is labeled by𝐭𝐫𝐮𝐞\\mathbf\{true\}or𝐟𝐚𝐥𝐬𝐞\\mathbf\{false\}and \(ii\) every non\-leaf node is labeled by a feature in\{1,…,n\}\\\{1,\.\.\.,n\\\}and has exactly two outgoing edges, one labeled by0and the other by11\.

LetBBbe a binary decision diagram and let𝐞\\mathbf\{e\}be an instance of dimensionnn\. The valueB​\(𝐞\)B\(\\mathbf\{e\}\)is defined as the Boolean value of the leaf obtained by starting at the root and following the path such that, at each non\-leaf node labeled byii, the outgoing edge labeled by𝐞​\[i\]\\mathbf\{e\}\[i\]is chosen\.

#### Decision Trees\.

A*decision tree*\(𝖣𝖳𝗋𝖾𝖾\\mathsf\{DTree\}\) over instances of dimensionnnis a binary decision diagram of dimensionnnsuch that \(i\) its underlying graph is a tree and \(ii\) no feature appears more than once on any root\-to\-leaf path\. An example is shown in[Figure 2](https://arxiv.org/html/2607.06407#S2.F2)\.

𝐭𝐫𝐮𝐞\\mathbf\{true\}𝐟𝐚𝐥𝐬𝐞\\mathbf\{false\}𝐭𝐫𝐮𝐞\\mathbf\{true\}𝐟𝐚𝐥𝐬𝐞\\mathbf\{false\}x4x\_\{4\}x4x\_\{4\}x3x\_\{3\}x2x\_\{2\}x1x\_\{1\}𝐟𝐚𝐥𝐬𝐞\\mathbf\{false\}𝐭𝐫𝐮𝐞\\mathbf\{true\}011011011011011Figure 2\.A decision tree of dimension 4\. For the sake of readability, we label nodes byxix\_\{i\}rather than by indices\.Example of a decision tree of dimension 4\.The figure shows a decision tree with root x2\. If x2 equals 0, the tree moves to x1\. From x1, edge 0 leads to a false leaf and edge 1 leads to a true leaf\. If x2 equals 1, the tree moves to x3\. Both outgoing edges of x3 lead to a node labeled x4\. In each x4 node, edge 0 leads to a true leaf and edge 1 leads to a false leaf\.

### 2\.2\.Explainability queries

We now define the explainability queries studied in this work\.

#### Weak Abductive Explanation\.

Given an instance𝐞\\mathbf\{e\}and a modelℳ\\mathcal\{M\}, a partial instance𝐞1\\mathbf\{e\}\_\{1\}is a*weak abductive explanation*\(𝗐𝖠𝖷𝗉\\mathsf\{wAXp\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}if𝐞1⊆𝐞\\mathbf\{e\}\_\{1\}\\subseteq\\mathbf\{e\}and, for every𝐞2∈comp​\(𝐞1\)\\mathbf\{e\}\_\{2\}\\in\\textit\{comp\}\(\\mathbf\{e\}\_\{1\}\), the conditionℳ​\(𝐞\)=ℳ​\(𝐞2\)\\mathcal\{M\}\(\\mathbf\{e\}\)=\\mathcal\{M\}\(\\mathbf\{e\}\_\{2\}\)holds\(Huanget al\.,[2023](https://arxiv.org/html/2607.06407#bib.bib64)\)\. This notion is also known as*sufficient reason*in the literature\(Arenaset al\.,[2024](https://arxiv.org/html/2607.06407#bib.bib316)\)\. For example, in[Figure 1](https://arxiv.org/html/2607.06407#S2.F1),\(1,1,1,⊥\)\(1,1,1,\\bot\)is a weak abductive explanation for the instance\(1,1,1,1\)\(1,1,1,1\)\.

#### Abductive Explanation\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), a partial instance𝐞1\\mathbf\{e\}\_\{1\}is an*abductive explanation*\(𝖠𝖷𝗉\\mathsf\{AXp\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}if it is a weak abductive explanation for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}and there is no weak abductive explanation𝐞2\\mathbf\{e\}\_\{2\}such that𝐞2⊂𝐞1\\mathbf\{e\}\_\{2\}\\subset\\mathbf\{e\}\_\{1\}\(Ignatievet al\.,[2019](https://arxiv.org/html/2607.06407#bib.bib18)\)\. This notion of explanation has been extensively studied and it can be found in the literature under names such as*sufficient reason*\(Lindner and Möllney,[2019](https://arxiv.org/html/2607.06407#bib.bib315); Darwiche and Hirth,[2020](https://arxiv.org/html/2607.06407#bib.bib66)\),*prime implicant*\(Shihet al\.,[2018](https://arxiv.org/html/2607.06407#bib.bib23)\), and*minimal sufficient reason*\(Arenaset al\.,[2024](https://arxiv.org/html/2607.06407#bib.bib316)\)\. Modeling abduction using propositional logic or first\-order logic, and the complexity of computing such an explanation has been studied for many decades\(Marquis,[1991](https://arxiv.org/html/2607.06407#bib.bib19)\)\. In[Figure 1](https://arxiv.org/html/2607.06407#S2.F1),\(1,1,1,⊥\)\(1,1,1,\\bot\)is a weak abductive explanation for\(1,1,1,1\)\(1,1,1,1\), but not an abductive explanation\. By contrast, one can check that\(1,1,⊥,⊥\)\(1,1,\\bot,\\bot\)is indeed an abductive explanation\.

#### Minimum Abductive Explanation\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), a partial instance𝐞1\\mathbf\{e\}\_\{1\}is a*minimum abductive explanation*\(𝗆𝖠𝖷𝗉\\mathsf\{mAXp\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}if it is a weak abductive explanation for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}and there is no weak abductive explanation𝐞2\\mathbf\{e\}\_\{2\}such that𝐞2≺𝐞1\\mathbf\{e\}\_\{2\}\\prec\\mathbf\{e\}\_\{1\}\. Our definition is based on the*minimum sufficient reason*explainability query by\(Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41)\)and\(Arenaset al\.,[2024](https://arxiv.org/html/2607.06407#bib.bib316)\)\. In[Figure 1](https://arxiv.org/html/2607.06407#S2.F1),\(1,1,⊥,⊥\)\(1,1,\\bot,\\bot\)is also a minimum abductive explanation for\(1,1,1,1\)\(1,1,1,1\)\.

#### Weak Contrastive Explanation\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), the partial instance𝐞1\\mathbf\{e\}\_\{1\}is a*weak contrastive explanation*\(𝗐𝖢𝖷𝗉\\mathsf\{wCXp\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}if𝐞1⊆𝐞\\mathbf\{e\}\_\{1\}\\subseteq\\mathbf\{e\}and there is an instance𝐞2∈comp​\(𝐞1\)\\mathbf\{e\}\_\{2\}\\in\\textit\{comp\}\(\\mathbf\{e\}\_\{1\}\)such that the conditionℳ​\(𝐞\)≠ℳ​\(𝐞2\)\\mathcal\{M\}\(\\mathbf\{e\}\)\\not=\\mathcal\{M\}\(\\mathbf\{e\}\_\{2\}\)holds\(Marques\-Silva,[2024](https://arxiv.org/html/2607.06407#bib.bib317)\)\.

#### Contrastive Explanation\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), the partial instance𝐞1\\mathbf\{e\}\_\{1\}is a*contrastive explanation*\(𝖢𝖷𝗉\\mathsf\{CXp\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}if it is a weak contrastive explanation for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}such that there is no weak contrastive explanation𝐞2\\mathbf\{e\}\_\{2\}satisfying𝐞1⊂𝐞2\\mathbf\{e\}\_\{1\}\\subset\\mathbf\{e\}\_\{2\}\. It can be shown that this definition is equivalent to the one by\(Marques\-Silva,[2024](https://arxiv.org/html/2607.06407#bib.bib317)\)\.

#### Maximum Contrastive Explanation\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), the partial instance𝐞1\\mathbf\{e\}\_\{1\}is a*maximum contrastive explanation*\(𝗆𝖢𝖷𝗉\\mathsf\{mCXp\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}if it is a weak contrastive explanation for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}such that there is no weak contrastive explanation𝐞2\\mathbf\{e\}\_\{2\}satisfying𝐞1≺𝐞2\\mathbf\{e\}\_\{1\}\\prec\\mathbf\{e\}\_\{2\}\. In[Figure 1](https://arxiv.org/html/2607.06407#S2.F1), the partial instance\(⊥,1,1,1\)\(\\bot,1,1,1\)is a maximum contrastive explanation for\(1,1,1,1\)\(1,1,1,1\), thus also a𝖢𝖷𝗉\\mathsf\{CXp\}and a𝗐𝖢𝖷𝗉\\mathsf\{wCXp\}\.

#### Minimum Change Required\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), an instance𝐞1\\mathbf\{e\}\_\{1\}is a solution to the*minimum change required*query \(𝖬𝖢𝖱\\mathsf\{MCR\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}ifℳ​\(𝐞\)≠ℳ​\(𝐞1\)\\mathcal\{M\}\(\\mathbf\{e\}\)\\not=\\mathcal\{M\}\(\\mathbf\{e\}\_\{1\}\)andℳ​\(𝐞\)=ℳ​\(𝐞2\)\\mathcal\{M\}\(\\mathbf\{e\}\)=\\mathcal\{M\}\(\\mathbf\{e\}\_\{2\}\), for every instance𝐞2\\mathbf\{e\}\_\{2\}at smaller Hamming distance \(meaning the number of flipped features between two instances\) from𝐞\\mathbf\{e\}than𝐞1\\mathbf\{e\}\_\{1\}\. The instance𝐞1\\mathbf\{e\}\_\{1\}represents the minimum distance required to change the value on the model\. This notion is based on the query introduced by\(Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41)\)\. Considering the instance\(1,1,1,1\)\(1,1,1,1\)in[Figure 1](https://arxiv.org/html/2607.06407#S2.F1), one possible explanation for the query is\(0,1,1,1\)\(0,1,1,1\)\.

#### Maximum Change Allowed\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), an instance𝐞1\\mathbf\{e\}\_\{1\}is a solution to the*maximum change allowed*query \(𝖬𝖢𝖠\\mathsf\{MCA\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}ifℳ​\(𝐞\)=ℳ​\(𝐞1\)\\mathcal\{M\}\(\\mathbf\{e\}\)=\\mathcal\{M\}\(\\mathbf\{e\}\_\{1\}\)andℳ​\(𝐞\)≠ℳ​\(𝐞2\)\\mathcal\{M\}\(\\mathbf\{e\}\)\\not=\\mathcal\{M\}\(\\mathbf\{e\}\_\{2\}\), for every instance𝐞2\\mathbf\{e\}\_\{2\}at greater Hamming distance from𝐞\\mathbf\{e\}than𝐞1\\mathbf\{e\}\_\{1\}\. This notion is based on the query introduced by\(Alfanoet al\.,[2024](https://arxiv.org/html/2607.06407#bib.bib42)\)\. As the authors argue, only studying counterfactual queries like minimum change required may not capture the whole picture for explaining certain situations\. That is why we include their semifactual version of the problem\. In[Figure 1](https://arxiv.org/html/2607.06407#S2.F1), the maximum change allowed for the negative input\(0,1,1,1\)\(0,1,1,1\)is the instance\(0,0,0,0\)\(0,0,0,0\)\.

The original versions of𝖬𝖢𝖱\\mathsf\{MCR\}and𝖬𝖢𝖠\\mathsf\{MCA\}\(Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41); Alfanoet al\.,[2024](https://arxiv.org/html/2607.06407#bib.bib42)\)are defined similarly with respect to each other, but have very different interpretations\. Given a distancekkfor𝖬𝖢𝖱\\mathsf\{MCR\}it is not trivially easier to decide any distancek′k^\{\\prime\}for𝖬𝖢𝖠\\mathsf\{MCA\}; and vice versa\. We only know that the inequalityk≤k′\+1k\\leq k^\{\\prime\}\+1holds\. Thus, presenting both queries has additional value and lets us present another useful case of maximization\.

#### Necessary Feature\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), a partial instance𝐞1\\mathbf\{e\}\_\{1\}with exactly one defined feature is a*necessary feature*\(𝖭𝖥\\mathsf\{NF\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}if, for every weak abductive explanation𝐞2\\mathbf\{e\}\_\{2\}for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\},𝐞1⊆𝐞2\\mathbf\{e\}\_\{1\}\\subseteq\\mathbf\{e\}\_\{2\}holds\. An equivalent formulation appears in\(Huanget al\.,[2023](https://arxiv.org/html/2607.06407#bib.bib64)\), where the authors define the property*feature necessity*on their own framework using abductive explanations instead of weak abductive explanations\. It is easy to see that both definitions are equivalent\. Considering\(1,1,1,1\)\(1,1,1,1\)in[Figure 1](https://arxiv.org/html/2607.06407#S2.F1), the instance has two abductive explanations:\(1,1,⊥,⊥\)\(1,1,\\bot,\\bot\)and\(1,⊥,1,1\)\(1,\\bot,1,1\)\. Therefore, the featurex1=1x\_\{1\}=1, represented by the partial instance\(1,⊥,⊥,⊥\)\(1,\\bot,\\bot,\\bot\), is a necessary feature\.

#### Relevant Feature\.

Given a pair\(𝐞,ℳ\)\(\\mathbf\{e\},\\mathcal\{M\}\), a partial instance𝐞1\\mathbf\{e\}\_\{1\}with exactly one defined feature is a*relevant feature*\(𝖱𝖥\\mathsf\{RF\}\) for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}if there exists an abductive explanation𝐞2\\mathbf\{e\}\_\{2\}for𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}such that𝐞1⊆𝐞2\\mathbf\{e\}\_\{1\}\\subseteq\\mathbf\{e\}\_\{2\}\. This notion appears in the literature as the*AXp membership problem*\(Huanget al\.,[2021](https://arxiv.org/html/2607.06407#bib.bib16)\)and as*feature relevancy*\(Huanget al\.,[2023](https://arxiv.org/html/2607.06407#bib.bib64)\)\. Considering\(1,1,1,1\)\(1,1,1,1\)and its abductive explanations\(1,1,⊥,⊥\)\(1,1,\\bot,\\bot\)and\(1,⊥,1,1\)\(1,\\bot,1,1\)in[Figure 1](https://arxiv.org/html/2607.06407#S2.F1),x2=1x\_\{2\}=1is one of the relevant features, and can be represented by the partial instance\(⊥,1,⊥,⊥\)\(\\bot,1,\\bot,\\bot\)\.

## 3\.First Order Interpretability Logic

In this section, we introduce an initial interpretability logic designed for expressing queries\. We demonstrate that it faces limitations in expressive power and exhibits high computational complexity for its evaluation problem\.

Our work is inspired by thefirst\-order interpretability logic\(FOIL\)\(Arenaset al\.,[2021a](https://arxiv.org/html/2607.06407#bib.bib10)\), which is a simple explainability language rooted in first\-order logic\.FOILis simply first\-order logic over two relations on the set of partial instances of a given dimension: a unary relation𝖯𝗈𝗌\\mathsf\{Pos\}whose interpretation is the set of instances that are positively classified by the model, and a binary relation⊆\\subseteqthat represents the subsumption relation among partial instances\.

Given a vocabularyσ\\sigmaconsisting of relationsR1R\_\{1\},…\\ldots,RℓR\_\{\\ell\}, recall that a structure𝔄\\mathfrak\{A\}overσ\\sigmaconsists of a domain over which quantifiers range, and an interpretation for each relationRiR\_\{i\}\. Moreover, given a first\-order formulaφ\\varphidefined over the vocabularyσ\\sigma, we writeφ​\(x1,…,xk\)\\varphi\(x\_\{1\},\\ldots,x\_\{k\}\)to indicate that the free variables ofφ\\varphiare among\{x1,…,xk\}\\\{x\_\{1\},\\ldots,x\_\{k\}\\\}\. Finally, given a structure𝔄\\mathfrak\{A\}over the vocabularyσ\\sigmaand elementsa1a\_\{1\},…\\ldots,aka\_\{k\}in the domain of𝔄\\mathfrak\{A\}, we use𝔄⊧φ​\(a1,…,ak\)\\mathfrak\{A\}\\models\\varphi\(a\_\{1\},\\ldots,a\_\{k\}\)to indicate that the formulaφ\\varphiis satisfied by𝔄\\mathfrak\{A\}when each variablexix\_\{i\}is replaced by elementaia\_\{i\}\(1≤i≤k1\\leq i\\leq k\)\.

Consider a modelℳ\\mathcal\{M\}withd​i​m​\(ℳ\)=n\{\\textit\{d\}im\}\(\\mathcal\{M\}\)=n\. The structure𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}representingℳ\\mathcal\{M\}over the vocabulary formed by𝖯𝗈𝗌\\mathsf\{Pos\}and⊆\\subseteqis defined as follows\. The domain of𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}is the set\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}of all partial instances of dimensionnn\. A partial instance𝐞∈\{0,1,⊥\}n\\mathbf\{e\}\\in\\\{0,1,\\bot\\\}^\{n\}belongs to the interpretation of𝖯𝗈𝗌\\mathsf\{Pos\}in𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}if and only if𝐞∈\{0,1\}n\\mathbf\{e\}\\in\\\{0,1\\\}^\{n\}andℳ​\(𝐞\)=1\\mathcal\{M\}\(\\mathbf\{e\}\)=1\. Moreover, a pair\(𝐞1,𝐞2\)\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)is in the interpretation of relation⊆\\subseteqin𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}if and only if𝐞1\\mathbf\{e\}\_\{1\}is subsumed by𝐞2\\mathbf\{e\}\_\{2\}\. Finally, given a formulaφ​\(x1,…,xk\)\\varphi\(x\_\{1\},\\ldots,x\_\{k\}\)inFOILand partial instances𝐞1\\mathbf\{e\}\_\{1\},…\\ldots,𝐞k\\mathbf\{e\}\_\{k\}of dimensionnn, the modelℳ\\mathcal\{M\}is said tosatisfyφ​\(𝐞1,…,𝐞k\)\\varphi\(\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k\}\), denoted byℳ⊧φ​\(𝐞1,…,𝐞k\)\\mathcal\{M\}\\models\\varphi\(\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k\}\), if𝔄ℳ⊧φ​\(𝐞1,…,𝐞k\)\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\models\\varphi\(\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k\}\)\.

Notice that for a succinctly\-represented modelℳ\\mathcal\{M\}, the structure𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}can be exponentially larger than the representation ofℳ\\mathcal\{M\}\. Hence,𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}is a theoretical construction needed to formally define the semantics ofFOIL, but it should not be constructed explicitly when checking in practice if a formulaφ\\varphiis satisfied byℳ\\mathcal\{M\}\.

### 3\.1\.Expressing interpretability queries inFOIL

It will be instructive for the rest of our presentation to see a few examples of howFOILcan be used to express some natural explainability queries on models\. In these examples, we make use of the followingFOILformula:

𝖥𝗎𝗅𝗅​\(x\):=∀y​\(x⊆y→y⊆x\)\.\\mathsf\{Full\}\(x\)\\ :=\\ \\forall y\\,\(x\\subseteq y\\,\\rightarrow\\,y\\subseteq x\)\.Notice that ifℳ\\mathcal\{M\}is a model and𝐞\\mathbf\{e\}is a partial instance, thenℳ⊧𝖥𝗎𝗅𝗅​\(𝐞\)\\mathcal\{M\}\\models\\mathsf\{Full\}\(\\mathbf\{e\}\)if and only if𝐞\\mathbf\{e\}is also an instance \(i\.e\., it has no undefined features\)\. We also use the formula

𝖠𝗅𝗅𝖯𝗈𝗌​\(x\):=∀y​\(\(x⊆y∧𝖥𝗎𝗅𝗅​\(y\)\)→𝖯𝗈𝗌​\(y\)\),\\mathsf\{AllPos\}\(x\)\\ :=\\ \\forall y\\,\\big\(\(x\\subseteq y\\wedge\\mathsf\{Full\}\(y\)\)\\,\\rightarrow\\,\\mathsf\{Pos\}\(y\)\\big\),such thatℳ⊧𝖠𝗅𝗅𝖯𝗈𝗌​\(𝐞\)\\mathcal\{M\}\\models\\mathsf\{AllPos\}\(\\mathbf\{e\}\)if and only if every instance incomp​\(𝐞\)\\textit\{comp\}\(\\mathbf\{e\}\)is classified as positive byℳ\\mathcal\{M\}\. Analogously, we define a formula𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)\\mathsf\{AllNeg\}\(x\)\. The definitions of both predicates are inspired by important knowledge compilation queries such as a conditioning transformation of the input and a consistency or validity check on the conditioned model\(Darwiche and Marquis,[2011](https://arxiv.org/html/2607.06407#bib.bib24)\)\.𝖠𝗅𝗅𝖯𝗈𝗌​\(x\)\\mathsf\{AllPos\}\(x\)and𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)\\mathsf\{AllNeg\}\(x\)will be important components of a new logic for explainability defined in[Section4](https://arxiv.org/html/2607.06407#S4)since they enable the expression of a wide range of explainability queries\. For example, we can now define weak abductive explanations \(refer to[Section2\.2](https://arxiv.org/html/2607.06407#S2.SS2)\) inFOILas follows:

𝗐𝖠𝖷𝗉​\(x,y\):=𝖥𝗎𝗅𝗅​\(x\)∧y⊆x∧\(𝖯𝗈𝗌​\(x\)→𝖠𝗅𝗅𝖯𝗈𝗌​\(y\)\)∧\(¬𝖯𝗈𝗌​\(x\)→𝖠𝗅𝗅𝖭𝖾𝗀​\(y\)\)\.\\mathsf\{wAXp\}\(x,y\):=\\mathsf\{Full\}\(x\)\\wedge y\\subseteq x\\ \\wedge\\\\ \(\\mathsf\{Pos\}\(x\)\\to\\mathsf\{AllPos\}\(y\)\)\\wedge\(\\neg\\mathsf\{Pos\}\(x\)\\to\\mathsf\{AllNeg\}\(y\)\)\.
In fact, it is easy to see thatℳ⊧𝗐𝖠𝖷𝗉​\(𝐞,𝐞′\)\\mathcal\{M\}\\models\\mathsf\{wAXp\}\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)if and only if𝐞′\\mathbf\{e\}^\{\\prime\}is a weak abductive explanation for𝐞\\mathbf\{e\}overℳ\\mathcal\{M\}\. Notice that𝐞\\mathbf\{e\}is always a weak abductive explanation for itself\. However, we are typically interested in explanations that satisfy some optimality criterion\. A common such criterion is that of beingminimal\(Shihet al\.,[2018](https://arxiv.org/html/2607.06407#bib.bib23); Izzaet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib43); Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41)\)\. Let us writex⊂yx\\subset yforx⊆y∧¬\(y⊆x\)x\\subseteq y\\wedge\\neg\(y\\subseteq x\)\. Then, for

𝖠𝖷𝗉​\(x,y\):=𝗐𝖠𝖷𝗉​\(x,y\)∧∀z​\(z⊂y→¬𝗐𝖠𝖷𝗉​\(x,z\)\),\\mathsf\{AXp\}\(x,y\):=\\mathsf\{wAXp\}\(x,y\)\\land\\forall z\\,\(z\\subset y\\,\\rightarrow\\neg\\mathsf\{wAXp\}\(x,z\)\),we have thatℳ⊧𝖠𝖷𝗉​\(𝐞,𝐞′\)\\mathcal\{M\}\\models\\mathsf\{AXp\}\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)if and only if𝐞′\\mathbf\{e\}^\{\\prime\}is an abductive explanation for𝐞\\mathbf\{e\}overℳ\\mathcal\{M\}\. We could similarly express local explainability queries like contrastive explanations through the same approach usingFOIL\.

As we show below,FOILfails to meet either of the two criteria we are looking for in a practical language that provides explanations\. The first issue is its limited expressiveness: there are important notions of explanations that cannot be expressed in this language, even when restricted to decision trees, which are traditionally deemed to be easily interpretable\. The second issue is its high computational complexity: there are queries inFOILthat cannot be evaluated with a polynomial number of calls to an NP oracle\. Both facts firmly establish the inadequacy ofFOILas a practical language\.

### 3\.2\.FOILpresents limited expressiveness

In some scenarios we want to express a stronger condition for abductive and contrastive explanations: not only that they are minimal, but also that they areminimum\(see[Section2\.2](https://arxiv.org/html/2607.06407#S2.SS2)\)\. In the case of abductive explanations, they can be minimal without being minimum\. The following theorem shows thatFOILcannot express the query that asks whether a partial instance𝐞′\\mathbf\{e\}^\{\\prime\}is a minimum abductive explanation for a given instance𝐞\\mathbf\{e\}over decision trees\.

###### Theorem 3\.1\.

There is no formula𝗆𝖠𝖷𝗉​\(x,y\)\\mathsf\{mAXp\}\(x,y\)inFOILsuch that, for every decision tree𝒯\\mathcal\{T\}, instance𝐞\\mathbf\{e\}and partial instance𝐞′\\mathbf\{e\}^\{\\prime\}, we have that𝒯⊧𝗆𝖠𝖷𝗉​\(𝐞,𝐞′\)⇔𝐞′\\mathcal\{T\}\\models\\mathsf\{mAXp\}\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)\\Leftrightarrow\\mathbf\{e\}^\{\\prime\}is a minimum abductive explanation for𝐞\\mathbf\{e\}over𝒯\\mathcal\{T\}\.

###### Proof\.

The proof extends techniques from\(Libkin,[2004](https://arxiv.org/html/2607.06407#bib.bib8)\)such as the games for FO distinguishability\. We now present notions that will be used in this and the following arguments throughout this paper\.

Thequantifier rankof anFO\{\\rm FO\}formulaφ\\varphi, denoted byqr​\(φ\)\{\\rm qr\}\(\\varphi\), is the maximum depth of quantifier nesting in it\. For a structure𝔄\\mathfrak\{A\}, we writedom​\(𝔄\)\{\\rm dom\}\(\\mathfrak\{A\}\)to denote its domain\. AnEhrenfeucht\-Fraïssé\(EF\) game is played in two structures,𝔄1\\mathfrak\{A\}\_\{1\}and𝔄2\\mathfrak\{A\}\_\{2\}, of the same schema, by two players, thespoilerand theduplicator\. In roundiithe spoiler selects a structure, say𝔄1\\mathfrak\{A\}\_\{1\}, and an elementcic\_\{i\}indom​\(𝔄1\)\{\\rm dom\}\(\\mathfrak\{A\}\_\{1\}\); the duplicator responds by selecting an elementeie\_\{i\}indom​\(𝔄2\)\{\\rm dom\}\(\\mathfrak\{A\}\_\{2\}\)\. The duplicatorwinsinkkrounds, fork≥0k\\geq 0, if\{\(ci,ei\)∣i≤k\}\\\{\(c\_\{i\},e\_\{i\}\)\\mid i\\leq k\\\}defines a partial isomorphism between𝔄1\\mathfrak\{A\}\_\{1\}and𝔄2\\mathfrak\{A\}\_\{2\}\. If the duplicator wins no matter how the spoiler plays, we write𝔄1≡k𝔄2\\mathfrak\{A\}\_\{1\}\\equiv\_\{k\}\\mathfrak\{A\}\_\{2\}\. A classical result states that𝔄1≡k𝔄2\\mathfrak\{A\}\_\{1\}\\equiv\_\{k\}\\mathfrak\{A\}\_\{2\}iff𝔄1\\mathfrak\{A\}\_\{1\}and𝔄2\\mathfrak\{A\}\_\{2\}agree on allFO\{\\rm FO\}sentences of quantifier rank≤k\\leq k\(cf\.\(Libkin,[2004](https://arxiv.org/html/2607.06407#bib.bib8)\)\)\.

Also, ifa¯\\bar\{a\}is anmm\-tuple indom​\(𝔄1\)\{\\rm dom\}\(\\mathfrak\{A\}\_\{1\}\)andb¯\\bar\{b\}is anmm\-tuple indom​\(𝔄2\)\{\\rm dom\}\(\\mathfrak\{A\}\_\{2\}\), wherem≥0m\\geq 0, we write\(𝔄1,a¯\)≡k\(𝔄2,b¯\)\(\\mathfrak\{A\}\_\{1\},\\bar\{a\}\)\\equiv\_\{k\}\(\\mathfrak\{A\}\_\{2\},\\bar\{b\}\)whenever the duplicator wins inkkrounds no matter how the spoiler plays, but starting from position\(a¯,b¯\)\(\\bar\{a\},\\bar\{b\}\)\. In the same way,\(𝔄1,a¯\)≡k\(𝔄2,b¯\)\(\\mathfrak\{A\}\_\{1\},\\bar\{a\}\)\\equiv\_\{k\}\(\\mathfrak\{A\}\_\{2\},\\bar\{b\}\)iff for everyFO\{\\rm FO\}formulaφ​\(x¯\)\\varphi\(\\bar\{x\}\)of quantifier rank≤k\\leq k, it holds that𝔄1⊧φ​\(a¯\)⇔𝔄2⊧φ​\(b¯\)\\mathfrak\{A\}\_\{1\}\\models\\varphi\(\\bar\{a\}\)\\Leftrightarrow\\mathfrak\{A\}\_\{2\}\\models\\varphi\(\\bar\{b\}\)\.

It is well\-known \(cf\.\(Libkin,[2004](https://arxiv.org/html/2607.06407#bib.bib8)\)\) that there are only finitely manyFO\{\\rm FO\}formulae of quantifier rankkk, up to logical equivalence\. Therank\-kktypeof anmm\-tuplea¯\\bar\{a\}in a structure𝔄\\mathfrak\{A\}is the set of all formulaeφ​\(x¯\)\\varphi\(\\bar\{x\}\)of quantifier rank≤k\\leq ksuch that𝔄⊧φ​\(a¯\)\\mathfrak\{A\}\\models\\varphi\(\\bar\{a\}\)\. Given the above, there are only finitely many rank\-kktypes, and each one of them is definable by anFO\{\\rm FO\}formulaτk\(𝔄,a¯\)​\(x¯\)\\tau\_\{k\}^\{\(\\mathfrak\{A\},\\bar\{a\}\)\}\(\\bar\{x\}\)of quantifier rankkk\.

We now introduce some terminology necessary for the proof\.

Letℳ\\mathcal\{M\},ℳ′\\mathcal\{M\}^\{\\prime\}be models of dimensionnnandpp, respectively, and consider the structures𝔄ℳ=⟨\{0,1,⊥\}n,⊆𝔄ℳ,𝖯𝗈𝗌𝔄ℳ⟩\\mathfrak\{A\}\_\{\\mathcal\{M\}\}=\\langle\\\{0,1,\\bot\\\}^\{n\},\\subseteq^\{\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\},\\mathsf\{Pos\}^\{\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\}\\rangleand𝔄ℳ′=⟨\{0,1,⊥\}p,⊆𝔄ℳ′,𝖯𝗈𝗌𝔄ℳ′⟩\\mathfrak\{A\}\_\{\\mathcal\{M\}^\{\\prime\}\}=\\langle\\\{0,1,\\bot\\\}^\{p\},\\subseteq^\{\\mathfrak\{A\}\_\{\\mathcal\{M\}^\{\\prime\}\}\},\\mathsf\{Pos\}^\{\\mathfrak\{A\}\_\{\\mathcal\{M\}^\{\\prime\}\}\}\\rangle\. We write𝔄ℳ⊕𝔄ℳ′\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}^\{\\prime\}\}for the structure over the same vocabulary that satisfies the following:

- •The domain of𝔄ℳ⊕𝔄ℳ′\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}^\{\\prime\}\}is\{0,1,⊥\}n\+p\\\{0,1,\\bot\\\}^\{n\+p\}\.
- •The interpretation of⊆\\subseteqon𝔄ℳ⊕𝔄ℳ′\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}^\{\\prime\}\}is the usual subsumption relation on\{0,1,⊥\}n\+p\\\{0,1,\\bot\\\}^\{n\+p\}\.
- •The interpretation of𝖯𝗈𝗌\\mathsf\{Pos\}on𝔄ℳ⊕𝔄ℳ′\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}^\{\\prime\}\}is the set of instances𝐞∈\{0,1\}n\+p\\mathbf\{e\}\\in\\\{0,1\\\}^\{n\+p\}such that\(𝐞​\[1\],⋯,𝐞​\[n\]\)∈𝖯𝗈𝗌𝔄ℳ\(\\mathbf\{e\}\[1\],\\cdots,\\mathbf\{e\}\[n\]\)\\in\\mathsf\{Pos\}^\{\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\}or\(𝐞​\[n\+1\],⋯,𝐞​\[n\+p\]\)∈𝖯𝗈𝗌𝔄ℳ′\(\\mathbf\{e\}\[n\+1\],\\cdots,\\mathbf\{e\}\[n\+p\]\)\\in\\mathsf\{Pos\}^\{\\mathfrak\{A\}\_\{\\mathcal\{M\}^\{\\prime\}\}\}\.

We will also consider structures of the form𝔄n=⟨\{0,1,⊥\}n,⊆𝔄⟩\\mathfrak\{A\}\_\{n\}=\\langle\\\{0,1,\\bot\\\}^\{n\},\\subseteq^\{\\mathfrak\{A\}\}\\rangle, where⊆\\subseteqis interpreted as the subsumption relation over\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}\. For any such structure, we write𝔄n\+\\mathfrak\{A\}\_\{n\}^\{\+\}for the structure over the vocabulary\{⊆,𝖯𝗈𝗌\}\\\{\\subseteq,\\mathsf\{Pos\}\\\}that extends𝔄\\mathfrak\{A\}by adding only the tuple\{1\}n\\\{1\\\}^\{n\}to the interpretation of𝖯𝗈𝗌\\mathsf\{Pos\}\.

We now state two crucial lemmas, whose proofs can be found in the appendix of this work \(refer to Sections[A\.1](https://arxiv.org/html/2607.06407#A1.SS1)and[A\.2](https://arxiv.org/html/2607.06407#A1.SS2)\)\.

###### Lemma 3\.2\.

Ifn,p≥3kn,p\\geq 3^\{k\}, then\(𝔄n,\{1\}n\)≡k\(𝔄p,\{1\}p\)\(\\mathfrak\{A\}\_\{n\},\\\{1\\\}^\{n\}\)\\ \\equiv\_\{k\}\\ \(\\mathfrak\{A\}\_\{p\},\\\{1\\\}^\{p\}\)\. In particular,𝔄n\+≡k𝔄p\+\\mathfrak\{A\}\_\{n\}^\{\+\}\\equiv\_\{k\}\\mathfrak\{A\}\_\{p\}^\{\+\}\.

###### Lemma 3\.3\.

Consider modelsℳ\\mathcal\{M\},ℳ1\\mathcal\{M\}\_\{1\}, andℳ2\\mathcal\{M\}\_\{2\}of dimensionnn,pp, andqq, respectively, and assume that\(𝔄ℳ1,\{1\}p\)≡k\(𝔄ℳ2,\{1\}q\)\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\},\\\{1\\\}^\{p\}\)\\equiv\_\{k\}\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\},\\\{1\\\}^\{q\}\)\. Then it is the case that

\(𝔄ℳ⊕𝔄ℳ1,\{1\}n\+p,\{⊥\}n⋅\{1\}p\)≡k\(𝔄ℳ⊕𝔄ℳ2,\{1\}n\+q,\{⊥\}n⋅\{1\}q\)\.\\displaystyle\\big\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\},\\\{1\\\}^\{n\+p\},\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{p\}\\big\)\\ \\equiv\_\{k\}\\ \\big\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\},\\\{1\\\}^\{n\+q\},\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{q\}\\big\)\.

