LiFTER: A Grounded Neuro-Symbolic Microscope for Continuous-Time Dynamic Graph Forecasting

# LiFTER: A Grounded Neuro-Symbolic Microscope for Continuous-Time Dynamic Graph Forecasting
Source: [https://arxiv.org/html/2608.06765](https://arxiv.org/html/2608.06765)
Minwoo Yu and Young\-guk HaM\. Yu and Y\.\-g\. Ha are with the Smart Computing Laboratory, Department of Computer Science & Engineering, Konkuk University, Seoul 05029, Republic of Korea\. E\-mail: \{snowypainter, ygha\}@konkuk\.ac\.kr\. Y\.\-g\. Ha is the corresponding author\.

###### Abstract

Continuous\-time dynamic graph models predict future links by compressing past interactions into neural states\. Although effective for forecasting, this computation obscures which entities are shared across events and how temporal patterns contribute to a prediction\. We treat this gap as a property of the predictive architecture rather than a problem to be addressed after prediction\. Link\-Fact Temporal Rule Inducer \(LiFTER\) is a neuro\-symbolic predictor that preserves observed interactions as grounded temporal facts and applies executable temporal rules to pre\-query facts\. Each score is a signed sum of rule executions whose historical facts, entity bindings, and temporal order are explicitly satisfied\. The evidence and rules responsible for a prediction can therefore be inspected, independently recomputed, and intervened upon\. Across four CTDG benchmarks, LiFTER achieves competitive historical\-negative forecasting and the highest macro explanation accuracy and deletion fidelity\. The same architecture also serves as a microscope that separates the contributions of recurrence, history position, and transition across datasets and traces them to individual facts\. Independent execution reconstructs all logits for 19,664 test predictions with a maximum error of 0\.0000131\. LiFTER turns future\-link forecasting into a verifiable grounded computation\.

###### Index Terms:

Continuous\-time dynamic graph, neuro\-symbolic learning, temporal link prediction, rule induction, explainable artificial intelligence\.

## 1Introduction

A link in a continuous\-time dynamic graph \(CTDG\) is an event in time\. Records of a user editing a page, a student accessing course content, or a listener playing a song identify who interacted with whom and when\. Future interactions depend not only on event counts, but also on repeated pairs, recurrence intervals, the positions of recent destinations, and the temporal state formed by successive choices\.

TGN, TGAT, GraphMixer, and DyGFormer compress history into memory, temporal neighborhoods, or event\-sequence representations\[[8](https://arxiv.org/html/2608.06765#bib.bib1),[13](https://arxiv.org/html/2608.06765#bib.bib2),[3](https://arxiv.org/html/2608.06765#bib.bib3),[17](https://arxiv.org/html/2608.06765#bib.bib4)\]\. They forecast well, but their final scores do not state how historical facts were combined by temporal relations and entity bindings\. T\-GNNExplainer and TempME search for an event subset or motif that preserves a trained predictor’s output\[[11](https://arxiv.org/html/2608.06765#bib.bib6),[2](https://arxiv.org/html/2608.06765#bib.bib7)\]; TGIB and SIG couple an explanation mask or subgraph to the predictor\[[9](https://arxiv.org/html/2608.06765#bib.bib8),[5](https://arxiv.org/html/2608.06765#bib.bib9)\]\. These methods identify important evidence, but the selected facts do not themselves execute a rule whose signed contribution is a unit of prediction\. Figure[1](https://arxiv.org/html/2608.06765#S1.F1)contrasts these approaches at the levels of input, inference, and output\.

Neuro\-symbolic learning on temporal knowledge graphs \(TKGs\) starts from a different representation\. Each observed fact already contains a semantic predicaterrin\(h,r,o,t\)\(h,r,o,t\)\. Neural\-LP can consequently learn differentiable first\-order programs, while TLogic and TILP learn temporal rules over observed predicate sequences\[[14](https://arxiv.org/html/2608.06765#bib.bib10),[6](https://arxiv.org/html/2608.06765#bib.bib11),[12](https://arxiv.org/html/2608.06765#bib.bib12)\]\. Their strength lies in grounding rule variables to entities and executing walks and temporal conditions that support a candidate\.

The CTDG benchmarks studied here record every event under a single interaction type, eliminating the relation\-chain identity available in a TKG\. Repeating one predicate cannot distinguish how a historical interaction binds to the query source, destination, or an existential entity; where a pair recurs in the source history; or how recent destinations lead to a candidate\. LiFTER instead defines rule identity through query\-relative entity bindings, pair renewal, history position, temporal order, and candidate\-conditioned transitions\.

These primitives are not dataset\-specific motifs named after inspecting scores\. Entity bindings express equality between historical and query arguments; pair renewal binds both arguments simultaneously; and order and position preserve the event sequence induced by timestamps\. All remain invariant under a consistent renaming of entity identifiers\. Candidate\-conditioned transitions distinguish differentYYvalues under the same source history\. Independent CTDG studies have repeatedly established the importance of recurrence, recency, sequential dynamics, higher\-order interaction, and target\-aware matching\[[7](https://arxiv.org/html/2608.06765#bib.bib5),[4](https://arxiv.org/html/2608.06765#bib.bib17),[16](https://arxiv.org/html/2608.06765#bib.bib15),[15](https://arxiv.org/html/2608.06765#bib.bib16),[1](https://arxiv.org/html/2608.06765#bib.bib14)\]\. LiFTER expresses them in one finite language executed directly by historical facts, rather than attaching them as separate heuristic features\.

Link\-Fact Temporal Rule Inducer \(LiFTER\) preserves each raw interaction as a grounded factLink⁡\(u,v,t\)\\operatorname\{Link\}\(u,v,t\)and treats the pre\-query history as an executable fact database\. The designer specifies admissible argument bindings and temporal operators\. The future\-link objective learns which groundings are predictive, which temporal compatibility applies, and how strongly each execution supports or inhibits a candidate\. Here, rule induction means selecting and parameterizing predictive clauses and constructing query\-specific executions within this finite hypothesis language, rather than unconstrained grammar discovery\.

Every logit term except the prior must identify a concrete fact, a satisfied binding, a temporal condition, and a signed contribution\. Prediction and explanation are therefore the same forward execution\. An independent verifier recomputes the grounding set and logit from raw history and frozen parameters; editing a cited fact re\-executes the program, including alternative groundings\. This property turns LiFTER into a*neuro\-symbolic microscope*: its units of observation are individual facts, rule executions, and signed responsibility, rather than opaque representations\.

The microscope enables a finer diagnosis than ordinary component ablation\. We compute exact Shapley values over all272^\{7\}coalitions of seven execution components, then trace dataset\-level performance to individual evidence through query\-regime deletion and grounded\-fact intervention\. The analysis attributes the largest predictive value to pair renewal on Wikipedia and Reddit, two\-event transitions on MOOC, and one\-event transitions on LastFM\.

Our contributions are as follows\.

- •We introduce a grounded temporal rule language for CTDGs without a semantic relation vocabulary\.
- •We express the entire candidate logit as a signed sum of concrete grounded executions that can be independently verified and edited\.
- •LiFTER competes directly with neural CTDG models on four historical\-negative forecasting benchmarks and achieves the highest macro ACC\-AUC and AUFSC among five explanation systems\.
- •Exact coalition decomposition, query\-regime analysis, and grounded\-fact intervention diagnose predictive mechanisms down to individual facts\.

