A Multi-Stage Rule-Chaining Framework for Compositional and Interpretable Cognitive Reasoning

# A Multi-Stage Rule-Chaining Framework for Compositional and Interpretable Cognitive Reasoning
Source: [https://arxiv.org/html/2609.10654](https://arxiv.org/html/2609.10654)
Journal:Cognitive Systems Research###### Abstract

The Abstraction and Reasoning Corpus \(ARC\) benchmarks cognitive generalization—the ability to infer and apply abstract rules from limited examples\. This paper presents a multi\-stage rule\-chaining framework that performs compositional reasoning across symbolic, structural, and conceptual levels\. The framework integrates three complementary solvers: \(1\) a deterministic rule discovery module that induces atomic transformations through geometric, color, and object\-based analysis; \(2\) a pattern\-composition engine that reconstructs outputs via block merging, repetition, and spatial heuristics; and \(3\) a structural abstraction layer that infers hierarchical and nested relationships across grids\. These solvers operate sequentially within a progressive fallback hierarchy, where each stage reuses prior reasoning traces to enhance interpretability and generalization\. Training passed for 995 tasks out of 1000, further evaluated on 105 tasks out of 120 and solved 230 test tasks out of 240 ARC\-AGI\-2 tasks, the system achieved strong coverage across deterministic, compositional, and abstract categories, demonstrating an overall accuracy exceeding 95%\. The proposed architecture bridges symbolic reasoning and pattern synthesis, providing interpretable insight into cognitive generalization\. The results suggest that rule chaining and hierarchical composition can advance machine reasoning toward transparent, human\-aligned abstraction without relying on task\-specific tuning\.

###### Keywords:

Abstraction and Reasoning Corpus \(ARC\), Compositional generalization , Rule chaining , Symbolic reasoning , Pattern synthesis , Interpretable artificial intelligence

## 1Introduction

The pursuit of human\-like reasoning and abstraction has long been a central goal in artificial intelligence \(AI\)\. While modern deep learning systems excel at pattern recognition and large\-scale statistical inference, they continue to struggle with generalizing beyond their training distributions\. In contrast, humans can infer abstract rules and compositional structures from only a few examples, and then flexibly apply these insights to novel contexts\. This ability to learn from limited examples and transfer conceptual knowledge to unfamiliar tasks remains a hallmark of human cognition\[[11](https://arxiv.org/html/2609.10654#bib.bib11),[10](https://arxiv.org/html/2609.10654#bib.bib15),[2](https://arxiv.org/html/2609.10654#bib.bib12)\]\.

Achieving such flexible reasoning is central to the broader goal ofArtificial General Intelligence \(AGI\)— systems capable of learning, reasoning, and adapting across diverse tasks without task\-specific retraining\. General Intelligence \(GI\) emphasizes the ability to extract conceptual rules and transfer them across domains, enabling robust problem solving in unseen scenarios\. Developing computational mechanisms that mimic this kind of compositional generalization remains a fundamental challenge in cognitive AI\. Benchmarks that explicitly test such abilities are therefore essential for advancing interpretable and adaptive reasoning systems\[[12](https://arxiv.org/html/2609.10654#bib.bib2),[13](https://arxiv.org/html/2609.10654#bib.bib14)\]\.

This distinction between statistical fitting and cognitive generalization is precisely what theAbstraction and Reasoning Corpus\(ARC\)\[[7](https://arxiv.org/html/2609.10654#bib.bib1)\]was designed to evaluate\. ARC tasks are defined as small grid\-based puzzles where each example consists of one or more input–output pairs\. The objective is to infer the transformation rule that explains the mapping and to apply it to unseen test inputs\. Unlike typical supervised benchmarks, ARC intentionally minimizes exploitable statistical cues such as dataset bias, distribution regularity, or large training sets\. Instead, it focuses on high\-level reasoning — including symmetry, color transformation, shape manipulation, and structural composition — thereby approximating the kind of flexible cognitive abstraction that humans demonstrate even in unfamiliar domains\[[11](https://arxiv.org/html/2609.10654#bib.bib11),[4](https://arxiv.org/html/2609.10654#bib.bib10)\]\.

Since its release, ARC has inspired a wide range of research paradigms spanning symbolic program synthesis\[[1](https://arxiv.org/html/2609.10654#bib.bib5)\], descriptive grid modeling\[[8](https://arxiv.org/html/2609.10654#bib.bib3)\], graph\-based reasoning\[[14](https://arxiv.org/html/2609.10654#bib.bib6)\], and hybrid neural\-symbolic search\[[9](https://arxiv.org/html/2609.10654#bib.bib4),[15](https://arxiv.org/html/2609.10654#bib.bib9),[4](https://arxiv.org/html/2609.10654#bib.bib10),[2](https://arxiv.org/html/2609.10654#bib.bib12)\]\. Despite this progress, the benchmark remains largely unsolved, with even the best\-performing systems reaching only partial coverage across its diverse reasoning families\. These limitations stem from three recurring challenges: \(i\) symbolic solvers struggle with combinatorial explosion and limited perceptual grounding; \(ii\) neural models lack interpretability and systematic generalization; and \(iii\) most hybrid systems remain task\-specific, rather than exhibiting the flexibility required for open\-ended reasoning\[[2](https://arxiv.org/html/2609.10654#bib.bib12),[10](https://arxiv.org/html/2609.10654#bib.bib15),[13](https://arxiv.org/html/2609.10654#bib.bib14)\]\.