We now proceed with the proof of Theorem[3\.1](https://arxiv.org/html/2607.06407#S3.Thmtheorem1)\. Assume, for the sake of contradiction, that there is in fact a formula𝗆𝖠𝖷𝗉​\(x,y\)\\mathsf\{mAXp\}\(x,y\)inFOILsuch that, for every decision treeℳ\\mathcal\{M\}, instance𝐞\\mathbf\{e\}, and partial instance𝐞′\\mathbf\{e\}^\{\\prime\}, we have that𝔄ℳ⊧𝗆𝖠𝖷𝗉​\(𝐞,𝐞′\)\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\models\\mathsf\{mAXp\}\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)iff𝐞′\\mathbf\{e\}^\{\\prime\}is a minimum abductive explanation for𝐞\\mathbf\{e\}overℳ\\mathcal\{M\}\. Letk≥0k\\geq 0be the quantifier rank of this formula\. We show that there exist decision treesℳ1\\mathcal\{M\}\_\{1\}andℳ2\\mathcal\{M\}\_\{2\}, instances𝐞1\\mathbf\{e\}\_\{1\}and𝐞2\\mathbf\{e\}\_\{2\}overℳ1\\mathcal\{M\}\_\{1\}andℳ2\\mathcal\{M\}\_\{2\}, respectively, and partial instances𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}and𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}overℳ1\\mathcal\{M\}\_\{1\}andℳ2\\mathcal\{M\}\_\{2\}, respectively, for which the following holds:

- •\(ℳ1,𝐞1,𝐞1′\)≡k\(ℳ2,𝐞2,𝐞2′\)\(\\mathcal\{M\}\_\{1\},\\mathbf\{e\}\_\{1\},\\mathbf\{e\}^\{\\prime\}\_\{1\}\)\\equiv\_\{k\}\(\\mathcal\{M\}\_\{2\},\\mathbf\{e\}\_\{2\},\\mathbf\{e\}^\{\\prime\}\_\{2\}\), and hence ℳ1⊧𝗆𝖠𝖷𝗉​\(𝐞1,𝐞1′\)⇔ℳ2⊧𝗆𝖠𝖷𝗉​\(𝐞2,𝐞2′\)\.\\displaystyle\\mathcal\{M\}\_\{1\}\\models\\mathsf\{mAXp\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}^\{\\prime\}\_\{1\}\)\\ \\Leftrightarrow\\ \\mathcal\{M\}\_\{2\}\\models\\mathsf\{mAXp\}\(\\mathbf\{e\}\_\{2\},\\mathbf\{e\}^\{\\prime\}\_\{2\}\)\.
- •It is the case that𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}is a minimum abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}underℳ1\\mathcal\{M\}\_\{1\}, but𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}is not a minimum abductive explanation for𝐞2\\mathbf\{e\}\_\{2\}underℳ2\\mathcal\{M\}\_\{2\}\.

This is our desired contradiction\.

Letℳn,p\\mathcal\{M\}\_\{n,p\}be a decision tree of dimensionn\+pn\+psuch that, for every instance𝐞∈\{0,1\}n\+p\\mathbf\{e\}\\in\\\{0,1\\\}^\{n\+p\}, we have thatℳn,p​\(𝐞\)=1\\mathcal\{M\}\_\{n,p\}\(\\mathbf\{e\}\)=1iff𝐞\\mathbf\{e\}is of the form\{1\}n⋅\{0,1\}p\\\{1\\\}^\{n\}\\cdot\\\{0,1\\\}^\{p\}, i\.e\., the firstnnfeatures of𝐞\\mathbf\{e\}are set to 1, or𝐞\\mathbf\{e\}is of the form\{0,1\}n⋅\{1\}p\\\{0,1\\\}^\{n\}\\cdot\\\{1\\\}^\{p\}, i\.e\., the lastppfeatures of𝐞\\mathbf\{e\}are set to 1\. Take the instance𝐞=\{1\}n\+p\\mathbf\{e\}=\\\{1\\\}^\{n\+p\}\. It is easy to see that𝐞\\mathbf\{e\}only has two abductive explanations inℳn,p\\mathcal\{M\}\_\{n,p\}; namely,𝐞1=\{1\}n⋅\{⊥\}p\\mathbf\{e\}\_\{1\}=\\\{1\\\}^\{n\}\\cdot\\\{\\bot\\\}^\{p\}and𝐞2=\{⊥\}n⋅\{1\}p\\mathbf\{e\}\_\{2\}=\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{p\}\.

We define the following:

- •ℳ1:=ℳ2k,2k\\mathcal\{M\}\_\{1\}:=\\mathcal\{M\}\_\{2^\{k\},2^\{k\}\}andℳ2:=ℳ2k,2k\+1\\mathcal\{M\}\_\{2\}:=\\mathcal\{M\}\_\{2^\{k\},2^\{k\}\+1\}\.
- •𝐞1:=\{1\}2k\+2k\\mathbf\{e\}\_\{1\}:=\\\{1\\\}^\{2^\{k\}\+2^\{k\}\}and𝐞2:=\{1\}2k\+2k\+1\\mathbf\{e\}\_\{2\}:=\\\{1\\\}^\{2^\{k\}\+2^\{k\}\+1\}\.
- •𝐞1′:=\{⊥\}2k⋅\{1\}2k\\mathbf\{e\}^\{\\prime\}\_\{1\}:=\\\{\\bot\\\}^\{2^\{k\}\}\\cdot\\\{1\\\}^\{2^\{k\}\}and𝐞2′:=\{⊥\}2k⋅\{1\}2k\+1\\mathbf\{e\}^\{\\prime\}\_\{2\}:=\\\{\\bot\\\}^\{2^\{k\}\}\\cdot\\\{1\\\}^\{2^\{k\}\+1\}\.

From our previous observation,𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}is an abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}overℳ1\\mathcal\{M\}\_\{1\}and𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}is an abductive explanation for𝐞2\\mathbf\{e\}\_\{2\}overℳ2\\mathcal\{M\}\_\{2\}\.

We show first that\(𝔄ℳ1,𝐞1,𝐞1′\)≡k\(𝔄ℳ2,𝐞2,𝐞2′\)\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\},\\mathbf\{e\}\_\{1\},\\mathbf\{e\}^\{\\prime\}\_\{1\}\)\\equiv\_\{k\}\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\},\\mathbf\{e\}\_\{2\},\\mathbf\{e\}^\{\\prime\}\_\{2\}\)\. It can be observed that𝔄ℳ1\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\}is of the form𝔄N⊕𝔄N1\\mathfrak\{A\}\_\{N\}\\oplus\\mathfrak\{A\}\_\{N\_\{1\}\}, whereNNis a model of dimension2k2^\{k\}that only accepts the tuple\{1\}2k\\\{1\\\}^\{2^\{k\}\}and the same holds forN1N\_\{1\}\. Analogously,𝔄ℳ2\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\}is of the form𝔄N⊕𝔄N2\\mathfrak\{A\}\_\{N\}\\oplus\\mathfrak\{A\}\_\{N\_\{2\}\}, whereN2N\_\{2\}is a model of dimension2k\+12^\{k\}\+1that only accepts the tuple\{1\}2k\+1\\\{1\\\}^\{2^\{k\}\+1\}\. From Lemma[3\.2](https://arxiv.org/html/2607.06407#S3.Thmtheorem2), we have that

\(𝔄N1,\{1\}2k\)≡k\(𝔄N2,\{1\}2k\+1\)\.\(\\mathfrak\{A\}\_\{N\_\{1\}\},\\\{1\\\}^\{2^\{k\}\}\)\\ \\equiv\_\{k\}\\ \(\\mathfrak\{A\}\_\{N\_\{2\}\},\\\{1\\\}^\{2^\{k\}\+1\}\)\.Notice that indeed any winning strategy for the Duplicator on this game must map the tuples\{1\}2k\\\{1\\\}^\{2^\{k\}\}in𝔄N1\\mathfrak\{A\}\_\{N\_\{1\}\}and\{1\}2k\+1\\\{1\\\}^\{2^\{k\}\+1\}into each other\.

Now, from Lemma[3\.3](https://arxiv.org/html/2607.06407#S3.Thmtheorem3), we obtain that

\(𝔄N⊕𝔄N1,\{1\}2k\+2k,\{⊥\}2k⋅\{1\}2k\)≡k\(𝔄N⊕𝔄N2,\{1\}2k\+2k\+1,\{⊥\}2k⋅\{1\}2k\+1\)\.\\displaystyle\\big\(\\mathfrak\{A\}\_\{N\}\\oplus\\mathfrak\{A\}\_\{N\_\{1\}\},\\\{1\\\}^\{2^\{k\}\+2^\{k\}\},\\\{\\bot\\\}^\{2^\{k\}\}\\cdot\\\{1\\\}^\{2^\{k\}\}\\big\)\\ \\equiv\_\{k\}\\ \\big\(\\mathfrak\{A\}\_\{N\}\\oplus\\mathfrak\{A\}\_\{N\_\{2\}\},\\\{1\\\}^\{2^\{k\}\+2^\{k\}\+1\},\\\{\\bot\\\}^\{2^\{k\}\}\\cdot\\\{1\\\}^\{2^\{k\}\+1\}\\big\)\.We can then conclude that\(𝔄ℳ1,𝐞1,𝐞1′\)≡k\(𝔄ℳ2,𝐞2,𝐞2′\)\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\},\\mathbf\{e\}\_\{1\},\\mathbf\{e\}^\{\\prime\}\_\{1\}\)\\equiv\_\{k\}\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\},\\mathbf\{e\}\_\{2\},\\mathbf\{e\}^\{\\prime\}\_\{2\}\), as desired\.

Notice now that𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}is a minimum abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}overℳ1\\mathcal\{M\}\_\{1\}\. In fact, by our previous observations, the only other abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}overℳ1\\mathcal\{M\}\_\{1\}is𝐞1′′=\{1\}2k⋅\{⊥\}2k\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\}=\\\{1\\\}^\{2^\{k\}\}\\cdot\\\{\\bot\\\}^\{2^\{k\}\}, which has the same number of undefined features as𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}\. In turn,𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}is not a minimum abductive explanation for𝐞2\\mathbf\{e\}\_\{2\}overℳ2\\mathcal\{M\}\_\{2\}\. This is because𝐞2′′=\{1\}2k⋅\{⊥\}2k\+1\\mathbf\{e\}^\{\\prime\\prime\}\_\{2\}=\\\{1\\\}^\{2^\{k\}\}\\cdot\\\{\\bot\\\}^\{2^\{k\}\+1\}is also an abductive explanation for𝐞2\\mathbf\{e\}\_\{2\}overℳ2\\mathcal\{M\}\_\{2\}, and𝐞2′′\\mathbf\{e\}^\{\\prime\\prime\}\_\{2\}has more undefined features than𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}\. ∎

### 3\.3\.EvaluatingFOILis intractable

For each queryφ​\(x1,…,xk\)\\varphi\(x\_\{1\},\\ldots,x\_\{k\}\)inFOILand𝒞\\mathcal\{C\}a class of models, we define its associated problemEval\(φ,𝒞\)\(\\varphi,\\mathcal\{C\}\)as follows \(we assume models and instances have the same dimension\):

It is known that there exists a formulaφ​\(x\)\\varphi\(x\)inFOILfor which its evaluation problem over the class of decision trees isNP\{\\rm NP\}\-hard\(Arenaset al\.,[2021a](https://arxiv.org/html/2607.06407#bib.bib10)\)\. We want to determine whether the languageFOILis appropriate for implementation using SAT encodings\. Thus, it is natural to ask whether the evaluation problem for formulas in this logic can always be decided in polynomial time by using aNP\{\\rm NP\}oracle\. However, we prove that this is not always the case\. Although the evaluation ofFOILformulas is always in the polynomial hierarchy \(PH\), there exist formulas inFOILfor which their corresponding evaluation problems are hard for every level of PH\. Based on widely held complexity assumptions, we can conclude thatFOILcontains formulas whose evaluations cannot be decided in polynomial time by using aNP\{\\rm NP\}oracle even on decision trees \(𝖣𝖳𝗋𝖾𝖾\\mathsf\{DTree\}\)\.

###### Theorem 3\.4\.

The following statements hold:

1. \(1\)Letφ\\varphibe aFOILformula\. Then, there existsk≥0k\\geq 0such thatEval\(φ,𝖭𝖭𝖥\)\(\\varphi,\\mathsf\{NNF\}\)is in theΣkP\\Sigma\_\{k\}^\{\\rm\{P\}\}complexity class\.
2. \(2\)For everyk≥0k\\geq 0, there is anFOIL\-formulaφk\\varphi\_\{k\}such thatEval\(φk,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi\_\{k\},\\mathsf\{DTree\}\)isΣkP\\Sigma\_\{k\}^\{\\rm\{P\}\}\-hard\.

###### Proof\.

For the first item, consider a fixedFOILformulaφ​\(x1,…,xm\)\\varphi\(x\_\{1\},\\dots,x\_\{m\}\)\. We assume without loss of generality thatφ\\varphiis in prenex normal form, i\.e\., it is of the form

∃y¯1​∀y¯2​⋯​Qk​y¯k​ψ​\(x1,…,xm,y¯1,…,y¯k\),\(k≥0\)\\exists\\bar\{y\}\_\{1\}\\forall\\bar\{y\}\_\{2\}\\cdots Q\_\{k\}\\bar\{y\}\_\{k\}\\,\\psi\(x\_\{1\},\\dots,x\_\{m\},\\bar\{y\}\_\{1\},\\dots,\\bar\{y\}\_\{k\}\),\\quad\\quad\(k\\geq 0\)whereQk=∃Q\_\{k\}=\\existsifkkis odd andQk=∀Q\_\{k\}=\\forallotherwise, andψ\\psiis a quantifier\-free formula\. A FOIL formula of this form is called aΣk\\Sigma\_\{k\}\-FOILformula\. Consider thatℳ\\mathcal\{M\}is a negation normal form of dimensionnn, and assume that we want to check whetherℳ⊧φ​\(𝐞1,…,𝐞m\)\\mathcal\{M\}\\models\\varphi\(\\mathbf\{e\}\_\{1\},\\dots,\\mathbf\{e\}\_\{m\}\), for𝐞1,…,𝐞m\\mathbf\{e\}\_\{1\},\\dots,\\mathbf\{e\}\_\{m\}given partial instances of dimensionnn\. We know that the predicates𝖯𝗈𝗌\\mathsf\{Pos\}and⊆\\subseteqcan be decided in polynomial time onℳ\\mathcal\{M\}\. Additionally, the formulaφ\\varphiis fixed, and thus the length of each tupley¯i\\bar\{y\}\_\{i\}, fori≤ki\\leq k, is constant\. Therefore, we can decide this problem in polynomial time by using aΣk\\Sigma\_\{k\}\-alternating Turing machine \(as the fixed size quantifier\-free formulaψ\\psican be evaluated in polynomial time overℳ\\mathcal\{M\}\)\.

We now deal with the second item\. We start by studying the complexity of the well\-knownquantified Boolean formula\(QBF\) problem for the case when the underlying formula \(or, more precisely, the underlying Boolean function\) is defined by a decision tree\. More precisely, suppose thatℳ\\mathcal\{M\}is a decision tree over instances of dimensionnn\. AΣk\\Sigma\_\{k\}\-QBF overℳ\\mathcal\{M\}, fork\>1k\>1, is an expression

∃P1​∀P2​⋯​Qk​Pk​ℳ,\\exists P\_\{1\}\\forall P\_\{2\}\\cdots Q\_\{k\}P\_\{k\}\\,\\mathcal\{M\},whereQk=∃Q\_\{k\}=\\existsifkkis odd andQk=∀Q\_\{k\}=\\forallotherwise, andP1,…,PkP\_\{1\},\\dots,P\_\{k\}is a partition of\{1,…,n\}\\\{1,\\dots,n\\\}intokkequivalence classes\. As an example, ifℳ\\mathcal\{M\}is of dimension 3 then∃\{2,1\}​∀\{3\}​ℳ\\exists\\\{2,1\\\}\\forall\\\{3\\\}\\,\\mathcal\{M\}is aΣ2\\Sigma\_\{2\}\-QBF overℳ\\mathcal\{M\}\. The semantics of these expressions is standard\. For instance,∃\{1,2\}​∀\{3\}​ℳ\\exists\\\{1,2\\\}\\forall\\\{3\\\}\\,\\mathcal\{M\}holds if there exists a partial instance\(b1,b2,⊥\)∈\{0,1\}×\{0,1\}×\{⊥\}\(b\_\{1\},b\_\{2\},\\bot\)\\in\\\{0,1\\\}\\times\\\{0,1\\\}\\times\\\{\\bot\\\}such that bothℳ​\(b1,b2,0\)=1\\mathcal\{M\}\(b\_\{1\},b\_\{2\},0\)=1andℳ​\(b1,b2,1\)=1\\mathcal\{M\}\(b\_\{1\},b\_\{2\},1\)=1\.

For a fixedk\>1k\>1, we introduce then the problemΣk\\Sigma\_\{k\}\-QBF\(𝖣𝖳𝗋𝖾𝖾\)\(\\mathsf\{DTree\}\)\. It takes as input aΣk\\Sigma\_\{k\}\-QBFα\\alphaoverℳ\\mathcal\{M\}, forℳ\\mathcal\{M\}a decision tree, and asks whetherα\\alphaholds\. We establish the following result, which we believe of independent interest, as \(to the best of our knowledge\) the complexity of the QBF problem over decision trees has not been studied in the literature \(refer to Section[A\.3](https://arxiv.org/html/2607.06407#A1.SS3)for the proof\)\.

###### Lemma 3\.5\.

For every oddk≥1k\\geq 1, the problemΣk\+1\\Sigma\_\{k\+1\}\-QBF\(𝖣𝖳𝗋𝖾𝖾\)\(\\mathsf\{DTree\}\)isΣkP\\Sigma\_\{k\}^\{\\text\{P\}\}\-complete\.

For the second item, we can now finish the proof of the theorem with the help of Lemma[3\.5](https://arxiv.org/html/2607.06407#S3.Thmtheorem5)and a reduction fromΣk\+1\\Sigma\_\{k\+1\}\-QBF\(𝖣𝖳𝗋𝖾𝖾\)\(\\mathsf\{DTree\}\)\. We can assume thatkkis odd because for proving that there areFOIL\-formulasφk′\\varphi\_\{k^\{\\prime\}\}such thatEval\(φk′,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi\_\{k^\{\\prime\}\},\\mathsf\{DTree\}\)isΣk′P\\Sigma\_\{k^\{\\prime\}\}^\{\\rm\{P\}\}\-hard it is enough to show that there areFOIL\-formulasφk\\varphi\_\{k\}such thatEval\(φk,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi\_\{k\},\\mathsf\{DTree\}\)isΣk′P\\Sigma\_\{k^\{\\prime\}\}^\{\\rm\{P\}\}\-hard for somek≥k′k\\geq k^\{\\prime\}\. The input toΣk\+1\\Sigma\_\{k\+1\}\-QBF\(𝖣𝖳𝗋𝖾𝖾\)\(\\mathsf\{DTree\}\)is given by an expressionα\\alphaof the form

∃P1​∀P2​⋯​∃Pk​∀Pk\+1​ℳ,\\exists P\_\{1\}\\forall P\_\{2\}\\cdots\\exists P\_\{k\}\\forall P\_\{k\+1\}\\,\\mathcal\{M\},forℳ\\mathcal\{M\}a decision tree of dimensionnnandP1,…,Pk\+1P\_\{1\},\\dots,P\_\{k\+1\}a partition of\{1,…,n\}\\\{1,\\dots,n\\\}\. We explain next how the formulaφk​\(x1,…,xk\+1\)\\varphi\_\{k\}\(x\_\{1\},\\dots,x\_\{k\+1\}\)is defined\.

We start by defining some auxiliary terminology\. We usex​\[i\]x\[i\]to denote theii\-th feature of the partial instance that is assigned to variablexx\. We define the following formulas\.

- •𝖴𝗇𝖽𝖾𝖿​\(x\):=¬∃y​\(y⊂x\)\{\\sf Undef\}\(x\):=\\neg\\exists y\(y\\subset x\)\. That is,𝖴𝗇𝖽𝖾𝖿\{\\sf Undef\}defines the set that only consists of the partial instance\{⊥\}n\\\{\\bot\\\}^\{n\}in which all components are undefined\.
- •𝖲𝗂𝗇𝗀𝗅𝖾​\(x\):=∃y​\(y⊂x\)∧∀y​\(y⊂x→𝖴𝗇𝖽𝖾𝖿​\(y\)\)\{\\sf Single\}\(x\):=\\exists y\(y\\subset x\)\\wedge\\forall y\(y\\subset x\\,\\rightarrow\\,\{\\sf Undef\}\(y\)\)\. That is,𝖲𝗂𝗇𝗀𝗅𝖾\{\\sf Single\}defines the set that consists precisely of those partial instances in\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}which have exactly one defined component\.
- •\(x⊔y=z\):=\(x⊆z\)∧\(y⊆z\)∧¬∃w​\(\(x⊆w\)∧\(y⊆w\)∧\(w⊂z\)\)\(x\\sqcup y=z\):=\(x\\subseteq z\)\\wedge\(y\\subseteq z\)\\wedge\\neg\\exists w\\big\(\(x\\subseteq w\)\\wedge\(y\\subseteq w\)\\wedge\(w\\subset z\)\\big\)\. That is,zz, if it exists, is thejoinofxxandyy\. In other words,zzis defined if every feature that is defined overxxandyytakes the same value in both partial instances, and, in such case, for each1≤i≤n1\\leq i\\leq nwe have thatz​\[i\]=x​\[i\]⊔y​\[i\]z\[i\]=x\[i\]\\sqcup y\[i\], where⊔\\sqcupis the commutative and idempotent binary operation that satisfies⊥⊔0=0\\bot\\sqcup 0=0and⊥⊔1=1\\bot\\sqcup 1=1\. As an example,\(1,0,⊥,⊥\)⊔\(1,⊥,⊥,1\)=\(1,0,⊥,1\)\(1,0,\\bot,\\bot\)\\sqcup\(1,\\bot,\\bot,1\)=\(1,0,\\bot,1\), while\(1,⊥\)⊔\(0,0\)\(1,\\bot\)\\sqcup\(0,0\)is undefined\.
- •\(x⊓y=z\):=\(z⊆x\)∧\(z⊆y\)∧¬∃w​\(\(w⊆x\)∧\(w⊆y\)∧\(z⊂w\)\)\(x\\sqcap y=z\):=\(z\\subseteq x\)\\wedge\(z\\subseteq y\)\\wedge\\neg\\exists w\\big\(\(w\\subseteq x\)\\wedge\(w\\subseteq y\)\\wedge\(z\\subset w\)\\big\)\. That is,zzis themeetofxxandyy\(which always exists\)\. In other words, for each1≤i≤n1\\leq i\\leq nwe have thatz​\[i\]=x​\[i\]⊓y​\[i\]z\[i\]=x\[i\]\\sqcap y\[i\], where⊓\\sqcapis the commutative and idempotent binary operation that satisfies⊥⊓0=⊥⊓1=0⊓1=⊥\\bot\\sqcap 0=\\bot\\sqcap 1=0\\sqcap 1=\\bot\. As an example,\(1,0,⊥,⊥\)⊓\(1,⊥,⊥,1\)=\(1,⊥,⊥,⊥\)\(1,0,\\bot,\\bot\)\\sqcap\(1,\\bot,\\bot,1\)=\(1,\\bot,\\bot,\\bot\), while\(1,⊥\)⊓\(0,0\)=\(⊥,⊥\)\(1,\\bot\)\\sqcap\(0,0\)=\(\\bot,\\bot\)\.
- •𝖢𝗈𝗆𝗉​\(x,y\):=∃w​∃z​\(𝖴𝗇𝖽𝖾𝖿​\(z\)∧x⊔y=w∧x⊓y=z\)\{\\sf Comp\}\(x,y\):=\\exists w\\exists z\(\{\\sf Undef\}\(z\)\\,\\wedge\\,x\\sqcup y=w\\,\\wedge\\,x\\sqcap y=z\)\. That is,𝖢𝗈𝗆𝗉\{\\sf Comp\}defines the pairs\(𝐞1,𝐞2\)\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)of partial instances in\{0,1,⊥\}n×\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}\\times\\\{0,1,\\bot\\\}^\{n\}such that no feature that is defined in𝐞1\\mathbf\{e\}\_\{1\}is also defined in𝐞2\\mathbf\{e\}\_\{2\}, and vice versa\. In fact, assume for the sake of contradiction that this is not the case\. By symmetry, we only have to consider the following two cases\. - –There is ani≤ni\\leq nwith𝐞1​\[i\]=1\\mathbf\{e\}\_\{1\}\[i\]=1and𝐞2​\[i\]=0\\mathbf\{e\}\_\{2\}\[i\]=0\. Then the join of𝐞1\\mathbf\{e\}\_\{1\}and𝐞2\\mathbf\{e\}\_\{2\}does not exist\. - –There is ani≤ni\\leq nwith𝐞1​\[i\]=𝐞2​\[i\]=1\\mathbf\{e\}\_\{1\}\[i\]=\\mathbf\{e\}\_\{2\}\[i\]=1\. Then theii\-th component of the meet of𝐞1\\mathbf\{e\}\_\{1\}and𝐞2\\mathbf\{e\}\_\{2\}takes value 1, and hence𝐞1⊓𝐞2≠\{⊥\}n\\mathbf\{e\}\_\{1\}\\sqcap\\mathbf\{e\}\_\{2\}\\neq\\\{\\bot\\\}^\{n\}\.
- •𝖬𝖺𝗑𝖢𝗈𝗆𝗉​\(x,y\):=𝖢𝗈𝗆𝗉​\(x,y\)∧¬∃z​\(\(y⊂z\)∧𝖢𝗈𝗆𝗉​\(x,z\)\)\{\\sf MaxComp\}\(x,y\):=\{\\sf Comp\}\(x,y\)\\,\\wedge\\,\\neg\\exists z\\big\(\(y\\subset z\)\\wedge\{\\sf Comp\}\(x,z\)\\big\)\. That is,𝖬𝖺𝗑𝖢𝗈𝗆𝗉\{\\sf MaxComp\}defines the pairs\(𝐞1,𝐞2\)\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)such that the components that are defined in𝐞1\\mathbf\{e\}\_\{1\}are precisely the ones that are undefined in𝐞2\\mathbf\{e\}\_\{2\}, and vice versa\.
- •𝖱𝖾𝗅​\(x,y\):=¬∃z​\(\(z⊆y\)∧𝖲𝗂𝗇𝗀𝗅𝖾​\(z\)∧𝖢𝗈𝗆𝗉​\(x,z\)\)\{\\sf Rel\}\(x,y\):=\\neg\\exists z\\big\(\(z\\subseteq y\)\\,\\wedge\\,\{\\sf Single\}\(z\)\\,\\wedge\\,\{\\sf Comp\}\(x,z\)\\big\)\. That is,𝖱𝖾𝗅\{\\sf Rel\}defines the pairs\(𝐞1,𝐞2\)\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)of partial instances in\{0,1,⊥\}n×\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}\\times\\\{0,1,\\bot\\\}^\{n\}such that every feature that is defined in𝐞1\\mathbf\{e\}\_\{1\}is also defined in𝐞2\\mathbf\{e\}\_\{2\}\.
- •𝖬𝖺𝗑𝖱𝖾𝗅​\(x,y\):=𝖱𝖾𝗅​\(x,y\)∧¬∃z​\(\(z⊂y\)∧𝖱𝖾𝗅​\(x,z\)\)\{\\sf MaxRel\}\(x,y\):=\{\\sf Rel\}\(x,y\)\\,\\wedge\\,\\neg\\exists z\\big\(\(z\\subset y\)\\wedge\{\\sf Rel\}\(x,z\)\\big\)\. That is,𝖬𝖺𝗑𝖱𝖾𝗅\{\\sf MaxRel\}defines the pairs\(𝐞1,𝐞2\)\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)such that the features defined in𝐞1\\mathbf\{e\}\_\{1\}and in𝐞2\\mathbf\{e\}\_\{2\}are the same\.

For defining the formulaφk​\(x1,…,xk\+1\)\\varphi\_\{k\}\(x\_\{1\},\\dots,x\_\{k\+1\}\)we will use guarded quantifiers\. For eachiiwith1≤i≤k\+11\\leq i\\leq k\+1consider

∃𝒢​\(xi\)yi​ψ\\displaystyle\\exists^\{\\mathcal\{G\}\(x\_\{i\}\)\}y\_\{i\}\\,\\psi\\=∃yi​\(𝖬𝖺𝗑𝖱𝖾𝗅​\(xi,yi\)∧ψ\)\\displaystyle=\\ \\exists y\_\{i\}\\,\\big\(\{\\sf MaxRel\}\(x\_\{i\},y\_\{i\}\)\\ \\wedge\\ \\psi\\big\)∀𝒢​\(xi\)yi​ψ\\displaystyle\\forall^\{\\mathcal\{G\}\(x\_\{i\}\)\}y\_\{i\}\\,\\psi\\=∀yi​\(𝖬𝖺𝗑𝖱𝖾𝗅​\(xi,yi\)→ψ\)\\displaystyle=\\ \\forall y\_\{i\}\\,\\big\(\{\\sf MaxRel\}\(x\_\{i\},y\_\{i\}\)\\ \\rightarrow\\ \\psi\\big\)
We now define the formulaφk​\(x1,…,xk\+1\)\\varphi\_\{k\}\(x\_\{1\},\\dots,x\_\{k\+1\}\)as

∃𝒢​\(x1\)y1​∀𝒢​\(x2\)y2​⋯​∃𝒢​\(xk\)yk​∀𝒢​\(xk\+1\)yk\+1​∀z​\(z=y1⊔y2⊔⋯⊔yk\+1→𝖯𝗈𝗌​\(z\)\)\\displaystyle\\exists^\{\\mathcal\{G\}\(x\_\{1\}\)\}y\_\{1\}\\forall^\{\\mathcal\{G\}\(x\_\{2\}\)\}y\_\{2\}\\cdots\\exists^\{\\mathcal\{G\}\(x\_\{k\}\)\}y\_\{k\}\\forall^\{\\mathcal\{G\}\(x\_\{k\+1\}\)\}y\_\{k\+1\}\\ \\forall z\\big\(z=y\_\{1\}\\sqcup y\_\{2\}\\sqcup\\cdots\\sqcup y\_\{k\+1\}\\,\\rightarrow\\,\\mathsf\{Pos\}\(z\)\\big\)
For eachiiwith1≤i≤k\+11\\leq i\\leq k\+1, let𝐞i\\mathbf\{e\}\_\{i\}be the partial instance of dimensionnnsuch that

𝐞i​\[j\]=\{1ifj∈Pi,⊥otherwise\.\\mathbf\{e\}\_\{i\}\[j\]\\ =\\ \\begin\{cases\}1\\quad\\quad&\\text\{if $j\\in P\_\{i\}$,\}\\\\ \\bot&\\text\{otherwise\.\}\\end\{cases\}That is,𝐞i\\mathbf\{e\}\_\{i\}takes value 1 over the features inPiP\_\{i\}and it is undefined over all other features\. We claim thatα\\alphaholds if, and only if,ℳ⊧φk​\(𝐞1,…,𝐞k\+1\)\\mathcal\{M\}\\models\\varphi\_\{k\}\(\\mathbf\{e\}\_\{1\},\\dots,\\mathbf\{e\}\_\{k\+1\}\)\. The result then follows sinceℳ\\mathcal\{M\}is a decision tree\.

For the sake of presentation we only prove the aforementioned equivalence for the case whenk=1k=1, since the extension tok\>1k\>1is standard \(but cumbersome\)\. That is, we consider the case whenα=∃P1​∀P2​ℳ\\alpha=\\exists P\_\{1\}\\forall P\_\{2\}\\mathcal\{M\}and, therefore,

φ2​\(x1,x2\)=∃𝒢​\(x1\)y1​∀𝒢​\(x2\)y2​∀z​\(z=y1⊔y2→𝖯𝗈𝗌​\(z\)\)\.\\displaystyle\\varphi\_\{2\}\(x\_\{1\},x\_\{2\}\)=\\exists^\{\\mathcal\{G\}\(x\_\{1\}\)\}y\_\{1\}\\forall^\{\\mathcal\{G\}\(x\_\{2\}\)\}y\_\{2\}\\ \\forall z\\big\(z=y\_\{1\}\\sqcup y\_\{2\}\\,\\rightarrow\\,\\mathsf\{Pos\}\(z\)\\big\)\.
- \(⇐\)\(\\Leftarrow\)Assume first thatℳ⊧φ2​\(𝐞1,𝐞2\)\\mathcal\{M\}\\models\\varphi\_\{2\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\. Hence, there exists a partial instance𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}such that \(1\)ℳ⊧\(𝖬𝖺𝗑𝖱𝖾𝗅​\(𝐞1,𝐞1′\)∧∀𝒢​\(𝐞2\)y2​∀z​\(z=𝐞1′⊔y2→𝖯𝗈𝗌​\(z\)\)\)\.\\mathcal\{M\}\\,\\models\\,\\big\(\{\\sf MaxRel\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}^\{\\prime\}\_\{1\}\)\\ \\wedge\\ \\forall^\{\\mathcal\{G\}\(\\mathbf\{e\}\_\{2\}\)\}y\_\{2\}\\ \\forall z\\big\(z=\\mathbf\{e\}^\{\\prime\}\_\{1\}\\sqcup y\_\{2\}\\,\\rightarrow\\,\\mathsf\{Pos\}\(z\)\\big\)\\big\)\.This means that the features defined in𝐞1\\mathbf\{e\}\_\{1\}and𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}are exactly the same, and hence𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}is a partial instance that is defined precisely over the features inP1P\_\{1\}\. We claim that every instance𝐞\\mathbf\{e\}that is a completion of𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}satisfiesℳ​\(𝐞\)=1\\mathcal\{M\}\(\\mathbf\{e\}\)=1, thus showing thatα\\alphaholds\. In fact, take𝐞\\mathbf\{e\}to be an arbitrary completion\. By definition,𝐞\\mathbf\{e\}can be written as𝐞1′⊔𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{1\}\\sqcup\\mathbf\{e\}^\{\\prime\}\_\{2\}, where𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}is a partial instance that is defined precisely over those features not inP1P\_\{1\}, i\.e\., over the features inP2P\_\{2\}\. Thus in the formula \([1](https://arxiv.org/html/2607.06407#S3.E1)\) we can assign the partial instance𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}to the variabley2y\_\{2\}and the instance𝐞\\mathbf\{e\}to the variablezz, which allows us to conclude thatℳ⊧𝖯𝗈𝗌​\(𝐞\)\\mathcal\{M\}\\models\\mathsf\{Pos\}\(\\mathbf\{e\}\)\. This tells us thatℳ​\(𝐞\)=1\\mathcal\{M\}\(\\mathbf\{e\}\)=1\.
- \(⇒\)\(\\Rightarrow\)Assume in turn thatα\\alphaholds, and hence that there is a partial instance𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}that is defined precisely over the features inP1P\_\{1\}such that every instance𝐞\\mathbf\{e\}that is a completion of𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}satisfiesℳ​\(𝐞\)=1\\mathcal\{M\}\(\\mathbf\{e\}\)=1\. We claim that ℳ⊧\(𝖬𝖺𝗑𝖱𝖾𝗅​\(𝐞1,𝐞1′\)∧∀𝒢​\(𝐞2\)y2​∀z​\(z=𝐞1′⊔y2→𝖯𝗈𝗌​\(z\)\)\),\\displaystyle\\mathcal\{M\}\\,\\models\\,\\big\(\{\\sf MaxRel\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}^\{\\prime\}\_\{1\}\)\\ \\wedge\\ \\forall^\{\\mathcal\{G\}\(\\mathbf\{e\}\_\{2\}\)\}y\_\{2\}\\ \\forall z\\big\(z=\\mathbf\{e\}^\{\\prime\}\_\{1\}\\sqcup y\_\{2\}\\,\\rightarrow\\,\\mathsf\{Pos\}\(z\)\\big\)\\big\),which implies thatℳ⊧φ2​\(𝐞1,𝐞2\)\\mathcal\{M\}\\models\\varphi\_\{2\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\. In fact, let𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}be an arbitrary instance such that𝖬𝖺𝗑𝖱𝖾𝗅​\(𝐞2,𝐞2′\)\{\\sf MaxRel\}\(\\mathbf\{e\}\_\{2\},\\mathbf\{e\}^\{\\prime\}\_\{2\}\)holds\. By definition,𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}is defined precisely over the features inP2P\_\{2\}\. Let𝐞=𝐞1′⊔𝐞2′\\mathbf\{e\}=\\mathbf\{e\}^\{\\prime\}\_\{1\}\\sqcup\\mathbf\{e\}^\{\\prime\}\_\{2\}\. Notice that𝐞\\mathbf\{e\}is well\-defined since the sets of features defined in𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}and𝐞2′\\mathbf\{e\}^\{\\prime\}\_\{2\}, respectively, are disjoint\. Moreover,𝐞\\mathbf\{e\}is a completion of𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\}asP1∪P2=\{1,…,n\}P\_\{1\}\\cup P\_\{2\}=\\\{1,\\dots,n\\\}\. We then have thatℳ​\(𝐞\)=1\\mathcal\{M\}\(\\mathbf\{e\}\)=1asα\\alphaholds\. This allows us to conclude thatℳ⊧𝖯𝗈𝗌​\(𝐞\)\\mathcal\{M\}\\models\\mathsf\{Pos\}\(\\mathbf\{e\}\), and hence thatℳ⊧φ2​\(𝐞1,𝐞2\)\\mathcal\{M\}\\models\\varphi\_\{2\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\.