![Refer to caption](https://arxiv.org/html/2608.06765v1/x1.png)Figure 1:Neural CTDG models compress history, and post\-hoc explainers select evidence after prediction\. TILP executes rules over observed TKG predicates\. LiFTER constructs predictions directly from grounded CTDG facts and signed rule executions\.
## 2Related Work

### 2\.1Neural CTDG Link Prediction

TGN updates temporal state through node memory and message passing, while TGAT uses time encoding and temporal attention\. GraphMixer processes neighborhoods with temporal link encoding and MLP mixing, and DyGFormer compresses interaction sequences into patches\[[8](https://arxiv.org/html/2608.06765#bib.bib1),[13](https://arxiv.org/html/2608.06765#bib.bib2),[3](https://arxiv.org/html/2608.06765#bib.bib3),[17](https://arxiv.org/html/2608.06765#bib.bib4)\]\. CRAFT applies cross\-attention to source history with the candidate destination as the query; CRAFT\-R further incorporates repeat counts\[[15](https://arxiv.org/html/2608.06765#bib.bib16)\]\. These approaches offer strong GPU\-friendly forecasting with continuous representations\. LiFTER is evaluated under the same protocol but constrains the score to a sum of grounded rule executions\.

EdgeBank showed that repeated\-edge memory can be strong under random negatives\[[7](https://arxiv.org/html/2608.06765#bib.bib5)\]\. Heuristics based only on recency and popularity also compete with neural models on several temporal benchmarks\[[4](https://arxiv.org/html/2608.06765#bib.bib17)\], and representative models deteriorate sharply on the sequential dynamics of TGB\-Seq, which reduces repeated edges\[[16](https://arxiv.org/html/2608.06765#bib.bib15)\]\. These findings establish recurrence and order as central structures that a prediction language should represent, rather than incidental metadata\.

### 2\.2Explainable Temporal Graph Prediction

T\-GNNExplainer searches pre\-query events for a prediction\-preserving subset, and TempME uses temporal motifs as explanation units\[[11](https://arxiv.org/html/2608.06765#bib.bib6),[2](https://arxiv.org/html/2608.06765#bib.bib7)\]\. TGIB learns sparse evidence through an information bottleneck, while SIG constructs a self\-interpretable temporal subgraph\[[9](https://arxiv.org/html/2608.06765#bib.bib8),[5](https://arxiv.org/html/2608.06765#bib.bib9)\]\. Their outputs are importance scores, masks, or subgraphs\. LiFTER instead returns the historical facts, variable bindings, temporal conditions, and signed contributions actually used in score computation\. This output is not a surrogate and accounts for the entire score apart from the prior\.

### 2\.3Differentiable and Temporal Rule Learning

Neural\-LP learns differentiable inference over a finite first\-order rule space\[[14](https://arxiv.org/html/2608.06765#bib.bib10)\]\. TLogic converts temporal random walks that preserve timestamp order into rules, and TILP learns temporal intervals and confidence on relation\-typed TKGs\[[6](https://arxiv.org/html/2608.06765#bib.bib11),[12](https://arxiv.org/html/2608.06765#bib.bib12)\]\. CAW also structures temporal walks through recurring entity positions\[[10](https://arxiv.org/html/2608.06765#bib.bib13)\]\. In this line of work, the designer supplies a language bias that specifies admissible predicates, variable bindings, and maximum body structure, rather than dataset\-specific individual rules\. LiFTER follows the same principle: it enumerates all admissible combinations and learns their weights and temporal compatibility\.

The symbolic primitives differ\. A TILP rule is identified by an observed semantic predicate sequence and temporal constraints; in a single\-relation CTDG, that sequence collapses to repetitions of one predicate\. LiFTER fills this gap with argument roles relative to query endpoints, same\-pair recurrence, source\-local position, and transitions from recent destination states to a candidate\. Preserving a raw interaction as a ground atom does not invent semantic relations; it supplies the representation on which this language executes\.

The output contract also differs\. In LiFTER, every final logit term other than the prior must be an enumerated execution with cited facts and a signed contribution, and it must be independently replayable from raw history\. A direct numerical comparison with TILP would require an artificial conversion from CTDG interactions to a relation\-typed interval TKG, changing both the input semantics and prediction protocol\. The formulation\-level distinction is therefore more faithful than such an adaptation\.

## 3Problem Formulation

A CTDG is a chronologically ordered sequence of interactions\{\(ui,vi,ti\)\}i=1n\\\{\(u\_\{i\},v\_\{i\},t\_\{i\}\)\\\}\_\{i=1\}^\{n\}\. A queryq=Link⁡\(X,Y,Tq\)q=\\operatorname\{Link\}\(X,Y,T\_\{q\}\)asks whether sourceXXwill interact with candidate destinationYY\. LiFTER preserves every pre\-query interaction as a fact whose source, destination, and timestamp are grounded, and executes a finite rule setℛ\\mathcal\{R\}over these facts\. For rulerr,Gr​\(q,ℱ<Tq\)G\_\{r\}\(q,\\mathcal\{F\}\_\{<T\_\{q\}\}\)contains the historical fact substitutions that satisfy its argument bindings and temporal guards, ander​\(g,q\)e\_\{r\}\(g,q\)denotes the temporal evidence of groundinggg\. With a rule\-specific existential aggregatorArA\_\{r\}and signed weightwrw\_\{r\}, the formulation is

ℱ<Tq\\displaystyle\\mathcal\{F\}\_\{<T\_\{q\}\}=\{Link⁡\(ui,vi,ti\)∣ti<Tq\},\\displaystyle=\\\{\\operatorname\{Link\}\(u\_\{i\},v\_\{i\},t\_\{i\}\)\\mid t\_\{i\}<T\_\{q\}\\\},\(1a\)Er​\(q\)\\displaystyle E\_\{r\}\(q\)=Ar​\(\{er​\(g,q\)∣g∈Gr​\(q,ℱ<Tq\)\}\),\\displaystyle=A\_\{r\}\\\!\\left\(\\\{e\_\{r\}\(g,q\)\\mid g\\in G\_\{r\}\(q,\\mathcal\{F\}\_\{<T\_\{q\}\}\)\\\}\\right\),\(1b\)s​\(q\)\\displaystyle s\(q\)=b\+∑r∈ℛwr​Er​\(q\)\+spos​\(q\)\+str​\(q\)\.\\displaystyle=b\+\\sum\_\{r\\in\\mathcal\{R\}\}w\_\{r\}E\_\{r\}\(q\)\+s\_\{\\mathrm\{pos\}\}\(q\)\+s\_\{\\mathrm\{tr\}\}\(q\)\.\(1c\)
Equation \([1a](https://arxiv.org/html/2608.06765#S3.E1.1)\) excludes future information and defines the fact database available at prediction time\. Grounding uses the database’s actual integer entity identifiers, so a shared variable can bind only to the same entity\. Equation \([1b](https://arxiv.org/html/2608.06765#S3.E1.2)\) aggregates temporal evidence from valid groundings into clause evidence;Er​\(q\)=0E\_\{r\}\(q\)=0when no grounding exists\.

In Eq\. \([1c](https://arxiv.org/html/2608.06765#S3.E1.3)\),bbis a scalar prior,sposs\_\{\\mathrm\{pos\}\}sums positioned\-recurrence executions, andstrs\_\{\\mathrm\{tr\}\}sums one\- and two\-event transition contributions\. Every non\-prior term, includingwr​Er​\(q\)w\_\{r\}E\_\{r\}\(q\), is associated with concrete historical facts and a signed contribution\. Equation \([1c](https://arxiv.org/html/2608.06765#S3.E1.3)\) therefore defines the candidate logit and its execution trace without a separate explanation surrogate\. Predicate assignments are added to the execution only when the optional typed vocabulary is used\.