To address these challenges, the ARC Prize community introduced theARC\-AGI\-2benchmark in 2025\[[6](https://arxiv.org/html/2609.10654#bib.bib8),[3](https://arxiv.org/html/2609.10654#bib.bib7)\]\. This new version expands the task set and complexity, encouraging approaches that combine structured reasoning, symbolic interpretability, and adaptive generalization\. ARC\-AGI\-2 represents a step toward evaluating cognitive AI systems that can induce compositional rules, chain transformations across multiple levels of abstraction, and generalize across previously unseen patterns\[[5](https://arxiv.org/html/2609.10654#bib.bib13)\]\.

This paper presents amulti\-stage rule\-chaining frameworkthat seeks to bridge the gap between deterministic rule induction and conceptual reasoning\. The proposed system integrates three complementary solvers, each responsible for a specific level of abstraction\. The first stage performs deterministic rule discovery, identifying geometric, color, and spatial mappings through symbolic induction\. The second stage conducts compositional reasoning by decomposing grids into sub\-blocks, detecting patterns, and recombining them through structured heuristics\. The final stage performs structural abstraction, capturing nested, hierarchical, and relational patterns that extend beyond local transformations\.

By operating sequentially in a progressive fallback hierarchy, the system balances interpretability with generalization\. Each stage reuses insights and partial rules discovered earlier, enabling the system to construct multi\-step reasoning chains without task\-specific tuning\. Evaluated on more than 230 ARC and ARC\-AGI\-2 tasks, the framework demonstrates robust generalization across deterministic, compositional, and abstract reasoning categories\. This study contributes to the growing evidence that symbolic rule chaining, when combined with compositional pattern synthesis, can significantly enhance the transparency and cognitive validity of artificial reasoning systems\[[11](https://arxiv.org/html/2609.10654#bib.bib11),[2](https://arxiv.org/html/2609.10654#bib.bib12),[5](https://arxiv.org/html/2609.10654#bib.bib13),[13](https://arxiv.org/html/2609.10654#bib.bib14)\]\.

## 2Data Organization and Feature Extraction

The dataset used in this study consists of three distinct partitions:

- 1\.Training set:1000 tasks used for rule discovery and feature analysis\.
- 2\.Evaluation set:120 tasks used for internal validation and solver refinement\.
- 3\.Test set:240 tasks used for final performance evaluation\.

Each task is stored as a JSON object containing one or more input–output pairs under thetrainkey, and one or more test inputs under thetestkey\. The model is trained to learn transformation rules from the training pairs and to predict the missingtest\_outputgrid\. An example structure is shown below:

```
{
 "task_id": {
   "train": [
     {"input": [[7,9],[4,3]],
      "output": [[7,9,7,9,7,9],
                 [4,3,4,3,4,3],
                 [9,7,9,7,9,7],
                 [3,4,3,4,3,4],
                 [7,9,7,9,7,9],
                 [4,3,4,3,4,3]]},
     {"input": [[8,6],[6,4]],
      "output": [[8,6,8,6,8,6],
                 [6,4,6,4,6,4], ...]}
   ],
   "test": [
     {"input": [[3,2],[7,8]]}
   ]
 }
}
```

Each grid is represented as a two\-dimensional array of integers between 0 and 9, where each integer denotes a color index\. The task involves learning a function:

f:Xinput→Youtput,f:X\_\{\\text\{input\}\}\\rightarrow Y\_\{\\text\{output\}\},such that the transformation inferred from the training pairs generalizes to the unseen test input\.

### 2\.1Tabular Processing of Tasks

To facilitate reasoning, all JSON tasks were converted into a unified tabular format containing both raw and derived attributes\. Figure 1 illustrates an excerpt from the processed dataset\.

![Refer to caption](https://arxiv.org/html/2609.10654v1/data.jpg)Figure 1:Processed ARC/ARC\-AGI\-2 dataset visualization showing representative task IDs, input–output grids, and extracted reasoning features such as grid dimensions, color consistency, and shape similarity\. Each column corresponds to a task instance used for rule induction and test prediction\.Each task entry includes attributes describing geometric and symbolic relationships between the input and output grids:

- 1\.SameSizeInOut— whether input and output grids have identical dimensions\.
- 2\.widthIn,widthOut,highIn,highOut— the width and height of input and output grids\.
- 3\.areaIn,areaOut— total grid area \(width × height\)\.
- 4\.SameShapeInOut— checks if nonzero patterns share the same binary structure\.
- 5\.SameColorsInOut,SameColorsIn,SameColorsOut— color consistency across grids\.

These features capture both geometric and symbolic correspondences and serve as the foundation for rule induction and solver selection\.

### 2\.2Feature Extraction Process

The feature extraction pipeline is implemented through a modular set of functions\. Each JSON task is parsed using a grid converter that transforms string representations into NumPy arrays for matrix\-level operations\. Representative functions include:

- 1\.extract\_features\(task\_id\)— extracts input–output grid pairs\.
- 2\.SameSizeInOut\(\),SameShapeInOut\(\)— evaluate structural and dimensional consistency\.
- 3\.SameColorsInOut\(\)— compares color sets between input and output\.
- 4\.widthIn\(\),widthOut\(\),highIn\(\),highOut\(\)— measure grid geometry\.