This concludes the proof of the theorem\. ∎

## 4\.ExplAIner: a tractable logic for explainability

In the previous section we identified two limitations ofFOILthat must be addressed in order to build a practical logic for explanations\. On one hand, we must extendFOILto increase its expressive power, and on the other hand, we must constrain the resulting logic to ensure that its evaluation complexity is appropriate\. In this section we defineExplAIner, a logic that takes both criteria into account and in which explainability notions can be expressed naturally\.

### 4\.1\.The atomic layer ofExplAIner

FOILcannot express properties such as minimum abductive explanations that involve comparing cardinalities of sets of features\. As a first step, we solve this issue by extending the vocabulary ofFOILwith a simple binary relation⪯\\preceqdefined as:

ℳ⊧𝐞⪯𝐞′⟺\|𝐞⊥\|≥\|𝐞⊥′\|\.\\mathcal\{M\}\\models\\mathbf\{e\}\\preceq\\mathbf\{e\}^\{\\prime\}\\ \\ \\Longleftrightarrow\\ \\ \|\\mathbf\{e\}\_\{\\bot\}\|\\geq\|\\mathbf\{e\}^\{\\prime\}\_\{\\bot\}\|\.As we will show later, the use of this predicate indeed allows us to express many notions of explanations\. Note that we could not simply keep only one of⊆\\subseteqand⪯\\preceqwhen defining the new logic, as we show that they cannot be defined in terms of each other\. First, we show that predicate⪯\\preceqcannot be defined in terms of predicate⊆\\subseteq\.

###### Proposition 4\.1\.

There is no formulaφ​\(x,y\)\\varphi\(x,y\)inFOILdefined over the vocabulary\{⊆\}\\\{\\subseteq\\\}such that, for every decision tree𝒯\\mathcal\{T\}and pair of partial instances𝐞\\mathbf\{e\},𝐞′\\mathbf\{e\}^\{\\prime\}, we have that

𝒯⊧φ​\(𝐞,𝐞′\)⟺\|𝐞⊥\|≥\|𝐞⊥′\|\.\\mathcal\{T\}\\models\\varphi\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)\\ \\Longleftrightarrow\\ \|\\mathbf\{e\}\_\{\\bot\}\|\\geq\|\\mathbf\{e\}^\{\\prime\}\_\{\\bot\}\|\.

###### Proof\.

For the sake of contradiction, assume thatφ​\(x,y\)\\varphi\(x,y\)is definable inFOILover the vocabulary\{⊆\}\\\{\\subseteq\\\}\. Then the following are formulas inFOIL:

𝗐𝖠𝖷𝗉​\(x,y\)\\displaystyle\\mathsf\{wAXp\}\(x,y\):=\\displaystyle:=𝖥𝗎𝗅𝗅\(x\)∧y⊆x∧∀z\(y⊆z∧𝖥𝗎𝗅𝗅\(z\)→\(𝖯𝗈𝗌\(z\)↔𝖯𝗈𝗌\(x\)\)\),\\displaystyle\\mathsf\{Full\}\(x\)\\wedge\\,y\\subseteq x\\wedge\\forall z\\,\\big\(y\\subseteq z\\wedge\\mathsf\{Full\}\(z\)\\,\\rightarrow\\,\(\\mathsf\{Pos\}\(z\)\\leftrightarrow\\mathsf\{Pos\}\(\\ x\)\)\\big\),𝗆𝖠𝖷𝗉​\(x,y\)\\displaystyle\\mathsf\{mAXp\}\(x,y\):=\\displaystyle:=𝗐𝖠𝖷𝗉​\(x,y\)∧∀z​\(𝗐𝖠𝖷𝗉​\(x,z\)→\(φ​\(z,y\)→φ​\(y,z\)\)\)\.\\displaystyle\\mathsf\{wAXp\}\(x,y\)\\wedge\\forall z\\,\\big\(\\mathsf\{wAXp\}\(x,z\)\\to\(\\varphi\(z,y\)\\to\\varphi\(y,z\)\)\\big\)\.But the second formula verifies if a partial instanceyyis a minimum abductive explanation for a given instancexx, which contradicts the inexpressibility result of Theorem[3\.1](https://arxiv.org/html/2607.06407#S3.Thmtheorem1), and hence concludes the proof of the proposition\. ∎

Second, we show that predicate⊆\\subseteqcannot be defined in terms of predicate⪯\\preceq\.

###### Proposition 4\.2\.

There is no formulaψ​\(x,y\)\\psi\(x,y\)inFOILdefined over the vocabulary\{⪯\}\\\{\\preceq\\\}such that, for every decision tree𝒯\\mathcal\{T\}222Naturally, this statement does not rely on decision trees at all since it concerns only⊆\\subseteqand⪯\\preceq; we only state it in these terms for consistency\.and pair of partial instances𝐞\\mathbf\{e\},𝐞′\\mathbf\{e\}^\{\\prime\}, we have that

𝒯⊧ψ​\(𝐞,𝐞′\)⟺𝐞​is subsumed by​𝐞′\.\\mathcal\{T\}\\models\\psi\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)\\ \\Longleftrightarrow\\ \\mathbf\{e\}\\text\{ is subsumed by \}\\mathbf\{e\}^\{\\prime\}\.

###### Proof\.

Intuitively,⪯\\preceqis invariant under any bijection of partial instances that preserves the number of⊥\\bot’s, whereas subsumption is not\. We formalize this as follows\.

For the sake of contradiction, assume thatψ​\(x,y\)\\psi\(x,y\)is definable inFOILover the vocabulary\{⪯\}\\\{\\preceq\\\}, and letn≥3n\\geq 3\. Moreover, for everyk∈\{0,…,n\}k\\in\\\{0,\\ldots,n\\\}, defineLkL\_\{k\}as the following set of partial instances:

Lk\\displaystyle L\_\{k\}=\\displaystyle=\{𝐞∈\{0,1,⊥\}n∣\|𝐞⊥\|=k\},\\displaystyle\\\{\\mathbf\{e\}\\in\\\{0,1,\\bot\\\}^\{n\}\\mid\|\\mathbf\{e\}\_\{\\bot\}\|=k\\\},and letfk:Lk→Lkf\_\{k\}:L\_\{k\}\\to L\_\{k\}be an arbitrary bijection fromLkL\_\{k\}to itself\. Finally, letf:\{0,1,⊥\}n→\{0,1,⊥\}nf:\\\{0,1,\\bot\\\}^\{n\}\\to\\\{0,1,\\bot\\\}^\{n\}be defined asf​\(𝐞\)=fi​\(𝐞\)f\(\\mathbf\{e\}\)=f\_\{i\}\(\\mathbf\{e\}\)if𝐞∈Li\\mathbf\{e\}\\in L\_\{i\}\. Clearly,ffis a bijection from\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}to\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}\.

For a decision tree𝒯\\mathcal\{T\}of dimensionnn, define𝔄𝒯′\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}as the restriction of𝔄𝒯\\mathfrak\{A\}\_\{\\mathcal\{T\}\}to the vocabulary\{⪯\}\\\{\\preceq\\\}\. Then functionffis an automorphism of𝔄𝒯′\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}sinceffis a bijection from\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}to\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}, and for every pair of partial instances𝐞1,𝐞2\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}:

𝔄𝒯′⊧𝐞1⪯𝐞2if and only if𝔄𝒯′⊧f​\(𝐞1\)⪯f​\(𝐞2\)\.\\displaystyle\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}\\models\\mathbf\{e\}\_\{1\}\\preceq\\mathbf\{e\}\_\{2\}\\quad\\text\{ if and only if \}\\quad\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}\\models f\(\\mathbf\{e\}\_\{1\}\)\\preceq f\(\\mathbf\{e\}\_\{2\}\)\.Then given thatψ​\(x,y\)\\psi\(x,y\)is definable in first\-order logic over the vocabulary\{⪯\}\\\{\\preceq\\\}, we have that for every pair of partial instances𝐞1,𝐞2\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}:

\(2\)𝔄𝒯′⊧ψ​\(𝐞1,𝐞2\)if and only if𝔄𝒯′⊧ψ​\(f​\(𝐞1\),f​\(𝐞2\)\)\.\\displaystyle\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}\\models\\psi\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\\quad\\text\{ if and only if \}\\quad\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}\\models\\psi\(f\(\\mathbf\{e\}\_\{1\}\),f\(\\mathbf\{e\}\_\{2\}\)\)\.But now assume thatgk:Lk→Lkg\_\{k\}:L\_\{k\}\\to L\_\{k\}is defined as the identity function for everyk∈\{0,…,n\}∖\{1\}k\\in\\\{0,\\ldots,n\\\}\\setminus\\\{1\\\}, and assume thatg1g\_\{1\}is defined as follows for every partial instance𝐞\\mathbf\{e\}:

g1​\(𝐞\)\\displaystyle g\_\{1\}\(\\mathbf\{e\}\)=\\displaystyle=\{\(⊥,0,…,0\)if​𝐞=\(0,…,0,⊥\)\(0,…,0,⊥\)if​𝐞=\(⊥,0,…,0\)𝐞otherwise\\displaystyle\\begin\{cases\}\(\\bot,0,\\ldots,0\)&\\text\{if \}\\mathbf\{e\}=\(0,\\ldots,0,\\bot\)\\\\ \(0,\\ldots,0,\\bot\)&\\text\{if \}\\mathbf\{e\}=\(\\bot,0,\\ldots,0\)\\\\ \\mathbf\{e\}&\\text\{otherwise\}\\end\{cases\}Clearly, each functiongig\_\{i\}is a bijection\. Moreover, letg:\{0,1,⊥\}n→\{0,1,⊥\}ng:\\\{0,1,\\bot\\\}^\{n\}\\to\\\{0,1,\\bot\\\}^\{n\}be defined asg​\(𝐞\)=gi​\(𝐞\)g\(\\mathbf\{e\}\)=g\_\{i\}\(\\mathbf\{e\}\)if𝐞∈Li\\mathbf\{e\}\\in L\_\{i\}\. Then we have by \([2](https://arxiv.org/html/2607.06407#S4.E2)\) that for every pair of partial instances𝐞1\\mathbf\{e\}\_\{1\},𝐞2\\mathbf\{e\}\_\{2\}:

𝔄𝒯′⊧ψ​\(𝐞1,𝐞2\)if and only if𝔄𝒯′⊧ψ​\(g​\(𝐞1\),g​\(𝐞2\)\)\.\\displaystyle\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}\\models\\psi\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\\quad\\text\{ if and only if \}\\quad\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}\\models\\psi\(g\(\\mathbf\{e\}\_\{1\}\),g\(\\mathbf\{e\}\_\{2\}\)\)\.Hence, taking𝐞1=\(⊥,⊥,0,…,0\)\\mathbf\{e\}\_\{1\}=\(\\bot,\\bot,0,\\ldots,0\)and𝐞2=\(⊥,0,…,0\)\\mathbf\{e\}\_\{2\}=\(\\bot,0,\\ldots,0\), given thatg​\(𝐞1\)=\(⊥,⊥,0,…,0\)g\(\\mathbf\{e\}\_\{1\}\)=\(\\bot,\\bot,0,\\ldots,0\)andg​\(𝐞2\)=\(0,…,0,⊥\)\{g\(\\mathbf\{e\}\_\{2\}\)=\(0,\\ldots,0,\\bot\)\}, we conclude that:

𝔄𝒯′⊧ψ​\(\(⊥,⊥,0,…,0\),\(⊥,0,…,0\)\)if and only if𝔄𝒯′⊧ψ​\(\(⊥,⊥,0,…,0\),\(0,…,0,⊥\)\)\.\\displaystyle\\begin\{gathered\}\\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}\\models\\psi\(\(\\bot,\\bot,0,\\ldots,0\),\\,\(\\bot,0,\\ldots,0\)\)\\\\ \\text\{ if and only if \}\\\\ \\mathfrak\{A\}^\{\\prime\}\_\{\\mathcal\{T\}\}\\models\\psi\(\(\\bot,\\bot,0,\\ldots,0\),\\,\(0,\\ldots,0,\\bot\)\)\.\\end\{gathered\}But this leads to a contradiction, since\(⊥,⊥,0,…,0\)\(\\bot,\\bot,0,\\ldots,0\)is subsumed by\(⊥,0,…,0\)\(\\bot,0,\\ldots,0\), but\(⊥,⊥,0,…,0\)\(\\bot,\\bot,0,\\ldots,0\)is not subsumed by\(0,…,0,⊥\)\(0,\\ldots,0,\\bot\)\. This concludes the proof of the proposition\. ∎

However, adding⪯\\preceqtoFOILcan only add extra complexity\. Therefore, our second step is to define the logicExplAInerexpressive enough to capture important notions, but keeping the evaluation tractable usingSATsolvers\. Our logicExplAInerconsists of three hierarchical layers, where the first layer does not depend on the structure of the model\.

Predicates⊆\\subseteqand⪯\\preceq, as well as predicate𝖥𝗎𝗅𝗅\\mathsf\{Full\}used in Section[3\.1](https://arxiv.org/html/2607.06407#S3.SS1), can be regarded assyntacticin the sense that they refer to the values of the features of partial instances, and they do not make reference to classification models\. It turns out that all the syntactic predicates needed in our logical formalism can be expressed as first\-order formulas over the predicates⊆\\subseteqand⪯\\preceq\. Theatomic formulasofExplAInerare defined as first\-order formulas over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}\. We now prove that such formulas can be evaluated in polynomial time\. We also prove that in the case of*sentences*, that is, formulas without free variables, it is decidable whether a sentence is true in every structure𝔅n\\mathfrak\{B\}\_\{n\}\.

Givenn≥0n\\geq 0and a modelℳ\\mathcal\{M\}of dimensionnn, define𝔅ℳ\\mathfrak\{B\}\_\{\\mathcal\{M\}\}as a structure over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}generated from𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}by removing the interpretation of predicate𝖯𝗈𝗌\\mathsf\{Pos\}, and adding the interpretation of predicate⪯\\preceq\. Notice that, given two modelsℳ1\\mathcal\{M\}\_\{1\}andℳ2\\mathcal\{M\}\_\{2\}of dimensionnn, we have that𝔅ℳ1=𝔅ℳ2\\mathfrak\{B\}\_\{\\mathcal\{M\}\_\{1\}\}=\\mathfrak\{B\}\_\{\\mathcal\{M\}\_\{2\}\}, so we define simply𝔅n\\mathfrak\{B\}\_\{n\}as𝔅ℳ\\mathfrak\{B\}\_\{\\mathcal\{M\}\}for an arbitrary model of dimensionnn\. Therefore, when measuring the complexity of evaluating formulas in the atomic layer, we takennin unary as part of the input, sincennis the size of the partial instances\. Hence, for each formulaφ​\(x1,…,xk\)\\varphi\(x\_\{1\},\\ldots,x\_\{k\}\)in the atomic layer ofExplAIner, we define its associated problemEval\(φ\)\(\\varphi\)as follows:

Recall also that the*width*of a first\-order formulaφ\\varphi, denotedwd​\(φ\)\\rm\{wd\}\(\\varphi\), is defined as the maximum number of free variables among all subformulas ofφ\\varphi\(see\(Grädelet al\.,[2007](https://arxiv.org/html/2607.06407#bib.bib321)\)for a reference\)\.

###### Theorem 4\.3\.

The following statements hold:

1. \(1\)Letφ\\varphibe a first\-order formula defined over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}\. ThenEval\(φ\)∈P\(\\varphi\)\\in\{\\rm P\}\.
2. \(2\)It is decidable whether a given first\-order sentenceφ\\varphidefined over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}is true in every structure𝔅n\\mathfrak\{B\}\_\{n\}\. In particular, it can be solved in22poly​\(\|φ\|⋅3wd​\(φ\)\)2^\{2^\{\\rm\{poly\}\\big\(\|\\varphi\|\\cdot 3^\{\\rm\{wd\}\(\\varphi\)\}\\big\)\}\}space, and, hence, in222poly​\(\|φ\|⋅3wd​\(φ\)\)2^\{2^\{2^\{\\rm\{poly\}\\big\(\|\\varphi\|\\cdot 3^\{\\rm\{wd\}\(\\varphi\)\}\\big\)\}\}\}time\.

###### Proof\.

We will prove both claims by a reduction to Presburger arithmetic\. One standard presentation of Presburger arithmetic consists of two constants,0and11, a binary relation<<and a binary function\+\+\. We consider the modelℕ\\mathbb\{N\}of the non\-negative integers with the usual interpretations\. We will use the following two well\-known facts about Presburger arithmetic:

1. I\.Presburger arithmetic admits quantifier elimination, that is, for every Presburger formulaφ​\(y1,…,ym\)\\varphi\(y\_\{1\},\\dots,y\_\{m\}\)there exists a quantifier\-free formulaψ​\(y1,…,ym\)\\psi\(y\_\{1\},\\dots,y\_\{m\}\)such thatφ\\varphiandψ\\psiare logically equivalent\(Presburger,[1991](https://arxiv.org/html/2607.06407#bib.bib319)\)\.
2. II\.The problem of determining the truth of sentences in Presburger Arithmetic with respect to the modelℕ\\mathbb\{N\}can be solved in double exponential space with respect to the size of the sentence\(Berman,[1980](https://arxiv.org/html/2607.06407#bib.bib320)\)\.

First, we introduce some terminology\. Letn∈ℕn\\in\\mathbb\{N\}be the dimension\. Then, for a tupleΓ=\(x1,…,xk\)\\Gamma=\(x\_\{1\},\\dots,x\_\{k\}\)of \(distinct\) variables, an assignments:Γ→\{0,1,⊥\}ns\\colon\\Gamma\\to\\\{0,1,\\bot\\\}^\{n\}, and a coordinatei∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\}, we define the “pattern”patiΓ,s\{\\operatorname\{pat\}\}^\{\\Gamma,s\}\_\{i\}as the tuple\(s​\(x1\)i,…,s​\(xk\)i\)∈\{0,1,⊥\}k\.\\bigl\(s\(x\_\{1\}\)\_\{i\},\\,\\dots,\\,s\(x\_\{k\}\)\_\{i\}\\bigr\)\\in\\\{0,1,\\bot\\\}^\{k\}\.Given a patternρ∈\{0,1,⊥\}k\\rho\\in\\\{0,1,\\bot\\\}^\{k\}, we define its*pattern count*in\(Γ,s\)\(\\Gamma,s\)by

cρΓ,s:=\|\{i∈\{1,…,n\}:patiΓ,s=ρ\}\|\.c^\{\\Gamma,s\}\_\{\\rho\}:=\\bigl\|\\\{\\,i\\in\\\{1,\\dots,n\\\}\\,:\\,\{\\operatorname\{pat\}\}^\{\\Gamma,s\}\_\{i\}=\\rho\\,\\\}\\bigr\|\.Clearly,∑ρ∈\{0,1,⊥\}kcρΓ,s=n\.\\displaystyle\\sum\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}c^\{\\Gamma,s\}\_\{\\rho\}=n\.For example, letΓ=\(x1,x2\)\\Gamma=\(x\_\{1\},x\_\{2\}\)and letn=4n=4\. Suppose thats​\(x1\)=\(1,⊥,0,0\)s\(x\_\{1\}\)=\(1,\\bot,0,0\)ands​\(x2\)=\(1,1,⊥,⊥\)s\(x\_\{2\}\)=\(1,1,\\bot,\\bot\)\. Then the four coordinates have patterns

pat1Γ,s=\(1,1\),pat2Γ,s=\(⊥,1\),pat3Γ,s=\(0,⊥\),pat4Γ,s=\(0,⊥\)\.\\operatorname\{pat\}^\{\\Gamma,s\}\_\{1\}=\(1,1\),\\qquad\\operatorname\{pat\}^\{\\Gamma,s\}\_\{2\}=\(\\bot,1\),\\qquad\\operatorname\{pat\}^\{\\Gamma,s\}\_\{3\}=\(0,\\bot\),\\qquad\\operatorname\{pat\}^\{\\Gamma,s\}\_\{4\}=\(0,\\bot\)\.Hence

c\(1,1\)Γ,s=c\(⊥,1\)Γ,s=1,c\(0,⊥\)Γ,s=2,c^\{\\Gamma,s\}\_\{\(1,1\)\}=c^\{\\Gamma,s\}\_\{\(\\bot,1\)\}=1,\\qquad c^\{\\Gamma,s\}\_\{\(0,\\bot\)\}=2,and every other pattern in\{0,1,⊥\}2\\\{0,1,\\bot\\\}^\{2\}has count0\.

We are now ready to state the reduction lemma \(refer to Section[A\.4](https://arxiv.org/html/2607.06407#A1.SS4)for the proof\):

###### Lemma 4\.4\.

Letφ​\(x1,…,xℓ\)\\varphi\(x\_\{1\},\\dots,x\_\{\\ell\}\)be a first\-order formula defined over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}, and letΓ:\-\(x1,…,xk\)\\Gamma\\coloneq\(x\_\{1\},\\dots,x\_\{k\}\)be a tuple of distinct variables that contains all free variables ofφ\\varphi\. Then there exists a Presburger formula

TΓ⁡\(φ\)​\(\(zρ\)ρ∈\{0,1,⊥\}k\)\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}\\bigr\)such that for everyn∈ℕn\\in\\mathbb\{N\}and every assignments:Γ→\{0,1,⊥\}ns:\\Gamma\\to\\\{0,1,\\bot\\\}^\{n\},

𝔅n⊧φ​\(s​\(x1\),…,s​\(xℓ\)\)⟺ℕ⊧TΓ⁡\(φ\)​\(\(cρΓ,s\)ρ\)\.\\mathfrak\{B\}\_\{n\}\\models\\varphi\(s\(x\_\{1\}\),\\dots,s\(x\_\{\\ell\}\)\)\\quad\\Longleftrightarrow\\quad\\mathbb\{N\}\\models\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(c^\{\\Gamma,s\}\_\{\\rho\}\)\_\{\\rho\}\\bigr\)\.Moreover, ifk=ℓk=\\ell\(that is, ifΓ\\Gammacontains exactly the free variables ofφ\\varphi\), then we have that

\|TΓ⁡\(φ\)​\(\(zρ\)ρ∈\{0,1,⊥\}k\)\|=O​\(\|φ\|⋅3wd​\(φ\)⋅poly​\(wd​\(φ\)\)\),\\bigg\|\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}\\bigr\)\\bigg\|=O\\bigg\(\|\\varphi\|\\cdot 3^\{\\rm\{wd\}\(\\varphi\)\}\\cdot\\mathrm\{poly\}\(\\rm\{wd\}\(\\varphi\)\)\\bigg\),and the reduction can be computed using the same space\.

We now show how Lemma[4\.4](https://arxiv.org/html/2607.06407#S4.Thmtheorem4)implies both statements of the theorem\.

For the first part, letφ​\(x1,…,xk\)\\varphi\(x\_\{1\},\\dots,x\_\{k\}\)be a first\-order formula defined over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}, and letΓ=\(x1,…,xk\)\\Gamma=\(x\_\{1\},\\dots,x\_\{k\}\)\. Because Presburger arithmetic admits quantifier elimination, we know that there exists a quantifier\-free Presburger formulaψ​\(\(zρ\)ρ∈\{0,1,⊥\}k\)\\psi\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}\\bigr\)that is logically equivalent toTΓ⁡\(φ\)​\(\(zρ\)ρ∈\{0,1,⊥\}k\)\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}\\bigr\)\. From Lemma[4\.4](https://arxiv.org/html/2607.06407#S4.Thmtheorem4)we know that, for everyn∈ℕn\\in\\mathbb\{N\}and every assignments:Γ→\{0,1,⊥\}ns:\\Gamma\\to\\\{0,1,\\bot\\\}^\{n\},

𝔅n⊧φ​\(s​\(x1\),…,s​\(xℓ\)\)⟺ℕ⊧ψ​\(\(cρΓ,s\)ρ\)\.\\mathfrak\{B\}\_\{n\}\\models\\varphi\(s\(x\_\{1\}\),\\dots,s\(x\_\{\\ell\}\)\)\\quad\\Longleftrightarrow\\quad\\mathbb\{N\}\\models\\psi\\bigl\(\(c^\{\\Gamma,s\}\_\{\\rho\}\)\_\{\\rho\}\\bigr\)\.Now suppose we are given an integern∈ℕn\\in\\mathbb\{N\}in unary and partial instances𝐞1,…,𝐞k\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k\}\. Lets:Γ→\{0,1,⊥\}ns:\\Gamma\\to\\\{0,1,\\bot\\\}^\{n\}be the assignment such thats​\(xi\)=𝐞is\(x\_\{i\}\)=\\mathbf\{e\}\_\{i\}for everyi∈\{1,…,k\}i\\in\\\{1,\\dots,k\\\}\. Notice that we can compute in linear time all the values\{cρΓ,s\}ρ∈\{0,1,⊥\}k\\\{c^\{\\Gamma,s\}\_\{\\rho\}\\\}\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}, as3k3^\{k\}is constant with respect to the input size ofEval\(φ\)\(\\varphi\)\. Notice that each variablecρΓ,sc^\{\\Gamma,s\}\_\{\\rho\}has a value less than or equal tonn\. Becauseψ\\psiis a fixed, quantifier\-free formula, we can evaluate it onto the values\{cρΓ,s\}ρ∈\{0,1,⊥\}k\\\{c^\{\\Gamma,s\}\_\{\\rho\}\\\}\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}in polynomial time\. This shows thatEval\(φ\)∈P\(\\varphi\)\\in\{\\rm P\}\.

For the second part, letφ\\varphibe a first\-order sentence defined over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}\. Because we can takeΓ\\Gammaas an empty context, there is only one possible pattern\. Therefore, the formulaTΓ⁡\(φ\)​\(\(z\)\)\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(z\)\\bigr\)has exactly one free variable\. Recall that, in general, given an assignment forΓ\\Gammain𝔅n\\mathfrak\{B\}\_\{n\}, the sum over all pattern counts must equalnn\. Hence, in this case,φ\\varphiis true in every structure𝔅n\\mathfrak\{B\}\_\{n\}if and only if∀m​TΓ⁡\(φ\)​\(\(m\)\)\\forall m\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(m\)\\bigr\)is a true Presburger sentence\. We know from Lemma[4\.4](https://arxiv.org/html/2607.06407#S4.Thmtheorem4)that the sentence∀m​TΓ⁡\(φ\)​\(\(m\)\)\\forall m\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(m\)\\bigr\)has sizeO​\(\|φ\|⋅3wd​\(φ\)⋅poly​\(wd​\(φ\)\)\)O\\bigg\(\|\\varphi\|\\cdot 3^\{\\rm\{wd\}\(\\varphi\)\}\\cdot\\mathrm\{poly\}\(\\rm\{wd\}\(\\varphi\)\)\\bigg\)and it can be constructed in at most the same space\. Because the problem of determining the truth value of a Presburger sentence can be solved in double exponential space, we conclude that we can determine ifφ\\varphiis true in every structure𝔅n\\mathfrak\{B\}\_\{n\}in22poly​\(\|φ\|⋅3wd​\(φ\)\)2^\{2^\{\\rm\{poly\}\\big\(\|\\varphi\|\\cdot 3^\{\\rm\{wd\}\(\\varphi\)\}\\big\)\}\}space\. This concludes the proof of the theorem\.

∎

### 4\.2\.The quantified layer ofExplAIner

In this layer we introduce predicates whose interpretation does depend on the model\. The vocabulary of this layer is\{⊆,⪯,𝖠𝗅𝗅𝖯𝗈𝗌,𝖠𝗅𝗅𝖭𝖾𝗀\}\\\{\\subseteq,\\preceq,\\mathsf\{AllPos\},\\mathsf\{AllNeg\}\\\}, whereℳ⊧𝖠𝗅𝗅𝖯𝗈𝗌​\(𝐞\)\\mathcal\{M\}\\models\\mathsf\{AllPos\}\(\\mathbf\{e\}\)if and only if all instances incomp​\(𝐞\)\\textit\{comp\}\(\\mathbf\{e\}\)are classified positively byℳ\\mathcal\{M\}, and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}is defined analogously\. As we will show, at this point we will already be able to express properties over polynomial\-size sets of instances and minimality/minimum conditions\.

We will need two auxiliary formulas from the atomic layer that we already defined during the proof of Theorem[3\.4](https://arxiv.org/html/2607.06407#S3.Thmtheorem4)\. The first is

𝖴𝗇𝖽𝖾𝖿​\(x\):=¬∃y​\(y⊂x\),\{\\sf Undef\}\(x\):=\\neg\\exists y\(y\\subset x\),which defines the set that only contains the partial instance\{⊥\}n\\\{\\bot\\\}^\{n\}\. The second is

𝖲𝗂𝗇𝗀𝗅𝖾​\(x\):=∃y​\(y⊂x\)∧∀y​\(y⊂x→𝖴𝗇𝖽𝖾𝖿​\(y\)\),\{\\sf Single\}\(x\):=\\exists y\(y\\subset x\)\\wedge\\forall y\(y\\subset x\\,\\rightarrow\\,\{\\sf Undef\}\(y\)\),which defines the set of partial instances with exactly one defined feature\.

The*quantified layer*is recursively defined as follows:

1. \(1\)Boolean combinations of formulas from the atomic layer, together with𝖠𝗅𝗅𝖯𝗈𝗌​\(x\)\\mathsf\{AllPos\}\(x\)and𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)\\mathsf\{AllNeg\}\(x\), are formulas from the quantified layer\.
2. \(2\)Ifφ\\varphiis a formula from the quantified layer, then∃x​φ\\exists x\\ \\varphiis a formula from the quantified layer\.
3. \(3\)Ifφ\\varphiis a formula from the quantified layer, then∀x​\(𝖲𝗂𝗇𝗀𝗅𝖾​\(x\)→φ\)\\forall x\\ \\left\(\{\\sf Single\}\(x\)\\to\\varphi\\right\)is a formula from the quantified layer\.

Using only the first rule we can already express some basic explainability properties\. For example, we can express the query for weak abductive explanations as follows:

𝗐𝖠𝖷𝗉​\(x,y\):=𝖥𝗎𝗅𝗅​\(x\)∧y⊆x∧\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\)→𝖠𝗅𝗅𝖯𝗈𝗌​\(y\)\)∧\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)→𝖠𝗅𝗅𝖭𝖾𝗀​\(y\)\)\.\\mathsf\{wAXp\}\(x,y\):=\\mathsf\{Full\}\(x\)\\wedge y\\subseteq x\\wedge\(\\mathsf\{AllPos\}\(x\)\\to\\mathsf\{AllPos\}\(y\)\)\\wedge\(\\mathsf\{AllNeg\}\(x\)\\to\\mathsf\{AllNeg\}\(y\)\)\.
The third rule involves the concept ofguardedquantification\. In that case we only quantify over partial instances with exactly one defined feature, which naturally correspond to assignments of a value to a single feature\. On any class of models𝒞\\mathcal\{C\}, the number of partial instances with one defined feature is at most twice the dimension of the model, so we cannot express universal properties over superpolynomial\-size sets of partial instances\. Notice that our rules do not allow us to define unguarded universal quantifiers because the first rule only allows us to take Boolean combinations of unquantified formulas\. In particular, in this layer we are not allowed to negate formulas that were produced using the second \(or third\) rule\.

###### Theorem 4\.5\.

The following statements hold:

1. \(1\)Let𝒞\\mathcal\{C\}be a class of models such thatEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∈P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}andEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∈P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}\. ThenEval\(φ,𝒞\)∈NP\(\\varphi,\\mathcal\{C\}\)\\in\{\\rm NP\}for every formulaφ\\varphifrom the quantified layer ofExplAIner\.
2. \(2\)There exists a formulaφ\\varphifrom the quantified layer ofExplAInersuch thatEval\(φ,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi,\\mathsf\{DTree\}\)isNP\{\\rm NP\}\-hard\.

###### Proof\.

For the first item, let𝒞\\mathcal\{C\}be a class of models such thatEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∈P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}andEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∈P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}, and letφ\\varphibe a fixed formula from the quantified layer ofExplAIner\. The algorithm is the following\. For each existential quantifier we nondeterministically guess a partial instance as a polynomial\-size witness\. Each guarded universal quantifier ranges only over the set of partial instances with exactly one defined feature, whose size is linear in the dimensionnn\. Since the formula is fixed, unfolding all guarded universal quantifiers yields only polynomially many cases \(to be more precise, at mostncn^\{c\}cases, whereccis the quantifier rank ofφ\\varphi\)\. At every computation path of this process, we are left with a Boolean combination of formulas from the atomic layer, together with𝖠𝗅𝗅𝖯𝗈𝗌​\(x\)\\mathsf\{AllPos\}\(x\)and𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)\\mathsf\{AllNeg\}\(x\)\. Thanks to Theorem[4\.3](https://arxiv.org/html/2607.06407#S4.Thmtheorem3)and to the hypothesis thatEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∈P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}andEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∈P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}, and considering that the formulaφ\\varphiis fixed, we can do that evaluation in polynomial time\.

For the second item, we consider the following formula from the quantified layer ofExplAIner:

𝗇𝗆𝖠𝖷𝗉​\(x,y\):=∃z​\(¬𝗐𝖠𝖷𝗉​\(x,y\)∨\[z≺y∧𝗐𝖠𝖷𝗉​\(x,z\)\]\)\.\\mathsf\{nmAXp\}\(x,y\)\\ :=\\ \\exists z\\ \\big\(\\neg\\mathsf\{wAXp\}\(x,y\)\\vee\[z\\prec y\\wedge\\mathsf\{wAXp\}\(x,z\)\]\\big\)\.𝗇𝗆𝖠𝖷𝗉\\mathsf\{nmAXp\}defines the pairs\(𝐞1,𝐞2\)\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)such that the partial instance𝐞2\\mathbf\{e\}\_\{2\}is not a minimum abductive explanation for the instance𝐞1\\mathbf\{e\}\_\{1\}, that is, it is logically equivalent to¬𝗆𝖠𝖷𝗉​\(x,y\)\\neg\\mathsf\{mAXp\}\(x,y\)\. We conclude the proof using the following intermediate result, whose proof can be found in the appendix of this work \(refer to Section[A\.5](https://arxiv.org/html/2607.06407#A1.SS5)\)\.

###### Lemma 4\.6\.

Eval\(𝗆𝖠𝖷𝗉​\(x,y\),𝖣𝖳𝗋𝖾𝖾\)\(\\mathsf\{mAXp\}\(x,y\),\\mathsf\{DTree\}\)iscoNP\{\\rm coNP\}\-hard\.

∎

### 4\.3\.TheExplAInerlogic

We defineExplAIneras the logic obtained by taking Boolean combinations of formulas from the quantified layer\. In particular, since we can negate quantified formulas, in this third layer we are allowed to use unguarded universal quantifiers\. Nevertheless, notice that a necessary condition for a formula to have a valid syntax according to theExplAInerlogic is that alternations between unguarded quantifiers cannot occur\.

We now provide a precise characterization of the complexity of the evaluation problem forExplAIner\. More specifically, we establish that this problem can always be solved in theBoolean Hierarchy overNP\{\\rm NP\}\(Wechsung,[1985](https://arxiv.org/html/2607.06407#bib.bib5); Caiet al\.,[1988](https://arxiv.org/html/2607.06407#bib.bib2)\), i\.e\., in the class consisting of Boolean combinations ofNP\{\\rm NP\}languages\. In fact, we will show that theExplAInerlogic captures the entire Boolean Hierarchy\.

For the following theorem, we denote the levels of the Boolean Hierarchy byBHk\{\\rm BH\}\_\{k\}, and we denote byBH\{\\rm BH\}the Boolean Hierarchy consisting of all these levels\.

###### Theorem 4\.7\.

The following statements hold:

1. \(1\)Letφ\\varphibe anExplAInerformula\. Then there exists ak≥1k\\geq 1such that, for every class of models𝒞\\mathcal\{C\}such thatEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∈P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}andEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∈P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}, it holds thatEval\(φ,𝒞\)∈BHk\(\\varphi,\\mathcal\{C\}\)\\in\{\\rm BH\}\_\{k\}\.
2. \(2\)For everyk≥1k\\geq 1, there exists anExplAInerformulaφ\\varphisuch thatEval\(φ,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi,\\mathsf\{DTree\}\)isBHk\{\\rm BH\}\_\{k\}\-hard\.