## 4LiFTER

### 4\.1Interaction\-to\-Fact Lifting

An interaction\(ui,vi,ti\)\(u\_\{i\},v\_\{i\},t\_\{i\}\)records entityuiu\_\{i\}interacting with entityviv\_\{i\}at timetit\_\{i\}\. LiFTER represents it asfi=Link⁡\(ui,vi,ti\)f\_\{i\}=\\operatorname\{Link\}\(u\_\{i\},v\_\{i\},t\_\{i\}\), with source, destination, and timestamp as the first, second, and temporal arguments\. For example, user 5619 editing page 949 at timestamp 2352512 becomes the factLink⁡\(5619,949,2352512\)\\operatorname\{Link\}\(5619,949,2352512\)\.

This representation permits direct substitution of historical entities for rule variables\. For queryLink⁡\(X,Y,Tq\)\\operatorname\{Link\}\(X,Y,T\_\{q\}\), the factLink⁡\(5619,949,t1\)\\operatorname\{Link\}\(5619,949,t\_\{1\}\)bindsX=5619X=5619andY=949Y=949\. If the same user edited page 103, the factLink⁡\(5619,103,t2\)\\operatorname\{Link\}\(5619,103,t\_\{2\}\)bindsX=5619X=5619andZ=103Z=103\. The shared variableXXreceives the same integer identifier 5619 in both facts, preserving the structure that the two interactions share a source\.

The local fact database for a query contains the most recentHHfacts adjacent to its source and candidate endpoints\. ThusHHbounds the number of entity substitutions examined per query\. The selected facts retain their timestamps, from which the executor computes event order, time to query, and recurrence intervals\.

### 4\.2Grounded Temporal Rule Language

LiFTER rules specify which entities must be equal across a query and historical facts, and in what temporal order the events must occur\. In the forecasting experiments of Section[5\.1](https://arxiv.org/html/2608.06765#S5.SS1), all interactions share one observed predicateP1P\_\{1\}\. The typed extension generalizes it toPkP\_\{k\},k∈\{1,…,K\}k\\in\\\{1,\\ldots,K\\\}\. Figure[2](https://arxiv.org/html/2608.06765#S4.F2)illustrates four rule schemas\.

Endpoint\-bound unary rules\.These rules test the six orientations in which one historical fact shares an endpoint with the query source or destination\.

Pk​\(X,Y,T\),Pk​\(Y,X,T\),Pk​\(X,Z,T\),Pk​\(Z,X,T\),Pk​\(Y,Z,T\),Pk​\(Z,Y,T\),T<Tq\.\\begin\{split\}P\_\{k\}\(X,Y,T\),\\ P\_\{k\}\(Y,X,T\),\\ P\_\{k\}\(X,Z,T\),\\\\ P\_\{k\}\(Z,X,T\),\\ P\_\{k\}\(Y,Z,T\),\\ P\_\{k\}\(Z,Y,T\),\\quad T<T\_\{q\}\.\\end\{split\}\(2\)Equation \([2](https://arxiv.org/html/2608.06765#S4.E2)\) distinguishes, for example, a historicalX→YX\\rightarrow Yinteraction fromX→ZX\\rightarrow ZandZ→YZ\\rightarrow Yroles\. Equality with a query argument, rather than the entity identifier itself, determines the role, yielding6​K6Kclauses\.

Pair\-renewal rules\.A query pair that occurred twice is represented as

Pa​\(X,Y,T1\)∧Pb​\(X,Y,T2\)\\displaystyle P\_\{a\}\(X,Y,T\_\{1\}\)\\land P\_\{b\}\(X,Y,T\_\{2\}\)⇒Link⁡\(X,Y,Tq\),\\displaystyle\\Rightarrow\\operatorname\{Link\}\(X,Y,T\_\{q\}\),\(3\)T1<T2<Tq\.\\displaystyle T\_\{1\}<T\_\{2\}<T\_\{q\}\.The rule compares the previous recurrence intervalT2−T1T\_\{2\}\-T\_\{1\}with the gap from the last interaction to the query,Tq−T2T\_\{q\}\-T\_\{2\}\. The language includes allK2K^\{2\}predicate pairs\.

Positioned recurrence rules\.These rules retain the position of a historical interaction with the same pair in the source history\.

Pk​\(X,Y,T\)∧PosX⁡\(T\)=j⇒Link⁡\(X,Y,Tq\),1≤j≤H,P\_\{k\}\(X,Y,T\)\\land\\operatorname\{Pos\}\_\{X\}\(T\)=j\\Rightarrow\\operatorname\{Link\}\(X,Y,T\_\{q\}\),\\quad 1\\leq j\\leq H,\(4\)In Eq\. \([4](https://arxiv.org/html/2608.06765#S4.E4)\),j=1j=1denotes the most recent fact\. Rather than preselecting a position, the language includes every1≤j≤H1\\leq j\\leq H, and training determines its signed weight\.

Ordered grounded transitions\.These rules evaluate the next candidate from the destinations reached by recent outgoing facts of sourceXX\. The two\-event body is

Pa​\(X,Z1,T1\)∧Pb​\(X,Z2,T2\),T1≤T2<Tq\.P\_\{a\}\(X,Z\_\{1\},T\_\{1\}\)\\land P\_\{b\}\(X,Z\_\{2\},T\_\{2\}\),\\qquad T\_\{1\}\\leq T\_\{2\}<T\_\{q\}\.\(5\)The one\-event schema uses only the second fact in Eq\. \([5](https://arxiv.org/html/2608.06765#S4.E5)\)\. Distinct timestamps determine temporal order; equal timestamps are treated as simultaneous recent facts\. The executor first finds actual facts satisfying Eq\. \([5](https://arxiv.org/html/2608.06765#S4.E5)\), and only then computes the compatibility from grounded destinationsZ1,Z2Z\_\{1\},Z\_\{2\}to candidateYY\.

![Refer to caption](https://arxiv.org/html/2608.06765v1/x2.png)Figure 2:Four grounded temporal rule schemas\. A historical fact’s argument role is determined by its binding to query endpoints\. All admissible predicates and history positions are enumerated in the same finite language\.A transition score is the learned compatibility between grounded destinations and a candidate\. If sourceXXrecently interacted withZZ, the one\-event transition asks whetherYYfollowsZZ\. If recent interactions occurred in the orderZ1,Z2Z\_\{1\},Z\_\{2\}, the two\-event transition asks whetherYYfollows that ordered pair\. Hereaza\_\{z\}andbyb\_\{y\}aredd\-dimensional vectors for a historical destination and candidate, andpkp\_\{k\}is a predicate factor\. The two\-event schema multiplies three vectors so that its value is large when both historical destinations support the candidate in the same latent dimension\. Equations \([6a](https://arxiv.org/html/2608.06765#S4.E6.1)\) and \([6b](https://arxiv.org/html/2608.06765#S4.E6.2)\) define these computations\.