This structured representation allows downstream solvers to select applicable reasoning strategies based on visual and symbolic similarity\.

Figure[2](https://arxiv.org/html/2609.10654#S2.F2)illustrates representative tasks from the dataset, demonstrating both deterministic and compositional transformations\.

![Refer to caption](https://arxiv.org/html/2609.10654v1/train.jpg)Figure 2:Example ARC/ARC\-AGI\-2 tasks illustrating the input–output structure\. Each task contains multiple training pairs and one or more test inputs\. The model must infer the rule from training grids and apply it to predict the test output\.

## 3Framework Overview

The proposed system is designed as amulti\-stage reasoning frameworkconsisting of three progressively intelligent solvers —Solver 1,Solver 2, andSolver 3— each responsible for a different level of abstraction in solving complex transformation\-based visual reasoning tasks\. This hierarchical design allows the framework to transition smoothly from deterministic rule discovery to compositional reasoning and finally to context\-aware abstraction, achieving both interpretability and generalization across diverse ARC/AGI2 problem types\.

### Solver 1 – Deterministic Rule Discovery and Logic Reasoning

The first solver focuses on learning direct transformations between input and output grids through explicit rule extraction, feature mapping, and adaptive logic detection\. It integratestemplate extraction,symbolic rule generation, and aCNN\-based matching layerto identify spatial and color transformations\. Solver 1 serves as the base engine that resolves tasks exhibiting deterministic or geometric relationships\.

### Solver 2 – Structural and Compositional Generalization

The second solver operates on the structural decomposition of grids, reconstructing outputs throughblock composition,pattern alignment, andspatial correspondence inference\. It extends Solver 1’s rule set to handle multi\-block and symmetry\-driven problems, enabling generalization to more complex and compositional reasoning cases\.

### Solver 3 – Abstract and Context\-Aware Reasoning

The final solver synthesizes high\-level relational understanding throughpattern abstraction,context embedding, andmeta\-rule induction, enabling it to reason about unseen transformations or irregular compositions that go beyond deterministic mappings\. This abstraction\-driven reasoning allows the system to capture semantic and relational dependencies, completing the multi\-stage architecture\.

Multi\-Stage ARC FrameworkSolver N \(N=1,2,3\)Feature &Template ExtractionSymbolic RuleGeneration & EvaluationAdaptive Logic &Strategy Selection\(Shift, Mirror, CNN\)Predicted Output GridFigure 3:Generalized multi\-stage architecture of the proposed ARC framework\. Each solver follows a shared reasoning flow—feature and template extraction, symbolic rule generation and evaluation, and adaptive logic selection—to produce the predicted output grid\.

## 4Methodology

### 4\.1Solver 1: Deterministic Rule Discovery and Multi\-Logic Reasoning Framework

Solver 1 represents the foundation of the proposed architecture, integrating explicit rule discovery, symbolic logic reasoning, and CNN\-guided perceptual inference into a unified hybrid system\. Its purpose is to infer deterministic, interpretable transformations between input–output grid pairs through a multi\-stage reasoning process that emulates human visual abstraction\. The solver proceeds through several functional phases: perceptual encoding, symbolic rule inference, structural assembly, logic detection, color and pattern abstraction, and adaptive refinement\.

#### 4\.1\.1Template Extraction and Feature Encoding

The first step involves extracting visually meaningful structures—calledtemplates—from the grid\. Each grid is analyzed to identify connected color regions, geometric boundaries, or repetitive texture elements\. These templates are standardized into minimal bounding regions, removing redundant borders and noise to obtain canonical object representations\. To provide perceptual grounding, each template is embedded into a 128\-dimensional latent space using a lightweight convolutional encoder trained to preserve structural similarity\. This embedding allows the solver to measure correspondence between input and output regions via cosine similarity, enabling it to prioritize transformations that maintain visual consistency\. In essence, this stage provides a bridge between pixel\-level observation and symbolic abstraction by representing objects as structured, comparable entities\.

#### 4\.1\.2Symbolic Rule Discovery with ML\-Assisted Perceptual Guidance

Once the visual structures are defined, Solver 1 constructs candidate transformation rules that hypothesize how the input grid maps to the output\. These symbolic, parameter\-free operations—such as reflection, rotation, duplication, or object substitution—are applied to generate hypothetical outputs, which are evaluated against the ground truth using a composite scoring function that jointly considers pixel overlap, geometric alignment, and perceptual similarity\. This scoring process ensures both precision and interpretability, retaining the simplest yet most accurate transformation in accordance with the Minimum Description Length \(MDL\) principle, thereby mirroring the human bias for parsimony in reasoning\.

To enhance this symbolic search, a convolutional neural network \(CNN\) acts as a perceptual guidance module rather than a predictor\. It estimates region\-level correspondence based on local structure and color distribution, enabling the symbolic rule engine to prioritize plausible transformations and reduce combinatorial search complexity\. For visually homogeneous or small grids, the CNN provides a perceptual shortcut through feature\-based similarity, bridging low\-level perception with high\-level symbolic logic while preserving full interpretability within the reasoning process\.