This result tells us thatExplAInermeets one of the fundamental criteria for an interpretability logic, namely that we can evaluate anExplAInerformula over a tuple of partial instances in polynomial time with a polynomial number of calls to anNP\{\\rm NP\}oracle\. In fact, by definition of the Boolean hierarchy, the evaluation of a fixedExplAInerformula can be done with a constant number of calls to anNP\{\\rm NP\}oracle\. Thus, we argue that the technology ofSATsolvers will allow us to tractably evaluateExplAInerover classes that support consistency and validity checks in polynomial time\. More precisely, Theorem[4\.7](https://arxiv.org/html/2607.06407#S4.Thmtheorem7)requires the class of models𝒞\\mathcal\{C\}to satisfy thatEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∈P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}andEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∈P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}in order forExplAInerto be able to tractably solve its evaluation problem over that class\. This includes decision trees, but also richer representation classes like𝖽​\-​𝖣𝖭𝖭𝖥\\mathsf\{d\}\\text\{\-\}\\mathsf\{DNNF\}circuits\. Moreover, this includes fragments of the class of circuits corresponding to propositional formulas in conjunctive normal form \(𝖢𝖭𝖥\\mathsf\{CNF\}\) whose satisfiability can be decided in polynomial time, such as the class of circuits corresponding to CNF formulas in which each clause contains at most two literals \(2​𝖢𝖭𝖥2\\mathsf\{CNF\}\), and the class of circuits corresponding to Horn CNF formulas \(𝖧𝖮𝖱𝖭\\mathsf\{HORN\}\)\. We formally state these results in the following corollary\.

###### Corollary 4\.8\.

Letφ\\varphibe anExplAInerformula\. ThenEval\(φ,𝖽​\-​𝖣𝖭𝖭𝖥\)∈BH\(\\varphi,\\mathsf\{d\}\\text\{\-\}\\mathsf\{DNNF\}\)\\in\{\\rm BH\},Eval\(φ,2​𝖢𝖭𝖥\)∈BH\(\\varphi,2\\mathsf\{CNF\}\)\\in\{\\rm BH\}, andEval\(φ,𝖧𝖮𝖱𝖭\)∈BH\(\\varphi,\\mathsf\{HORN\}\)\\in\{\\rm BH\}\.

In what follows, we provide a proof of Theorem[4\.7](https://arxiv.org/html/2607.06407#S4.Thmtheorem7)\.

###### Proof of Theorem[4\.7](https://arxiv.org/html/2607.06407#S4.Thmtheorem7)\.

We consider languages over a finite alphabetΣ\\Sigma\. First, we introduce Boolean operations between complexity classes as follows\(Wechsung,[1985](https://arxiv.org/html/2607.06407#bib.bib5)\):

1. \(1\)A∨B=\{LA∪LB\|LA∈A​and​LB∈B\}A\\vee B=\\\{L\_\{A\}\\cup L\_\{B\}\\ \|\\ L\_\{A\}\\in A\\text\{ and \}L\_\{B\}\\in B\\\};
2. \(2\)A∧B=\{LA∩LB\|LA∈A​and​LB∈B\}A\\wedge B=\\\{L\_\{A\}\\cap L\_\{B\}\\ \|\\ L\_\{A\}\\in A\\text\{ and \}L\_\{B\}\\in B\\\};
3. \(3\)coA=\{L¯\|L∈A\}\\rm\{co\}A=\\\{\\overline\{L\}\\ \|\\ L\\in A\\\}\.

Then, the Boolean HierarchyBH\{\\rm BH\}is defined as the union∪k≥1BHk\\cup\_\{k\\geq 1\}\{\\rm BH\}\_\{k\}\(Caiet al\.,[1988](https://arxiv.org/html/2607.06407#bib.bib2)\), where:

1. \(1\)BH1=NP\{\\rm BH\}\_\{1\}=\{\\rm NP\};
2. \(2\)BH2​i=BH2​i−1∧coNP\{\\rm BH\}\_\{2i\}=\{\\rm BH\}\_\{2i\-1\}\\wedge\{\\rm coNP\};
3. \(3\)BH2​i\+1=BH2​i∨NP\{\\rm BH\}\_\{2i\+1\}=\{\\rm BH\}\_\{2i\}\\vee\{\\rm NP\}\.

Note thatcoNP⊆BH2\{\\rm coNP\}\\subseteq\{\\rm BH\}\_\{2\}\. In fact, letL∈coNPL\\in\{\\rm coNP\}and note thatL=Σ∗∩LL=\\Sigma^\{\\ast\}\\cap L, whereΣ∗∈BH1\\Sigma^\{\\ast\}\\in\{\\rm BH\}\_\{1\}\.

With this definition, every Boolean combination ofNP\{\\rm NP\}andcoNP\{\\rm coNP\}languages is contained inBHk\{\\rm BH\}\_\{k\}for some positive integerkk\.

For the first item of the theorem, letφ\\varphibe anExplAInerformula\. We know thatφ\\varphiis a fixed Boolean combination of formulas from the quantified layer ofExplAIner\. Now let𝒞\\mathcal\{C\}be a class of models such thatEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∈P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}andEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∈P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}\. Thanks to the first part of Theorem[4\.5](https://arxiv.org/html/2607.06407#S4.Thmtheorem5)we know thatEval\(ψ,𝒞\)∈NP\(\\psi,\\mathcal\{C\}\)\\in\{\\rm NP\}for every formulaψ\\psifrom the quantified layer ofExplAInerthat appears as a subformula ofφ\\varphi\. This means that the evaluation problemEval\(φ,𝒞\)\(\\varphi,\\mathcal\{C\}\)corresponds to a fixed Boolean combination of languages inNP\{\\rm NP\}, and so it must be contained inBHk\{\\rm BH\}\_\{k\}for somek≥1k\\geq 1\. Notice that such akkdoes depend onφ\\varphibut not on𝒞\\mathcal\{C\}\.

We now turn our attention to the second item of the theorem\. We will first describe a family of decision problems known to be complete for every level of the Boolean hierarchy\. As usual, let us denote bySATthe language of propositional formulas that are satisfiable, and byUNSATthe language of propositional formulas that are not satisfiable\. For eachk≥1k\\geq 1we define the languageSAT​\(k\)\\textsc\{SAT\}\(k\)recursively as follows:

1. \(1\)SAT​\(1\):=\{\(φ1\)∣φ1∈SAT\}\\textsc\{SAT\}\(1\)\\ :=\\ \\\{\(\\varphi\_\{1\}\)\\ \\mid\\ \\varphi\_\{1\}\\in\\textsc\{SAT\}\\\};
2. \(2\)SAT​\(2​i\):=\{\(φ1,…​φ2​i\)∣\(φ1,…​φ2​i−1\)∈SAT​\(2​i−1\)∧φ2​i∈UNSAT\}\\textsc\{SAT\}\(2i\)\\ :=\\ \\\{\(\\varphi\_\{1\},\\dots\\varphi\_\{2i\}\)\\ \\mid\\ \(\\varphi\_\{1\},\\dots\\varphi\_\{2i\-1\}\)\\in\\textsc\{SAT\}\(2i\-1\)\\,\\wedge\\,\\varphi\_\{2i\}\\in\\textsc\{UNSAT\}\\\};
3. \(3\)SAT​\(2​i\+1\):=\{\(φ1,…​φ2​i\+1\)∣\(φ1,…​φ2​i\)∈SAT​\(2​i\)∨φ2​i\+1∈SAT\}\\textsc\{SAT\}\(2i\+1\)\\ :=\\ \\\{\(\\varphi\_\{1\},\\dots\\varphi\_\{2i\+1\}\)\\ \\mid\\ \(\\varphi\_\{1\},\\dots\\varphi\_\{2i\}\)\\in\\textsc\{SAT\}\(2i\)\\,\\vee\\,\\varphi\_\{2i\+1\}\\in\\textsc\{SAT\}\\\}\.

It is known that, for everyk≥1k\\geq 1,SAT​\(k\)\\textsc\{SAT\}\(k\)isBHk\{\\rm BH\}\_\{k\}\-complete\(Caiet al\.,[1988](https://arxiv.org/html/2607.06407#bib.bib2)\)\. We will now fix ak≥1k\\geq 1and prove that there exists anExplAInerformulaφk\\varphi\_\{k\}such thatSAT​\(k\)\\textsc\{SAT\}\(k\)can be reduced in polynomial time toEval\(φk,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi\_\{k\},\\mathsf\{DTree\}\), thus concluding the hardness item of the theorem\.

We know from Lemma[4\.6](https://arxiv.org/html/2607.06407#S4.Thmtheorem6)that the following language isNP\{\\rm NP\}\-hard:

L=\{\(𝒯,𝐞,𝐞′\)∣𝒯​is a decision tree,​𝐞​and​𝐞′​are partial instances and​𝒯⊧¬𝗆𝖠𝖷𝗉​\(𝐞,𝐞′\)\}\.L=\\\{\(\\mathcal\{T\},\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)\\ \\mid\\ \\mathcal\{T\}\\text\{ is a decision tree, \}\\mathbf\{e\}\\text\{ and \}\\mathbf\{e\}^\{\\prime\}\\text\{ are partial instances and \}\\mathcal\{T\}\\models\\neg\\mathsf\{mAXp\}\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)\\\}\.Also, because the class of decision trees satisfies the hypothesis of the first item of Theorem[4\.5](https://arxiv.org/html/2607.06407#S4.Thmtheorem5), we actually know thatLLisNP\{\\rm NP\}\-complete\. Hence, we have a polynomial\-time algorithm that, given a propositional formulaψ\\psi, constructs a decision tree𝒯ψ\\mathcal\{T\}\_\{\\psi\}and partial instances𝐞ψ\\mathbf\{e\}\_\{\\psi\},𝐞ψ′\\mathbf\{e\}\_\{\\psi\}^\{\\prime\}such that:

\(3\)ψ∈SAT⇔𝒯ψ⊧¬𝗆𝖠𝖷𝗉​\(𝐞ψ,𝐞ψ′\)\.\\psi\\in\\textsc\{SAT\}\\quad\\iff\\quad\\mathcal\{T\}\_\{\\psi\}\\models\\neg\\mathsf\{mAXp\}\(\\mathbf\{e\}\_\{\\psi\},\\mathbf\{e\}\_\{\\psi\}^\{\\prime\}\)\.
Let\(ψ1,…,ψk\)\(\\psi\_\{1\},\\ldots,\\psi\_\{k\}\)be a tuple ofkkpropositional formulas, and assume that, for eachi∈\{1,…,k\}i\\in\\\{1,\\dots,k\\\}, the decision tree𝒯ψi\\mathcal\{T\}\_\{\\psi\_\{i\}\}has dimensionnin\_\{i\}\. Then a decision tree𝒯\\mathcal\{T\}of dimensiond=k\+∑ℓ=1knℓd=k\+\\sum\_\{\\ell=1\}^\{k\}\{n\_\{\\ell\}\}is defined as follows:

11𝒯ψ1\\mathcal\{T\}\_\{\\psi\_\{1\}\}22𝒯ψ2\\mathcal\{T\}\_\{\\psi\_\{2\}\}33𝒯ψ3\\mathcal\{T\}\_\{\\psi\_\{3\}\}⋯\\cdotskk𝒯ψk\\mathcal\{T\}\_\{\\psi\_\{k\}\}𝐭𝐫𝐮𝐞\\mathbf\{true\}1101101100110

where, for eachi∈\{1,…,k\}i\\in\\\{1,\\dots,k\\\},𝒯ψi\\mathcal\{T\}\_\{\\psi\_\{i\}\}mentions the features:

Bi:=\{si\+1,si\+2,…​si\+ni\}where​si=k\+∑ℓ=1i−1nℓ\.B\_\{i\}\\ :=\\ \\\{s\_\{i\}\+1,\\,s\_\{i\}\+2,\\,\\dots s\_\{i\}\+n\_\{i\}\\\}\\qquad\\text\{where \}s\_\{i\}=k\+\\sum\_\{\\ell=1\}^\{i\-1\}\{n\_\{\\ell\}\}\.This way, we ensure that, for everyi≠ji\\neq j,𝒯ψi\\mathcal\{T\}\_\{\\psi\_\{i\}\}and𝒯ψj\\mathcal\{T\}\_\{\\psi\_\{j\}\}are defined over disjoint sets of features\. Moreover, define the following partial instances of dimensiondd:

- •For eachi∈\{1,…,k\}i\\in\\\{1,\\ldots,k\\\}, the partial instance𝐞i\\mathbf\{e\}\_\{i\}is defined as\{0\}i−1⋅\{1\}⋅\{0\}k−i⋅\{⊥\}d−k\\\{0\\\}^\{i\-1\}\\cdot\\\{1\\\}\\cdot\\\{0\\\}^\{k\-i\}\\cdot\\\{\\bot\\\}^\{d\-k\}\.
- •For eachi∈\{1,…,k\}i\\in\\\{1,\\ldots,k\\\}, the partial instances𝐞i,1\\mathbf\{e\}\_\{i,1\}and𝐞i,2\\mathbf\{e\}\_\{i,2\}are defined as follows: - –𝐞i,1​\[i\]=𝐞i,2​\[i\]=1\\mathbf\{e\}\_\{i,1\}\[i\]=\\mathbf\{e\}\_\{i,2\}\[i\]=1; - –𝐞i,1​\[j\]=𝐞i,2​\[j\]=0\\mathbf\{e\}\_\{i,1\}\[j\]=\\mathbf\{e\}\_\{i,2\}\[j\]=0for everyj∈\{1,…,k\}∖\{i\}j\\in\\\{1,\\ldots,k\\\}\\setminus\\\{i\\\}; - –𝐞i,1​\[j\]=𝐞ψi​\[j−si\]\\mathbf\{e\}\_\{i,1\}\[j\]=\\mathbf\{e\}\_\{\\psi\_\{i\}\}\[j\-s\_\{i\}\]for everyj∈Bij\\in B\_\{i\}; - –𝐞i,2​\[j\]=𝐞ψi′​\[j−si\]\\mathbf\{e\}\_\{i,2\}\[j\]=\\mathbf\{e\}\_\{\\psi\_\{i\}\}^\{\\prime\}\[j\-s\_\{i\}\]for everyj∈Bij\\in B\_\{i\}; - –𝐞i,1​\[j\]=0\\mathbf\{e\}\_\{i,1\}\[j\]=0for everyj∈\{k\+1,…,d\}∖Bij\\in\\\{k\+1,\\ldots,d\\\}\\setminus B\_\{i\}; - –𝐞i,2​\[j\]=⊥\\mathbf\{e\}\_\{i,2\}\[j\]=\\botfor everyj∈\{k\+1,…,d\}∖Bij\\in\\\{k\+1,\\ldots,d\\\}\\setminus B\_\{i\}\.

Moreover, consider the followingExplAInerformulas:

𝗋𝗐𝖠𝖷𝗉​\(x,y,w\)\\displaystyle\\mathsf\{rwAXp\}\(x,y,w\):=\\displaystyle:=w⊆x∧w⊆y∧𝖥𝗎𝗅𝗅​\(x\)∧y⊆x∧\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\)→𝖠𝗅𝗅𝖯𝗈𝗌​\(y\)\)∧\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)→𝖠𝗅𝗅𝖭𝖾𝗀​\(y\)\);\\displaystyle w\\subseteq x\\wedge w\\subseteq y\\wedge\\mathsf\{Full\}\(x\)\\wedge y\\subseteq x\\wedge\(\\mathsf\{AllPos\}\(x\)\\to\\mathsf\{AllPos\}\(y\)\)\\wedge\(\\mathsf\{AllNeg\}\(x\)\\to\\mathsf\{AllNeg\}\(y\)\);𝗋𝗆𝖠𝖷𝗉​\(x,y,w\)\\displaystyle\\mathsf\{rmAXp\}\(x,y,w\):=\\displaystyle:=𝗋𝗐𝖠𝖷𝗉​\(x,y,w\)∧¬∃z​\(w⊆z∧𝗋𝗐𝖠𝖷𝗉​\(x,z,w\)∧z≺y\)\.\\displaystyle\\mathsf\{rwAXp\}\(x,y,w\)\\ \\wedge\\ \\neg\\exists z\\,\\big\(w\\subseteq z\\,\\wedge\\,\\mathsf\{rwAXp\}\(x,z,w\)\\,\\wedge\\,z\\prec y\\big\)\.These formulas should be interpreted as the usual predicates, but relativized to one branch of𝒯\\mathcal\{T\}\. More concretely, the partial instancewwwill serve to select in which of the branches𝒯ψ1,𝒯ψ2,…,𝒯ψk\\mathcal\{T\}\_\{\\psi\_\{1\}\},\\mathcal\{T\}\_\{\\psi\_\{2\}\},\\dots,\\mathcal\{T\}\_\{\\psi\_\{k\}\}we will look at\.

Notice that the decision tree𝒯\\mathcal\{T\}and the partial instances𝐞1\\mathbf\{e\}\_\{1\},…\\ldots,𝐞k\\mathbf\{e\}\_\{k\},𝐞1,1\\mathbf\{e\}\_\{1,1\},𝐞1,2\\mathbf\{e\}\_\{1,2\},…\\ldots,𝐞k,1\\mathbf\{e\}\_\{k,1\},𝐞k,2\\mathbf\{e\}\_\{k,2\}can be constructed in polynomial time in the size of\(ψ1,…,ψk\)\(\\psi\_\{1\},\\ldots,\\psi\_\{k\}\)\. Besides, from the definition of these elements, for everyi∈\{1,…,k\}i\\in\\\{1,\\ldots,k\\\}it holds that

\(4\)𝒯ψi⊧𝗆𝖠𝖷𝗉​\(𝐞ψi,𝐞ψi′\)⇔𝒯⊧𝗋𝗆𝖠𝖷𝗉​\(𝐞i,1,𝐞i,2,𝐞i\)\.\\mathcal\{T\}\_\{\\psi\_\{i\}\}\\models\\ \\mathsf\{mAXp\}\(\\mathbf\{e\}\_\{\\psi\_\{i\}\},\\mathbf\{e\}\_\{\\psi\_\{i\}\}^\{\\prime\}\)\\quad\\iff\\quad\\mathcal\{T\}\\models\\ \\mathsf\{rmAXp\}\(\\mathbf\{e\}\_\{i,1\},\\mathbf\{e\}\_\{i,2\},\\mathbf\{e\}\_\{i\}\)\.Finally, letφk\\varphi\_\{k\}be theExplAInerformula obtained by constructing the following sequences of formulasα1,…,αk\\alpha\_\{1\},\\dots,\\alpha\_\{k\}, and then definingφk​\(x1,1,x1,2,x1,…,xk,1,xk,2,xk\):=αk​\(x1,1,x1,2,x1,…,xk,1,xk,2,xk\)\\varphi\_\{k\}\(x\_\{1,1\},x\_\{1,2\},x\_\{1\},\\ldots,x\_\{k,1\},x\_\{k,2\},x\_\{k\}\):=\\alpha\_\{k\}\(x\_\{1,1\},x\_\{1,2\},x\_\{1\},\\ldots,x\_\{k,1\},x\_\{k,2\},x\_\{k\}\):

α1\\displaystyle\\alpha\_\{1\}:=\\displaystyle:=¬𝗋𝗆𝖠𝖷𝗉​\(x1,1,x1,2,x1\);\\displaystyle\\neg\\mathsf\{rmAXp\}\(x\_\{1,1\},x\_\{1,2\},x\_\{1\}\);α2​ℓ\\displaystyle\\alpha\_\{2\\ell\}:=\\displaystyle:=\(α2​ℓ−1∧𝗋𝗆𝖠𝖷𝗉​\(x2​ℓ,1,x2​ℓ,2,x2​ℓ\)\);\\displaystyle\(\\alpha\_\{2\\ell\-1\}\\wedge\\mathsf\{rmAXp\}\(x\_\{2\\ell,1\},x\_\{2\\ell,2\},x\_\{2\\ell\}\)\);α2​ℓ\+1\\displaystyle\\alpha\_\{2\\ell\+1\}:=\\displaystyle:=\(α2​ℓ∨¬𝗋𝗆𝖠𝖷𝗉​\(x2​ℓ\+1,1,x2​ℓ\+1,2,x2​ℓ\+1\)\)\.\\displaystyle\(\\alpha\_\{2\\ell\}\\vee\\neg\\mathsf\{rmAXp\}\(x\_\{2\\ell\+1,1\},x\_\{2\\ell\+1,2\},x\_\{2\\ell\+1\}\)\)\.For example, we have that:

α2\\displaystyle\\alpha\_\{2\}=\\displaystyle=\(¬𝗋𝗆𝖠𝖷𝗉​\(x1,1,x1,2,x1\)∧𝗋𝗆𝖠𝖷𝗉​\(x2,1,x2,2,x2\)\)\\displaystyle\(\\neg\\mathsf\{rmAXp\}\(x\_\{1,1\},x\_\{1,2\},x\_\{1\}\)\\wedge\\mathsf\{rmAXp\}\(x\_\{2,1\},x\_\{2,2\},x\_\{2\}\)\)α3\\displaystyle\\alpha\_\{3\}=\\displaystyle=\(¬𝗋𝗆𝖠𝖷𝗉​\(x1,1,x1,2,x1\)∧𝗋𝗆𝖠𝖷𝗉​\(x2,1,x2,2,x2\)\)∨¬𝗋𝗆𝖠𝖷𝗉​\(x3,1,x3,2,x3\)\\displaystyle\(\\neg\\mathsf\{rmAXp\}\(x\_\{1,1\},x\_\{1,2\},x\_\{1\}\)\\wedge\\mathsf\{rmAXp\}\(x\_\{2,1\},x\_\{2,2\},x\_\{2\}\)\)\\vee\\neg\\mathsf\{rmAXp\}\(x\_\{3,1\},x\_\{3,2\},x\_\{3\}\)Combining conditions \([3](https://arxiv.org/html/2607.06407#S4.E3)\) and \([4](https://arxiv.org/html/2607.06407#S4.E4)\) with the definitionφk​\(x1,1,x1,2,x1,…,xk,1,xk,2,xk\)\\varphi\_\{k\}\(x\_\{1,1\},x\_\{1,2\},x\_\{1\},\\ldots,x\_\{k,1\},x\_\{k,2\},x\_\{k\}\), we conclude that\(ψ1,…,ψk\)∈SAT​\(k\)\(\\psi\_\{1\},\\ldots,\\psi\_\{k\}\)\\in\\textsc\{SAT\}\(k\)if and only if𝒯⊧φk​\(𝐞1,1,𝐞1,2,𝐞1,…,𝐞k,1,𝐞k,2,𝐞k\)\\mathcal\{T\}\\models\\varphi\_\{k\}\(\\mathbf\{e\}\_\{1,1\},\\mathbf\{e\}\_\{1,2\},\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k,1\},\\mathbf\{e\}\_\{k,2\},\\mathbf\{e\}\_\{k\}\)\. Given thatSAT​\(k\)\\textsc\{SAT\}\(k\)isBHk\{\\rm BH\}\_\{k\}\-complete and that the decision tree𝒯\\mathcal\{T\}and the partial instances𝐞1,1\\mathbf\{e\}\_\{1,1\},𝐞1,2\\mathbf\{e\}\_\{1,2\},𝐞1\\mathbf\{e\}\_\{1\},…\\ldots,𝐞k,1\\mathbf\{e\}\_\{k,1\},𝐞k,2\\mathbf\{e\}\_\{k,2\},𝐞k\\mathbf\{e\}\_\{k\}can be constructed in polynomial time in the size of\(ψ1,…,ψk\)\(\\psi\_\{1\},\\ldots,\\psi\_\{k\}\), we conclude thatEval\(φk,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi\_\{k\},\\mathsf\{DTree\}\)isBHk\{\\rm BH\}\_\{k\}\-hard\. This completes the proof of the theorem\. ∎

### 4\.4\.On the expressiveness ofExplAIner

ExplAInerallows us to express in a simple way the basic notions of explainability studied in this paper\. Moreover, its evaluation problem is tractable given access toSATsolvers\.[Figure 3](https://arxiv.org/html/2607.06407#S4.F3)shows how all the queries defined in Section[2\.2](https://arxiv.org/html/2607.06407#S2.SS2)can be expressed in theExplAInerlogic\. Just for clarity we use unguarded universal quantifiers, which are not allowed according to theExplAInersyntax, because in these cases they can be rewritten as negations of unguarded existential quantifiers, as we discussed in Section[4\.3](https://arxiv.org/html/2607.06407#S4.SS3)\. We also make use of some auxiliary predicates defined in the appendix \(refer to Section[A\.7](https://arxiv.org/html/2607.06407#A1.SS7)\)\.

Probably the most complicated formula of this section is the one used to express the query of relevant feature𝖱𝖥​\(x,y\)\\mathsf\{RF\}\(x,y\)\. The idea there is to guess a weak abductive explanationwwcontaining the assigned feature under consideration and verify that undefining the feature makeswwlose the property of weak abductiveness\. In fact, suppose first thatyyis indeed a relevant feature forxx\. Then some abductive explanationwwcontainsyy\. Becausewwis minimal, undefiningyyfromwwcannot produce another weak abductive explanation, so the witness pairw,zw,zexists\. Conversely, ifyyis not a relevant feature forxx, then it cannot be contained in any minimal abductive explanation forxx\. Assume toward a contradiction that there exist witnesseswwandzzfor the formula, wherewwis a weak abductive explanation containingyy,zzis obtained fromwwby undefiningyy, andzzis not a weak abductive explanation\. Now letu⊆wu\\subseteq wbe an abductive explanation\. Asyyis not relevant,uucannot containyy, and thereforeu⊆zu\\subseteq z\. Since every completion ofzzis also a completion ofuu, it would follow thatzzis a weak abductive explanation, a contradiction\. Hence no such witness pairw,zw,zcan exist\.

𝗐𝖠𝖷𝗉​\(x,y\)=\\displaystyle\\mathsf\{wAXp\}\(x,y\)=\\𝖥𝗎𝗅𝗅​\(x\)∧y⊆x∧\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\)→𝖠𝗅𝗅𝖯𝗈𝗌​\(y\)\)∧\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)→𝖠𝗅𝗅𝖭𝖾𝗀​\(y\)\)\\displaystyle\\mathsf\{Full\}\(x\)\\wedge y\\subseteq x\\wedge\(\\mathsf\{AllPos\}\(x\)\\to\\mathsf\{AllPos\}\(y\)\)\\wedge\(\\mathsf\{AllNeg\}\(x\)\\to\\mathsf\{AllNeg\}\(y\)\)𝖠𝖷𝗉​\(x,y\)=\\displaystyle\\mathsf\{AXp\}\(x,y\)=\\𝗐𝖠𝖷𝗉​\(x,y\)∧∀z​\[z⊂y→¬𝗐𝖠𝖷𝗉​\(x,z\)\]\\displaystyle\\mathsf\{wAXp\}\(x,y\)\\wedge\\forall z\\ \[z\\subset y\\to\\neg\\mathsf\{wAXp\}\(x,z\)\]𝗆𝖠𝖷𝗉​\(x,y\)=\\displaystyle\\mathsf\{mAXp\}\(x,y\)=\\𝗐𝖠𝖷𝗉​\(x,y\)∧∀z​\[z≺y→¬𝗐𝖠𝖷𝗉​\(x,z\)\]\\displaystyle\\mathsf\{wAXp\}\(x,y\)\\wedge\\forall z\\ \[z\\prec y\\to\\neg\\mathsf\{wAXp\}\(x,z\)\]𝗐𝖢𝖷𝗉​\(x,y\)=\\displaystyle\\mathsf\{wCXp\}\(x,y\)=\\𝖥𝗎𝗅𝗅​\(x\)∧y⊆x∧\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\)→¬𝖠𝗅𝗅𝖯𝗈𝗌​\(y\)\)∧\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)→¬𝖠𝗅𝗅𝖭𝖾𝗀​\(y\)\)\\displaystyle\\mathsf\{Full\}\(x\)\\wedge y\\subseteq x\\wedge\(\\mathsf\{AllPos\}\(x\)\\to\\neg\\mathsf\{AllPos\}\(y\)\)\\wedge\(\\mathsf\{AllNeg\}\(x\)\\to\\neg\\mathsf\{AllNeg\}\(y\)\)𝖢𝖷𝗉​\(x,y\)=\\displaystyle\\mathsf\{CXp\}\(x,y\)=\\𝗐𝖢𝖷𝗉​\(x,y\)∧∀z​\[y⊂z→¬𝗐𝖢𝖷𝗉​\(x,z\)\]\\displaystyle\\mathsf\{wCXp\}\(x,y\)\\wedge\\forall z\\ \[y\\subset z\\to\\neg\\mathsf\{wCXp\}\(x,z\)\]𝗆𝖢𝖷𝗉​\(x,y\)=\\displaystyle\\mathsf\{mCXp\}\(x,y\)=\\𝗐𝖢𝖷𝗉​\(x,y\)∧∀z​\[y≺z→¬𝗐𝖢𝖷𝗉​\(x,z\)\]\\displaystyle\\mathsf\{wCXp\}\(x,y\)\\wedge\\forall z\\ \[y\\prec z\\to\\neg\\mathsf\{wCXp\}\(x,z\)\]𝖬𝖢𝖱​\(x,y\)=\\displaystyle\\mathsf\{MCR\}\(x,y\)=\\𝖥𝗎𝗅𝗅\(x\)∧𝖥𝗎𝗅𝗅\(y\)∧¬\(𝖠𝗅𝗅𝖯𝗈𝗌\(x\)↔𝖠𝗅𝗅𝖯𝗈𝗌\(y\)\)∧\\displaystyle\\mathsf\{Full\}\(x\)\\wedge\\mathsf\{Full\}\(y\)\\wedge\\neg\(\\mathsf\{AllPos\}\(x\)\\leftrightarrow\\mathsf\{AllPos\}\(y\)\)\\ \\wedge∀z\(\[𝖥𝗎𝗅𝗅\(z\)∧¬\(𝖠𝗅𝗅𝖯𝗈𝗌\(x\)↔𝖠𝗅𝗅𝖯𝗈𝗌\(z\)\)\]→𝖫𝖤𝖧\(x,y,z\)\)\\displaystyle\\hskip 101\.00021pt\\forall z\\ \\big\(\[\\mathsf\{Full\}\(z\)\\wedge\\neg\(\\mathsf\{AllPos\}\(x\)\\leftrightarrow\\mathsf\{AllPos\}\(z\)\)\]\\to\\mathsf\{LEH\}\(x,y,z\)\\big\)𝖬𝖢𝖠​\(x,y\)=\\displaystyle\\mathsf\{MCA\}\(x,y\)=\\𝖥𝗎𝗅𝗅\(x\)∧𝖥𝗎𝗅𝗅\(y\)∧\(𝖠𝗅𝗅𝖯𝗈𝗌\(x\)↔𝖠𝗅𝗅𝖯𝗈𝗌\(y\)\)∧\\displaystyle\\mathsf\{Full\}\(x\)\\wedge\\mathsf\{Full\}\(y\)\\wedge\(\\mathsf\{AllPos\}\(x\)\\leftrightarrow\\mathsf\{AllPos\}\(y\)\)\\ \\wedge∀z\(\[𝖥𝗎𝗅𝗅\(z\)∧\(𝖠𝗅𝗅𝖯𝗈𝗌\(x\)↔𝖠𝗅𝗅𝖯𝗈𝗌\(z\)\)\]→𝖫𝖤𝖧\(x,z,y\)\)\\displaystyle\\hskip 101\.00021pt\\forall z\\ \\big\(\[\\mathsf\{Full\}\(z\)\\wedge\(\\mathsf\{AllPos\}\(x\)\\leftrightarrow\\mathsf\{AllPos\}\(z\)\)\]\\to\\mathsf\{LEH\}\(x,z,y\)\\big\)𝖭𝖥​\(x,y\)=\\displaystyle\\mathsf\{NF\}\(x,y\)=\\𝖲𝗂𝗇𝗀𝗅𝖾​\(y\)∧𝖥𝗎𝗅𝗅​\(x\)∧y⊆x∧¬∃z​\[𝗐𝖠𝖷𝗉​\(x,z\)∧¬\(y⊆z\)\]\\displaystyle\{\\sf Single\}\(y\)\\wedge\\mathsf\{Full\}\(x\)\\wedge y\\subseteq x\\wedge\\neg\\exists z\\ \[\\mathsf\{wAXp\}\(x,z\)\\wedge\\neg\(y\\subseteq z\)\]𝖱𝖥​\(x,y\)=\\displaystyle\\mathsf\{RF\}\(x,y\)=\\𝖲𝗂𝗇𝗀𝗅𝖾​\(y\)∧𝖥𝗎𝗅𝗅​\(x\)∧y⊆x∧∃w,z​\[𝗐𝖠𝖷𝗉​\(x,w\)∧𝖠𝖽𝖽​\(z,y,w\)∧¬𝗐𝖠𝖷𝗉​\(x,z\)\]\\displaystyle\{\\sf Single\}\(y\)\\wedge\\mathsf\{Full\}\(x\)\\wedge y\\subseteq x\\wedge\\exists w,z\\ \[\\mathsf\{wAXp\}\(x,w\)\\wedge\\mathsf\{Add\}\(z,y,w\)\\wedge\\neg\\mathsf\{wAXp\}\(x,z\)\]Figure 3\.Formulas that express all queries in Section[2\.2](https://arxiv.org/html/2607.06407#S2.SS2)usingExplAIner\.ExplAIner formulas for Explainability queries\.The figure lists ExplAIner formulas for the explainability queries defined in Section 2\.2: weak abductive explanation, abductive explanation, minimum abductive explanation, weak contrastive explanation, contrastive explanation, maximum contrastive explanation, minimum change required, maximum change allowed, necessary feature, and relevant feature\.

It should be noted that if𝒞\\mathcal\{C\}is a class of models such thatEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∈P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}butEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∉P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\not\\in\{\\rm P\}\(under standard complexity\-theoretic assumptions\), then we still have that the problemEval\(ψ,𝒞\)\(\\psi,\\mathcal\{C\}\)is in the Boolean hierarchy for theExplAInerformulasψ\\psithat do not mention the predicate𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)\\mathsf\{AllNeg\}\(x\)\. Hence, the evaluation problem for this restricted fragment is still in the Boolean hierarchy, thus satisfying our criteria for an interpretability logic\. This is the case, for example, for𝖢𝖭𝖥\\mathsf\{CNF\}formulas, for which validity checks can be done in polynomial time but checking unsatisfiability iscoNP\{\\rm coNP\}\-complete\. In this case, we can still express queries such as minimum change required and maximum change allowed, since they can be expressed inExplAInerwithout mentioning the predicate𝖠𝗅𝗅𝖭𝖾𝗀​\(x\)\\mathsf\{AllNeg\}\(x\)\. Similarly, if𝒞\\mathcal\{C\}is a class of models such thatEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∈P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}butEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∉P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\not\\in\{\\rm P\}\(for example,𝖣𝖭𝖥\\mathsf\{DNF\}formulas\), then we still have that the evaluation problem is in the Boolean hierarchy forExplAInerformulasψ\\psithat do not mention the predicate𝖠𝗅𝗅𝖯𝗈𝗌​\(x\)\\mathsf\{AllPos\}\(x\)\.

Despite all the virtues ofExplAIner, unfortunately it is in general not able to solve computation problems efficiently\. In fact, note that the ability to tractably evaluate queries over concrete partial instances does not directly imply that positive answers to the query can be constructed efficiently\. We devote the rest of the paper to addressing this problem and propose a third logic that resolves it\.

## 5\.Opt\-FOIL: computing explanations efficiently

Given anExplAInerformulaφ​\(x,u1,…,uk\)\\varphi\(x,u\_\{1\},\\ldots,u\_\{k\}\), we use the notationφ​\[u1,…,uk\]​\(x\)\\varphi\[u\_\{1\},\\ldots,u\_\{k\}\]\(x\)to indicate thatxxis a distinguished variable andu1,…,uku\_\{1\},\\ldots,u\_\{k\}are parameters that define the possible values forxx\. In general, we use this syntax whenxxstores an explanation given an assignment for the variablesu1u\_\{1\},…\\ldots,uℓu\_\{\\ell\}\. For example, we writeφ​\[u\]​\(x\)=𝖠𝖷𝗉​\(u,x\)\\varphi\[u\]\(x\)=\\mathsf\{AXp\}\(u,x\)to indicate thatxxis an abductive explanation given an assignment for the variableuu\(that is,xxis an abductive explanation foruu\)\.

For each queryφ​\[x1,…,xk\]​\(x\)\\varphi\[x\_\{1\},\\ldots,x\_\{k\}\]\(x\)inExplAInerand𝒞\\mathcal\{C\}a class of models, we define the computation problem𝖢𝗈𝗆𝗉​\(φ,𝒞\)\\mathsf\{Comp\}\(\\varphi,\\mathcal\{C\}\)as follows:

### 5\.1\.The computational drawback ofExplAIner

We proved in Section[4\.3](https://arxiv.org/html/2607.06407#S4.SS3)that theExplAInerlogic admits tractable evaluations over adequate classes of models, which allows us to check if a partial instance is an answer for some explainability query\. The next step in the study ofExplAIneris to establish the complexity of actually computing such answers\. Unfortunately, the following result tells us that this problem cannot be solved with a polynomial number of calls to anNP\{\\rm NP\}oracle, showing an important limitation ofExplAIner\.