ctr,1\\displaystyle c\_\{\\mathrm\{tr\},1\}=\(aZ⊙pk\)⊤​bYd​s1,rH​\(X,Y\),\\displaystyle=\\frac\{\{\(a\_\{Z\}\\odot p\_\{k\}\)\}^\{\\top\}b\_\{Y\}\}\{\\sqrt\{d\}\}\\,s\_\{1,r\_\{H\}\(X,Y\)\},\(6a\)ctr,2\\displaystyle c\_\{\\mathrm\{tr\},2\}=s2,rH​\(X,Y\)d​∑ℓ=1d\(aZ1,ℓ\(1\)​pa,ℓ\(1\)\)​\(aZ2,ℓ\(2\)​pb,ℓ\(2\)\)​bY,ℓ\(2\)\.\\displaystyle=\\frac\{s\_\{2,r\_\{H\}\(X,Y\)\}\}\{\\sqrt\{d\}\}\\sum\_\{\\ell=1\}^\{d\}\(a^\{\(1\)\}\_\{Z\_\{1\},\\ell\}p^\{\(1\)\}\_\{a,\\ell\}\)\(a^\{\(2\)\}\_\{Z\_\{2\},\\ell\}p^\{\(2\)\}\_\{b,\\ell\}\)b^\{\(2\)\}\_\{Y,\\ell\}\.\(6b\)HererH​\(X,Y\)r\_\{H\}\(X,Y\)indicates whether the query pair appears in the recent source history, andsm,r=softplus⁡\(ρm\)​softplus⁡\(ηm,r\)/log⁡2s\_\{m,r\}=\\operatorname\{softplus\}\(\\rho\_\{m\}\)\\operatorname\{softplus\}\(\\eta\_\{m,r\}\)/\\log 2is a learned positive scale\. Equation \([6a](https://arxiv.org/html/2608.06765#S4.E6.1)\) evaluates a transition from one destination, while Eq\. \([6b](https://arxiv.org/html/2608.06765#S4.E6.2)\) evaluates a transition from an ordered pair\. These vectors are accessed only when a grounding satisfies Eq\. \([5](https://arxiv.org/html/2608.06765#S4.E5)\); neural compatibility alone cannot create an execution\. The two potentials contain5​N​d\+3​K​d\+65Nd\+3Kd\+6parameters\.

### 4\.3Grounded Execution and Score Accounting

For each clause, the executor first finds historical facts that satisfy its body\. Because facts satisfying the same rule can have different gaps to the query, Eq\. \([7](https://arxiv.org/html/2608.06765#S4.E7)\) measures how closely each grounding matches the rule’s learned preferred gap\.

er​\(g,q\)=exp⁡\[−12​\(log⁡\(1\+Tq−t\)−μrsoftplus⁡\(λr\)\+0\.05\)2\]\.e\_\{r\}\(g,q\)=\\exp\\\!\\left\[\-\\frac\{1\}\{2\}\\Biggl\(\\frac\{\\log\(1\+T\_\{q\}\-t\)\-\\mu\_\{r\}\}\{\\operatorname\{softplus\}\(\\lambda\_\{r\}\)\+0\.05\}\\Biggr\)^\{\\\!2\}\\right\]\.\(7\)The kernel lies in\[0,1\]\[0,1\]and approaches one at the preferred gap\. Parameterμr\\mu\_\{r\}is rulerr’s preferred log\-time gap, andsoftplus⁡\(λr\)\+0\.05\\operatorname\{softplus\}\(\\lambda\_\{r\}\)\+0\.05is its tolerance\. Renewal rules receive the difference between the previous recurrence interval and the current gap instead of absolute recency\. Aggregating all grounding values yieldsErE\_\{r\}\. Direct\-pair unary rules use a maximum to avoid repeatedly counting the same pair; other rules use the capped sumEr=min⁡\(e4,∑ger​\(g,q\)\)E\_\{r\}=\\min\(e^\{4\},\\sum\_\{g\}e\_\{r\}\(g,q\)\)\. If no valid grounding exists,Er=0E\_\{r\}=0\.

The candidate logit contains no hidden residual scorer beyond this grounded evidence and is computed as

s​\(X,Y,Tq\)=b\+∑r∈ℛwr​Er\+∑j=1H∑k=1Kuj,k​Rj,k\+∑m=12ctr,m\.s\(X,Y,T\_\{q\}\)=b\+\\sum\_\{r\\in\\mathcal\{R\}\}w\_\{r\}E\_\{r\}\+\\sum\_\{j=1\}^\{H\}\\sum\_\{k=1\}^\{K\}u\_\{j,k\}R\_\{j,k\}\+\\sum\_\{m=1\}^\{2\}c\_\{\\mathrm\{tr\},m\}\.\(8\)In Eq\. \([8](https://arxiv.org/html/2608.06765#S4.E8)\),bbis the global prior,wr​Erw\_\{r\}E\_\{r\}represents unary and renewal rules,uj,k​Rj,ku\_\{j,k\}R\_\{j,k\}represents positioned recurrence, andctr,mc\_\{\\mathrm\{tr\},m\}is an ordered\-transition contribution\. The weightswrw\_\{r\}anduj,ku\_\{j,k\}and the transition terms may be positive or negative, thereby supporting or inhibiting a candidate\. Every nonzero term identifies the facts and conditions that produced it\. Summing the prior and all signed contributions exactly reconstructs the logit\. Figure[3](https://arxiv.org/html/2608.06765#S4.F3)follows this accounting from facts to score\.

![Refer to caption](https://arxiv.org/html/2608.06765v1/x3.png)Figure 3:A prediction retains its grounded facts, satisfied conditions, and signed score ledger\. The prior and ledger exactly reconstruct the candidate logit\.
### 4\.4Learning and Efficient Execution

All rule parameters are trained jointly through future\-link prediction\. The objective raises the positive\-candidate logits\+s^\{\+\}and lowers the sampled\-negative logits−s^\{\-\}:

ℒpred=−log⁡σ​\(s\+\)−log⁡σ​\(−s−\)\.\\mathcal\{L\}\_\{\\mathrm\{pred\}\}=\-\\log\\sigma\(s^\{\+\}\)\-\\log\\sigma\(\-s^\{\-\}\)\.\(9\)Rather than assigning every clause a wholly independent weight, clauses with the same structural schema and predicates share the parameters in Eq\. \([10](https://arxiv.org/html/2608.06765#S4.E10)\)\.

wr=tanh⁡\[tτ​\(r\)⊤​\(⨀s∈body​\(r\)qks\)\+θr\]\.w\_\{r\}=\\tanh\\\!\\left\[t\_\{\\tau\(r\)\}^\{\\top\}\\left\(\\bigodot\_\{s\\in\\mathrm\{body\}\(r\)\}q\_\{k\_\{s\}\}\\right\)\+\\theta\_\{r\}\\right\]\.\(10\)Heretτ​\(r\)t\_\{\\tau\(r\)\}represents a schema,qksq\_\{k\_\{s\}\}represents a predicate slot in the body, andθr\\theta\_\{r\}is a clause\-specific residual\. We addλs​∑r\|θr\|\\lambda\_\{s\}\\sum\_\{r\}\|\\theta\_\{r\}\|to the prediction loss and use a type\-usage entropy regularizer only whenK\>1K\>1\.

Finite histories and batched grounding bound execution cost\. After indexing the global event stream once, each batch gathers recent fact tensors once\. Unary and renewal clauses use parallel tensor operations, while Eqs\. \([6a](https://arxiv.org/html/2608.06765#S4.E6.1)\) and \([6b](https://arxiv.org/html/2608.06765#S4.E6.2)\) use batched embedding operations\. The full stream is never rescanned per query\.