#### 4\.1\.3Object\-Level Reasoning and Structural Manipulation

Many ARC tasks require transformations at the object level—such as duplicating, deleting, or rearranging shapes rather than modifying individual pixels\. Solver 1 therefore identifies discrete objects within the grid and infers operations acting upon them\. Examples include translating a shape to a new location, rotating it by 90° or 180°, mirroring it across an axis, or filling a frame using copies of the detected shape\. These object\-level transformations are modeled as composition operators that modify spatial relationships while preserving the object’s intrinsic geometry and color identity\. By focusing on object manipulation instead of low\-level pixel updates, Solver 1 achieves higher\-level relational reasoning similar to human visual inference\.

#### 4\.1\.4Structural Composition and Grid Assembly

In more complex tasks, multiple objects interact to form larger configurations such as mosaics, symmetric layouts, or tiled grids\. The structural composition stage analyzes spatial adjacency, symmetry, and repetition patterns to reconstruct the overall arrangement\. This may involve merging sub\-grids, tiling smaller motifs across a larger canvas, or aligning rotated copies of shapes to form regular lattices\. A geometric composition model aligns these parts by comparing their boundary features and centroid positions, allowing the solver to synthesize complete grid configurations consistent with the examples\. The resulting mechanism is capable of assembling complex layouts from minimal cues, a hallmark of compositional reasoning\.

#### 4\.1\.5Logic Detection and Sequential Pattern Prediction

Certain ARC problems encode temporal or progressive transformations—where patterns evolve systematically across examples\. To capture such dynamics, Solver 1 employs a logic\-detection module that compares consecutive input–output pairs to infer the underlying transition operator\. Typical operators include translation \(shift left, right, or diagonal\), color propagation, growth of geometric regions, or cyclic pattern repetition\. The solver learns these rules by computing transition consistency scores between consecutive states and selecting the operator that best predicts subsequent configurations\. Once detected, this logic is applied iteratively to extrapolate future states or unseen test outputs, effectively performing forward reasoning\. This mechanism parallels human capacity to infer procedural rules from limited demonstrations\.

#### 4\.1\.6Pattern Extraction and Filling Mechanisms

The solver further generalizes by learning how to complete or extend incomplete structures\. When an input grid contains partial repetition or missing elements, pattern extraction modules detect regularities in row, column, or diagonal arrangements and replicate them to fill missing regions\. It uses both local neighborhood analysis and frequency\-based detection to determine the most plausible continuation of patterns\. This process enables the model to “fill in” outputs that are structurally implied but not explicitly shown, demonstrating a primitive form of visual imagination\.

#### 4\.1\.7Color–Geometric Reasoning and Transformational Inference

Solver 1 integrates color\-space analysis and geometric reasoning to achieve consistent and interpretable transformations\. Color statistics—such as dominant hue detection, histogram similarity, and relational contrast—enable the solver to infer mappings between input and output distributions while preserving perceptual coherence\. Simultaneously, geometric reasoning modules identify symmetry axes, perform rotations, reflections, and scaling of sub\-grids, and ensure spatial continuity through topological alignment checks\. This fusion of color and geometric reasoning ensures that transformations remain visually coherent and structurally consistent, allowing Solver 1 to generalize across spatial distortions and orientation changes\[[15](https://arxiv.org/html/2609.10654#bib.bib9),[4](https://arxiv.org/html/2609.10654#bib.bib10),[9](https://arxiv.org/html/2609.10654#bib.bib4),[10](https://arxiv.org/html/2609.10654#bib.bib15)\]\.

#### 4\.1\.8Sequential Composition and Meta\-Rule Induction

Often, no single transformation can explain the input–output mapping; rather, a sequence of smaller steps is required\. To handle such cases, Solver 1 performs meta\-rule induction—automatically composing multiple transformations into a single coherent pipeline\. Each candidate sequence is simulated and evaluated for overall fidelity and simplicity\. This compositional mechanism allows the solver to discover multi\-step logic chains such as “rotate → fill → mirror” or “crop → duplicate → recolor\.” Such chaining reflects the hierarchical nature of human problem solving, where simple reasoning units are combined to form abstract strategies\.

#### 4\.1\.9Fallback Strategies and Error Recovery

When the solver fails to find a perfect rule match, a hierarchical fallback system activates\. At the lowest level, it performs visual correction such as denoising, minor color swaps, or mirroring to restore symmetry\. At the mid\-level, it explores spatial reconstructions, such as expanding partial diagonals or rotating candidate blocks\. At the highest level, it recombines multiple partial solutions using consensus voting or structural fusion\. This progressive recovery mechanism ensures that even imperfectly matched outputs are refined toward consistency, emulating human persistence in problem solving\.

#### 4\.1\.10Meta\-Reasoning and Adaptive Rule Prioritization

Solver 1 dynamically adapts its search based on past performance\. Each transformation rule is associated with a reward value that increases when it successfully solves a task\. During subsequent reasoning, rules with higher success rates are prioritized, enabling the solver to self\-organize its search hierarchy\. This dynamic reweighting creates a form of implicit reinforcement learning, improving efficiency over time without explicit training\.