###### Theorem 5\.1\.

There exists anExplAInerformulaφ​\[y,z\]​\(x\)\\varphi\[y,z\]\(x\)such that𝖢𝗈𝗆𝗉​\(φ​\[y,z\]​\(x\),𝖣𝖳𝗋𝖾𝖾\)∉FPNP\\mathsf\{Comp\}\(\\varphi\[y,z\]\(x\),\\,\\mathsf\{DTree\}\)\\not\\in\{\\rm FP\}^\{\{\\rm NP\}\}unlessPH\{\\rm PH\}collapses toPNP\{\\rm P\}^\{\{\\rm NP\}\}\.

###### Proof\.

Consider the followingExplAInerformula:

φ​\(x,y,z\):=𝖬𝖺𝗑𝖱𝖾𝗅​\(x,y\)∧¬∃w​\[𝖬𝖺𝗑𝖱𝖾𝗅​\(w,z\)∧x⊆w∧𝖠𝗅𝗅𝖭𝖾𝗀​\(w\)\],\\varphi\(x,y,z\)\\ :=\\ \{\\sf MaxRel\}\(x,y\)\\wedge\\neg\\exists w\\ \[\{\\sf MaxRel\}\(w,z\)\\wedge x\\subseteq w\\wedge\\mathsf\{AllNeg\}\(w\)\],where𝖬𝖺𝗑𝖱𝖾𝗅​\(x,y\)\{\\sf MaxRel\}\(x,y\), defined during the proof of Theorem[3\.4](https://arxiv.org/html/2607.06407#S3.Thmtheorem4), is a formula from the atomic layer ofExplAInersuch thatℳ⊧𝖬𝖺𝗑𝖱𝖾𝗅​\(𝐞,𝐞′\)\\mathcal\{M\}\\models\{\\sf MaxRel\}\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\)if and only if𝐞⊥=𝐞⊥′\\mathbf\{e\}\_\{\\bot\}=\\mathbf\{e\}^\{\\prime\}\_\{\\bot\}, i\.e\., the sets of undefined features in𝐞\\mathbf\{e\}and𝐞′\\mathbf\{e\}^\{\\prime\}are the same\. Now consider the formulaφ​\(y,z\):=∃x​φ​\(x,y,z\)\\varphi\(y,z\):=\\exists x\\ \\varphi\(x,y,z\), which is aFOILformula with predicates\{⊆,⪯,𝖠𝗅𝗅𝖯𝗈𝗌,𝖠𝗅𝗅𝖭𝖾𝗀\}\\\{\\subseteq,\\preceq,\\mathsf\{AllPos\},\\mathsf\{AllNeg\}\\\}\. We will show thatEval\(φ,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi,\\mathsf\{DTree\}\)isNPNP\{\\rm NP\}^\{\{\\rm NP\}\}\-hard by a Karp reduction from the following well\-knownNPNP\{\\rm NP\}^\{\{\\rm NP\}\}\-complete problem \(see\(Arora and Barak,[2006](https://arxiv.org/html/2607.06407#bib.bib15)\)for a reference\): given a propositional formulaα​\(p,q\)\\alpha\(\\textbf\{p\},\\textbf\{q\}\)in𝖣𝖭𝖥\\mathsf\{DNF\}, wherepandqare sets of variables, decide if the quantified propositional formula∃p​∀q​α​\(p,q\)\\exists\\textbf\{p\}\\forall\\textbf\{q\}\\ \\alpha\(\\textbf\{p\},\\textbf\{q\}\)is true\. We will describe a polynomial\-time reduction that constructs a decision tree𝒯α\\mathcal\{T\}\_\{\\alpha\}and partial instances𝐞α,𝐞α′\\mathbf\{e\}\_\{\\alpha\},\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}such that∃p​∀q​α​\(p,q\)\\exists\\textbf\{p\}\\forall\\textbf\{q\}\\ \\alpha\(\\textbf\{p\},\\textbf\{q\}\)is true if and only if𝒯α⊧φ​\(𝐞α,𝐞α′\)\\mathcal\{T\}\_\{\\alpha\}\\models\\varphi\(\\mathbf\{e\}\_\{\\alpha\},\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}\)\.

Letnnbe the number of terms inα\\alpha\. We construct a decision tree𝒯α\\mathcal\{T\}\_\{\\alpha\}of dimensionn\+\|p\|\+\|q\|n\+\|\\textbf\{p\}\|\+\|\\textbf\{q\}\|\. We partition the features of𝒯α\\mathcal\{T\}\_\{\\alpha\}into three consecutive blocksHH,EEandUU, where\|H\|=n\|H\|=n,\|E\|=\|p\|\|E\|=\|\\textbf\{p\}\|, and\|U\|=\|q\|\|U\|=\|\\textbf\{q\}\|\. The blockHHwill be used to select a term fromα\\alpha, whileEEandUUwill encode truth assignments to the existentialpand to the universal variablesq, respectively\.

![Refer to caption](https://arxiv.org/html/2607.06407v1/figures/reduction_comp.png)Figure 4\.Construction used in the proof of Theorem[5\.1](https://arxiv.org/html/2607.06407#S5.Thmtheorem1)\. On the left, the decision tree𝒯α\\mathcal\{T\}\_\{\\alpha\}\. On the right, the gadget for the termo∧p∧¬q\\textnormal\{o\}\\wedge\\textnormal\{p\}\\wedge\\neg\\textnormal\{q\}\.Illustration of the construction of the decision tree T\_alpha from a DNF formula, together with the gadget for one term\.The figure has two panels\. In the left panel, the tree starts at node h\_1\. Edge 0 leads to the term gadget C\_1, while edge 1 continues to h\_2\. The same pattern repeats: from h\_2, edge 0 leads to C\_2 and edge 1 continues to h\_3, and so on until h\_n, where edge 0 leads to C\_n and edge 1 leads to a false leaf\. Thus, the h\-variables are used to select one term gadget\. In the right panel, the gadget for the term o and p and not q is shown\. The root is o\. Edge 0 leads to a false leaf and edge 1 leads to p\. From p, edge 0 leads to a false leaf and edge 1 leads to q\. From q, edge 0 leads to a true leaf and edge 1 leads to a false leaf\. Therefore, the gadget accepts exactly the assignments that satisfy the term o and p and not q\.

For every termhrh\_\{r\}\(1≤r≤n1\\leq r\\leq n\) we construct a decision treeCrC\_\{r\}overEEandUUfeatures in such a way that an input to that tree encoding a truth assignment reaches a𝐭𝐫𝐮𝐞\\mathbf\{true\}leaf if and only if the termhrh\_\{r\}evaluated over that assignment is𝐭𝐫𝐮𝐞\\mathbf\{true\}\. We now explain how to construct𝒯α\\mathcal\{T\}\_\{\\alpha\}\. We use the feature corresponding to termh1h\_\{1\}as the root\. For everyi<ni<n, the outgoing edge ofhih\_\{i\}labeled by11is connected tohi\+1h\_\{i\+1\}, and the outgoing edge ofhnh\_\{n\}labeled by11is connected to a𝐟𝐚𝐥𝐬𝐞\\mathbf\{false\}leaf\. Also, for everyii, we connect the outgoing edge labeled by0ofhih\_\{i\}to a copy of the treeCiC\_\{i\}\. An example is shown in[Figure 4](https://arxiv.org/html/2607.06407#S5.F4)\.

Set𝐞α=\{⊥\}n⋅\{0\}\|p\|⋅\{⊥\}\|q\|\\mathbf\{e\}\_\{\\alpha\}=\\\{\\bot\\\}^\{n\}\\cdot\\\{0\\\}^\{\|\\textbf\{p\}\|\}\\cdot\\\{\\bot\\\}^\{\|\\textbf\{q\}\|\}and𝐞α′=\{⊥\}n⋅\{0\}\|p\|\+\|q\|\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}=\\\{\\bot\\\}^\{n\}\\cdot\\\{0\\\}^\{\|\\textbf\{p\}\|\+\|\\textbf\{q\}\|\}\. We now show that the reduction is correct\.

First suppose that∃p​∀q​α​\(p,q\)\\exists\\textbf\{p\}\\forall\\textbf\{q\}\\ \\alpha\(\\textbf\{p\},\\textbf\{q\}\)is true\. Letν\\nube a truth assignment for the variablespsuch that for every truth assignmentσ\\sigmafor the variablesqit holds thatα​\(ν​\(p\),σ​\(q\)\)\\alpha\(\\nu\(\\textbf\{p\}\),\\sigma\(\\textbf\{q\}\)\)is true\. Let𝐞α′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\}be a partial instance with just itsEEfeatures defined according toν\\nu\. Notice that𝐞α′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\}and𝐞α\\mathbf\{e\}\_\{\\alpha\}have the same defined features\. We claim thatφ​\(𝐞α′′,𝐞α,𝐞α′\)\\varphi\(\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\},\\mathbf\{e\}\_\{\\alpha\},\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}\)\. In fact, let𝐞α′′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\\prime\}be a partial instance with the same defined features as𝐞α′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}and such that𝐞α′′⊆𝐞α′′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\}\\subseteq\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\\prime\}\. Notice that𝐞α′′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\\prime\}naturally encodes a truth assignmentσ\\sigmafor the variablesqtogether withν\\nu\. By taking a completion of𝐞α′′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\\prime\}that has a0in the feature corresponding to a true term under the truth assignment\(ν,σ\)\(\\nu,\\sigma\)and a11in the features corresponding to all previous terms, we can see that¬𝖠𝗅𝗅𝖭𝖾𝗀​\(𝐞α′′′\)\\neg\\mathsf\{AllNeg\}\(\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\\prime\}\)\. This shows thatφ​\(𝐞α′′,𝐞α,𝐞α′\)\\varphi\(\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\},\\mathbf\{e\}\_\{\\alpha\},\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}\), and therefore𝒯α⊧φ​\(𝐞α,𝐞α′\)\\mathcal\{T\}\_\{\\alpha\}\\models\\varphi\(\\mathbf\{e\}\_\{\\alpha\},\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}\)\.

Now suppose that𝒯α⊧φ​\(𝐞α,𝐞α′\)\\mathcal\{T\}\_\{\\alpha\}\\models\\varphi\(\\mathbf\{e\}\_\{\\alpha\},\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}\), so that there exists a partial instance𝐞α′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\}such thatφ​\(𝐞α′′,𝐞α,𝐞α′\)\\varphi\(\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\},\\mathbf\{e\}\_\{\\alpha\},\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}\)\. Because𝐞α′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\}has the same defined features as𝐞α\\mathbf\{e\}\_\{\\alpha\}, we can define a truth assignmentν\\nufor the variablespaccording to𝐞α′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\}\. Now letσ\\sigmabe any truth assignment for the variablesq\. We claim thatα​\(ν​\(p\),σ​\(q\)\)\\alpha\(\\nu\(\\textbf\{p\}\),\\sigma\(\\textbf\{q\}\)\)is true\. In fact, let𝐞α′′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\\prime\}be the partial instance with the same defined features as𝐞α′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}and which corresponds to the pair\(ν,σ\)\(\\nu,\\sigma\)\. Because we have thatφ​\(𝐞α′′,𝐞α,𝐞α′\)\\varphi\(\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\},\\mathbf\{e\}\_\{\\alpha\},\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\}\), it must be the case that¬𝖠𝗅𝗅𝖭𝖾𝗀​\(𝐞α′′′\)\\neg\\mathsf\{AllNeg\}\(\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\\prime\}\)\. That means that there exists a completion of𝐞α′′′\\mathbf\{e\}\_\{\\alpha\}^\{\\prime\\prime\\prime\}that is evaluated as𝐭𝐫𝐮𝐞\\mathbf\{true\}by𝒯α\\mathcal\{T\}\_\{\\alpha\}\. That necessarily means that there is a term inα​\(ν​\(p\),σ​\(q\)\)\\alpha\(\\nu\(\\textbf\{p\}\),\\sigma\(\\textbf\{q\}\)\)that is being satisfied\. Hence∃p​∀q​α​\(p,q\)\\exists\\textbf\{p\}\\forall\\textbf\{q\}\\ \\alpha\(\\textbf\{p\},\\textbf\{q\}\)is true\.

This reduction shows thatEval\(φ,𝖣𝖳𝗋𝖾𝖾\)\(\\varphi,\\mathsf\{DTree\}\)isNPNP\{\\rm NP\}^\{\{\\rm NP\}\}\-hard\. To conclude the proof of the theorem, assume for the sake of contradiction that𝖢𝗈𝗆𝗉​\(φ​\[y,z\]​\(x\),𝖣𝖳𝗋𝖾𝖾\)∈FPNP\\mathsf\{Comp\}\(\\varphi\[y,z\]\(x\),\\,\\mathsf\{DTree\}\)\\in\{\\rm FP\}^\{\{\\rm NP\}\}\. Then, it is clear that we would haveEval\(φ,𝖣𝖳𝗋𝖾𝖾\)∈PNP\(\\varphi,\\mathsf\{DTree\}\)\\in\{\\rm P\}^\{\{\\rm NP\}\}\. Finally,PNP=NPNP\{\\rm P\}^\{\{\\rm NP\}\}=\{\\rm NP\}^\{\{\\rm NP\}\}implies thatPH=PNP\{\\rm PH\}=\{\\rm P\}^\{\{\\rm NP\}\}\.

∎

To solve the problem that Theorem[5\.1](https://arxiv.org/html/2607.06407#S5.Thmtheorem1)signifies, we now proposeOpt\-FOIL, a logic that is defined by introducing a minimality operator over a subset ofExplAIner\. As we will show in the next section,Opt\-FOILmeets all the criteria for an appropriate interpretability logic\.

### 5\.2\.TheOpt\-FOILlogic

Our aim is to capture the right subset ofExplAInerthat meets all the criteria for an interpretability logic\. For this, we will define a third logic calledOpt\-FOIL\. We will show that the computation problem for this logic can be solved in polynomial time with a polynomial number of calls to anNP\{\\rm NP\}oracle\.

We will say that a formulaρ​\(x,y,v1,…,vℓ\)\\rho\(x,y,v\_\{1\},\\ldots,v\_\{\\ell\}\)from the atomic layer ofExplAInerrepresents a*strict partial order*if, for every natural numbernnand assignment of partial instances of dimensionnnto the variablesv1v\_\{1\},…\\ldots,vℓv\_\{\\ell\}, the resulting binary relation over the variablesxxandyyis a strict partial order over the partial instances of dimensionnn\. Formally,ρ​\(x,y,v1,…,vℓ\)\\rho\(x,y,v\_\{1\},\\ldots,v\_\{\\ell\}\)represents a strict partial order if, for everyn∈ℕn\\in\\mathbb\{N\},

𝔅n⊧∀v1​⋯​∀vℓ​\[∀x​¬ρ​\(x,x,v1,…,vℓ\)∧∀x​∀y​∀z​\(\(ρ​\(x,y,v1,…,vℓ\)∧ρ​\(y,z,v1,…,vℓ\)\)→ρ​\(x,z,v1,…,vℓ\)\)\]\.\\mathfrak\{B\}\_\{n\}\\models\\ \\forall v\_\{1\}\\cdots\\forall v\_\{\\ell\}\\,\\big\[\\forall x\\,\\neg\\rho\(x,x,v\_\{1\},\\ldots,v\_\{\\ell\}\)\\wedge\\forall x\\forall y\\forall z\\,\\big\(\(\\rho\(x,y,v\_\{1\},\\ldots,v\_\{\\ell\}\)\\wedge\\rho\(y,z,v\_\{1\},\\ldots,v\_\{\\ell\}\)\)\\to\\rho\(x,z,v\_\{1\},\\ldots,v\_\{\\ell\}\)\\big\)\\big\]\.
The variablesv1v\_\{1\},…\\ldots,vℓv\_\{\\ell\}in the formulaρ​\(x,y,v1,…,vℓ\)\\rho\(x,y,v\_\{1\},\\ldots,v\_\{\\ell\}\)are considered as parameters that define a strict partial order\. In fact, different assignments for these variables can give rise to different orders\. Hence, we use the notationρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\)to make explicit the distinction between the parametersv1v\_\{1\},…\\ldots,vℓv\_\{\\ell\}and the variablesxx,yythat are instantiated with partial instances\. For example, the strict partial order determined by the subsumption relation is defined by the formulaρ1​\(x,y\)=x⊂y\\rho\_\{1\}\(x,y\)=x\\subset y\.

As a second example, consider the case where a certain feature must be disregarded when defining an order on partial instances \(for instance, it is often undesirable to use the featuregenderfor comparisons\)\. Such an order can be defined as follows\. Notice that, with the appropriate values for the variablesv1v\_\{1\}andv2v\_\{2\}, the following formula checks whether theii\-th feature ofxxis undefined:

𝖴𝖥​\(x,v1,v2\):=¬\(v1⊆x\)∧¬\(v2⊆x\)\.\\mathsf\{UF\}\(x,v\_\{1\},v\_\{2\}\)\\ :=\\ \\neg\(v\_\{1\}\\subseteq x\)\\wedge\\neg\(v\_\{2\}\\subseteq x\)\.For example, if we are considering partial instances of dimension55and we need to check whether instancexxhas value⊥\\botin the first feature, then we can use the valuesc1=\(0,⊥,⊥,⊥,⊥\)c\_\{1\}=\(0,\\bot,\\bot,\\bot,\\bot\)andc2=\(1,⊥,⊥,⊥,⊥\)c\_\{2\}=\(1,\\bot,\\bot,\\bot,\\bot\)for the variablesv1v\_\{1\}andv2v\_\{2\}, respectively\. Moreover, let

𝖯𝗋​\(x,y\):=x⊂y∧¬∃z​\(x⊂z∧z⊂y\)\\mathsf\{Pr\}\(x,y\)\\ :=\\ x\\subset y\\wedge\\neg\\exists z\\,\(x\\subset z\\wedge z\\subset y\)be a formula that checks whetherxxis a predecessor ofyyunder the order⊂\\subset\. Now define

𝖲𝗍𝗋𝗂𝗉​\[v1,v2\]​\(x,y\):=\(𝖴𝖥​\(x,v1,v2\)∧x=y\)∨\(¬𝖴𝖥​\(x,v1,v2\)∧𝖯𝗋​\(y,x\)∧𝖴𝖥​\(y,v1,v2\)\)\.\\mathsf\{Strip\}\[v\_\{1\},v\_\{2\}\]\(x,y\)\\ :=\\ \(\\mathsf\{UF\}\(x,v\_\{1\},v\_\{2\}\)\\wedge x=y\)\\ \\vee\\ \(\\neg\\mathsf\{UF\}\(x,v\_\{1\},v\_\{2\}\)\\wedge\\mathsf\{Pr\}\(y,x\)\\wedge\\mathsf\{UF\}\(y,v\_\{1\},v\_\{2\}\)\)\.Notice that, with the appropriate values forv1v\_\{1\}andv2v\_\{2\},𝖲𝗍𝗋𝗂𝗉​\[v1,v2\]​\(x,y\)\\mathsf\{Strip\}\[v\_\{1\},v\_\{2\}\]\(x,y\)holds if and only ifyyis obtained fromxxby undefining the distinguished feature when necessary\. Then, the following formula defines a strict partial order based on⊂\\subsetbut that disregards theii\-th feature when comparing partial instances:

ρ2​\[v1,v2\]​\(x,y\):=∃x′​∃y′​\(𝖲𝗍𝗋𝗂𝗉​\[v1,v2\]​\(x,x′\)∧𝖲𝗍𝗋𝗂𝗉​\[v1,v2\]​\(y,y′\)∧x′⊂y′\)\.\\rho\_\{2\}\[v\_\{1\},v\_\{2\}\]\(x,y\)\\ :=\\ \\exists x^\{\\prime\}\\exists y^\{\\prime\}\\ \\bigl\(\\mathsf\{Strip\}\[v\_\{1\},v\_\{2\}\]\(x,x^\{\\prime\}\)\\,\\wedge\\,\\mathsf\{Strip\}\[v\_\{1\},v\_\{2\}\]\(y,y^\{\\prime\}\)\\,\\wedge\\,x^\{\\prime\}\\subset y^\{\\prime\}\\bigr\)\.
For example,ρ2​\[c1,c2\]​\(x,y\)\\rho\_\{2\}\[c\_\{1\},c\_\{2\}\]\(x,y\)with constantsc1c\_\{1\}andc2c\_\{2\}mentioned above defines a strict partial order that disregards the first feature when comparing partial instances of dimension55\.

Formulas from the atomic layer ofExplAInerrepresenting strict partial orders will be used in the definition ofOpt\-FOIL\. Hence, it is necessary to have an algorithm that verifies whether this condition is satisfied in order to have a decidable syntax forOpt\-FOIL\. We will now prove that such an algorithm exists\.

###### Proposition 5\.2\.

The problem of verifying, given a formulaρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\)from the atomic layer ofExplAIner, whether it represents a strict partial order can be solved in22poly​\(\|ρ\|⋅3wd​\(ρ\)\)2^\{2^\{\\rm\{poly\}\\big\(\|\\rho\|\\cdot 3^\{\\rm\{wd\}\(\\rho\)\}\\big\)\}\}space, and, hence, in222poly​\(\|ρ\|⋅3wd​\(ρ\)\)2^\{2^\{2^\{\\rm\{poly\}\\big\(\|\\rho\|\\cdot 3^\{\\rm\{wd\}\(\\rho\)\}\\big\)\}\}\}time\.

###### Proof\.

Letρ​\(x,y,v1,…,vℓ\)\\rho\(x,y,v\_\{1\},\\ldots,v\_\{\\ell\}\)be an arbitrary formula from the atomic layer ofExplAIner, that is, a formula over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}\. Then, we consider the following sentence:

φ:=∀v1​⋯​∀vℓ​\[∀x​¬ρ​\(x,x,v1,…,vℓ\)∧∀x​∀y​∀z​\(\(ρ​\(x,y,v1,…,vℓ\)∧ρ​\(y,z,v1,…,vℓ\)\)→ρ​\(x,z,v1,…,vℓ\)\)\]\.\\displaystyle\\varphi\\ :=\\ \\forall v\_\{1\}\\cdots\\forall v\_\{\\ell\}\\,\\big\[\\forall x\\,\\neg\\rho\(x,x,v\_\{1\},\\ldots,v\_\{\\ell\}\)\\ \\wedge\\forall x\\forall y\\forall z\\,\\big\(\(\\rho\(x,y,v\_\{1\},\\ldots,v\_\{\\ell\}\)\\wedge\\rho\(y,z,v\_\{1\},\\ldots,v\_\{\\ell\}\)\)\\to\\rho\(x,z,v\_\{1\},\\ldots,v\_\{\\ell\}\)\\big\)\\big\]\.Notice thatφ\\varphihas widthwd​\(φ\)≤wd​\(ρ\)\+1\\rm\{wd\}\(\\varphi\)\\leq\\rm\{wd\}\(\\rho\)\+1\. By definition, determining ifρ\\rhocorresponds to a strict partial order is equivalent to checking if for every structure𝔅n\\mathfrak\{B\}\_\{n\}it holds that𝔅n⊧φ\\mathfrak\{B\}\_\{n\}\\models\\varphi\. Thanks to the second part of Theorem[4\.3](https://arxiv.org/html/2607.06407#S4.Thmtheorem3), we know that this can be done in the stated space\.

∎

Proposition[5\.2](https://arxiv.org/html/2607.06407#S5.Thmtheorem2)serves as a theoretical upper bound to prove thatOpt\-FOILhas a decidable syntax\. Observe that if we restrict ourselves to formulas of bounded width, then the space complexity falls from triple exponential to double exponential in\|ρ\|\|\\rho\|\(which implies that the time complexity is triple exponential in\|ρ\|\|\\rho\|\)\. In practice, we expect formulas representing strict partial orders to be small and to have a simple structure, so we do not expect this theoretical high computational complexity to pose an actual implementation obstacle\.

We now explain how strict partial orders will be used in the logicOpt\-FOIL\.

Given a formulaφ​\[u1,…,uk\]​\(x\)\\varphi\[u\_\{1\},\\ldots,u\_\{k\}\]\(x\)from the quantified layer ofExplAInerand another formulaρ​\[v1,…,vℓ\]​\(y,z\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(y,z\)from the atomic layer ofExplAInerthat represents a strict partial order, anOpt\-FOILformulais an expression of the following form:

Ψ​\[u1,…,uk,v1,…,vℓ\]​\(x\)=min​\[φ​\[u1,…,uk\]​\(x\),ρ​\[v1,…,vℓ\]​\(y,z\)\]\.\\Psi\[u\_\{1\},\\ldots,u\_\{k\},v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x\)\\ =\\ \\text\{\\rm min\}\[\\varphi\[u\_\{1\},\\ldots,u\_\{k\}\]\(x\),\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(y,z\)\]\.
Notice thatxx,u1,…,uku\_\{1\},\\ldots,u\_\{k\},v1,…,vℓv\_\{1\},\\ldots,v\_\{\\ell\}are the free variables of this expression, while the variablesyy,zzwill be quantified\. In particular,u1,…,uku\_\{1\},\\ldots,u\_\{k\}are the parameters that define the notion of explanation,v1,…,vℓv\_\{1\},\\ldots,v\_\{\\ell\}are the parameters that define the strict partial order, andxxis a variable used to store an explanation that is minimal in the sense given by the strict partial order\. The semantics ofΨ​\[u1,…,uk,v1,…,vℓ\]​\(x\)\\Psi\[u\_\{1\},\\ldots,u\_\{k\},v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x\)is defined by considering the followingExplAInerformula:

θmin​\(x,u1,…,uk,v1,…,vℓ\):=φ​\(x,u1,…,uk\)∧¬∃y​\(φ​\(y,u1,…,uk\)∧ρ​\(y,x,v1,…,vℓ\)\)\.\\theta\_\{\\text\{\\rm min\}\}\(x,u\_\{1\},\\ldots,u\_\{k\},v\_\{1\},\\ldots,v\_\{\\ell\}\)\\ :=\\ \\varphi\(x,u\_\{1\},\\ldots,u\_\{k\}\)\\ \\wedge\\ \\neg\\exists y\\ \\big\(\\varphi\(y,u\_\{1\},\\ldots,u\_\{k\}\)\\,\\wedge\\,\\rho\(y,x,v\_\{1\},\\ldots,v\_\{\\ell\}\)\\big\)\.More precisely, given a modelℳ\\mathcal\{M\}of dimensionnnand partial instances𝐞\\mathbf\{e\},𝐞1′,…,𝐞k′\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\},𝐞1′′,…,𝐞ℓ′′\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\\prime\}\_\{\\ell\}of dimensionnn, we define thatℳ⊧Ψ​\[𝐞1′,…,𝐞k′,𝐞1′′,…,𝐞ℓ′′\]​\(𝐞\)\\mathcal\{M\}\\models\\Psi\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\},\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\)if and only ifℳ⊧θmin​\(𝐞,𝐞1′,…,𝐞k′,𝐞1′′,…,𝐞ℓ′′\)\\mathcal\{M\}\\models\\theta\_\{\\text\{\\rm min\}\}\(\\mathbf\{e\},\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\},\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\\prime\}\_\{\\ell\}\)\.

The computation problem forOpt\-FOILhas to be defined considering the different roles of the variables in the formulaΨ​\[u1,…,uk,v1,…,vℓ\]​\(x\)\\Psi\[u\_\{1\},\\ldots,u\_\{k\},v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x\)\. In particular, the parametersu1,…,uk,v1,…,vℓu\_\{1\},\\ldots,u\_\{k\},v\_\{1\},\\ldots,v\_\{\\ell\}should be given as input, while the value ofxxis the explanation to be computed\. The following definition takes these considerations into account\. As usual, we write𝒞\\mathcal\{C\}to denote some class of models\.

We now show thatOpt\-FOILfulfills our criteria by establishing that the computation problem forOpt\-FOILcan be solved in polynomial time with a polynomial number of calls to anNP\{\\rm NP\}oracle:

###### Theorem 5\.3\.

Let𝒞\\mathcal\{C\}be a class of models such thatEval\(𝖠𝗅𝗅𝖯𝗈𝗌​\(x\),𝒞\)∈P\(\\mathsf\{AllPos\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}andEval\(𝖠𝗅𝗅𝖭𝖾𝗀​\(x\),𝒞\)∈P\(\\mathsf\{AllNeg\}\(x\),\\mathcal\{C\}\)\\in\{\\rm P\}\. ThenComp\(Ψ,𝒞\)∈FPNP\(\\Psi,\\mathcal\{C\}\)\\in\{\\rm FP\}^\{\{\\rm NP\}\}for every formulaΨ\\PsiinOpt\-FOIL\.

As a corollary of this result, we obtain thatOpt\-FOILcan be used to compute explanations in polynomial time using a polynomial number of calls to anNP\{\\rm NP\}oracle for the class of decision trees\. Moreover, the same holds for more expressive representation classes, including𝖽​\-​𝖣𝖭𝖭𝖥\\mathsf\{d\}\\text\{\-\}\\mathsf\{DNNF\}circuits and fragments of the class of circuits corresponding to propositional formulas in conjunctive normal form \(𝖢𝖭𝖥\\mathsf\{CNF\}\) whose satisfiability is decidable in polynomial time, such as 2\-CNF formulas \(2​𝖢𝖭𝖥2\\mathsf\{CNF\}\) and Horn CNF formulas \(𝖧𝖮𝖱𝖭\\mathsf\{HORN\}\)\. We formally state these results in the following corollary\.

###### Corollary 5\.4\.

Letφ\\varphibe anOpt\-FOILformula\. ThenComp\(φ,𝖽​\-​𝖣𝖭𝖭𝖥\)∈FPNP\(\\varphi,\\mathsf\{d\}\\text\{\-\}\\mathsf\{DNNF\}\)\\in\{\\rm FP\}^\{\{\\rm NP\}\},Comp\(φ,2​𝖢𝖭𝖥\)∈FPNP\(\\varphi,2\\mathsf\{CNF\}\)\\in\{\\rm FP\}^\{\{\\rm NP\}\}, andComp\(φ,𝖧𝖮𝖱𝖭\)∈FPNP\(\\varphi,\\mathsf\{HORN\}\)\\in\{\\rm FP\}^\{\{\\rm NP\}\}\.

###### Proof of Theorem[5\.3](https://arxiv.org/html/2607.06407#S5.Thmtheorem3)\.

Letρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\)be a formula from the atomic layer ofExplAInerthat represents a strict partial order\. We say that a sequence\(𝐞1,…,𝐞k\)\(\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k\}\)of partial instances of dimensionnnis a*path*of dimensionnninρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\)if there exist partial instances𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\},…\\ldots,𝐞ℓ′\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}of dimensionnnsuch that, for everyi∈\{1,…,k−1\}i\\in\\\{1,\\ldots,k\-1\\\}, it holds that

𝔅n⊧ρ​\[𝐞1′,…,𝐞ℓ′\]​\(𝐞i,𝐞i\+1\)\.\\mathfrak\{B\}\_\{n\}\\models\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{i\},\\mathbf\{e\}\_\{i\+1\}\)\.
The following lemma shows that, for a fixed formulaρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\), the lengths of such paths are polynomially bounded with respect tonn\(refer to Section[A\.6](https://arxiv.org/html/2607.06407#A1.SS6)for the proof\)\.

###### Lemma 5\.5\.

Letρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\)be a formula from the atomic layer ofExplAInerthat represents a strict partial order\. Then there exists a fixed polynomialppsuch that for every path\(𝐞1,…,𝐞k\)\(\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k\}\)of dimensionnninρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\), it holds thatk≤p​\(n\)k\\leq p\(n\)\.

Lemma[5\.5](https://arxiv.org/html/2607.06407#S5.Thmtheorem5)gives us a simple algorithm to compute a solution for anOpt\-FOILformula

min​\[φ​\[u1,…,uk\]​\(x\),ρ​\[v1,…,vℓ\]​\(y,z\)\],\\displaystyle\\text\{\\rm min\}\[\\varphi\[u\_\{1\},\\ldots,u\_\{k\}\]\(x\),\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(y,z\)\],given as input a modelℳ∈𝒞\\mathcal\{M\}\\in\\mathcal\{C\}of dimensionnnand partial instances𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\},…\\ldots,𝐞k′\\mathbf\{e\}^\{\\prime\}\_\{k\},𝐞1′′\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\},…\\ldots,𝐞ℓ′′\\mathbf\{e\}^\{\\prime\\prime\}\_\{\\ell\}of dimensionnn\. We first use anNP\{\\rm NP\}oracle to verify whetherℳ⊧∃x​φ​\[𝐞1′,…,𝐞k′\]​\(x\)\\mathcal\{M\}\\models\\exists x\\,\\varphi\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\}\]\(x\), which is a formula from the quantified layer ofExplAIner\. Ifℳ⊧̸∃x​φ​\[𝐞1′,…,𝐞k′\]​\(x\)\\mathcal\{M\}\\not\\models\\exists x\\,\\varphi\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\}\]\(x\), then the answer isNo\. Otherwise, theNP\{\\rm NP\}oracle can be used to construct an initial partial instance𝐞0\\mathbf\{e\}\_\{0\}such thatℳ⊧φ​\[𝐞1′,…,𝐞k′\]​\(𝐞0\)\\mathcal\{M\}\\models\\varphi\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\}\]\(\\mathbf\{e\}\_\{0\}\)\. The idea is to maintain a current partial assignment𝐞′′\\mathbf\{e\}^\{\\prime\\prime\}\(originally set to\{⊥\}n\\\{\\bot\\\}^\{n\}\) of the features that is known to extend to some witness\. For each feature, we query whether there exists a witness extending𝐞′′\\mathbf\{e\}^\{\\prime\\prime\}but with that feature fixed to0\. If the answer is positive, we keep that feature as0in𝐞′′\\mathbf\{e\}^\{\\prime\\prime\}, otherwise we query whether there exists one extending𝐞′′\\mathbf\{e\}^\{\\prime\\prime\}but with that feature fixed to11\. If that answer is positive, we keep that feature as11, and if both answers are negative, then we leave the feature undefined\. This way, the invariant is preserved at every step, and after at most2​n2noracle queries we obtain the partial instance𝐞0\\mathbf\{e\}\_\{0\}\.

We then use theNP\{\\rm NP\}oracle to verify whether

ℳ⊧∃x​\(φ​\[𝐞1′,…,𝐞k′\]​\(x\)∧ρ​\[𝐞1′′,…,𝐞ℓ′′\]​\(x,𝐞0\)\);\\displaystyle\\mathcal\{M\}\\ \\models\\ \\exists x\\,\\big\(\\varphi\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\}\]\(x\)\\ \\wedge\\ \\rho\[\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\\prime\}\_\{\\ell\}\]\(x,\\mathbf\{e\}\_\{0\}\)\\big\);which can be written as a formula from the quantified layer ofExplAInerby appending ‘∧ρ​\[𝐞1′′,…,𝐞ℓ′′\]​\(x,𝐞0\)\\wedge\\,\\rho\[\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\\prime\}\_\{\\ell\}\]\(x,\\mathbf\{e\}\_\{0\}\)’ within all the quantifiers ofφ​\[𝐞1′,…,𝐞k′\]​\(x\)\\varphi\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\}\]\(x\)\. If the answer is positive, then again we use theNP\{\\rm NP\}oracle as described before to construct a partial instance𝐞1\\mathbf\{e\}\_\{1\}such that

ℳ⊧φ​\[𝐞1′,…,𝐞k′\]​\(𝐞1\)∧ρ​\[𝐞1′′,…,𝐞ℓ′′\]​\(𝐞1,𝐞0\)\.\\displaystyle\\mathcal\{M\}\\ \\models\\ \\varphi\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\}\]\(\\mathbf\{e\}\_\{1\}\)\\ \\wedge\\ \\rho\[\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{0\}\)\.The algorithm continues in this way, constructing a sequence of partial instances\(𝐞i,𝐞i−1,…,𝐞0\)\(\\mathbf\{e\}\_\{i\},\\mathbf\{e\}\_\{i\-1\},\\ldots,\\mathbf\{e\}\_\{0\}\)that constitutes a path of dimensionnninρ​\[v1,…,vℓ\]​\(y,z\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(y,z\)\. The algorithm stops when the condition

ℳ⊧∃x​\(φ​\[𝐞1′,…,𝐞k′\]​\(x\)∧ρ​\[𝐞1′′,…,𝐞ℓ′′\]​\(x,𝐞i\)\)\\displaystyle\\mathcal\{M\}\\ \\models\\ \\exists x\\,\\big\(\\varphi\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{k\}\]\(x\)\\wedge\\rho\[\\mathbf\{e\}^\{\\prime\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\\prime\}\_\{\\ell\}\]\(x,\\mathbf\{e\}\_\{i\}\)\\big\)does not hold, which by construction guarantees that𝐞i\\mathbf\{e\}\_\{i\}is a minimal instance\. Lemma[5\.5](https://arxiv.org/html/2607.06407#S5.Thmtheorem5)guarantees that𝐞i\\mathbf\{e\}\_\{i\}will be found in a polynomial number of steps\. Since in each step we call theNP\{\\rm NP\}oracle a polynomial number of times, this concludes the proof of the theorem\. ∎

We now begin the study of the expressiveness ofOpt\-FOIL\. As is customary, we say that a logicℒ1\\mathcal\{L\}\_\{1\}iscontainedin a logicℒ2\\mathcal\{L\}\_\{2\}if for every formula inℒ1\\mathcal\{L\}\_\{1\}there exists an equivalent formula inℒ2\\mathcal\{L\}\_\{2\}\. Moreover,ℒ1\\mathcal\{L\}\_\{1\}isproperly containedinℒ2\\mathcal\{L\}\_\{2\}ifℒ1\\mathcal\{L\}\_\{1\}is contained inℒ2\\mathcal\{L\}\_\{2\}andℒ2\\mathcal\{L\}\_\{2\}is not contained inℒ1\\mathcal\{L\}\_\{1\}\. The following proposition shows that the expressive power ofOpt\-FOILis less than that ofExplAIner, which in turn has less expressive power thanFOILwith predicates\{⊆,⪯,𝖠𝗅𝗅𝖯𝗈𝗌,𝖠𝗅𝗅𝖭𝖾𝗀\}\\\{\\subseteq,\\preceq,\\mathsf\{AllPos\},\\mathsf\{AllNeg\}\\\}\.