### 4\.5Optional Typed Predicate Vocabulary

The JODIE streams evaluated in Section[5\.1](https://arxiv.org/html/2608.06765#S5.SS1)provide one observed interaction type, so LiFTER usesK=1K=1\. All facts then share a predicate but remain distinct ground atoms through their entity arguments and timestamps\.

For datasets in which multiple event semantics are distinguishable from observed context, LiFTER also supports an optionalK\>1K\>1vocabulary\. A fact encoder maps the pre\-event context of eventiitoziz\_\{i\}and selects a predicate by similarity to learned prototypesckc\_\{k\}:

πi​k\\displaystyle\\pi\_\{ik\}=softmaxk⁡\(sim⁡\(zi,ck\)/τ\),\\displaystyle=\\operatorname\{softmax\}\_\{k\}\\left\(\\operatorname\{sim\}\(z\_\{i\},c\_\{k\}\)/\\tau\\right\),ki∗\\displaystyle k\_\{i\}^\{\*\}=arg⁡maxk⁡πi​k,\\displaystyle=\\arg\\max\_\{k\}\\pi\_\{ik\},\(11a\)π^i\\displaystyle\\widehat\{\\pi\}\_\{i\}=onehot⁡\(ki∗\)\+πi−stopgrad⁡\(πi\)\.\\displaystyle=\\operatorname\{onehot\}\(k\_\{i\}^\{\*\}\)\+\\pi\_\{i\}\-\\operatorname\{stopgrad\}\(\\pi\_\{i\}\)\.\(11b\)Forward execution uses the single discrete atomPki∗​\(ui,vi,ti\)P\_\{k\_\{i\}^\{\*\}\}\(u\_\{i\},v\_\{i\},t\_\{i\}\)selected by Eq\. \([11a](https://arxiv.org/html/2608.06765#S4.E11.1)\)\. Equation \([11b](https://arxiv.org/html/2608.06765#S4.E11.2)\) preserves this discrete choice while passing gradients from Eq\. \([9](https://arxiv.org/html/2608.06765#S4.E9)\) to the encoder and prototypes\. Section[5\.3](https://arxiv.org/html/2608.06765#S5.SS3)and[AppendixA](https://arxiv.org/html/2608.06765#A1)establish the conditions under whichK\>1K\>1is useful\.

## 5Experiments

### 5\.1Experimental Setup

We chronologically split the first 32,768 events of Wikipedia, Reddit, MOOC, and LastFM into 85% training and 15% testing\. All forecasting models are evaluated for 10 epochs with seeds 7, 17, and 27\. Across all four datasets, LiFTER uses a predicate vocabularyK=1K=1, hidden dimension 64, transition rank 32, dropout 0\.1, and history length 128\. The batch size is 512; AdamW uses a learning rate of 0\.004 and weight decay10−510^\{\-5\}\. Forecasting, component, and explanation results report the mean and sample standard deviation across three seeds\. Deterministic certificate verification covers the complete test split of the seed\-7 model for each dataset; hardware latency is reported separately as the mean of repeated runs\.

Grounding capacityHHis selected by chronological validation disjoint from the held\-out test interval\. After reserving the final 15% for testing, we use the last 15% of the preceding 85% development prefix for validation\. We apply eachH∈\{10,20,40,80\}H\\in\\\{10,20,40,80\\\}to all datasets and seeds and select the value with the highest macro mean validation Historical AUC, breaking ties in favor of smallerHH\. Table[I](https://arxiv.org/html/2608.06765#S5.T1)selectsH=10H=10, which is fixed for every test evaluation\. Although MOOC alone peaks atH=20H=20, the difference is only 0\.02 percentage points\. A shared capacity controls both predicate\-position grammar size and per\-query grounding cost\.

TABLE I:Chronological\-validation Historical AUC \(%\) on an interval disjoint from the held\-out test set\. Bold marks the shared capacity selected by the macro criterion\.Historical\-negative evaluation\.For query\(X,Y,Tq\)\(X,Y,T\_\{q\}\), we uniformly sample a historical negative from destinations thatXXinteracted with beforeTqT\_\{q\}, excluding the current positiveYY\. A random negative is often an entity never observed with the source and is therefore easily separated through source preference, exposure, or direct recurrence\. Historical positives and negatives both appeared in the source history; the task asks whether recency, intervals, history position, and transitions provide stronger current evidence for the positive\[[7](https://arxiv.org/html/2608.06765#bib.bib5)\]\.

When no historical destination is available, we draw a random negative from the common destination pool\. Historical\-candidate coverage is 51\.1% on Wikipedia, 48\.9% on Reddit, 97\.7% on MOOC, and 99\.7% on LastFM\. Candidates and fallbacks are identical across models and seeds\. We report AUC and AP for both historical and random candidates\.

### 5\.2Forecasting Results

TABLE II:Historical\- and random\-negative forecasting \(%\)\. Parentheses on LiFTER report historical rank and relative change from the strongest neural baseline\.Table[II](https://arxiv.org/html/2608.06765#S5.T2)compares LiFTER and EdgeBank with six neural CTDG baselines under both negative protocols\. LiFTER ranks first on both historical metrics for Reddit and LastFM, second on both for Wikipedia, and second in AUC and first in AP for MOOC\. Relative to the strongest neural baseline, its Reddit AUC/AP is 2\.05%/2\.01% higher and its LastFM AUC/AP is 5\.69%/9\.22% higher\. It trails TGN on Wikipedia by 2\.75%/2\.34%; on MOOC it trails GraphMixer by 0\.43% in AUC but leads by 0\.28% in AP\.

Under random negatives, LiFTER is within one percentage point of the best AUC and AP on Wikipedia, Reddit, and MOOC\. Grounded execution limits paths that reject an irrelevant random destination solely through unconstrained identity or popularity, yet the difference remains small on these datasets\. On LastFM, random AUC/AP is 9\.87/8\.15 points below the best baseline while historical AUC/AP ranks first\. This contrast localizes LiFTER’s strength to temporal discrimination among plausible alternatives previously present in the interaction history\.

EdgeBank often assigns the same memory score to several candidates because a historical alternative is, by definition, an observed pair\. LiFTER exceeds EdgeBank by 20\.44–56\.30 AUC points across the datasets\. Its performance therefore cannot be reduced to the lookup “this pair existed before”; recurrence intervals, source\-local position, and candidate transitions determine the ranking through a single grounded program\.

### 5\.3What Does Fact\-Level Typing Reveal?

LiFTER permits a latent vocabulary withK\>1K\>1to represent multiple functional roles within one observed relation\. We test whether this expressivity aids forecasting by comparingK=1,2,4,8K=1,2,4,8under the same architecture and protocol and by shuffling learned assignments across facts\. Grounded binding, recurrence, temporal order, and transition execution remain intact atK=1K=1\.

TABLE III:Best matchedK\>1K\>1change overK=1K=1on JODIE datasets and the effect of shuffling learned assignments \(percentage points\)\.The changes fromK\>1K\>1in Table[III](https://arxiv.org/html/2608.06765#S5.T3)are small and inconsistent\. Wikipedia and Reddit changes are comparable to seed variation, andK=1K=1obtains the best AP on MOOC\. LastFM’s 1\.08\-point AP change nearly survives shuffling the fact–predicate correspondence and is therefore not an effect of functional typing\. One predicate is the most concise executable representation for these four JODIE streams\. This result, in which recurrence, temporal order, and exposure matter more than finer relation semantics, is consistent with findings on recency/popularity heuristics and TGB\-Seq\[[4](https://arxiv.org/html/2608.06765#bib.bib17),[16](https://arxiv.org/html/2608.06765#bib.bib15)\]\.