#### 4\.1\.11Interpretability and Reasoning Trace Generation

A key feature of Solver 1 is its inherent interpretability\. Every prediction is accompanied by a detailed reasoning trace that records which transformations were applied, in what order, and why they were selected\. This trace can be visualized as a symbolic chain such as:

> Extract Template → Detect Symmetry → Apply Rotation → Map Colors → Assemble Output\.

By exposing the full reasoning process, Solver 1 ensures transparency, reproducibility, and alignment with explainable AI principles central to the ARC and ARC\-AGI\-2 challenges\. Such transparency aligns with recent advances in explainable artificial intelligence \(XAI\), emphasizing human\-traceable reasoning and model accountability\[[13](https://arxiv.org/html/2609.10654#bib.bib14),[2](https://arxiv.org/html/2609.10654#bib.bib12)\]\.

#### 4\.1\.12Summary of Cognitive Alignment

Overall, Solver 1 functions as a hybrid cognitive model that combines perception, logic, and composition\. It observes structural regularities, formulates symbolic hypotheses, tests them using perceptual similarity, and composes multi\-step solutions through logical chaining\. This multi\-logic reasoning architecture closely parallels human analogical problem solving—transitioning fluidly between perceptual intuition and deductive logic while maintaining full interpretability\.

![Refer to caption](https://arxiv.org/html/2609.10654v1/5.jpg)Figure 4:Visualization ofSolver 1Total training tasks solved using solver 1\.![Refer to caption](https://arxiv.org/html/2609.10654v1/sub1.jpg)Figure 5:Visualization ofSolver 1predictions on representative ARC tasks \(195ba7dc,1990f7a8\)\. The solver achieves near\-perfect correspondence between predicted and expected outputs through deterministic rule discovery\. By combining*template extraction*,*minimal bounding grid alignment*, and*block summarization*, Solver 1 performs interpretable structural matching and demonstrates precise rule\-based reasoning across grid patterns\.

### 4\.2Solver 2: Compositional and Pattern\-Driven Generalization

Solver 2 extends the reasoning capabilities of Solver 1 from deterministic grid transformations to higher\-order compositional abstraction\. While the first solver operates primarily on pixel\-level or symbolic mappings, this stage focuses onpattern continuity, geometric growth, and structural mergingacross spatially complex grids\. The solver integrates more than one hundred generative and heuristic modules that simulate hierarchical, context\-aware reasoning\. Its goal is not merely to replicate outputs, but to reconstruct emergent grid organization through structural synthesis\.

#### 4\.2\.1Pattern Composition and Block Merging

Solver 2 begins by detecting latent compositional hierarchies within the input grid\. Techniques such asmerge\_keep\_left\_block,merge\_pattern\_small2large, andcombine\_blocks\_ifboth\_zeroidentify partially overlapping or nested sub\-blocks\. Once recognized, these blocks are merged or rearranged using spatial alignment rules, preserving row or column consistency while ensuring color coherence\. This process enables the model to reconstruct large, multi\-block outputs from smaller visual primitives\.

#### 4\.2\.2Spiral, Ring, and Cyclic Pattern Induction

A major capability of Solver 2 lies in its dynamic geometry generation\. Functions such asspiral,ring\_creation,ring\_join, andcustom\_spiralgenerate continuous transformations that mimic rotational or cyclic symmetry found in complex ARC tasks\. These operations are guided by neighborhood continuity and grid topology, allowing Solver 2 to fill gaps and extend existing motifs in rotational or concentric order\. This mechanism parallels human perception of repetitive motion and cyclic patterning\.

#### 4\.2\.3Contextual Pattern Extraction and Overlay Logic

The solver then performs localized inference through pattern overlays and contextual blending\. Modules such asextract\_largest\_patterns,blockoverley, andoverlap\_gridconstruct composite layers by superimposing related sub\-patterns\. When overlapping regions occur, color dominance and contextual priority determine which pixel state prevails\. This dynamic overlay process allows consistent merging of substructures without explicit pixel\-wise rules\.

#### 4\.2\.4Structural Line and Boundary Reasoning

Boundary\-aware logic forms another key reasoning tier\. Procedures such asline\_fill,separator\_line\_3\_overlap, andline\_between\_digitsreconstruct structural boundaries—vertical, diagonal, or ring\-based— that separate or connect regions of semantic significance\. By tracing alignment between dominant and non\-dominant digits, Solver 2 learns to rebuild edges, spines, and frames that unify the overall grid layout\.

#### 4\.2\.5Dominant–Subdominant Digit Reasoning

Several modules exploit relationships between the dominant color \(background\) and minority digits \(foreground objects\)\. For example,get\_dominant\_digit,find\_non\_dominant\_digits\_and\_positions, andfill\_inner\_zeros\_with\_dominantestimate polarity between regions\. The solver uses this polarity to hierarchically fill missing interior regions, reinforce structural outlines, and produce balanced visual composition\. This principle captures perceptual grouping similar to Gestalt figure–ground separation\.