###### Proposition 5\.6\.

Assuming that the polynomial hierarchy does not collapse,Opt\-FOILis strictly contained inExplAIner, andExplAIneris strictly contained inFOILwith extended predicates\.

###### Proof\.

For the first containment, let

Ψ​\[u1,…,uk,v1,…,vℓ\]​\(x\)=min​\[φ​\[u1,…,uk\]​\(x\),ρ​\[v1,…,vℓ\]​\(y,z\)\]\\Psi\[u\_\{1\},\\ldots,u\_\{k\},v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x\)=\\text\{\\rm min\}\[\\varphi\[u\_\{1\},\\ldots,u\_\{k\}\]\(x\),\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(y,z\)\]be anOpt\-FOILformula\. As we discussed before, we can consider the equivalentExplAInerformula

θmin​\(x,u1,…,uk,v1,…,vℓ\)\.\\theta\_\{\\text\{\\rm min\}\}\(x,u\_\{1\},\\ldots,u\_\{k\},v\_\{1\},\\ldots,v\_\{\\ell\}\)\.Now, for the sake of contradiction, suppose that the containment is not strict\. Letφ​\[y,z\]​\(x\)\\varphi\[y,z\]\(x\)be anExplAInerformula such that𝖢𝗈𝗆𝗉​\(φ​\[y,z\]​\(x\),𝖣𝖳𝗋𝖾𝖾\)∉FPNP\\mathsf\{Comp\}\(\\varphi\[y,z\]\(x\),\\,\\mathsf\{DTree\}\)\\not\\in\{\\rm FP\}^\{\{\\rm NP\}\}unlessPH\{\\rm PH\}collapses toPNP\{\\rm P\}^\{\{\\rm NP\}\}\(whose existence is guaranteed by Theorem[5\.1](https://arxiv.org/html/2607.06407#S5.Thmtheorem1)\)\. LetΨ​\[y,z\]​\(x\)\\Psi\[y,z\]\(x\)be its equivalent expression inOpt\-FOIL\. By Theorem[5\.3](https://arxiv.org/html/2607.06407#S5.Thmtheorem3), the problemComp\(Ψ​\[y,z\]​\(x\),𝖣𝖳𝗋𝖾𝖾\)\(\\Psi\[y,z\]\(x\),\\mathsf\{DTree\}\)is in FPNP\. Thus,Comp\(φ​\[y,z\]​\(x\),𝖣𝖳𝗋𝖾𝖾\)\(\\varphi\[y,z\]\(x\),\\mathsf\{DTree\}\)can be solved in FPNP\. This would imply the collapse of the polynomial hierarchy toPNP\{\\rm P\}^\{\{\\rm NP\}\}\. We conclude that the containment is strict\.

For the second containment, each formula inExplAIneris aFOILformula with extended predicates by definition\. It is strict because we can expressΣ2P\\Sigma\_\{2\}^\{\\rm\{P\}\}\-hard problems inFOILover decision trees \(Theorem[3\.4](https://arxiv.org/html/2607.06407#S3.Thmtheorem4)\), but noExplAInerformula can express aΣ2P\\Sigma\_\{2\}^\{\\rm\{P\}\}\-hard evaluation problem over decision trees unlessΣ2P⊆BH\\Sigma\_\{2\}^\{\\rm\{P\}\}\\subseteq\{\\rm BH\}\(Theorem[4\.7](https://arxiv.org/html/2607.06407#S4.Thmtheorem7)\)\. ∎

The logicOpt\-FOILallows us to express in a simple way all notions of explainability that we study in this paper\. For example, recall from Section[4\.2](https://arxiv.org/html/2607.06407#S4.SS2)that𝗐𝖠𝖷𝗉​\(u,x\)\\mathsf\{wAXp\}\(u,x\)can be expressed as a formula from the quantified layer ofExplAIner\. Therefore, takingφ​\[u\]​\(x\)=𝗐𝖠𝖷𝗉​\(u,x\)\\varphi\[u\]\(x\)=\\mathsf\{wAXp\}\(u,x\), the followingOpt\-FOILformulas encode the notions of minimal and minimum abductive explanations:

𝖠𝖷𝗉​\[u\]​\(x\)\\displaystyle\\mathsf\{AXp\}\[u\]\(x\)=\\displaystyle=min​\[φ​\[u\]​\(x\),y⊂z\],\\displaystyle\\text\{\\rm min\}\[\\varphi\[u\]\(x\),y\\subset z\],𝗆𝖠𝖷𝗉​\[u\]​\(x\)\\displaystyle\\mathsf\{mAXp\}\[u\]\(x\)=\\displaystyle=min​\[φ​\[u\]​\(x\),y≺z\]\.\\displaystyle\\text\{\\rm min\}\[\\varphi\[u\]\(x\),y\\prec z\]\.Likewise,min​\[φ​\[u\]​\(x\),ρ2​\[v1,v2\]​\(y,z\)\]\\text\{\\rm min\}\[\\varphi\[u\]\(x\),\\rho\_\{2\}\[v\_\{1\},v\_\{2\}\]\(y,z\)\]encodes the notion of abductive explanations for the orderρ2​\[v1,v2\]​\(y,z\)\\rho\_\{2\}\[v\_\{1\},v\_\{2\}\]\(y,z\)that disregards a feature\. The different variants of contrastive explanations can be expressed similarly\.

As a second example, consider the notion of minimum change required and the predicate𝖫𝖤𝖧\\mathsf\{LEH\}defined in Section[A\.7](https://arxiv.org/html/2607.06407#A1.SS7)\. Then, taking

φ\[u\]\(x\)=𝖥𝗎𝗅𝗅\(u\)∧𝖥𝗎𝗅𝗅\(x\)∧¬\(𝖠𝗅𝗅𝖯𝗈𝗌\(u\)↔𝖠𝗅𝗅𝖯𝗈𝗌\(x\)\)\\varphi\[u\]\(x\)=\\mathsf\{Full\}\(u\)\\wedge\\mathsf\{Full\}\(x\)\\wedge\\neg\(\\mathsf\{AllPos\}\(u\)\\leftrightarrow\\mathsf\{AllPos\}\(x\)\)andρ3​\[u\]​\(y,z\)=𝖫𝖤𝖧​\(u,y,z\)∧¬𝖫𝖤𝖧​\(u,z,y\)\\rho\_\{3\}\[u\]\(y,z\)=\\mathsf\{LEH\}\(u,y,z\)\\wedge\\neg\\mathsf\{LEH\}\(u,z,y\), we can express the notion of minimum change required inOpt\-FOILas follows:

𝖬𝖢𝖱​\[u\]​\(x\)=min​\[φ​\[u\]​\(x\),ρ3​\[u\]​\(y,z\)\]\.\\displaystyle\\mathsf\{MCR\}\[u\]\(x\)=\\text\{\\rm min\}\[\\varphi\[u\]\(x\),\\rho\_\{3\}\[u\]\(y,z\)\]\.By reversing the order, the logicOpt\-FOILcan also be used to express notions of explainability that involve maximality conditions\. For example, consider the query of maximum change allowed that asks for the maximum number of changes that can be made to an instance without changing the output of the classification model\. Taking

φ\[u\]\(x\)=𝖥𝗎𝗅𝗅\(u\)∧𝖥𝗎𝗅𝗅\(x\)∧\(𝖠𝗅𝗅𝖯𝗈𝗌\(u\)↔𝖠𝗅𝗅𝖯𝗈𝗌\(x\)\)\\varphi\[u\]\(x\)=\\mathsf\{Full\}\(u\)\\wedge\\mathsf\{Full\}\(x\)\\wedge\(\\mathsf\{AllPos\}\(u\)\\leftrightarrow\\mathsf\{AllPos\}\(x\)\)and defining the reverse orderρ4​\[u\]​\(y,z\)=ρ3​\[u\]​\(z,y\)\\rho\_\{4\}\[u\]\(y,z\)=\\rho\_\{3\}\[u\]\(z,y\), we can express the notion of maximum change allowed inOpt\-FOILas follows:

𝖬𝖢𝖠​\[u\]​\(x\)=min​\[φ​\[u\]​\(x\),ρ4​\[u\]​\(y,z\)\]\.\\displaystyle\\mathsf\{MCA\}\[u\]\(x\)=\\text\{\\rm min\}\[\\varphi\[u\]\(x\),\\rho\_\{4\}\[u\]\(y,z\)\]\.An important advantage ofOpt\-FOILis that it allows for the combination of explainability notions\. For example, given two instancesu1u\_\{1\}andu2u\_\{2\}of the same dimension, consider the query𝖢𝖠𝖷𝗉​\[u1,u2\]​\(x\)=𝗐𝖠𝖷𝗉​\(u1,x\)∧𝗐𝖠𝖷𝗉​\(u2,x\)\\mathsf\{CAXp\}\[u\_\{1\},u\_\{2\}\]\(x\)=\\mathsf\{wAXp\}\(u\_\{1\},x\)\\wedge\\mathsf\{wAXp\}\(u\_\{2\},x\)that checks whetherxxis a common weak abductive explanation foru1u\_\{1\}andu2u\_\{2\}\. Then the followingOpt\-FOILformula computes a common weak abductive explanation for two instances \(if such an explanation exists\):

Ψ1​\[u1,u2\]​\(x\)=min​\[𝖢𝖠𝖷𝗉​\[u1,u2\]​\(x\),y⊂z\]\.\\displaystyle\\Psi\_\{1\}\[u\_\{1\},u\_\{2\}\]\(x\)=\\text\{\\rm min\}\[\\mathsf\{CAXp\}\[u\_\{1\},u\_\{2\}\]\(x\),y\\subset z\]\.Note that an answer to this query is not necessarily minimal with respect to all weak abductive explanations for eitheru1u\_\{1\}oru2u\_\{2\}\.

Finally, another advantage ofOpt\-FOILis that it allows for the exploration of the space of explanations for a given classification\. For example, assume that we already have an abductive explanationu1u\_\{1\}for an instanceuu, which can be computed using theOpt\-FOILformula𝖠𝖷𝗉​\[u\]​\(x\)\\mathsf\{AXp\}\[u\]\(x\)\. Our aim is to compute a second abductive explanationu2u\_\{2\}foruu\. Consider the formula:

𝖲𝖠𝖷𝗉​\[u,u1\]​\(x\)=𝗐𝖠𝖷𝗉​\(u,x\)∧𝗐𝖠𝖷𝗉​\(u,u1\)∧¬\(u1⊆x\)\.\\displaystyle\\mathsf\{SAXp\}\[u,u\_\{1\}\]\(x\)=\\mathsf\{wAXp\}\(u,x\)\\wedge\\mathsf\{wAXp\}\(u,u\_\{1\}\)\\wedge\\neg\(u\_\{1\}\\subseteq x\)\.This formula checks whetherxxis a weak abductive explanation foruuthat does not subsume the abductive explanationu1u\_\{1\}\. Thus, an abductive explanation for the instanceuuthat is different fromu1u\_\{1\}can be computed using the followingOpt\-FOILformula:

Ψ2​\[u,u1\]​\(x\)=min​\[𝖲𝖠𝖷𝗉​\[u,u1\]​\(x\),y⊂z\]\.\\displaystyle\\Psi\_\{2\}\[u,u\_\{1\}\]\(x\)=\\text\{\\rm min\}\[\\mathsf\{SAXp\}\[u,u\_\{1\}\]\(x\),y\\subset z\]\.We can apply the same idea to other notions of explanation, such as the𝖬𝖢𝖱\\mathsf\{MCR\}explainability query, in order to compute multiple explanations for the output of a classification model\.

## 6\.Concluding remarks and future work

We have proposed a declarative approach to model interpretability based on query languages for explaining Boolean classification models\. The starting point of our work is the observation that the growing number of explanation notions studied in formal XAI calls for a uniform language in which such notions can be specified, combined, and analyzed\. This view is natural from a data management perspective: explanation notions become queries, models become the structures over which these queries are evaluated, and the main questions are those of expressiveness, evaluation complexity, and computation of answers\.

Our first contribution was to revisitFOILfrom this perspective\. We showed that, despite its foundational role,FOILis not well suited as a practical query language for explanations\. On the one hand, it cannot express some central optimality\-based notions, such as minimum abductive explanations, even over decision trees\. On the other hand, its evaluation problem over decision trees is hard for every level of the polynomial hierarchy\. These results show that a useful explainability language must carefully balance expressive power with controlled evaluation complexity\.

To address this challenge, we introducedExplAIner, a layered query language with an extended vocabulary for reasoning about partial instances and the behavior of Boolean models\. We showed thatExplAInercan express a broad family of explanation notions, including abductive, contrastive, feature\-based, and distance\-based queries\. At the same time, we proved that the evaluation problem for each fixedExplAInerquery belongs to the Boolean hierarchy over every class of Boolean models for which the predicates𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}can be evaluated in polynomial time\. This condition holds not only for decision trees, but also for more general representation classes such as deterministic and decomposable Boolean circuits\.

We also introducedOpt\-FOIL, an optimization\-oriented fragment ofExplAInerfor computing explanations that are minimal with respect to strict partial orders\. This fragment captures a wide range of optimality\-based explanation tasks while retaining controlled computational behavior: under the same assumptions on𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}, explanations specified inOpt\-FOILcan be computed inFPNP\\mathrm\{FP\}^\{\\mathrm\{NP\}\}\. Together, the results forExplAInerandOpt\-FOILshow that declarative specification and complexity\-theoretic analysis can provide a principled foundation for model interpretability\.

Several directions remain open\. A first direction is to extend the framework beyond Boolean classification models\. Although Boolean models are standard in formal XAI and already capture many explanation tasks, many applications involve multi\-class outputs, non\-Boolean features, or structured feature domains\. It would be interesting to understand which parts of the present framework extend directly to these richer settings, and which additional predicates or language constructs are needed\.

A second direction is to study further model representations\. Our upper bounds are stated for every class of Boolean models over which𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}can be evaluated in polynomial time, and this already includes decision trees and deterministic decomposable Boolean circuits\. A natural next step is to identify additional representation classes that satisfy this condition\. This would help clarify the connection between explainability languages and knowledge compilation more broadly\.

A third direction concerns query optimization\. One of the motivations for a declarative language is that different explanation notions can share common subqueries and operators\. This suggests the possibility of developing optimization techniques for explainability queries, in the same spirit as query optimization in databases\. Such techniques could exploit common subformulas, reuse calls to procedures for𝖠𝗅𝗅𝖯𝗈𝗌\\mathsf\{AllPos\}and𝖠𝗅𝗅𝖭𝖾𝗀\\mathsf\{AllNeg\}, or identify fragments with better evaluation strategies\.

Finally, it would be valuable to study richer answer mechanisms for explainability queries\. In this paper, explanations are treated as partial instances satisfying a logical specification, possibly optimized with respect to a strict partial order\. However, users may require different levels of detail, multiple alternative explanations, or rankings of explanations according to several criteria\. Extending the language with principled mechanisms for enumeration, ranking, and comparison of explanations is an important step toward a more complete declarative framework for model interpretability\.

## 7\.Acknowledgements

Part of this work has been funded by ANID \- Millennium Science Initiative Program \- Code ICN17002\. Diego Bustamante was partially funded by ANID \- Subdirección de Capital Humano \(Magíster Nacional, 2023, folio 22231282\)\. María Alejandra Schild was financially supported by ANID \(Doctorado Nacional, 2025, folio 21251617\)\. Bernardo Subercaseaux is \(partially\) supported by the DARPA expMath program through the DARPA CMO contract number HR0011262E028\.

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## Appendix ASupplementary Material

### A\.1\.Proof of Lemma[3\.2](https://arxiv.org/html/2607.06407#S3.Thmtheorem2)

We will first prove an auxiliary result\. We start by introducing some terminology\.

LetU=\{ai∣i\>0\}U=\\\{a\_\{i\}\\mid i\>0\\\}be a countably infinite set\. We take a disjoint copyU¯=\{a¯i∣i\>0\}\\overline\{U\}=\\\{\\overline\{a\}\_\{i\}\\mid i\>0\\\}ofUU\. For anX⊆U∪U¯X\\subseteq U\\cup\\overline\{U\}, we define

XU∖U¯\\displaystyle X\_\{U\\setminus\\overline\{U\}\}\\:=\{a∈U∣a∈X​and​a¯∉X\}\\displaystyle:=\\ \\\{a\\in U\\mid a\\in X\\text\{ and \}\\overline\{a\}\\not\\in X\\\}XU∩U¯\\displaystyle X\_\{U\\cap\\overline\{U\}\}\\:=\{a∈U∣a∈X​and​a¯∈X\}\\displaystyle:=\\ \\\{a\\in U\\mid a\\in X\\text\{ and \}\\overline\{a\}\\in X\\\}XU¯∖U\\displaystyle X\_\{\\overline\{U\}\\setminus U\}\\:=\{a¯∈U¯∣a¯∈X​and​a∉X\}\\displaystyle:=\\ \\\{\\overline\{a\}\\in\\overline\{U\}\\mid\\overline\{a\}\\in X\\text\{ and \}a\\not\\in X\\\}Theℓ\\ell\-typeofXX, forℓ≥0\\ell\\geq 0, is the tuple

\(min⁡\{ℓ,\|XU∖U¯\|\},min⁡\{ℓ,\|XU∩U¯\|\},min⁡\{ℓ,\|XU¯∖U\|\}\)\.\\big\(\\min\{\\\{\\ell,\|X\_\{U\\setminus\\overline\{U\}\}\|\\\}\},\\,\\min\{\\\{\\ell,\|X\_\{U\\cap\\overline\{U\}\}\|\\\}\},\\,\\min\{\\\{\\ell,\|X\_\{\\overline\{U\}\\setminus U\}\|\\\}\}\\big\)\.We writeX⇆ℓX′X\\leftrightarrows\_\{\\ell\}X^\{\\prime\}, forX,X′⊆U∪U¯X,X^\{\\prime\}\\subseteq U\\cup\\overline\{U\}, ifXXandX′X^\{\\prime\}have the sameℓ\\ell\-type\. IfX⊆U∪U¯X\\subseteq U\\cup\\overline\{U\}, thenXXiswell formed\(wf\) if for eachi\>0i\>0at most one element from\{ai,a¯i\}\\\{a\_\{i\},\\overline\{a\}\_\{i\}\\\}is inXX\.

###### Lemma A\.1\.

Assume thatX⇆3k\+1YX\\leftrightarrows\_\{3^\{k\+1\}\}Y, forX,Y⊆U∪U¯X,Y\\subseteq U\\cup\\overline\{U\}andk≥0k\\geq 0\. Then:

- •For every wfX1⊆XX\_\{1\}\\subseteq X, there exists a wfY1⊆YY\_\{1\}\\subseteq Ysuch that X1⇆3kY1andX∖X1⇆3kY∖Y1\.X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\_\{1\}\\ \\ \\text\{ and \}\\ \\ X\\setminus X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\\setminus Y\_\{1\}\.
- •For every wfY1⊆YY\_\{1\}\\subseteq Y, there exists a wfX1⊆XX\_\{1\}\\subseteq Xsuch that X1⇆3kY1andX∖X1⇆3kY∖Y1\.X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\_\{1\}\\ \\ \\text\{ and \}\\ \\ X\\setminus X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\\setminus Y\_\{1\}\.

###### Proof\.

GivenZ⊆UZ\\subseteq U, we useZ¯\\overline\{Z\}to denote the set\{a¯∈U¯∣a∈Z\}\\\{\\overline\{a\}\\in\\overline\{U\}\\mid a\\in Z\\\}, and givenW⊆U¯W\\subseteq\\overline\{U\}, we useW¯\\overline\{W\}to denote the set\{a∈U∣a¯∈W\}\\\{a\\in U\\mid\\overline\{a\}\\in W\\\}\. LetX1X\_\{1\}be a wf subset ofXX\. Then we have thatX1=X1,1∪X1,2∪X1,3∪X1,4X\_\{1\}=X\_\{1,1\}\\cup X\_\{1,2\}\\cup X\_\{1,3\}\\cup X\_\{1,4\}, where

X1,1\\displaystyle X\_\{1,1\}⊆\\displaystyle\\subseteqXU∖U¯,\\displaystyle X\_\{U\\setminus\\overline\{U\}\},X1,2\\displaystyle X\_\{1,2\}⊆\\displaystyle\\subseteq\{a∈U∣a∈XU∩U¯\},\\displaystyle\\\{a\\in U\\mid a\\in X\_\{U\\cap\\overline\{U\}\}\\\},X1,3\\displaystyle X\_\{1,3\}⊆\\displaystyle\\subseteq\{a¯∈U¯∣a∈XU∩U¯\},\\displaystyle\\\{\\overline\{a\}\\in\\overline\{U\}\\mid a\\in X\_\{U\\cap\\overline\{U\}\}\\\},X1,4\\displaystyle X\_\{1,4\}⊆\\displaystyle\\subseteqXU¯∖U,\\displaystyle X\_\{\\overline\{U\}\\setminus U\},andX1,2¯∩X1,3=∅\\overline\{X\_\{1,2\}\}\\cap X\_\{1,3\}=\\emptyset\(sinceX1X\_\{1\}is wf\)\. We construct a setY1=Y1,1∪Y1,2∪Y1,3∪Y1,4Y\_\{1\}=Y\_\{1,1\}\\cup Y\_\{1,2\}\\cup Y\_\{1,3\}\\cup Y\_\{1,4\}by considering the following rules\.

1. \(1\)If\|XU∖U¯\|<3k\+1\|X\_\{U\\setminus\\overline\{U\}\}\|<3^\{k\+1\}, then\|YU∖U¯\|=\|XU∖U¯\|\|Y\_\{U\\setminus\\overline\{U\}\}\|=\|X\_\{U\\setminus\\overline\{U\}\}\|sinceX⇆3k\+1YX\\leftrightarrows\_\{3^\{k\+1\}\}Y\. In this case, we chooseY1,1⊆YU∖U¯Y\_\{1,1\}\\subseteq Y\_\{U\\setminus\\overline\{U\}\}in such a way that\|Y1,1\|=\|X1,1\|\|Y\_\{1,1\}\|=\|X\_\{1,1\}\|and\|YU∖U¯∖Y1,1\|=\|XU∖U¯∖X1,1\|\|Y\_\{U\\setminus\\overline\{U\}\}\\setminus Y\_\{1,1\}\|=\|X\_\{U\\setminus\\overline\{U\}\}\\setminus X\_\{1,1\}\|\. If\|XU∖U¯\|≥3k\+1\|X\_\{U\\setminus\\overline\{U\}\}\|\\geq 3^\{k\+1\}, then\|YU∖U¯\|≥3k\+1\|Y\_\{U\\setminus\\overline\{U\}\}\|\\geq 3^\{k\+1\}sinceX⇆3k\+1YX\\leftrightarrows\_\{3^\{k\+1\}\}Y\. In this case, we chooseY1,1⊆YU∖U¯Y\_\{1,1\}\\subseteq Y\_\{U\\setminus\\overline\{U\}\}in the following way\. If\|X1,1\|<3k\|X\_\{1,1\}\|<3^\{k\}, then\|Y1,1\|=\|X1,1\|\|Y\_\{1,1\}\|=\|X\_\{1,1\}\|, and if\|XU∖U¯∖X1,1\|<3k\|X\_\{U\\setminus\\overline\{U\}\}\\setminus X\_\{1,1\}\|<3^\{k\}, then\|YU∖U¯∖Y1,1\|=\|XU∖U¯∖X1,1\|\|Y\_\{U\\setminus\\overline\{U\}\}\\setminus Y\_\{1,1\}\|=\|X\_\{U\\setminus\\overline\{U\}\}\\setminus X\_\{1,1\}\|\. Finally, if\|X1,1\|≥3k\|X\_\{1,1\}\|\\geq 3^\{k\}and\|XU∖U¯∖X1,1\|≥3k\|X\_\{U\\setminus\\overline\{U\}\}\\setminus X\_\{1,1\}\|\\geq 3^\{k\}, then\|Y1,1\|≥3k\|Y\_\{1,1\}\|\\geq 3^\{k\}and\|YU∖U¯∖Y1,1\|≥3k\|Y\_\{U\\setminus\\overline\{U\}\}\\setminus Y\_\{1,1\}\|\\geq 3^\{k\}\. Notice that we can choose such a setY1,1Y\_\{1,1\}since\|YU∖U¯\|≥3k\+1\|Y\_\{U\\setminus\\overline\{U\}\}\|\\geq 3^\{k\+1\}\.
2. \(2\)If\|XU∩U¯\|<3k\+1\|X\_\{U\\cap\\overline\{U\}\}\|<3^\{k\+1\}, then\|YU∩U¯\|=\|XU∩U¯\|\|Y\_\{U\\cap\\overline\{U\}\}\|=\|X\_\{U\\cap\\overline\{U\}\}\|sinceX⇆3k\+1YX\\leftrightarrows\_\{3^\{k\+1\}\}Y\. In this case, we chooseY1,2⊆\{a∈U∣a∈YU∩U¯\}Y\_\{1,2\}\\subseteq\\\{a\\in U\\mid a\\in Y\_\{U\\cap\\overline\{U\}\}\\\}andY1,3⊆\{a¯∈U¯∣a∈YU∩U¯\}Y\_\{1,3\}\\subseteq\\\{\\overline\{a\}\\in\\overline\{U\}\\mid a\\in Y\_\{U\\cap\\overline\{U\}\}\\\}in such a way thatY1,2¯∩Y1,3=∅\\overline\{Y\_\{1,2\}\}\\cap Y\_\{1,3\}=\\emptyset,\|Y1,2\|=\|X1,2\|\|Y\_\{1,2\}\|=\|X\_\{1,2\}\|,\|Y1,3\|=\|X1,3\|\|Y\_\{1,3\}\|=\|X\_\{1,3\}\|and\|YU∩U¯∖\(Y1,2∪Y1,3¯\)\|=\|XU∩U¯∖\(X1,2∪X1,3¯\)\|\|Y\_\{U\\cap\\overline\{U\}\}\\setminus\(Y\_\{1,2\}\\cup\\overline\{Y\_\{1,3\}\}\)\|=\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|\. If\|XU∩U¯\|≥3k\+1\|X\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}, then\|YU∩U¯\|≥3k\+1\|Y\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}sinceX⇆3k\+1YX\\leftrightarrows\_\{3^\{k\+1\}\}Y\. In this case, we chooseY1,2⊆\{a∈U∣a∈YU∩U¯\}Y\_\{1,2\}\\subseteq\\\{a\\in U\\mid a\\in Y\_\{U\\cap\\overline\{U\}\}\\\}andY1,3⊆\{a¯∈U¯∣a∈YU∩U¯\}Y\_\{1,3\}\\subseteq\\\{\\overline\{a\}\\in\\overline\{U\}\\mid a\\in Y\_\{U\\cap\\overline\{U\}\}\\\}in the following way\. 1. \(a\)If\|X1,2\|<3k\|X\_\{1,2\}\|<3^\{k\},\|X1,3\|<3k\|X\_\{1,3\}\|<3^\{k\}and\|XU∩U¯∖\(X1,2∪X1,3¯\)\|≥3k\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|\\geq 3^\{k\}, then\|Y1,2\|=\|X1,2\|\|Y\_\{1,2\}\|=\|X\_\{1,2\}\|,\|Y1,3\|=\|X1,3\|\|Y\_\{1,3\}\|=\|X\_\{1,3\}\|andY1,2¯∩Y1,3=∅\\overline\{Y\_\{1,2\}\}\\cap Y\_\{1,3\}=\\emptyset\. Notice that we can choose such setsY1,2Y\_\{1,2\}andY1,3Y\_\{1,3\}since\|YU∩U¯\|≥3k\+1\|Y\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}\. 2. \(b\)If\|X1,2\|<3k\|X\_\{1,2\}\|<3^\{k\},\|X1,3\|≥3k\|X\_\{1,3\}\|\\geq 3^\{k\}and\|XU∩U¯∖\(X1,2∪X1,3¯\)\|<3k\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|<3^\{k\}, then\|Y1,2\|=\|X1,2\|\|Y\_\{1,2\}\|=\|X\_\{1,2\}\|,\|YU∩U¯∖\(Y1,2∪Y1,3¯\)\|=\|XU∩U¯∖\(X1,2∪X1,3¯\)\|\|Y\_\{U\\cap\\overline\{U\}\}\\setminus\(Y\_\{1,2\}\\cup\\overline\{Y\_\{1,3\}\}\)\|=\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|andY1,2¯∩Y1,3=∅\\overline\{Y\_\{1,2\}\}\\cap Y\_\{1,3\}=\\emptyset\. Notice that we can choose such setsY1,2Y\_\{1,2\}andY1,3Y\_\{1,3\}since\|YU∩U¯\|≥3k\+1\|Y\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}\. 3. \(c\)If\|X1,2\|≥3k\|X\_\{1,2\}\|\\geq 3^\{k\},\|X1,3\|<3k\|X\_\{1,3\}\|<3^\{k\}and\|XU∩U¯∖\(X1,2∪X1,3¯\)\|<3k\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|<3^\{k\}, then\|Y1,3\|=\|X1,3\|\|Y\_\{1,3\}\|=\|X\_\{1,3\}\|,\|YU∩U¯∖\(Y1,2∪Y1,3¯\)\|=\|XU∩U¯∖\(X1,2∪X1,3¯\)\|\|Y\_\{U\\cap\\overline\{U\}\}\\setminus\(Y\_\{1,2\}\\cup\\overline\{Y\_\{1,3\}\}\)\|=\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|andY1,2¯∩Y1,3=∅\\overline\{Y\_\{1,2\}\}\\cap Y\_\{1,3\}=\\emptyset\. Notice that we can choose such setsY1,2Y\_\{1,2\}andY1,3Y\_\{1,3\}since\|YU∩U¯\|≥3k\+1\|Y\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}\. 4. \(d\)If\|X1,2\|<3k\|X\_\{1,2\}\|<3^\{k\},\|X1,3\|≥3k\|X\_\{1,3\}\|\\geq 3^\{k\}and\|XU∩U¯∖\(X1,2∪X1,3¯\)\|≥3k\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|\\geq 3^\{k\}, then\|Y1,2\|=\|X1,2\|\|Y\_\{1,2\}\|=\|X\_\{1,2\}\|,\|Y1,3\|≥3k\|Y\_\{1,3\}\|\\geq 3^\{k\},\|YU∩U¯∖\(Y1,2∪Y1,3¯\)\|≥3k\|Y\_\{U\\cap\\overline\{U\}\}\\setminus\(Y\_\{1,2\}\\cup\\overline\{Y\_\{1,3\}\}\)\|\\geq 3^\{k\}andY1,2¯∩Y1,3=∅\\overline\{Y\_\{1,2\}\}\\cap Y\_\{1,3\}=\\emptyset\. Notice that we can choose such setsY1,2Y\_\{1,2\}andY1,3Y\_\{1,3\}since\|YU∩U¯\|≥3k\+1\|Y\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}\. 5. \(e\)If\|X1,2\|≥3k\|X\_\{1,2\}\|\\geq 3^\{k\},\|X1,3\|<3k\|X\_\{1,3\}\|<3^\{k\}and\|XU∩U¯∖\(X1,2∪X1,3¯\)\|≥3k\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|\\geq 3^\{k\}, then\|Y1,3\|=\|X1,3\|\|Y\_\{1,3\}\|=\|X\_\{1,3\}\|,\|Y1,2\|≥3k\|Y\_\{1,2\}\|\\geq 3^\{k\},\|YU∩U¯∖\(Y1,2∪Y1,3¯\)\|≥3k\|Y\_\{U\\cap\\overline\{U\}\}\\setminus\(Y\_\{1,2\}\\cup\\overline\{Y\_\{1,3\}\}\)\|\\geq 3^\{k\}andY1,2¯∩Y1,3=∅\\overline\{Y\_\{1,2\}\}\\cap Y\_\{1,3\}=\\emptyset\. Notice that we can choose such setsY1,2Y\_\{1,2\}andY1,3Y\_\{1,3\}since\|YU∩U¯\|≥3k\+1\|Y\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}\. 6. \(f\)If\|X1,2\|≥3k\|X\_\{1,2\}\|\\geq 3^\{k\},\|X1,3\|≥3k\|X\_\{1,3\}\|\\geq 3^\{k\}and\|XU∩U¯∖\(X1,2∪X1,3¯\)\|<3k\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|<3^\{k\}, then\|YU∩U¯∖\(Y1,2∪Y1,3¯\)\|=\|XU∩U¯∖\(X1,2∪X1,3¯\)\|\|Y\_\{U\\cap\\overline\{U\}\}\\setminus\(Y\_\{1,2\}\\cup\\overline\{Y\_\{1,3\}\}\)\|=\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|,\|Y1,2\|≥3k\|Y\_\{1,2\}\|\\geq 3^\{k\},\|Y1,3\|≥3k\|Y\_\{1,3\}\|\\geq 3^\{k\}andY1,2¯∩Y1,3=∅\\overline\{Y\_\{1,2\}\}\\cap Y\_\{1,3\}=\\emptyset\. Notice that we can choose such setsY1,2Y\_\{1,2\}andY1,3Y\_\{1,3\}since\|YU∩U¯\|≥3k\+1\|Y\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}\. 7. \(g\)If\|X1,2\|≥3k\|X\_\{1,2\}\|\\geq 3^\{k\},\|X1,3\|≥3k\|X\_\{1,3\}\|\\geq 3^\{k\}and\|XU∩U¯∖\(X1,2∪X1,3¯\)\|≥3k\|X\_\{U\\cap\\overline\{U\}\}\\setminus\(X\_\{1,2\}\\cup\\overline\{X\_\{1,3\}\}\)\|\\geq 3^\{k\}, then\|Y1,2\|≥3k\|Y\_\{1,2\}\|\\geq 3^\{k\},\|Y1,3\|≥3k\|Y\_\{1,3\}\|\\geq 3^\{k\},\|YU∩U¯∖\(Y1,2∪Y1,3¯\)\|≥3k\|Y\_\{U\\cap\\overline\{U\}\}\\setminus\(Y\_\{1,2\}\\cup\\overline\{Y\_\{1,3\}\}\)\|\\geq 3^\{k\}andY1,2¯∩Y1,3=∅\\overline\{Y\_\{1,2\}\}\\cap Y\_\{1,3\}=\\emptyset\. Notice that we can choose such setsY1,2Y\_\{1,2\}andY1,3Y\_\{1,3\}since\|YU∩U¯\|≥3k\+1\|Y\_\{U\\cap\\overline\{U\}\}\|\\geq 3^\{k\+1\}\.
3. \(3\)If\|XU¯∖U\|<3k\+1\|X\_\{\\overline\{U\}\\setminus U\}\|<3^\{k\+1\}, then\|YU¯∖U\|=\|XU¯∖U\|\|Y\_\{\\overline\{U\}\\setminus U\}\|=\|X\_\{\\overline\{U\}\\setminus U\}\|sinceX⇆3k\+1YX\\leftrightarrows\_\{3^\{k\+1\}\}Y\. In this case, we chooseY1,4⊆YU¯∖UY\_\{1,4\}\\subseteq Y\_\{\\overline\{U\}\\setminus U\}in such a way that\|Y1,4\|=\|X1,4\|\|Y\_\{1,4\}\|=\|X\_\{1,4\}\|and\|YU¯∖U∖Y1,4\|=\|XU¯∖U∖X1,4\|\|Y\_\{\\overline\{U\}\\setminus U\}\\setminus Y\_\{1,4\}\|=\|X\_\{\\overline\{U\}\\setminus U\}\\setminus X\_\{1,4\}\|\. If\|XU¯∖U\|≥3k\+1\|X\_\{\\overline\{U\}\\setminus U\}\|\\geq 3^\{k\+1\}, then\|YU¯∖U\|≥3k\+1\|Y\_\{\\overline\{U\}\\setminus U\}\|\\geq 3^\{k\+1\}sinceX⇆3k\+1YX\\leftrightarrows\_\{3^\{k\+1\}\}Y\. In this case, we chooseY1,4⊆YU¯∖UY\_\{1,4\}\\subseteq Y\_\{\\overline\{U\}\\setminus U\}in the following way\. If\|X1,4\|<3k\|X\_\{1,4\}\|<3^\{k\}, then\|Y1,4\|=\|X1,4\|\|Y\_\{1,4\}\|=\|X\_\{1,4\}\|, and if\|XU¯∖U∖X1,4\|<3k\|X\_\{\\overline\{U\}\\setminus U\}\\setminus X\_\{1,4\}\|<3^\{k\}, then\|YU¯∖U∖Y1,4\|=\|XU¯∖U∖X1,4\|\|Y\_\{\\overline\{U\}\\setminus U\}\\setminus Y\_\{1,4\}\|=\|X\_\{\\overline\{U\}\\setminus U\}\\setminus X\_\{1,4\}\|\. Finally, if\|X1,4\|≥3k\|X\_\{1,4\}\|\\geq 3^\{k\}and\|XU¯∖U∖X1,4\|≥3k\|X\_\{\\overline\{U\}\\setminus U\}\\setminus X\_\{1,4\}\|\\geq 3^\{k\}, then\|Y1,4\|≥3k\|Y\_\{1,4\}\|\\geq 3^\{k\}and\|YU¯∖U∖Y1,4\|≥3k\|Y\_\{\\overline\{U\}\\setminus U\}\\setminus Y\_\{1,4\}\|\\geq 3^\{k\}\. Notice that we can choose such a setY1,4Y\_\{1,4\}since\|YU¯∖U\|≥3k\+1\|Y\_\{\\overline\{U\}\\setminus U\}\|\\geq 3^\{k\+1\}\.