Typed predicates can nevertheless add expressivity in multi\-action streams when event meaning is identifiable from pre\-event history and changes the future distribution\. The controlled experiment that isolates these two conditions and its complete heatmap appear in the semantic\-predicate phase diagram in[AppendixA](https://arxiv.org/html/2608.06765#A1)\.

### 5\.4Predictive\-Mechanism Decomposition

The learnedK=1K=1LiFTER logit decomposes into seven disjoint components: direct pair, source context, candidate context, pair renewal, positioned recurrence, one\-event transition, and two\-event transition\. If one fact executes multiple rules, each execution contribution is recorded under its component\. Letsm​\(q\)s\_\{m\}\(q\)be the signed contribution of componentmm; thens​\(q\)=b\+∑msm​\(q\)s\(q\)=b\+\\sum\_\{m\}s\_\{m\}\(q\)holds exactly\.

We evaluate all27=1282^\{7\}=128execution coalitions with the same historical candidates\. A component’s Shapley contribution averages the AUC or AP change produced by adding it over every possible addition order\. It therefore includes both isolated effects and interactions, and the seven contributions sum to the difference between the complete and empty programs\. The exact definition and computation are given in[AppendixC](https://arxiv.org/html/2608.06765#A3); the maximum measured residual of this identity is1\.11×10−161\.11\\times 10^\{\-16\}\.

![Refer to caption](https://arxiv.org/html/2608.06765v1/x4.png)Figure 4:Historical\-AP Shapley allocation over all272^\{7\}execution coalitions\. An outline marks the largest positive contribution for each dataset\.Figure[4](https://arxiv.org/html/2608.06765#S5.F4)assigns the largest predictive value to pair\-renewal intervals on Wikipedia and Reddit, the recent two\-destination state on MOOC, and the transition from the most recent destination to a candidate on LastFM\. On Wikipedia, pair renewal and one\-event transition exhibit \+4\.28 AP points of complementarity\. The one\-/two\-event interaction is−9\.27\-9\.27points on MOOC and−5\.10\-5\.10on LastFM, indicating competition between the two temporal resolutions on some queries\.

Query regimes\.We repeat component deletion after partitioning queries by direct recurrence, source activity, destination popularity, and early/late test interval\. Activity and popularity are measured from the complete pre\-query history, not window\-truncated counts, and divided at the dataset median\. Table[IV](https://arxiv.org/html/2608.06765#S5.T4)localizes Wikipedia renewal to recurrent queries\. Reddit uses renewal for high\-activity sources and candidate context for low\-activity sources\. MOOC’s two\-event transition contributes 20\.91 AP points without recurrence, while LastFM’s one\-event transition dominates in both regimes\.

TABLE IV:Dominant\-component deletion by query regime\. AP and its drop are in percentage points\.Grounded\-fact intervention\.We select 512 queries distributed throughout each dataset’s test interval and each seed\. The fact with the largest absolute contribution in the dominant component is removed from the available prefix, and all rules are grounded again\. The control removes a random fact matched on query, evidence bank, and recency quartile\.

TABLE V:Grounded\-fact intervention\. The complete program is re\-executed after deleting either the top\-contributing fact or a recency\-matched random fact\.In Table[V](https://arxiv.org/html/2608.06765#S5.T5), top\-fact deletion changes the MOOC and LastFM logits 3\.63 and 9\.29 times more than matched\-random deletion and reduces AP by 13\.38 and 9\.61 points\. The top Wikipedia fact also yields a larger logit change and AP drop\. On Reddit, individual renewal executions have larger local logit effects without producing a dataset\-level AP drop, separating local execution influence from global ranking responsibility\.

We also verify signed roles\. A supporting fact hasΔf=cf​\(q\+\)−cf​\(q−\)\>0\\Delta\_\{f\}=c\_\{f\}\(q^\{\+\}\)\-c\_\{f\}\(q^\{\-\}\)\>0and should reduce the margin when deleted; an opposing fact hasΔf<0\\Delta\_\{f\}<0and should increase it\. Direction agreement for the top supporting facts is 99\.4%/99\.5% on Wikipedia/Reddit and 99\.5%/93\.9% for opposing facts\. On MOOC/LastFM, supporting\-fact deletion decreases the mean margin 5\.22/8\.22 times more than matched\-random deletion\.

### 5\.5Neural and Symbolic Responsibilities

We retrain variants that alter exact binding, binding schema, renewal, and transition, and compare a matched\-capacity MLP receiving the same input\.

TABLE VI:Neural and symbolic responsibility \(Historical AUC/AP, %\)\.In Table[VI](https://arxiv.org/html/2608.06765#S5.T6), the full program on LastFM outperforms every restricted variant and exceeds the matched MLP by 6\.60 AP points\. MOOC’s signal is concentrated in short ordered transitions, so transition\-only execution reproduces full forecasting\. This component is not an arbitrary low\-rank recommender: removing exact source binding lowers AP by 1\.91 points, and randomizing the prescribed binding schema lowers it by 2\.23 points\. The symbolic executor decides which facts satisfy binding and order and thus instantiate an execution; the neural potential determines candidate compatibility within that valid execution\.

### 5\.6Explanation Quality and Independent Verification

Following TempME and TGIB, we retain only selected events at explanation ratiosr∈\{\.05,\.10,…,\.30\}r\\in\\\{\.05,\.10,\\ldots,\.30\\\}and measure agreement with the original decision\. ACC\-AUC is the normalized area under this accuracy–ratio curve\. Following SIG, Deletion AUFSC integrates the AP reduction caused by removing selected events at the same ratios\. ACC\-AUC measures explanation\-only sufficiency, whereas Deletion AUFSC measures ranking responsibility\. Exact definitions appear in[AppendixB](https://arxiv.org/html/2608.06765#A2)\.

For each dataset, we use the same 256 queries and three seeds\. Every method receives the same evidence bank formed by the recent 10 events of the query source and candidate, with identical selection counts at each ratio\. T\-GNNExplainer runs explorer–navigator MCTS for 40 rollouts using the coalition reward of a frozen TGN\. TempME uses the same TGN\. Search\-budget sensitivity at 100 and 200 rollouts and fixed budgetsk∈\{1,2,3,5,10\}k\\in\\\{1,2,3,5,10\\\}are reported in[AppendixB](https://arxiv.org/html/2608.06765#A2)\.

![Refer to caption](https://arxiv.org/html/2608.06765v1/x5.png)Figure 5:Macro ACC\-AUC, Deletion AUFSC, and runtime under a shared evidence universe\.TABLE VII:Explanation quality, cost, and perturbation stability, macro\-averaged over four datasets under a shared evidence universe\.Figure[5](https://arxiv.org/html/2608.06765#S5.F5)and Table[VII](https://arxiv.org/html/2608.06765#S5.T7)show that LiFTER ranks first in both macro ACC\-AUC \(0\.8671\) and AUFSC \(0\.0552\)\. It also achieves the best ACC and deletion fidelity for every fixed budgetk≤5k\\leq 5\. Evidence with high ACC\-AUC need not have high deletion fidelity: the former measures decision reproduction by a small set, while the latter measures disruption of the global ranking after deletion\. LiFTER’s signed execution and re\-grounding expose these properties separately\.