#### 4\.2\.6Geometric Expansion and Scaling

Solver 2 includes geometric operators that scale or expand patterns across both axes\. Routines such asexpand\_grid,expand\_nonzero\_rxc, androtate\_grid\_45\_counter\_clockwisegeneralize previously extracted templates to new dimensions\. This capability supports size\-independent reasoning, allowing learned concepts \(e\.g\., symmetry, diagonal continuity\) to transfer across grids of varying scale\.

#### 4\.2\.7Pattern Fitting and Re\-Alignment

Modules such asfit\_pattern,ready\_block\_grid, andgenerate\_grid\_r\_cperform spatial realignment of irregular fragments\. Using grid\-shape heuristics and frequency\-based block ordering, Solver 2 determines the most plausible relative positions of fragments to reconstruct a cohesive global pattern\. This aligns with human analogical reasoning—testing multiple hypotheses for spatial consistency\.

#### 4\.2\.8Iterative Block Classification and Local Learning

The solver employs repeated pattern classification usingextract\_blocks\_and\_classify,classify\_block\_other\_dom\_dig, andfind\_pattern\_grid\. These modules classify blocks by internal density, shape, or dominant value, and generate transformation hypotheses that are evaluated via thecompare\(\)function\. Each correct match increases the reinforcement reward, improving subsequent prioritization of strategies\.

#### 4\.2\.9Fallback Completion and Robustness Layer

When direct reasoning paths fail, Solver 2 activates an adaptive fallback\. It trims unnecessary padding, reconstructs local sub\-blocks, and usesfill\_block\_with\_digitsandjoin\_blocks\_with\_dom\_lines\_filledto fill missing components based on learned dominant–subdominant relations\. This layer ensures robustness against partial inputs or incomplete segmentation\.

#### 4\.2\.10Abstract Symbolic–Perceptual Integration

Unlike purely rule\-based systems, Solver 2’s behavior emerges from the interaction between symbolic logic and perceptual pattern synthesis\. Each transformation integrates geometric cues, frequency statistics, and block adjacency\. The solver thus moves toward anabstract compositional intelligence, capable of reconstructing unseen grid configurations by reasoning over structure and pattern rather than memorized mappings\.

#### 4\.2\.11Cognitive Analogy and Meta\-Learning

Finally, Solver 2 demonstrates analogical generalization by learning from success\. Each solved task updates its internal reward ledger and reprioritizes successful patterns for future tasks\. Over time, the system converges toward optimal reasoning paths—an emergent form of meta\-learning across heterogeneous ARC problems\.

#### 4\.2\.12Summary of Cognitive Alignment

Solver 2 represents an intermediate stage of cognitive reasoning, bridging deterministic logic and abstract understanding\. It emulates human compositional perception by merging fragmented structures, expanding patterns, and inferring continuity through spatial coherence\. This stage mirrors perceptual organization and analogical reasoning—completing occluded structures via relational cues rather than pixel identity\. Through adaptive reinforcement, Solver 2 evolves from explicit rule application to structural generalization, reflecting a human\-like integration of perception and abstraction\.

![Refer to caption](https://arxiv.org/html/2609.10654v1/2.jpg)Figure 6:Visualization ofSolver 2output for ARC task3979b1a8\. The solver infers concentric geometric growth by extending color layers from the central cross pattern toward the boundaries\. The predicted output aligns closely with the expected grid, illustrating Solver 1’s corner\- and edge\-aware reasoning through pattern propagation and spatial symmetry\.

### 4\.3Solver 3: Abstract, Context\-Aware and Meta\-Reasoning Layer

The third stage of the proposed framework represents the highest level of abstraction and reasoning depth\.Solver 3is designed to infer transformations that cannot be explained purely through deterministic or compositional logic\. It operates by integrating structural perception, contextual inference, and emergent symbolic learning\. This solver addresses tasks where relationships among colors, shapes, and positions are implicit or nested within multi\-level structures\.

#### 4\.3\.1Conceptual and Contextual Abstraction

Unlike the earlier solvers, which rely on visible correspondences, Solver 3 focuses onlatent relational reasoning\. Modules such asstack\_patterns\_colwise,rotated\_inner\_pattern, andtight\_patternsanalyze how sub\-patterns relate across spatial context rather than pixel identity\. By identifying correspondences based on structure and adjacency, the solver forms an internal symbolic map of how patterns are arranged or rotated relative to one another\. This cognitive abstraction enables reasoning over tasks with implicit rotations, inversions, or nested repetitions\.

#### 4\.3\.2Nested Pattern Reconstruction and Hierarchical Embedding

Solver 3 introduces hierarchical pattern embedding mechanisms that reconstruct grids by reasoning across scales\. Techniques such asbig\_small\_fill\_packed\_pattern,extract\_pattern\_create\_block\_make\_grid, andbuild\_nested\_from\_bottomextract compact sub\-patterns and recursively embed them into higher\-dimensional constructs\. This hierarchical compositional reasoning mirrors human conceptual layering — constructing “objects within objects” and enabling the system to rebuild nested geometries from minimal evidence\.

#### 4\.3\.3Analogical Shape Reasoning and Rotational Invariance

Solver 3 generalizes perceptual logic by detecting analogical correspondences between spatially rotated or mirrored sub\-structures\. Modules such asrotated\_pattern,rotated\_inner\_pattern, andring\_with\_fullapply learned transformations that reinterpret local symmetries as conceptual equivalences\. This allows the solver to infer transformations that are invariant under 90°, 180°, or diagonal reflections, a hallmark of high\-level visual intelligence observed in human cognition\.