By definition ofY1,1Y\_\{1,1\},Y1,2Y\_\{1,2\},Y1,3Y\_\{1,3\}andY1,4Y\_\{1,4\}, it is straightforward to conclude thatY1Y\_\{1\}is wf,X1⇆3kY1X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\_\{1\}and\(X∖X1\)⇆3k\(Y∖Y1\)\(X\\setminus X\_\{1\}\)\\leftrightarrows\_\{3^\{k\}\}\(Y\\setminus Y\_\{1\}\)\.

We have just proved that for every wfX1⊆XX\_\{1\}\\subseteq X, there exists a wfY1⊆YY\_\{1\}\\subseteq Ysuch thatX1⇆3kY1X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\_\{1\}andX∖X1⇆3kY∖Y1X\\setminus X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\\setminus Y\_\{1\}\. In the same way, it can be shown that for every wfY1⊆YY\_\{1\}\\subseteq Y, there exists a wfX1⊆XX\_\{1\}\\subseteq Xsuch thatX1⇆3kY1X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\_\{1\}andX∖X1⇆3kY∖Y1X\\setminus X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}Y\\setminus Y\_\{1\}\. This concludes the proof of the lemma\.

∎

We now consider structures of the form𝔄∗=⟨2X,⊆𝔄∗⟩\\mathfrak\{A\}^\{\*\}=\\langle 2^\{X\},\\subseteq^\{\\mathfrak\{A\}^\{\*\}\}\\rangle, whereX⊆U∪U¯X\\subseteq U\\cup\\overline\{U\}and⊆𝔄∗\\subseteq^\{\\mathfrak\{A\}^\{\*\}\}is the relation that contains all pairs\(Y,Z\)\(Y,Z\), forY,Z⊆XY,Z\\subseteq X, such thatY⊆ZY\\subseteq Z\. Given two structures𝔄1∗\\mathfrak\{A\}^\{\*\}\_\{1\}and𝔄2∗\\mathfrak\{A\}^\{\*\}\_\{2\}of this form, perhaps with constants, we write𝔄1∗≡kwf𝔄2∗\\mathfrak\{A\}^\{\*\}\_\{1\}\\equiv\_\{k\}^\{\{\\rm wf\}\}\\mathfrak\{A\}^\{\*\}\_\{2\}to denote that the Duplicator has a winning strategy in thekk\-round Ehrenfeucht\-Fraïssé game played on structures𝔄1∗\\mathfrak\{A\}^\{\*\}\_\{1\}and𝔄2∗\\mathfrak\{A\}^\{\*\}\_\{2\}, but where Spoiler and Duplicator are forced to play wf subsets ofU∪U¯U\\cup\\overline\{U\}only\.

Consider structures𝔄1∗=⟨2X1,⊆𝔄1∗⟩\\mathfrak\{A\}^\{\*\}\_\{1\}=\\langle 2^\{X\_\{1\}\},\\subseteq^\{\\mathfrak\{A\}^\{\*\}\_\{1\}\}\\rangleand𝔄2∗=⟨2X2,⊆𝔄2∗⟩\\mathfrak\{A\}^\{\*\}\_\{2\}=\\langle 2^\{X\_\{2\}\},\\subseteq^\{\\mathfrak\{A\}^\{\*\}\_\{2\}\}\\rangleof the form described above\. We claim that, for everyk≥0k\\geq 0,

\(5\)X1⇆3kX2⟹\(𝔄1∗,\(X1∩U\)\)≡kwf\(𝔄2∗,\(X2∩U\)\)\.X\_\{1\}\\leftrightarrows\_\{3^\{k\}\}X\_\{2\}\\quad\\Longrightarrow\\quad\\big\(\\mathfrak\{A\}^\{\*\}\_\{1\},\(X\_\{1\}\\cap U\)\\big\)\\ \\equiv\_\{k\}^\{\{\\rm wf\}\}\\ \\big\(\\mathfrak\{A\}^\{\*\}\_\{2\},\(X\_\{2\}\\cap U\)\\big\)\.
Before proving the claim \([5](https://arxiv.org/html/2607.06407#A1.E5)\), we explain how it implies Lemma[3\.2](https://arxiv.org/html/2607.06407#S3.Thmtheorem2)\. Take a structure of the form𝔄n=⟨\{0,1,⊥\}n,⊆𝔄n⟩\{\\mathfrak\{A\}\_\{n\}=\\langle\\\{0,1,\\bot\\\}^\{n\},\\subseteq^\{\\mathfrak\{A\}\_\{n\}\}\\rangle\}, where⊆𝔄n\\subseteq^\{\\mathfrak\{A\}\_\{n\}\}is the subsumption relation over\{0,1,⊥\}n\\\{0,1,\\bot\\\}^\{n\}\. Take, on the other hand, the structure𝔄n∗=⟨2X,⊆𝔄n∗⟩\\mathfrak\{A\}\_\{n\}^\{\*\}=\\langle 2^\{X\},\\subseteq^\{\\mathfrak\{A\}\_\{n\}^\{\*\}\}\\rangle, whereX=\{a1,…,an,a¯1,…,a¯n\}X=\\\{a\_\{1\},\\dots,a\_\{n\},\\bar\{a\}\_\{1\},\\dots,\\bar\{a\}\_\{n\}\\\}\. It can be seen that there is an isomorphismffbetween𝔄n\\mathfrak\{A\}\_\{n\}and the substructure of𝔄n∗\\mathfrak\{A\}\_\{n\}^\{\*\}induced by the wf subsets ofXX\. The isomorphismfftakes an instance𝐞∈\{0,1,⊥\}n\\mathbf\{e\}\\in\\\{0,1,\\bot\\\}^\{n\}and maps it toY⊆XY\\subseteq Xsuch that for everyi∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\}, \(a\) if𝐞​\[i\]=1\\mathbf\{e\}\[i\]=1thenai∈Ya\_\{i\}\\in Y, \(b\) if𝐞​\[i\]=0\\mathbf\{e\}\[i\]=0thena¯i∈Y\\bar\{a\}\_\{i\}\\in Y, and \(c\) if𝐞​\[i\]=⊥\\mathbf\{e\}\[i\]=\\botthen neitheraia\_\{i\}nora¯i\\bar\{a\}\_\{i\}is inYY\. By definition, the isomorphismffmaps the tuple\{1\}n\\\{1\\\}^\{n\}in𝔄n\\mathfrak\{A\}\_\{n\}to the setX∩U=\{a1,…,an\}X\\cap U=\\\{a\_\{1\},\\dots,a\_\{n\}\\\}in𝔄n∗\\mathfrak\{A\}^\{\*\}\_\{n\}\.

From claim \([5](https://arxiv.org/html/2607.06407#A1.E5)\), it follows then that ifn,p≥3kn,p\\geq 3^\{k\}it is the case that

\(𝔄n∗,\{a1,…,an\}\)≡kwf\(𝔄p∗,\{a1,…,ap\}\)\.\(\\mathfrak\{A\}^\{\*\}\_\{n\},\\\{a\_\{1\},\\dots,a\_\{n\}\\\}\)\\ \\equiv\_\{k\}^\{\{\\rm wf\}\}\\ \(\\mathfrak\{A\}^\{\*\}\_\{p\},\\\{a\_\{1\},\\dots,a\_\{p\}\\\}\)\.From our previous observations, this implies that

\(𝔄n,\{1\}n\)≡k\(𝔄p,\{1\}p\)\.\(\\mathfrak\{A\}\_\{n\},\\\{1\\\}^\{n\}\)\\ \\equiv\_\{k\}\\ \(\\mathfrak\{A\}\_\{p\},\\\{1\\\}^\{p\}\)\.We conclude, in particular, that𝔄n\+≡k𝔄p\+\\mathfrak\{A\}^\{\+\}\_\{n\}\\equiv\_\{k\}\\mathfrak\{A\}^\{\+\}\_\{p\}, as desired\.

We now prove the claim in \([5](https://arxiv.org/html/2607.06407#A1.E5)\)\. We do it by induction onk≥0k\\geq 0\. The base casesk=0k=0andk=1k=1are immediate\. We now move to the induction case fork\+1k\+1\. Take structures𝔄1∗=⟨2X1,⊆𝔄1∗⟩\\mathfrak\{A\}^\{\*\}\_\{1\}=\\langle 2^\{X\_\{1\}\},\\subseteq^\{\\mathfrak\{A\}^\{\*\}\_\{1\}\}\\rangleand𝔄2∗=⟨2X2,⊆𝔄2∗⟩\\mathfrak\{A\}^\{\*\}\_\{2\}=\\langle 2^\{X\_\{2\}\},\\subseteq^\{\\mathfrak\{A\}^\{\*\}\_\{2\}\}\\rangleof the form described above, such thatX1⇆3k\+1X2X\_\{1\}\\leftrightarrows\_\{3^\{k\+1\}\}X\_\{2\}\. Assume, without loss of generality, that for the first round the Spoiler picks the well formed elementX1′⊆X1X^\{\\prime\}\_\{1\}\\subseteq X\_\{1\}in the structure𝔄1∗\\mathfrak\{A\}^\{\*\}\_\{1\}\. From Lemma[A\.1](https://arxiv.org/html/2607.06407#A1.Thmtheorem1), there existsX2′⊆X2X^\{\\prime\}\_\{2\}\\subseteq X\_\{2\}such that

X1′⇆3kX2′andX1∖X1′⇆3kX2∖X2′\.X^\{\\prime\}\_\{1\}\\leftrightarrows\_\{3^\{k\}\}X^\{\\prime\}\_\{2\}\\ \\ \\text\{ and \}\\ \\ X\_\{1\}\\setminus X^\{\\prime\}\_\{1\}\\leftrightarrows\_\{3^\{k\}\}X\_\{2\}\\setminus X^\{\\prime\}\_\{2\}\.By induction hypothesis, the following holds:

\(⟨2X1′,⊆⟩,\(X1′∩U\)\)\\displaystyle\\big\(\\langle 2^\{X^\{\\prime\}\_\{1\}\},\\subseteq\\rangle,\(X^\{\\prime\}\_\{1\}\\cap U\)\\big\)≡kwf\\displaystyle\\equiv^\{\\rm wf\}\_\{k\}\(⟨2X2′,⊆⟩,\(X2′∩U\)\)\\displaystyle\\big\(\\langle 2^\{X^\{\\prime\}\_\{2\}\},\\subseteq\\rangle,\(X^\{\\prime\}\_\{2\}\\cap U\)\\big\)\(⟨2X1∖X1′,⊆⟩,\(\(X1∖X1′\)∩U\)\)\\displaystyle\\big\(\\langle 2^\{X\_\{1\}\\setminus X^\{\\prime\}\_\{1\}\},\\subseteq\\rangle,\(\(X\_\{1\}\\setminus X^\{\\prime\}\_\{1\}\)\\cap U\)\\big\)≡kwf\\displaystyle\\equiv^\{\\rm wf\}\_\{k\}\(⟨2X2∖X2′,⊆⟩,\(\(X2∖X2′\)∩U\)\)\.\\displaystyle\\big\(\\langle 2^\{X\_\{2\}\\setminus X^\{\\prime\}\_\{2\}\},\\subseteq\\rangle,\(\(X\_\{2\}\\setminus X^\{\\prime\}\_\{2\}\)\\cap U\)\\big\)\.A simple composition argument allows to obtain the following from these two expressions:

\(6\)\(⟨2X1,⊆⟩,\(X1∩U\),X1′\)≡kwf\(⟨2X2,⊆⟩,\(X2∩U\),X2′\)\.\\big\(\\langle 2^\{X\_\{1\}\},\\subseteq\\rangle,\(X\_\{1\}\\cap U\),X^\{\\prime\}\_\{1\}\\big\)\\ \\equiv\_\{k\}^\{\{\\rm wf\}\}\\ \\big\(\\langle 2^\{X\_\{2\}\},\\subseteq\\rangle,\(X\_\{2\}\\cap U\),X^\{\\prime\}\_\{2\}\\big\)\.This holds becauseX1′=\(X1∩U\)X^\{\\prime\}\_\{1\}=\(X\_\{1\}\\cap U\)iffX2′=\(X2∩U\)X^\{\\prime\}\_\{2\}=\(X\_\{2\}\\cap U\)\. In fact, assume thatX1′=\(X1∩U\)X^\{\\prime\}\_\{1\}=\(X\_\{1\}\\cap U\), so thatX1′∩U¯=∅X^\{\\prime\}\_\{1\}\\cap\\overline\{U\}=\\emptyset\. SinceX1′⇆3kX2′X^\{\\prime\}\_\{1\}\\leftrightarrows\_\{3^\{k\}\}X^\{\\prime\}\_\{2\}, it follows thatX2′⊆X2∩UX^\{\\prime\}\_\{2\}\\subseteq X\_\{2\}\\cap U\. On the other hand,X1∖X1′=X1∩U¯X\_\{1\}\\setminus X\_\{1\}^\{\\prime\}=X\_\{1\}\\cap\\overline\{U\}\. AsX1∖X1′⇆3kX2∖X2′X\_\{1\}\\setminus X\_\{1\}^\{\\prime\}\\leftrightarrows\_\{3^\{k\}\}X\_\{2\}\\setminus X^\{\\prime\}\_\{2\}, we conclude thatX2∖X2′⊆U¯X\_\{2\}\\setminus X^\{\\prime\}\_\{2\}\\subseteq\\overline\{U\}, and henceX2∩U⊆X2′X\_\{2\}\\cap U\\subseteq X\_\{2\}^\{\\prime\}\. Combining both inclusions, we obtain thatX2′=\(X2∩U\)X^\{\\prime\}\_\{2\}=\(X\_\{2\}\\cap U\)\. The other direction is completely analogous\.

But Equation \([6](https://arxiv.org/html/2607.06407#A1.E6)\) is equivalent with the following fact:

\(⟨2X1,⊆⟩,\(X1∩U\)\)≡k\+1wf\(⟨2X2,⊆⟩,\(X2∩U\)\)\.\\big\(\\langle 2^\{X\_\{1\}\},\\subseteq\\rangle,\(X\_\{1\}\\cap U\)\\big\)\\ \\equiv\_\{k\+1\}^\{\{\\rm wf\}\}\\ \\big\(\\langle 2^\{X\_\{2\}\},\\subseteq\\rangle,\(X\_\{2\}\\cap U\)\\big\)\.This finishes the proof of Lemma[3\.2](https://arxiv.org/html/2607.06407#S3.Thmtheorem2)\.

### A\.2\.Proof of Lemma[3\.3](https://arxiv.org/html/2607.06407#S3.Thmtheorem3)

Let𝐞i\\mathbf\{e\}\_\{i\}and𝐞i′\\mathbf\{e\}^\{\\prime\}\_\{i\}be the moves played by Spoiler and Duplicator in𝔄ℳ⊕𝔄ℳ1\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\}and𝔄ℳ⊕𝔄ℳ2\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\}, respectively, for the firsti≤ki\\leq krounds of the Ehrenfeucht\-Fraïssé game

\(𝔄ℳ⊕𝔄ℳ1,\{1\}n\+p,\{⊥\}n⋅\{1\}p\)≡k\(𝔄ℳ⊕𝔄ℳ2,\{1\}n\+q,\{⊥\}n⋅\{1\}q\)\.\\displaystyle\\big\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\},\\\{1\\\}^\{n\+p\},\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{p\}\\big\)\\ \\equiv\_\{k\}\\ \\big\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\},\\\{1\\\}^\{n\+q\},\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{q\}\\big\)\.We write𝐞i=\(𝐞i​1,𝐞i​2\)\\mathbf\{e\}\_\{i\}=\(\\mathbf\{e\}\_\{i1\},\\mathbf\{e\}\_\{i2\}\)to denote that𝐞i​1\\mathbf\{e\}\_\{i1\}is the tuple formed by the firstnnfeatures of𝐞i\\mathbf\{e\}\_\{i\}and𝐞i​2\\mathbf\{e\}\_\{i2\}is the one formed by the lastppfeatures of𝐞i\\mathbf\{e\}\_\{i\}\. Similarly, we write𝐞i′=\(𝐞i​1′,𝐞i​2′\)\\mathbf\{e\}^\{\\prime\}\_\{i\}=\(\\mathbf\{e\}^\{\\prime\}\_\{i1\},\\mathbf\{e\}^\{\\prime\}\_\{i2\}\)to denote that𝐞i​1′\\mathbf\{e\}^\{\\prime\}\_\{i1\}is the tuple formed by the firstnnfeatures of𝐞i′\\mathbf\{e\}^\{\\prime\}\_\{i\}and𝐞i​2′\\mathbf\{e\}^\{\\prime\}\_\{i2\}is the one formed by the lastqqfeatures of𝐞i′\\mathbf\{e\}^\{\\prime\}\_\{i\}\.

The winning strategy for Duplicator is as follows\. Supposei−1i\-1rounds have been played, and for roundiithe Spoiler picks element𝐞i∈𝔄ℳ⊕𝔄ℳ1\\mathbf\{e\}\_\{i\}\\in\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\}\(the case when he picks an element in𝔄ℳ⊕𝔄ℳ2\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\}is symmetric\)\. Assume also that𝐞i=\(𝐞i​1,𝐞i​2\)\\mathbf\{e\}\_\{i\}=\(\\mathbf\{e\}\_\{i1\},\\mathbf\{e\}\_\{i2\}\)\. The duplicator then considers the position

\(\(𝐞12,…,𝐞\(i−1\)​2\),\(𝐞12′,…,𝐞\(i−1\)​2′\)\)\\big\(\(\\mathbf\{e\}\_\{12\},\\dots,\\mathbf\{e\}\_\{\(i\-1\)2\}\),\(\\mathbf\{e\}^\{\\prime\}\_\{12\},\\dots,\\mathbf\{e\}^\{\\prime\}\_\{\(i\-1\)2\}\)\\big\)on the game\(𝔄ℳ1,\{1\}p\)≡k\(𝔄ℳ2,\{1\}q\)\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\},\\\{1\\\}^\{p\}\)\\equiv\_\{k\}\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\},\\\{1\\\}^\{q\}\), and finds his response𝐞i​2′\\mathbf\{e\}^\{\\prime\}\_\{i2\}to𝐞i​2\\mathbf\{e\}\_\{i2\}in𝔄ℳ2\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\}\. The Duplicator then responds to the Spoiler’s move𝐞i∈𝔄ℳ⊕𝔄ℳ1\\mathbf\{e\}\_\{i\}\\in\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\}by choosing the element𝐞i′=\(𝐞i​1,𝐞i​2′\)∈𝔄ℳ⊕𝔄ℳ2\\mathbf\{e\}^\{\\prime\}\_\{i\}=\(\\mathbf\{e\}\_\{i1\},\\mathbf\{e\}^\{\\prime\}\_\{i2\}\)\\in\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\}\.

Notice, by definition, that𝐞i=\{1\}n\+p\\mathbf\{e\}\_\{i\}=\\\{1\\\}^\{n\+p\}iff𝐞i′=\{1\}n\+q\\mathbf\{e\}^\{\\prime\}\_\{i\}=\\\{1\\\}^\{n\+q\}\. Similarly,𝐞i=\{⊥\}n⋅\{1\}p\\mathbf\{e\}\_\{i\}=\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{p\}iff𝐞i′=\{⊥\}n⋅\{1\}q\\mathbf\{e\}^\{\\prime\}\_\{i\}=\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{q\}\. Moreover, it is easy to see that playing in this way the Duplicator preserves the subsumption relation\. Analogously, the strategy preserves the𝖯𝗈𝗌\\mathsf\{Pos\}relation\. In fact,𝐞i\\mathbf\{e\}\_\{i\}is a positive instance of𝔄ℳ⊕𝔄ℳ1\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\}iff𝐞i​1\\mathbf\{e\}\_\{i1\}is a positive instance of𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}or𝐞i​2\\mathbf\{e\}\_\{i2\}is a positive instance of𝔄ℳ1\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\}\. By definition, the latter follows if and only if𝐞i​1\\mathbf\{e\}\_\{i1\}is a positive instance of𝔄ℳ\\mathfrak\{A\}\_\{\\mathcal\{M\}\}or𝐞i​2′\\mathbf\{e\}^\{\\prime\}\_\{i2\}is a positive instance of𝔄ℳ2\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\}, which in turn is equivalent to𝐞i′\\mathbf\{e\}^\{\\prime\}\_\{i\}being a positive instance of𝔄ℳ⊕𝔄ℳ2\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\}\.

We conclude that this is a winning strategy for the Duplicator, and hence that

\(𝔄ℳ⊕𝔄ℳ1,\{1\}n\+p,\{⊥\}n⋅\{1\}p\)≡k\(𝔄ℳ⊕𝔄ℳ2,\{1\}n\+q,\{⊥\}n⋅\{1\}q\)\.\\displaystyle\\big\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{1\}\},\\\{1\\\}^\{n\+p\},\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{p\}\\big\)\\ \\equiv\_\{k\}\\ \\big\(\\mathfrak\{A\}\_\{\\mathcal\{M\}\}\\oplus\\mathfrak\{A\}\_\{\\mathcal\{M\}\_\{2\}\},\\\{1\\\}^\{n\+q\},\\\{\\bot\\\}^\{n\}\\cdot\\\{1\\\}^\{q\}\\big\)\.This finishes the proof of Lemma[3\.3](https://arxiv.org/html/2607.06407#S3.Thmtheorem3)\.

### A\.3\.Proof of Lemma[3\.5](https://arxiv.org/html/2607.06407#S3.Thmtheorem5)

To prove thatΣk\+1\\Sigma\_\{k\+1\}\-QBF\(𝖣𝖳𝗋𝖾𝖾\)\(\\mathsf\{DTree\}\)is inΣkP\\Sigma\_\{k\}^\{\\text\{P\}\}, note that we can decide in polynomial time if a given decision tree encodes a tautology\. Therefore, we can use aΣk\\Sigma\_\{k\}\-alternating Turing machine for guessing the values for the firstkkquantifiers, we prune the decision tree according to those guesses, and then we solve the remaining universal quantifier directly\.

For the hardness, we use a reduction from the following standardΣkP\\Sigma\_\{k\}^\{\\text\{P\}\}\-hard problem \(see\(Du and Ko,[2014](https://arxiv.org/html/2607.06407#bib.bib318)\)for a reference\): Given a 3CNF formulaφ\\varphiover the setX=\{x1,…,xm\}X=\\\{x\_\{1\},\\dots,x\_\{m\}\\\}of propositional variables, is it the case that the expressionψ=∃X1​∀X2​⋯​∃Xk​φ\\psi=\\exists X\_\{1\}\\forall X\_\{2\}\\cdots\\exists X\_\{k\}\\,\\varphiholds, whereX1,…,XkX\_\{1\},\\dots,X\_\{k\}is a partition ofXXinkkequivalence classes? Note that the hypothesis ofkkbeing odd is important here because if the last quantifier were universal, we could solve it directly as in the case of decision trees\. Fromψ\\psiwe build in polynomial time aΣk\+1\\Sigma\_\{k\+1\}\-QBFα\\alphaoverℳφ\\mathcal\{M\}\_\{\\varphi\}, whereℳφ\\mathcal\{M\}\_\{\\varphi\}is a decision tree that can be built in polynomial time fromφ\\varphi, such that

\(7\)ψholds⟺αholds\.\\text\{$\\psi$ holds\}\\quad\\Longleftrightarrow\\quad\\text\{$\\alpha$ holds\.\}
We now explain how to defineℳφ\\mathcal\{M\}\_\{\\varphi\}from the CNF formulaφ\\varphi\. Letφ=C1∧⋯∧Cn\\varphi=C\_\{1\}\\wedge\\cdots\\wedge C\_\{n\}be a propositional formula, where eachCiC\_\{i\}is a disjunction of three literals and does not contain repeated or complementary literals\. Moreover, assume that\{x1,…,xm\}\\\{x\_\{1\},\\ldots,x\_\{m\}\\\}is the set of variables occurring inφ\\varphi, and the proof will use partial instances of dimensionn\+mn\+m\. Notice that the lastmmfeatures of such a partial instance𝐞\\mathbf\{e\}naturally define a truth assignment for the propositional formulaφ\\varphi\. More precisely, for everyi∈\{1,…,n\}i\\in\\\{1,\\ldots,n\\\}, we use notation𝐞​\(Ci\)=1\\mathbf\{e\}\(C\_\{i\}\)=1to indicate that there is a disjunctℓ\\ellofCiC\_\{i\}such thatℓ=xj\\ell=x\_\{j\}and𝐞​\[n\+j\]=1\\mathbf\{e\}\[n\+j\]=1, orℓ=¬xj\\ell=\\neg x\_\{j\}and𝐞​\[n\+j\]=0\\mathbf\{e\}\[n\+j\]=0, for somej∈\{1,…,m\}j\\in\\\{1,\\ldots,m\\\}\. Furthermore, we write𝐞​\(φ\)=1\\mathbf\{e\}\(\\varphi\)=1if𝐞​\(Ci\)=1\\mathbf\{e\}\(C\_\{i\}\)=1for everyi∈\{1,…,n\}i\\in\\\{1,\\ldots,n\\\}\.

For each clauseCiC\_\{i\}\(i∈\{1,…,n\}i\\in\\\{1,\\ldots,n\\\}\), letℳCi\\mathcal\{M\}\_\{C\_\{i\}\}be a decision tree of dimensionn\+mn\+m\(but that will only use featuresn\+1,…,n\+mn\+1,\\ldots,n\+m\) such that for every instance𝐞\\mathbf\{e\}:ℳCi​\(𝐞\)=1\\mathcal\{M\}\_\{C\_\{i\}\}\(\\mathbf\{e\}\)=1if and only if𝐞​\(Ci\)=1\\mathbf\{e\}\(C\_\{i\}\)=1\. Notice thatℳCi\\mathcal\{M\}\_\{C\_\{i\}\}can be constructed in constant time as it only needs to contain at most eight paths of depth 3\. For example, assuming thatC=\(x1∨x2∨x3\)C=\(x\_\{1\}\\vee x\_\{2\}\\vee x\_\{3\}\), a possible decision treeℳC\\mathcal\{M\}\_\{C\}is depicted in the following figure:

n\+1n\+1n\+2n\+2n\+2n\+2n\+3n\+3n\+3n\+3n\+3n\+3n\+3n\+3𝐟𝐚𝐥𝐬𝐞\\mathbf\{false\}𝐭𝐫𝐮𝐞\\mathbf\{true\}𝐭𝐫𝐮𝐞\\mathbf\{true\}𝐭𝐫𝐮𝐞\\mathbf\{true\}𝐭𝐫𝐮𝐞\\mathbf\{true\}𝐭𝐫𝐮𝐞\\mathbf\{true\}𝐭𝐫𝐮𝐞\\mathbf\{true\}𝐭𝐫𝐮𝐞\\mathbf\{true\}011011011011011011011

Moreover, defineℳφ\\mathcal\{M\}\_\{\\varphi\}as the following decision tree\.

11ℳC1\\mathcal\{M\}\_\{C\_\{1\}\}22ℳC2\\mathcal\{M\}\_\{C\_\{2\}\}33ℳC3\\mathcal\{M\}\_\{C\_\{3\}\}⋯\\cdotsnnℳCn\\mathcal\{M\}\_\{C\_\{n\}\}𝐭𝐫𝐮𝐞\\mathbf\{true\}01101101111011

Recall that the set of features ofℳφ\\mathcal\{M\}\_\{\\varphi\}is\[1,n\+m\]\[1,n\+m\]\. The formulaα\\alphais defined as

∃P1​∀P2​⋯​∃Pk​∀P​ℳφ,\\exists P\_\{1\}\\forall P\_\{2\}\\cdots\\exists P\_\{k\}\\forall P\\,\\mathcal\{M\}\_\{\\varphi\},assuming thatPiP\_\{i\}, for1≤i≤k1\\leq i\\leq k, is the set\{n\+ℓ∣xℓ∈Xi\}\\\{n\+\\ell\\mid x\_\{\\ell\}\\in X\_\{i\}\\\}, andP=\{1,…,n\}P=\\\{1,\\dots,n\\\}\. That is,PiP\_\{i\}is the set of features fromℳφ\\mathcal\{M\}\_\{\\varphi\}that represent the variables inXiX\_\{i\}andPPis the set of features that are used to encode the clauses ofφ\\varphi\.

We show next that the equivalence stated in \([7](https://arxiv.org/html/2607.06407#A1.E7)\) holds\. For simplicity, we only do it for the casek=1k=1\. The proof fork\>1k\>1uses exactly the same ideas, only that it is slightly more cumbersome\.

Assume, on the one hand, thatψ=∃X1​φ\\psi=\\exists X\_\{1\}\\varphiholds\. That is, there exists an assignmentσ1:X1→\{0,1\}\\sigma\_\{1\}:X\_\{1\}\\to\\\{0,1\\\}such thatφ\\varphiholds when variables inX1X\_\{1\}are interpreted according toσ1\\sigma\_\{1\}\. We show next thatα=∃P1​∀P​ℳφ\\alpha=\\exists P\_\{1\}\\forall P\\mathcal\{M\}\_\{\\varphi\}holds, whereP1P\_\{1\}andPPare defined as above\. Take the partial instance𝐞σ1\\mathbf\{e\}\_\{\\sigma\_\{1\}\}of dimensionn\+mn\+mthat naturally “represents” the assignmentσ1\\sigma\_\{1\}; that is:

- •𝐞σ1​\[i\]=⊥\\mathbf\{e\}\_\{\\sigma\_\{1\}\}\[i\]=\\bot, for eachi∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\},
- •𝐞σ1​\[n\+i\]=σ1​\(xi\)\\mathbf\{e\}\_\{\\sigma\_\{1\}\}\[n\+i\]=\\sigma\_\{1\}\(x\_\{i\}\), for eachi∈\{1,…,m\}i\\in\\\{1,\\dots,m\\\}withxi∈X1x\_\{i\}\\in X\_\{1\}, and
- •𝐞σ1​\[n\+i\]=⊥\\mathbf\{e\}\_\{\\sigma\_\{1\}\}\[n\+i\]=\\bot, for eachi∈\{1,…,m\}i\\in\\\{1,\\dots,m\\\}withxi∉X1x\_\{i\}\\not\\in X\_\{1\}\.

To show thatα\\alphaholds, it suffices to show thatℳφ​\(𝐞\)=1\\mathcal\{M\}\_\{\\varphi\}\(\\mathbf\{e\}\)=1for every instance𝐞\\mathbf\{e\}of dimensionn\+mn\+mthat subsumes𝐞σ1\\mathbf\{e\}\_\{\\sigma\_\{1\}\}\. Take an arbitrary such an instance𝐞∈\{0,1\}n\+m\\mathbf\{e\}\\in\\\{0,1\\\}^\{n\+m\}\. Notice that if𝐞​\[i\]=1\\mathbf\{e\}\[i\]=1, for everyi∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\}, thenℳφ​\(𝐞\)=1\\mathcal\{M\}\_\{\\varphi\}\(\\mathbf\{e\}\)=1by definition ofℳφ\\mathcal\{M\}\_\{\\varphi\}\. Suppose then that there exists a minimum valuei∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\}such that𝐞​\[i\]=0\\mathbf\{e\}\[i\]=0\. Hence, to show thatℳφ​\(𝐞\)=1\\mathcal\{M\}\_\{\\varphi\}\(\\mathbf\{e\}\)=1we need to show thatℳCi​\(𝐞\)=1\\mathcal\{M\}\_\{C\_\{i\}\}\(\\mathbf\{e\}\)=1\. But this follows easily from the fact that𝐞\\mathbf\{e\}naturally represents an assignmentσ\\sigmaforφ\\varphisuch that the restriction ofσ\\sigmatoX1X\_\{1\}is preciselyσ1\\sigma\_\{1\}\. We know that any such an assignmentσ\\sigmasatisfiesφ\\varphi, and therefore it satisfiesCiC\_\{i\}\. It follows thatℳCi​\(𝐞\)=1\\mathcal\{M\}\_\{C\_\{i\}\}\(\\mathbf\{e\}\)=1\.

Assume, on the other hand, thatα=∃P1​∀P​ℳφ\\alpha=\\exists P\_\{1\}\\forall P\\mathcal\{M\}\_\{\\varphi\}holds\. Then there exists a partial instance𝐞\\mathbf\{e\}of dimensionn\+mn\+msuch that the following statements hold:

- •𝐞​\[i\]≠⊥\\mathbf\{e\}\[i\]\\neq\\botiff for somej∈\{1,…,m\}j\\in\\\{1,\\dots,m\\\}it is the case thati=n\+ji=n\+jandj∈P1j\\in P\_\{1\}, and
- •for every𝐞′∈comp​\(𝐞\)\\mathbf\{e\}^\{\\prime\}\\in\\textit\{comp\}\(\\mathbf\{e\}\)we have thatℳφ​\(𝐞′\)=1\\mathcal\{M\}\_\{\\varphi\}\(\\mathbf\{e\}^\{\\prime\}\)=1\.

We show next thatψ=∃X1​φ\\psi=\\exists X\_\{1\}\\varphiholds\. Letσ1:X1→\{0,1\}\\sigma\_\{1\}:X\_\{1\}\\to\\\{0,1\\\}be the assignment for the variables inX1X\_\{1\}that is naturally defined by𝐞\\mathbf\{e\}\. It suffices to show that each clauseCiC\_\{i\}ofφ\\varphi, fori∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\}, is satisfied by the assignment that interprets the variables inX1X\_\{1\}according toσ1\\sigma\_\{1\}\. Let us define a completion𝐞′\\mathbf\{e\}^\{\\prime\}of𝐞\\mathbf\{e\}that satisfies the following:

- •𝐞′​\[i\]=0\\mathbf\{e\}^\{\\prime\}\[i\]=0,
- •𝐞′​\[j\]=1\\mathbf\{e\}^\{\\prime\}\[j\]=1, for eachj∈\{1,…,n\}j\\in\\\{1,\\dots,n\\\}withi≠ji\\neq j, and
- •𝐞′​\[n\+j\]=σ1​\(xj\)\\mathbf\{e\}^\{\\prime\}\[n\+j\]=\\sigma\_\{1\}\(x\_\{j\}\), ifj∈\{1,…,m\}j\\in\\\{1,\\dots,m\\\}andj∈P1j\\in P\_\{1\}\.