Independent execution verification\.Explanation metrics quantify how selected evidence affects a prediction\. A separate verifier tests whether a LiFTER trace faithfully describes the claimed grounded execution\. For all 19,664 test predictions across four datasets, it checks raw\-fact existence, integer equality bindings,T<TqT<T\_\{q\}, outgoing\-fact selection for Q1/Q2, and temporal order\. It then recomputes groundings and signed contributions from raw history and learned parameters and compares them with the trace and candidate logit\. All predictions are reproduced within numerical tolerance2×10−52\\times 10^\{\-5\}\. Deliberate changes to a historical fact, timestamp, bound entity, execution, or signed contribution are all detected under the five corruption types in Table[VIII](https://arxiv.org/html/2608.06765#S5.T8)\.

TABLE VIII:Certificate\-corruption detection by the independent verifier\.
### 5\.7Full\-Stream Scalability

![Refer to caption](https://arxiv.org/html/2608.06765v1/x6.png)Figure 6:Fact\-index construction, forecasting throughput under a shared workload, and explanation latency\.As Reddit grows from 32,768 to 672,447 events, indexing sustains 31\.1–31\.5k events/s and payload grows linearly from 7\.5 to 153\.9 MiB\. The full index takes 21\.63 s to construct\. The query executor operates within fixedHHwithout scanning the global stream\. AtH=10,K=1H=10,K=1, it takes 0\.041 ms/query with batch 128 and reaches 101\.6k queries/s with batch 2,048\. Complete re\-execution after a fact edit takes 0\.138 ms/query; a human\-readable JSON certificate takes 1\.185 ms/query and averages 6\.1 KiB\.

Under architecture\-level timing with the same recent\-10 tensors and batch 512, neural predictors reach 234\.0k–487\.0k queries/s and LiFTER reaches 66\.9k\. Grounded execution incurs a cost but retains practical full\-stream throughput\. Intrinsic tracing takes 0\.024 ms/query, approximately 3,800 times faster than T\-GNNExplainer’s 91\.533\-ms search\. Feed\-forward attribution by TempME, TGIB, and SIG is faster at 0\.006, 0\.005, and 0\.008 ms/query\. Figure[6](https://arxiv.org/html/2608.06765#S5.F6)therefore does not claim the fastest attribution; it shows that LiFTER emits an exact executable trace with its prediction and without a separate search\.

## 6Conclusion

LiFTER provides a grounded temporal rule language for continuous\-time interaction streams without semantic relation chains\. It preserves source, destination, and timestamp in ground atoms and defines rule identity through endpoint bindings, pair renewal, history position, and ordered transitions\. Candidate scores and explanations arise from the same program trace; the prior and signed contributions in Eq\. \([8](https://arxiv.org/html/2608.06765#S4.E8)\) exactly reconstruct the logit\.

This constraint does not sacrifice forecasting performance\. In Table[II](https://arxiv.org/html/2608.06765#S5.T2), LiFTER ranks first in Historical AUC/AP on Reddit and LastFM, second on both metrics on Wikipedia, and second in AUC and first in AP on MOOC\. Figure[5](https://arxiv.org/html/2608.06765#S5.F5)and Table[VII](https://arxiv.org/html/2608.06765#S5.T7)show the highest macro explanation accuracy and deletion fidelity\. Table[VIII](https://arxiv.org/html/2608.06765#S5.T8)further establishes that each explanation is a prediction computation replayable from raw facts and frozen parameters, rather than a separate display\.

The neuro\-symbolic microscope goes beyond describing an output\. Exact Shapley decomposition in Figure[4](https://arxiv.org/html/2608.06765#S5.F4)separates pair renewal on Wikipedia/Reddit, two\-event transition on MOOC, and one\-event transition on LastFM\. Tables[IV](https://arxiv.org/html/2608.06765#S5.T4)and[V](https://arxiv.org/html/2608.06765#S5.T5)trace these structures to query regimes and individual historical evidence\. Dataset\-level performance, component interactions, and fact responsibility are observable within the same forward program\.

The fact\-level formulation is also distinct from semantic typing\. AlthoughK=1K=1is sufficient for the JODIE streams in Table[III](https://arxiv.org/html/2608.06765#S5.T3), the phase diagram in[AppendixA](https://arxiv.org/html/2608.06765#A1)activatesK\>1K\>1when event meaning is identifiable from pre\-event context and changes the future distribution\. LiFTER thus formulates CTDG forecasting from the outset as a verifiable and editable grounded temporal program, rather than an interpretation applied after event attribution\.

## Appendix ASemantic Predicate Phase Diagram

The JODIE results in Section[5\.3](https://arxiv.org/html/2608.06765#S5.SS3)do not establish whenK\>1K\>1becomes useful\. We isolate this condition in a controlled single\-relation stream over the complete5×55\\times 5gridα,β∈\{0,\.25,\.5,\.75,1\}\\alpha,\\beta\\in\\\{0,\.25,\.5,\.75,1\\\}\. Each query contains three facts with one of three hidden event types\. Cyclic sequences\(0,1,2\),\(1,2,0\),\(2,0,1\)\(0,1,2\),\(1,2,0\),\(2,0,1\)define the positive semantic class and all remaining sequences define the negative class, with exact 50:50 class balance\. The future label follows the semantic class with probability0\.5\+0\.5​α0\.5\+0\.5\\alpha\. Thus types are independent of the future atα=0\\alpha=0, whereas their sequence determines the label atα=1\\alpha=1\.

Each fact receives a 10\-dimensional pre\-event context\. The first three dimensions contain a type\-specific center scaled by3​β3\\betaplus unit Gaussian noise; the remaining seven contain independent unit Gaussian noise\. Parameterβ\\betachanges only the identifiability of hidden type from context, not the future\-transition distribution\. Training and testing use 6,000/3,000 queries with disjoint entity identifiers, 40 epochs, and seeds 7/17/27\. Type recovery is fact\-level accuracy after aligning learned identifiers by Hungarian matching; we also report NMI and ARI\. The oracle control supplies ground\-truth types, and the random control assigns fixed uniform predicates independent of context and label\.

![Refer to caption](https://arxiv.org/html/2608.06765v1/x7.png)Figure 7:Completeα×β\\alpha\\times\\betagrid\. A learned predicate becomes both recoverable and predictive only when its effect on the future and its identifiability from pre\-event context are simultaneously high\.In Figure[7](https://arxiv.org/html/2608.06765#A1.F7), the AUC gain of learnedK=3K=3overK=1K=1increases along both axes\. Atα=0,β=1\\alpha=0,\\beta=1, recovery reaches 79\.2% but gain is−0\.5\-0\.5percentage points: a distinguishable type is not predictive when future outcomes are identical\. Atα=1,β=0\\alpha=1,\\beta=0, semantic effect is maximal but recovery remains 33\.7%, limiting gain to 1\.8 points\. With both conditions at their maximum, recovery reaches 98\.9%, NMI 0\.943, ARI 0\.968, and AUC gain 49\.7 points; shuffling predicate assignments reduces AUC by 50\.3 points\. Oracle gain rises from 0\.3 to 50\.0 points withα\\alphaand is invariant toβ\\beta, while random\-predicate gain remains between−0\.4\-0\.4and0\.60\.6points in every cell\.

Consider a stream in which content views and add\-to\-cart actions are both recorded as the same interaction\. If earlier frequencies or intervals distinguish the two,β\\betais high\. If their subsequent destination distributions are identical,α\\alpharemains low and even exact recovery cannot improve forecasting\. Both axes become high when add\-to\-cart events lead to related products, views lead to other content, and the pre\-event history identifies the action\. Under this condition, a typed predicate is a grounded rule variable that changes the future\-link distribution, not a decorative relation label\.