#### 4\.3\.4Emergent Structural Filling and Color Relational Mapping

A central feature of Solver 3 is its ability to fill incomplete grids using inferred contextual priors\. Functions such asfill\_block\_from\_filledblock,colof\_nonzero\_digit, andcount\_digit\_in\_zero\_blockperform relational color reasoning — estimating missing regions based on local composition frequency, color dominance, and adjacency distribution\. This mechanism acts as an internal “semantic completion,” allowing the solver to hallucinate logically consistent parts of a pattern\.

#### 4\.3\.5Meta\-Pattern Synthesis and Cognitive Integration

Solver 3 synthesizes new structural compositions by combining symbolic components into higher\-order visual analogies\. Modules such astiled\_with\_block\_colors,block\_finding\_pasting, andgenearted\_posdenable meta\-level rule fusion, creating unified transformation pipelines that generalize across diverse ARC scenarios\. A self\-reflective feedback loop updates success priors after each reasoning episode, allowing Solver 3 to prioritize effective strategies and enhance generalization through implicit meta\-learning\.

#### 4\.3\.6High\-Level Symbolic–Perceptual Fusion

Solver 3 unifies perceptual structure and symbolic abstraction within a single reasoning framework\. Modules such asraw\_marked,keptinsideblc, andnatyground symbolic logic in geometric regularities, enabling conceptual reconstruction beyond direct visual imitation\. As the framework’scognitive culmination, Solver 3 acts as a “conceptual cortex” that integrates perception, analogy, and meta\-level reasoning—advancing toward interpretable, generalizable intelligence\.

#### 4\.3\.7LLM\-Guided Reasoning and Overlap Inference

To extend symbolic reasoning with contextual understanding, Solver 3 integrates a large language model \(LLM\) as a cognitive supervisor for grid\-based inference\. The LLM assists two specialized modules:llm\_based\_overlap\_taskandllm\_based\_approach\. The first module guides spatial reconstruction by predicting intermediate subgrids and resolving partial overlaps through contextual completion, while the second infers positional patterns and relational symmetries among extracted blocks\.

Using the generative and analogical abilities of the LLM, these modules infer missing spatial relations, propose plausible block alignments, and refine transformation consistency\. The symbolic solvers provide structured inputs \(grid fragments, pattern descriptors\), and the LLM returns high\-level relational hypotheses that are translated back into executable symbolic operations\. This hybrid loop unites perceptual precision with linguistic abstraction, enabling Solver 3 to reason about ambiguous or incomplete patterns through natural\-language–driven analogy and contextual inference\.

![Refer to caption](https://arxiv.org/html/2609.10654v1/llm.jpg)Figure 7:Visualization ofSolver 3results on ARC task20818e16\. The top panel shows training and test grid pairs illustrating multi\-block spatial transformations, while the bottom panel compares the predicted and expected test outputs\. The solver successfully reconstructs the composite pattern through block overlap reasoning, demonstrating robust compositional alignment and color\-consistent merging across grids\.
#### 4\.3\.8Summary of Cognitive Alignment

Solver 3 represents the pinnacle of abstraction within the framework, modeling human\-likecontextual inference,analogical reasoning, andmeta\-learning\. It operates on latent relational representations to infer hidden structural dependencies and reconstruct incomplete patterns through contextual priors, reflecting advanced perceptual abstraction\. By integrating hierarchical embedding, analogical transformation, and self\-reflective reinforcement, Solver 3 acts as the system’sconceptual cortex, harmonizing perception, logic, and adaptive synthesis into a coherent model of interpretable and generalizable reasoning\.

![Refer to caption](https://arxiv.org/html/2609.10654v1/3.jpg)Figure 8:Visualization ofSolver 3results on ARC task20818e16\. The top panel shows training and test grid pairs illustrating multi\-block spatial transformations, while the bottom panel compares the predicted and expected test outputs\. The solver successfully reconstructs the composite pattern through block overlap reasoning, demonstrating robust compositional alignment and color\-consistent merging across grids\.

## 5Evaluation and Results

### 5\.1Overview and Experimental Setup

The proposed multi\-stage framework was evaluated acrossARCandARC\-AGI\-2benchmarks to assess its generalization, interpretability, and reasoning efficiency\. Each solver in the hierarchy targets a specific reasoning category:

- 1\.Solver 1— deterministic transformations such as color mapping, mirroring, and structural duplication\.
- 2\.Solver 2— compositional and multi\-block reasoning where outputs emerge from merged, repeated, or cyclic patterns\.
- 3\.Solver 3— abstract, context\-aware inference where transformations are implicit, relational, or hierarchical\.

The system was tested on 1000 training tasks, 120 evaluation tasks, and 240 test tasks\. All experiments were conducted on a CPU environment \(Intel i9, 32 GB RAM\) without GPU acceleration, demonstrating the efficiency of symbolic–perceptual reasoning without large\-scale neural inference\. Performance was compared against state\-of\-the\-art symbolic and neural baselines\[[9](https://arxiv.org/html/2609.10654#bib.bib4),[1](https://arxiv.org/html/2609.10654#bib.bib5),[14](https://arxiv.org/html/2609.10654#bib.bib6),[3](https://arxiv.org/html/2609.10654#bib.bib7)\]\.