We know thatℳφ​\(𝐞′\)=1\\mathcal\{M\}\_\{\\varphi\}\(\\mathbf\{e\}^\{\\prime\}\)=1, which implies thatℳCi​\(𝐞′\)=1\\mathcal\{M\}\_\{C\_\{i\}\}\(\\mathbf\{e\}^\{\\prime\}\)=1\(since𝐞′\\mathbf\{e\}^\{\\prime\}takes value 0 for featureii\)\. We conclude thatCiC\_\{i\}is satisfied by the assignment which is naturally defined by𝐞′\\mathbf\{e\}^\{\\prime\}, which is precisely the one that interprets the variables inX1X\_\{1\}according toσ1\\sigma\_\{1\}\.

### A\.4\.Proof of Lemma[4\.4](https://arxiv.org/html/2607.06407#S4.Thmtheorem4)

We induct on the depth ofφ\\varphi\. First we will see the atomic cases:

- •For atomic formulas of the formxi=xjx\_\{i\}=x\_\{j\}, we take ∑ρ∈\{0,1,⊥\}kρi≠ρjzρ=0\.\\sum\_\{\\begin\{subarray\}\{c\}\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\\\\ \\rho\_\{i\}\\neq\\rho\_\{j\}\\end\{subarray\}\}\{z\_\{\\rho\}\}\\;=\\;0\.
- •For atomic formulas of the formxi⊆xjx\_\{i\}\\subseteq x\_\{j\}, we take ∑ρ∈\{0,1,⊥\}kρi≠ρj∧ρi≠⊥zρ=0\.\\sum\_\{\\begin\{subarray\}\{c\}\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\\\\ \\rho\_\{i\}\\neq\\rho\_\{j\}\\;\\land\\;\\rho\_\{i\}\\neq\\bot\\end\{subarray\}\}\{z\_\{\\rho\}\}\\;=\\;0\.
- •For atomic formulas of the formxi⪯xjx\_\{i\}\\preceq x\_\{j\}, we take ∑ρ∈\{0,1,⊥\}kρj=⊥zρ≤∑ρ∈\{0,1,⊥\}kρi=⊥zρ\.\\sum\_\{\\begin\{subarray\}\{c\}\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\\\\ \\rho\_\{j\}=\\bot\\end\{subarray\}\}\{z\_\{\\rho\}\}\\;\\leq\\;\\sum\_\{\\begin\{subarray\}\{c\}\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\\\\ \\rho\_\{i\}=\\bot\\end\{subarray\}\}\{z\_\{\\rho\}\}\.

Now we describe the structural induction\. For the negation, it is enough to takeTΓ⁡\(¬ψ\)​\(\(zρ\)ρ\)\\operatorname\{T\}\_\{\\Gamma\}\(\\neg\\psi\)\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\}\\bigr\)as¬TΓ⁡\(ψ\)​\(\(zρ\)ρ\)\\neg\\operatorname\{T\}\_\{\\Gamma\}\(\\psi\)\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\}\\bigr\)\. Ifφ\\varphiis of the formψ1∘ψ2\\psi\_\{1\}\\circ\\psi\_\{2\}, where∘\\circis a binary logical connective, then we takeTΓ⁡\(φ\)​\(\(zρ\)ρ\)\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\}\\bigr\)as

TΓ⁡\(ψ1\)​\(\(zρ\)\)∘TΓ⁡\(ψ2\)​\(\(zρ\)\)\.\\operatorname\{T\}\_\{\\Gamma\}\(\\psi\_\{1\}\)\\bigl\(\(z\_\{\\rho\}\)\\bigr\)\\;\\circ\\;\\operatorname\{T\}\_\{\\Gamma\}\(\\psi\_\{2\}\)\\bigl\(\(z\_\{\\rho\}\)\\bigr\)\.Note that here it is important to use the inductive hypothesis withΓ\\Gammathat contains both the free variables ofψ1\\psi\_\{1\}and ofψ2\\psi\_\{2\}\. We need to be more careful ifφ\\varphiis of the form∃u​ψ​\(y1,…,yℓ,u\)\\exists u\\,\\psi\(y\_\{1\},\\dots,y\_\{\\ell\},u\)\. By changing the name of the variable if necessary, we can assume thatu∉Γu\\not\\in\\Gamma\. ConsiderΛ=\(x1,…,xk,u\)\\Lambda=\(x\_\{1\},\\dots,x\_\{k\},u\)and take the formula

TΛ⁡\(ψ\)​\(\(wτ\)τ∈\{0,1,⊥\}k\+1\)\\operatorname\{T\}\_\{\\Lambda\}\(\\psi\)\\bigl\(\(w\_\{\\tau\}\)\_\{\\tau\\in\\\{0,1,\\bot\\\}^\{k\+1\}\}\\bigr\)given by the inductive hypothesis\. We takeTΓ⁡\(φ\)​\(\(zρ\)ρ∈\{0,1,⊥\}k\)\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}\\bigr\)to be

∃\(wτ\)τ∈\{0,1,⊥\}k\+1ProjΛ→Γ⁡\(\(wτ\)τ,\(zρ\)ρ\)∧TΛ⁡\(ψ\)​\(\(wτ\)τ∈\{0,1,⊥\}k\+1\),\\exists\(w\_\{\\tau\}\)\_\{\\tau\\in\\\{0,1,\\bot\\\}^\{k\+1\}\}\\quad\\operatorname\{Proj\}\_\{\\Lambda\\to\\Gamma\}\(\(w\_\{\\tau\}\)\_\{\\tau\},\\,\(z\_\{\\rho\}\)\_\{\\rho\}\)\\;\\land\\;\\operatorname\{T\}\_\{\\Lambda\}\(\\psi\)\\bigl\(\(w\_\{\\tau\}\)\_\{\\tau\\in\\\{0,1,\\bot\\\}^\{k\+1\}\}\\bigr\),whereProjΛ→Γ⁡\(\(wτ\)τ,\(zρ\)ρ\)\\operatorname\{Proj\}\_\{\\Lambda\\to\\Gamma\}\(\(w\_\{\\tau\}\)\_\{\\tau\},\\,\(z\_\{\\rho\}\)\_\{\\rho\}\)is defined to be

⋀ρ∈\{0,1,⊥\}kzρ=w\(ρ,0\)\+w\(ρ,1\)\+w\(ρ,⊥\)\.\\bigwedge\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}\{z\_\{\\rho\}=w\_\{\(\\rho,0\)\}\+w\_\{\(\\rho,1\)\}\+w\_\{\(\\rho,\\bot\)\}\}\.Finally, ifφ\\varphiis of the form∀u​ψ​\(y1,…,yℓ,u\)\\forall u\\,\\psi\(y\_\{1\},\\dots,y\_\{\\ell\},u\), we defineΛ\\Lambdaas before and takeTΓ⁡\(φ\)​\(\(zρ\)ρ∈\{0,1,⊥\}k\)\\operatorname\{T\}\_\{\\Gamma\}\(\\varphi\)\\bigl\(\(z\_\{\\rho\}\)\_\{\\rho\\in\\\{0,1,\\bot\\\}^\{k\}\}\\bigr\)to be

∀\(wτ\)τ∈\{0,1,⊥\}k\+1\(ProjΛ→Γ⁡\(\(wτ\)τ,\(zρ\)ρ\)→TΛ⁡\(ψ\)​\(\(wτ\)τ∈\{0,1,⊥\}k\+1\)\)\.\\forall\(w\_\{\\tau\}\)\_\{\\tau\\in\\\{0,1,\\bot\\\}^\{k\+1\}\}\\quad\\bigl\(\\operatorname\{Proj\}\_\{\\Lambda\\to\\Gamma\}\(\(w\_\{\\tau\}\)\_\{\\tau\},\\,\(z\_\{\\rho\}\)\_\{\\rho\}\)\\;\\rightarrow\\;\\operatorname\{T\}\_\{\\Lambda\}\(\\psi\)\\bigl\(\(w\_\{\\tau\}\)\_\{\\tau\\in\\\{0,1,\\bot\\\}^\{k\+1\}\}\\bigr\)\\bigr\)\.For the second part of the lemma, note that for a subformula translated in a contextΓ\\Gammaof sizekk, the corresponding Presburger formula has3k3^\{k\}variables\. In the atomic cases we just need to manage sums without repetitions over those variables\. Boolean connectives also do not cause any problems\. In the case of quantifiers, we need to increase the size of the context fromkktok\+1k\+1and we also add3k3^\{k\}projection formulas plus the recursive call\. But becausek\+1≤wd​\(φ\)k\+1\\leq\\rm\{wd\}\(\\varphi\)and there areO​\(\|φ\|\)O\(\|\\varphi\|\)subformulas, the total output size isO​\(\|φ\|⋅3wd​\(φ\)\)O\(\|\\varphi\|\\cdot 3^\{\\rm\{wd\}\(\\varphi\)\}\)up to polynomial factors ofwd​\(φ\)\\rm\{wd\}\(\\varphi\)\. Notice that the same argument applies for proving that the computation itself can be done using at most that same space\.

### A\.5\.Proof of Lemma[4\.6](https://arxiv.org/html/2607.06407#S4.Thmtheorem6)

Consider the following similar problem\. The input is a modelℳ′\\mathcal\{M\}^\{\\prime\}, an instance𝐞′\\mathbf\{e\}^\{\\prime\}and ak∈ℕk\\in\\mathbb\{N\}, and the question is whether there exists a partial instance𝐞\\mathbf\{e\}that is a weak abductive explanation for𝐞′\\mathbf\{e\}^\{\\prime\}onℳ′\\mathcal\{M\}^\{\\prime\}and whose number of defined features is at mostkk\. This problem was studied in\(Barcelóet al\.,[2020](https://arxiv.org/html/2607.06407#bib.bib41)\), where it was shown to beNP\{\\rm NP\}\-hard on decision trees\. We show a reduction from this problem\.

First assume thatℳ′​\(𝐞′\)=1\\mathcal\{M\}^\{\\prime\}\(\\mathbf\{e\}^\{\\prime\}\)=1\. We create new variablesXiX\_\{i\}fori∈\{0,1,…,k\}i\\in\\\{0,1,\.\.\.,k\\\}\. Letℳ\\mathcal\{M\}be a new decision tree such thatdim\(ℳ\)=k\+1\+dim\(ℳ′\)\\dim\(\\mathcal\{M\}\)=k\+1\+\\dim\(\\mathcal\{M\}^\{\\prime\}\), depicted in the following figure:

![[Uncaptioned image]](https://arxiv.org/html/2607.06407v1/figures/reduction_msr.png)Visual description of the reduction\.The figure shows the decision tree used in the reduction\. It is a chain of new variables X\_0, X\_1, …, X\_k, where each edge labeled 1 continues along the chain and the final 1\-edge reaches a leaf\. Every edge labeled 0 branches to a copy of the original decision tree M’\.

We useX0X\_\{0\}as the root ofℳ\\mathcal\{M\}\. For everyi<ki<k, the outgoing edge ofXiX\_\{i\}labeled by11is connected toXi\+1X\_\{i\+1\}, and the outgoing edge ofXkX\_\{k\}labeled by11is connected to a𝐭𝐫𝐮𝐞\\mathbf\{true\}leaf\. Connect all outgoing edges labeled by0to a copy ofℳ′\\mathcal\{M\}^\{\\prime\}\. Let𝐞1=\{1\}k\+1⋅𝐞′\\mathbf\{e\}\_\{1\}=\\\{1\\\}^\{k\+1\}\\cdot\\mathbf\{e\}^\{\\prime\}and𝐞2=\{1\}k\+1⋅\{⊥\}dim\(ℳ′\)\\mathbf\{e\}\_\{2\}=\\\{1\\\}^\{k\+1\}\\cdot\\\{\\bot\\\}^\{\\dim\(\\mathcal\{M\}^\{\\prime\}\)\}be partial instances of sized​i​m​\(ℳ\)\{\\textit\{d\}im\}\(\\mathcal\{M\}\)\. We claim that𝐞2\\mathbf\{e\}\_\{2\}is a minimum abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}onℳ\\mathcal\{M\}if and only if the answer to the original problem was negative\. For the caseℳ′​\(𝐞′\)=0\\mathcal\{M\}^\{\\prime\}\(\\mathbf\{e\}^\{\\prime\}\)=0we can just set the value of the new leaf to𝐟𝐚𝐥𝐬𝐞\\mathbf\{false\}and the same construction will work\.

We now discuss why the reduction works\. Suppose first that\(ℳ′,𝐞′,k\)\(\\mathcal\{M\}^\{\\prime\},\\mathbf\{e\}^\{\\prime\},k\)outputsYes\. It follows that there exists a partial instance𝐞\\mathbf\{e\}onℳ′\\mathcal\{M\}^\{\\prime\}that is a weak abductive explanation for𝐞′\\mathbf\{e\}^\{\\prime\}with at mostkkdefined features\. Now notice that the partial instance\{⊥\}k\+1⋅𝐞\\\{\\bot\\\}^\{k\+1\}\\cdot\\mathbf\{e\}is a weak abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}onℳ\\mathcal\{M\}and has at mostkkdefined features\. Because𝐞2\\mathbf\{e\}\_\{2\}hask\+1k\+1defined features, it follows that𝐞2\\mathbf\{e\}\_\{2\}is not a minimum abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}onℳ\\mathcal\{M\}\.

Now suppose that\(ℳ′,𝐞′,k\)\(\\mathcal\{M\}^\{\\prime\},\\mathbf\{e\}^\{\\prime\},k\)outputsNo\. Then there is no weak abductive explanation for𝐞′\\mathbf\{e\}^\{\\prime\}onℳ′\\mathcal\{M\}^\{\\prime\}with at mostkkdefined features\. This implies that there is also no partial instance𝐞\\mathbf\{e\}onℳ\\mathcal\{M\}that is a weak abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}with at mostkkdefined features\. This is because any candidate weak abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}with at mostkkdefined features leaves at least one of the new variables undefined, and therefore some completion of it reaches a copy ofℳ′\\mathcal\{M\}^\{\\prime\}\. Once it enters that copy ofℳ′\\mathcal\{M\}^\{\\prime\}, its restrictions on the old coordinates would induce a weak abductive explanation for𝐞′\\mathbf\{e\}^\{\\prime\}onℳ′\\mathcal\{M\}^\{\\prime\}with at mostkkdefined features, contradicting the assumption\. But we know that𝐞2\\mathbf\{e\}\_\{2\}is a weak abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}withk\+1k\+1defined features, so it is a minimum abductive explanation for𝐞1\\mathbf\{e\}\_\{1\}onℳ\\mathcal\{M\}, as we needed\.

### A\.6\.Proof of Lemma[5\.5](https://arxiv.org/html/2607.06407#S5.Thmtheorem5)

We first treat the parameter\-free caseρ​\(x,y\)\\rho\(x,y\), and then we extend the idea to the general case\.

Fix a dimensionnn\. Given a partial instance of dimensionnn, define\#0​\(𝐞\)\\\#\_\{0\}\(\\mathbf\{e\}\)as the number of occurrences of the symbol0in𝐞\\mathbf\{e\}, and likewise for\#1​\(𝐞\)\\\#\_\{1\}\(\\mathbf\{e\}\)and\#⊥​\(𝐞\)\\\#\_\{\\bot\}\(\\mathbf\{e\}\)\. Moreover, for every\(p,q,r\)∈ℕ3\(p,q,r\)\\in\\mathbb\{N\}^\{3\}such thatp\+q\+r=np\+q\+r=n, define

L\(p,q,r\):=\{𝐞∣𝐞​is a partial instance of dimension​n​such that​\#0​\(𝐞\)=p,\#1​\(𝐞\)=q​and​\#⊥​\(𝐞\)=r\}\.\\displaystyle L\_\{\(p,q,r\)\}\\ :=\\ \\\{\\mathbf\{e\}\\mid\\mathbf\{e\}\\text\{ is a partial instance of dimension \}n\\text\{ such that \}\\\#\_\{0\}\(\\mathbf\{e\}\)=p,\\,\\\#\_\{1\}\(\\mathbf\{e\}\)=q\\text\{ and \}\\\#\_\{\\bot\}\(\\mathbf\{e\}\)=r\\\}\.Notice that there are at most\(n\+22\)≤\(n\+1\)2\\binom\{n\+2\}\{2\}\\leq\(n\+1\)^\{2\}different setsL\(p,q,r\)L\_\{\(p,q,r\)\}\. We claim that if𝐞1\\mathbf\{e\}\_\{1\}and𝐞2\\mathbf\{e\}\_\{2\}are partial instances of dimensionnnsuch that𝐞1,𝐞2∈L\(p,q,r\)\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\\in L\_\{\(p,q,r\)\}for the same triple\(p,q,r\)\(p,q,r\), then𝔅n⊧̸ρ​\(𝐞1,𝐞2\)\\mathfrak\{B\}\_\{n\}\\not\\models\\rho\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\. From that we can conclude that the statement of the lemma holds forp​\(n\)=\(n\+1\)2p\(n\)=\(n\+1\)^\{2\}, since if\(𝐞1,…,𝐞k\)\(\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k\}\)is a path of dimensionnninρ​\(x,y\)\\rho\(x,y\), then each𝐞i\\mathbf\{e\}\_\{i\}must belong to a different setL\(p,q,r\)L\_\{\(p,q,r\)\}\.

For the sake of contradiction, suppose that𝐞1,𝐞2\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}are two different partial instances of dimensionnnthat belong to the same setL\(p,q,r\)L\_\{\(p,q,r\)\}and such that𝔅n⊧ρ​\(𝐞1,𝐞2\)\\mathfrak\{B\}\_\{n\}\\models\\rho\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\. Then there exists a permutationπ:\{1,…,n\}→\{1,…,n\}\\pi:\\\{1,\\ldots,n\\\}\\to\\\{1,\\ldots,n\\\}such thatπ​\(𝐞1\)=𝐞2\\pi\(\\mathbf\{e\}\_\{1\}\)=\\mathbf\{e\}\_\{2\}\. Notice that for every pair𝐞\\mathbf\{e\},𝐞′\\mathbf\{e\}^\{\\prime\}of partial instances of dimensionnnit holds that:

𝐞⊆𝐞′\\displaystyle\\mathbf\{e\}\\subseteq\\mathbf\{e\}^\{\\prime\}⇔π​\(𝐞\)⊆π​\(𝐞′\)\\displaystyle\\quad\\quad\\Leftrightarrow\\quad\\quad\\pi\(\\mathbf\{e\}\)\\subseteq\\pi\(\\mathbf\{e\}^\{\\prime\}\)𝐞⪯𝐞′\\displaystyle\\mathbf\{e\}\\preceq\\mathbf\{e\}^\{\\prime\}⇔π​\(𝐞\)⪯π​\(𝐞′\)\.\\displaystyle\\quad\\quad\\Leftrightarrow\\quad\\quad\\pi\(\\mathbf\{e\}\)\\preceq\\pi\(\\mathbf\{e\}^\{\\prime\}\)\.Thus,π\\piis an automorphism for the structure𝔅n\\mathfrak\{B\}\_\{n\}\. Because we have that𝔅n⊧ρ​\(𝐞1,𝐞2\)\\mathfrak\{B\}\_\{n\}\\models\\rho\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\), andρ\\rhois defined over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}, it follows𝔅n⊧ρ​\(π​\(𝐞1\),π​\(𝐞2\)\)\\mathfrak\{B\}\_\{n\}\\models\\rho\(\\pi\(\\mathbf\{e\}\_\{1\}\),\\pi\(\\mathbf\{e\}\_\{2\}\)\)\. But since𝐞2=π​\(𝐞1\)\\mathbf\{e\}\_\{2\}=\\pi\(\\mathbf\{e\}\_\{1\}\), we also have that𝔅n⊧ρ​\(𝐞2,π​\(π​\(𝐞1\)\)\)\\mathfrak\{B\}\_\{n\}\\models\\rho\(\\mathbf\{e\}\_\{2\},\\pi\(\\pi\(\\mathbf\{e\}\_\{1\}\)\)\)\. Becauseρ​\(x,y\)\\rho\(x,y\)is transitive, it follows that𝔅n⊧ρ​\(𝐞1,π2​\(𝐞1\)\)\\mathfrak\{B\}\_\{n\}\\models\\rho\(\\mathbf\{e\}\_\{1\},\\pi^\{2\}\(\\mathbf\{e\}\_\{1\}\)\)\. In the same way, we can conclude that𝔅n⊧ρ​\(𝐞1,πk​\(𝐞1\)\)\\mathfrak\{B\}\_\{n\}\\models\\rho\(\\mathbf\{e\}\_\{1\},\\pi^\{k\}\(\\mathbf\{e\}\_\{1\}\)\)for everyk≥1k\\geq 1\. Given that the set of permutations ofnnelements with the composition operator forms a group of ordern\!n\!, we know thatπn\!\\pi^\{n\!\}is the identity permutation, so thatπn\!​\(𝐞1\)=𝐞1\\pi^\{n\!\}\(\\mathbf\{e\}\_\{1\}\)=\\mathbf\{e\}\_\{1\}\. Therefore, we conclude that𝔅n⊧ρ​\(𝐞1,𝐞1\)\\mathfrak\{B\}\_\{n\}\\models\\rho\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{1\}\), which leads to a contradiction sinceρ​\(x,y\)\\rho\(x,y\)represents a strict partial order\.

Consider now a formulaρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\)with parameters\. As in the previous case, we fix a dimensionnn\. Moreover, we also fix a sequence𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\},…\\ldots,𝐞ℓ′\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}of partial instances of dimensionnn\(notice that the boundp​\(n\)p\(n\)should not depend on those partial instances\)\. Then, for every\(a1,…,aℓ\)∈\{0,1,⊥\}ℓ\(a\_\{1\},\\ldots,a\_\{\\ell\}\)\\in\\\{0,1,\\bot\\\}^\{\\ell\}, consider the set

P\(a1,…,aℓ\)=\{i∈\{1,…,n\}∣𝐞j′​\[i\]=aj​for all​j∈\{1,…,ℓ\}\},\\displaystyle P\_\{\(a\_\{1\},\\ldots,a\_\{\\ell\}\)\}\\ =\\ \\\{i\\in\\\{1,\\ldots,n\\\}\\ \\mid\\ \\mathbf\{e\}^\{\\prime\}\_\{j\}\[i\]=a\_\{j\}\\text\{ for all \}j\\in\\\{1,\\ldots,\\ell\\\}\\\},that is, all positions for which the sequence𝐞1′\\mathbf\{e\}^\{\\prime\}\_\{1\},…\\ldots,𝐞ℓ′\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}realizes the pattern\(a1,…,aℓ\)\(a\_\{1\},\\ldots,a\_\{\\ell\}\)\. Givens∈\{0,1,⊥\}s\\in\\\{0,1,\\bot\\\}, a patternt∈\{0,1,⊥\}ℓt\\in\\\{0,1,\\bot\\\}^\{\\ell\}and a partial instance𝐞\\mathbf\{e\}of dimensionnn, define\#s,t​\(𝐞\)\\\#\_\{s,t\}\(\\mathbf\{e\}\)as the number of indicesi∈Pti\\in P\_\{t\}such that𝐞​\[i\]=s\\mathbf\{e\}\[i\]=s\. Notice that the numbers\#s,t​\(𝐞\)\\\#\_\{s,t\}\(\\mathbf\{e\}\)are invariant under permutations of the features that map each pattern block onto itself\.

We define an equivalence relation as follows\. For two partial instances𝐞1\\mathbf\{e\}\_\{1\}and𝐞2\\mathbf\{e\}\_\{2\}of dimensionnn, we write𝐞1∼𝐞2\\mathbf\{e\}\_\{1\}\\sim\\mathbf\{e\}\_\{2\}if\#s,t​\(𝐞1\)=\#s,t​\(𝐞2\)\\\#\_\{s,t\}\(\\mathbf\{e\}\_\{1\}\)=\\\#\_\{s,t\}\(\\mathbf\{e\}\_\{2\}\)for everys∈\{0,1,⊥\}s\\in\\\{0,1,\\bot\\\}and everyt∈\{0,1,⊥\}ℓt\\in\\\{0,1,\\bot\\\}^\{\\ell\}\. Notice that there are at most

\(n\+3ℓ\+1−13ℓ\+1−1\)≤\(n\+3ℓ\+1−1\)3ℓ\+1−1\(3ℓ\+1−1\)\!\\binom\{n\+3^\{\\ell\+1\}\-1\}\{3^\{\\ell\+1\}\-1\}\\leq\\frac\{\\big\(n\+3^\{\\ell\+1\}\-1\\big\)^\{3^\{\\ell\+1\}\-1\}\}\{\\big\(3^\{\\ell\+1\}\-1\\big\)\!\}different equivalence classes\. Becauseℓ\\ellis fixed, we can consider that number as our polynomialp​\(n\)p\(n\)\.

We claim that if𝐞1∼𝐞2\\mathbf\{e\}\_\{1\}\\sim\\mathbf\{e\}\_\{2\}, then𝔅n⊧̸ρ​\[𝐞1′,…,𝐞ℓ′\]​\(𝐞1,𝐞2\)\\mathfrak\{B\}\_\{n\}\\not\\models\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\. From that we can conclude, as in the parameter\-free case, that the statement of the lemma holds\. In fact, if\(𝐞1,…,𝐞k\)\(\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{k\}\)is a path of dimensionnninρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\), then each𝐞i\\mathbf\{e\}\_\{i\}must belong to a different equivalence class\.

For the sake of contradiction, suppose that𝐞1,𝐞2\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}are two different partial instances of dimensionnnsuch that𝐞1∼𝐞2\\mathbf\{e\}\_\{1\}\\sim\\mathbf\{e\}\_\{2\}and𝔅n⊧ρ​\[𝐞1′,…,𝐞ℓ′\]​\(𝐞1,𝐞2\)\\mathfrak\{B\}\_\{n\}\\models\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)\. For each pattern blockPtP\_\{t\}, consider a permutationπt\\pi\_\{t\}ofPtP\_\{t\}sending the restriction of𝐞1\\mathbf\{e\}\_\{1\}onPtP\_\{t\}to the restriction of𝐞2\\mathbf\{e\}\_\{2\}onPtP\_\{t\}\. Combining these permutations yields a permutationπ:\{1,…,n\}→\{1,…,n\}\\pi:\\\{1,\\ldots,n\\\}\\to\\\{1,\\ldots,n\\\}such thatπ​\(𝐞1\)=𝐞2\\pi\(\\mathbf\{e\}\_\{1\}\)=\\mathbf\{e\}\_\{2\}\. Notice that by the way we constructed the permutationsπt\\pi\_\{t\}and the pattern blocksPtP\_\{t\}, we have thatπ​\(𝐞j′\)=𝐞j′\\pi\(\\mathbf\{e\}^\{\\prime\}\_\{j\}\)=\\mathbf\{e\}^\{\\prime\}\_\{j\}for everyj∈\{1,…,ℓ\}j\\in\\\{1,\\ldots,\\ell\\\}\.

As in the parameter\-free case, we have thatπ\\piis an automorphism for the structure𝔅n\\mathfrak\{B\}\_\{n\}\. Sinceρ\\rhois a formula defined over the vocabulary\{⊆,⪯\}\\\{\\subseteq,\\preceq\\\}, and𝔅n⊧ρ​\[𝐞1′,…,𝐞ℓ′\]​\(𝐞1,𝐞2\)\\mathfrak\{B\}\_\{n\}\\models\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\), it follows that𝔅n⊧ρ​\[π​\(𝐞1′\),…,π​\(𝐞ℓ′\)\]​\(π​\(𝐞1\),π​\(𝐞2\)\)\\mathfrak\{B\}\_\{n\}\\models\\rho\[\\pi\(\\mathbf\{e\}^\{\\prime\}\_\{1\}\),\\ldots,\\pi\(\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\)\]\(\\pi\(\\mathbf\{e\}\_\{1\}\),\\pi\(\\mathbf\{e\}\_\{2\}\)\)\. Becauseπ​\(𝐞j′\)=𝐞j′\\pi\(\\mathbf\{e\}^\{\\prime\}\_\{j\}\)=\\mathbf\{e\}^\{\\prime\}\_\{j\}for everyj∈\{1,…,ℓ\}j\\in\\\{1,\\ldots,\\ell\\\}and𝐞2=π​\(𝐞1\)\\mathbf\{e\}\_\{2\}=\\pi\(\\mathbf\{e\}\_\{1\}\), we obtain that𝔅n⊧ρ​\[𝐞1′,…,𝐞ℓ′\]​\(𝐞2,π​\(π​\(𝐞1\)\)\)\\mathfrak\{B\}\_\{n\}\\models\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{2\},\\pi\(\\pi\(\\mathbf\{e\}\_\{1\}\)\)\), and using thatρ​\[𝐞1′,…,𝐞ℓ′\]​\(x,y\)\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(x,y\)is transitive, we also have that𝔅n⊧ρ​\[𝐞1′,…,𝐞ℓ′\]​\(𝐞1,π2​\(𝐞1\)\)\\mathfrak\{B\}\_\{n\}\\models\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{1\},\\pi^\{2\}\(\\mathbf\{e\}\_\{1\}\)\)\. In the same way, it is possible to conclude that𝔅n⊧ρ​\[𝐞1′,…,𝐞ℓ′\]​\(𝐞1,πk​\(𝐞1\)\)\\mathfrak\{B\}\_\{n\}\\models\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{1\},\\pi^\{k\}\(\\mathbf\{e\}\_\{1\}\)\)for everyk≥1k\\geq 1\. Given that the set of permutations ofnnelements with the composition operator forms a group of ordern\!n\!, we know thatπn\!\\pi^\{n\!\}is the identity permutation, so thatπn\!​\(𝐞1\)=𝐞1\\pi^\{n\!\}\(\\mathbf\{e\}\_\{1\}\)=\\mathbf\{e\}\_\{1\}\. Therefore, we conclude that𝔅n⊧ρ​\[𝐞1′,…,𝐞ℓ′\]​\(𝐞1,𝐞1\)\\mathfrak\{B\}\_\{n\}\\models\\rho\[\\mathbf\{e\}^\{\\prime\}\_\{1\},\\ldots,\\mathbf\{e\}^\{\\prime\}\_\{\\ell\}\]\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{1\}\), which leads to a contradiction sinceρ​\[v1,…,vℓ\]​\(x,y\)\\rho\[v\_\{1\},\\ldots,v\_\{\\ell\}\]\(x,y\)represents a strict partial order\.

### A\.7\.Auxiliary predicates

#### Definition of the formula𝖫𝖤𝖧​\(x,y,z\)\\mathsf\{LEH\}\(x,y,z\)\.

Let𝖦𝖫𝖡\\mathsf\{GLB\}\(*Greatest Lower Bound*\) be the following formula:

𝖦𝖫𝖡​\(x,y,z\):=z⊆x∧z⊆y∧∀w​\(\(w⊆x∧w⊆y\)→w⊆z\)\.\\displaystyle\\mathsf\{GLB\}\(x,y,z\)\\ :=\\ z\\subseteq x\\wedge z\\subseteq y\\wedge\\forall w\\,\(\(w\\subseteq x\\wedge w\\subseteq y\)\\to w\\subseteq z\)\.The interpretation of this predicate is such that for every modelℳ\\mathcal\{M\}of dimensionnnand every sequence of instances𝐞1,𝐞2,𝐞3\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\},\\mathbf\{e\}\_\{3\}it holds thatℳ⊧𝖦𝖫𝖡​\(𝐞1,𝐞2,𝐞3\)\\mathcal\{M\}\\models\\mathsf\{GLB\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\},\\mathbf\{e\}\_\{3\}\)if and only if𝐞3\\mathbf\{e\}\_\{3\}is the greatest partial instance subsumed by𝐞1\\mathbf\{e\}\_\{1\}and𝐞2\\mathbf\{e\}\_\{2\}, i\.e\., the partial instance with most defined features subsumed by both\. This property allows us to measure the number of defined features on which the two instances agree\. By using this predicate, let𝖫𝖤𝖧\\mathsf\{LEH\}be a ternary predicate such thatℳ⊧𝖫𝖤𝖧​\(𝐞1,𝐞2,𝐞3\)\\mathcal\{M\}\\models\\mathsf\{LEH\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\},\\mathbf\{e\}\_\{3\}\)if and only if the Hamming distance between𝐞1\\mathbf\{e\}\_\{1\}and𝐞2\\mathbf\{e\}\_\{2\}is less than or equal to the Hamming distance between𝐞1\\mathbf\{e\}\_\{1\}and𝐞3\\mathbf\{e\}\_\{3\}\. The relation𝖫𝖤𝖧\\mathsf\{LEH\}can be expressed as a formula from the atomic layer ofExplAIneras follows:

𝖫𝖤𝖧​\(x,y,z\):=𝖥𝗎𝗅𝗅​\(x\)∧𝖥𝗎𝗅𝗅​\(y\)∧𝖥𝗎𝗅𝗅​\(z\)∧∃w1​∃w2​\(𝖦𝖫𝖡​\(x,y,w1\)∧𝖦𝖫𝖡​\(x,z,w2\)∧w2⪯w1\)\.\\displaystyle\\mathsf\{LEH\}\(x,y,z\)\\ :=\\ \\mathsf\{Full\}\(x\)\\wedge\\mathsf\{Full\}\(y\)\\wedge\\mathsf\{Full\}\(z\)\\ \\wedge\\exists w\_\{1\}\\exists w\_\{2\}\\,\(\\mathsf\{GLB\}\(x,y,w\_\{1\}\)\\wedge\\mathsf\{GLB\}\(x,z,w\_\{2\}\)\\wedge w\_\{2\}\\preceq w\_\{1\}\)\.

#### Definition of the formula𝖠𝖽𝖽​\(x,y,z\)\\mathsf\{Add\}\(x,y,z\)\.

Let𝖫𝖴\\mathsf\{LU\}\(*Level Up*\) be the following formula:

𝖫𝖴​\(x,y\):=x≺y∧¬∃z​\(x≺z∧z≺y\),\\displaystyle\\mathsf\{LU\}\(x,y\)\\ :=\\ x\\prec y\\wedge\\neg\\exists z\\ \(x\\prec z\\wedge z\\prec y\),such thatℳ⊨𝖫𝖴​\(𝐞1,𝐞2\)\\mathcal\{M\}\\vDash\\mathsf\{LU\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\}\)if and only if𝐞1\\mathbf\{e\}\_\{1\}has exactly one less defined feature than𝐞2\\mathbf\{e\}\_\{2\}\. By using this predicate, let𝖠𝖽𝖽\\mathsf\{Add\}be a ternary predicate such thatℳ⊨𝖠𝖽𝖽​\(𝐞1,𝐞2,𝐞3\)\\mathcal\{M\}\\vDash\\mathsf\{Add\}\(\\mathbf\{e\}\_\{1\},\\mathbf\{e\}\_\{2\},\\mathbf\{e\}\_\{3\}\)if and only if𝐞2\\mathbf\{e\}\_\{2\}is a feature subsumed by𝐞3\\mathbf\{e\}\_\{3\}and𝐞1\\mathbf\{e\}\_\{1\}is obtained from𝐞3\\mathbf\{e\}\_\{3\}by undefining the feature𝐞2\\mathbf\{e\}\_\{2\}\. The relation𝖠𝖽𝖽\\mathsf\{Add\}can be expressed as formula from the atomic layer ofExplAIneras follows:

𝖠𝖽𝖽​\(x,y,z\):=𝖲𝗂𝗇𝗀𝗅𝖾​\(y\)∧x⊆z∧𝖫𝖴​\(x,z\)∧y⊆z∧¬\(y⊆x\)\.\\displaystyle\\mathsf\{Add\}\(x,y,z\)\\ :=\\ \{\\sf Single\}\(y\)\\wedge x\\subseteq z\\wedge\\mathsf\{LU\}\(x,z\)\\wedge y\\subseteq z\\wedge\\neg\(y\\subseteq x\)\.Recall that𝖲𝗂𝗇𝗀𝗅𝖾​\(x\)\{\\sf Single\}\(x\)defines the set of partial instances with exactly one defined feature, and can be expressed as follows:

𝖲𝗂𝗇𝗀𝗅𝖾​\(x\):=∃y​\(y⊂x\)∧∀y​\(y⊂x→¬∃z​\(z⊂y\)\)\.\{\\sf Single\}\(x\):=\\exists y\(y\\subset x\)\\wedge\\forall y\(y\\subset x\\,\\rightarrow\\,\\neg\\exists z\(z\\subset y\)\)\.