## Appendix BComplete Explanation Results

### B\.1Ratio\-Controlled Evaluation

At explanation ratior∈\{\.05,\.10,…,\.30\}r\\in\\\{\.05,\.10,\\ldots,\.30\\\}, letEi\(r\)E\_\{i\}^\{\(r\)\}denote the events selected by a method andGiG\_\{i\}the complete evidence for queryii\. Explanation\-only agreement and its normalized area are

Accexp⁡\(r\)\\displaystyle\\operatorname\{Acc\}\_\{\\rm exp\}\(r\)=1N​∑i𝟏​\[y^i​\(Ei\(r\)\)=y^i​\(Gi\)\],\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{i\}\\mathbf\{1\}\[\\hat\{y\}\_\{i\}\(E\_\{i\}^\{\(r\)\}\)=\\hat\{y\}\_\{i\}\(G\_\{i\}\)\],\(12\)ACC​\-​AUC\\displaystyle\\operatorname\{ACC\\text\{\-\}AUC\}=1\.25​∫\.05\.30Accexp⁡\(r\)​𝑑r\.\\displaystyle=\\frac\{1\}\{\.25\}\\int\_\{\.05\}^\{\.30\}\\operatorname\{Acc\}\_\{\\rm exp\}\(r\)\\,dr\.\(13\)After deleting selected events from the complete evidence, AP degradation and its normalized area are

FIDA​P⁡\(r\)\\displaystyle\\operatorname\{FID\}^\{AP\}\(r\)=A​P​\(G\)−A​P​\(G−E\(r\)\),\\displaystyle=AP\(G\)\-AP\(G\-E^\{\(r\)\}\),\(14\)AUFSC\\displaystyle\\operatorname\{AUFSC\}=1\.25​∫\.05\.30FIDA​P⁡\(r\)​𝑑r\.\\displaystyle=\\frac\{1\}\{\.25\}\\int\_\{\.05\}^\{\.30\}\\operatorname\{FID\}^\{AP\}\(r\)\\,dr\.\(15\)Equations \([12](https://arxiv.org/html/2608.06765#A2.E12)\)–\([13](https://arxiv.org/html/2608.06765#A2.E13)\) measure sufficiency; Eqs\. \([14](https://arxiv.org/html/2608.06765#A2.E14)\)–\([15](https://arxiv.org/html/2608.06765#A2.E15)\) measure deletion responsibility\.

### B\.2T\-GNNExplainer Search Budget

T\-GNNExplainer’s MCTS cost grows with the number of rollouts, and its paper presents 100–200 rollouts as a practical runtime–fidelity range\[[11](https://arxiv.org/html/2608.06765#bib.bib6)\]\. Our ratio\-controlled comparison uses 40 rollouts over the shared recent\-10 evidence universe at each endpoint\. To test whether this budget limits search quality, we compare 40, 100, and 200 rollouts on 64 queries evenly distributed over each dataset’s test interval with seeds 7/17/27\.

TABLE IX:T\-GNNExplainer rollout sensitivity on 64 evenly spaced queries per dataset and three seeds\. Values are macro averages over four datasets\.Table[IX](https://arxiv.org/html/2608.06765#A2.T9)shows that 100 and 200 rollouts increase runtime by 2\.24 and 4\.03 times without improving macro ACC\-AUC or AUFSC\. Neither metric increases consistently within individual datasets\. Forty rollouts therefore provide a sufficient search budget for this shared evidence universe\.

Table[X](https://arxiv.org/html/2608.06765#A2.T10)expands the macro result in Section[5\.6](https://arxiv.org/html/2608.06765#S5.SS6)by dataset\. Base AP is predictor AP before explanation\. ACC\-AUC is the normalized area under the accuracy curve obtained by retaining 5–30% of available evidence; Deletion AUFSC is the corresponding AP\-degradation area after removal\. Available/selected reports the mean number of available and selected events per query and their ratio\. Stability measures preservation of top events after a small input perturbation\.

TABLE X:Ratio\-controlled explanation results by dataset\. Every method uses the same per\-query evidence bank and ratio budget\.
### B\.3Fixed\-Event\-Budget Evaluation

A ratio budget allows longer histories to select more events\. Tables[XI](https://arxiv.org/html/2608.06765#A2.T11)and[XII](https://arxiv.org/html/2608.06765#A2.T12)therefore give every method the same absolute budgetk∈\{1,2,3,5,10\}k\\in\\\{1,2,3,5,10\\\}per query\. ACC@kkis the fraction of queries whose original binary decision is reproduced using only the selectedkkevents\. For example, LiFTER ACC@3 of 0\.8774 means that three events reproduce 87\.74% of decisions\.

TABLE XI:Decision agreement under fixed\-event budgets\.Deletion FID@kkisA​P​\(G\)−A​P​\(G−E\(k\)\)AP\(G\)\-AP\(G\-E^\{\(k\)\}\)\. LiFTER FID@3 of 0\.0544 therefore means that deleting three selected events lowers AP by 5\.44 percentage points\. A negative value denotes a small AP increase after deletion\. Lowercasekkis the number of explanation events and is unrelated to predicate\-vocabulary sizeKK\.

TABLE XII:Deletion fidelity under fixed\-event budgets\.

## Appendix CShapley Predictive\-Mechanism Attribution

LetMMbe the set of execution components in a trained LiFTER andv​\(S\)v\(S\)the Historical AUC or AP obtained by executing only subsetS⊆MS\\subseteq M\. The Shapley contribution of componentmmis

ϕm​\(v\)=∑S⊆Mm∉S\|S\|\!​\(\|M\|−\|S\|−1\)\!\|M\|\!​\[v​\(S∪\{m\}\)−v​\(S\)\]\.\\phi\_\{m\}\(v\)=\\sum\_\{\\begin\{subarray\}\{c\}S\\subseteq M\\\\ m\\notin S\\end\{subarray\}\}\\frac\{\|S\|\!\(\|M\|\-\|S\|\-1\)\!\}\{\|M\|\!\}\\left\[v\(S\\cup\\\{m\\\}\)\-v\(S\)\\right\]\.\(16\)Equation \([16](https://arxiv.org/html/2608.06765#A3.E16)\) averages the marginal change from addingmmover all execution orders\. With seven components, all 128 subsets are evaluated and no sampling approximation is required\. The result satisfies∑m∈Mϕm​\(v\)=v​\(M\)−v​\(∅\)\\sum\_\{m\\in M\}\\phi\_\{m\}\(v\)=v\(M\)\-v\(\\varnothing\); the largest measured numerical residual is1\.11×10−161\.11\\times 10^\{\-16\}\.

## Acknowledgements

The authors have no acknowledgements to declare\.

## Funding

This research received no specific grant from funding agencies in the public, commercial, or not\-for\-profit sectors\.

## Data availability

This study uses the public Wikipedia, Reddit, MOOC, and LastFM temporal interaction datasets distributed with JODIE\. The raw data are available from the Stanford SNAP JODIE repository\. The accompanying materialization pipeline records the source metadata and produces the chronological splits used in this study\.

## Code availability

The implementation, experiment configurations, result summaries, and scripts for reproducing the tables and figures are publicly available at[https://github\.com/SnowyPainter/LiFTER\-public](https://github.com/SnowyPainter/LiFTER-public)\.

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