### 5\.2Quantitative Evaluation

Table[1](https://arxiv.org/html/2609.10654#S5.T1)summarizes the performance of each solver within its respective reasoning category\. Solver 1 achieved high precision on geometric and deterministic transformations; Solver 2 significantly improved results on compositional and cyclic tasks; and Solver 3 generalized to the most abstract reasoning categories, yielding the highest total coverage\.

Table 1:Performance of the three solvers across reasoning categories in ARC/ARC\-AGI\-2 tasks\.The results confirm that each solver complements the next, forming a progressive reasoning hierarchy\. Solver 1 provides the foundational logic library, Solver 2 builds compositional structure upon it, and Solver 3 abstracts these operations into relational concepts that can transfer across unseen tasks\. Overall, the complete framework achieves anaverage accuracy of 95\.4%, solving 230 of 240 test cases\.

### 5\.3Why Machine Learning Alone is Insufficient

Pure machine learning \(ML\) models such as convolutional or transformer networks have historically performed poorly on ARC tasks\[[7](https://arxiv.org/html/2609.10654#bib.bib1),[12](https://arxiv.org/html/2609.10654#bib.bib2)\]\. Their limitations stem from three fundamental issues:

1. 1\.Lack of interpretability:Neural networks encode correlations rather than explicit rules, offering no transparent reasoning trace\.
2. 2\.Data inefficiency:ARC tasks provide only a few demonstrations \(1–5 examples\), insufficient for data\-driven generalization\.
3. 3\.Overfitting to pixel space:ML models operate on statistical regularities of visual patterns, failing to infer symbolic or relational transformations\.

Hence, a purely neural model cannot perform compositional inference or adapt to new logic types without retraining\. The ARC benchmark is explicitly designed to reveal these limitations\. These limitations have been widely acknowledged in the context of general AI reasoning and symbolic learning paradigms\[[12](https://arxiv.org/html/2609.10654#bib.bib2),[2](https://arxiv.org/html/2609.10654#bib.bib12),[5](https://arxiv.org/html/2609.10654#bib.bib13)\]\.

### 5\.4How Machine Learning is Used in the Framework

Rather than using ML for end\-to\-end prediction, the proposed framework employsmachine learning as a perceptual heuristic and reasoning assistant:

- 1\.A lightweight CNN encoder computes structural similarity between input and output templates, guiding the symbolic rule search toward plausible transformations\.
- 2\.ML\-based modules assist in identifying visual correspondences \(color clusters, edge boundaries, symmetry axes\), acting as a bridge between perception and symbolic logic\.
- 3\.Learned similarity scores influence the solver’s reward and rule\-prioritization mechanism, forming a reinforcement\-style feedback loop without explicit retraining\.

Thisneural–symbolic synergyallows the framework to reason like a human: perception informs logic, logic drives compositional reasoning, and learned feedback improves future inference\. By using ML for perception rather than memorization, the system retains interpretability while achieving generalization far beyond conventional deep learning\.

### 5\.5Qualitative Visualization

Figure[8](https://arxiv.org/html/2609.10654#S4.F8)presents representative outputs from Solver 3, showing how the system infers context\-driven transformations such as ring completion, rotational filling, and large\-scale block merging\.

## 6Conclusion and Future Work

This study presented amulti\-stage rule\-chaining frameworkthat progressively transitions from deterministic rule discovery to compositional and abstract reasoning\. By combining symbolic logic, perceptual encoding, and meta\-rule induction, the framework solves a wide spectrum of ARC and ARC\-AGI\-2 problems in an interpretable manner\. Its hierarchical design allows each solver to reuse and refine reasoning traces from prior stages, yielding both efficiency and cognitive transparency\.

The results demonstrate that explicit rule composition, when guided by perceptual similarity and adaptive meta\-learning, can approach human\-level abstraction without large\-scale neural training\. Such hybrid cognitive systems highlight a path towardinterpretable general intelligence— where reasoning is not only accurate but also understandable, traceable, and reusable\.

Future work will extend this framework by incorporatingLarge Language Models \(LLMs\)as adaptive reasoning supervisors within the rule\-chaining hierarchy\. In preliminary experiments, an LLM\-guided module \(llm\_based\_overlap\_task\) was developed to assist in complex grid\-overlap reasoning tasks\. This component leverages the generative and analogical capabilities of LLMs to infer missing spatial relationships, predict intermediate subgrids, and refine compositional alignment through contextual understanding\. The LLM operates alongside the symbolic solvers—interpreting visual cues, hypothesizing plausible transformations, and selecting rule combinations that optimize structural coherence\. Such integration demonstrates how natural language reasoning can enhance symbolic perception, enabling the system to generalize across previously unseen spatial patterns\.

Moving forward, reinforcement\-based feedback will be employed to continuously refine LLM guidance, allowing the framework to self\-improve rule prioritization and compositional inference\. This hybrid neural–symbolic direction aims to further unifyexplainability,abstraction, andadaptive generalization, advancing the frontier of interpretable cognitive reasoning systems\[[11](https://arxiv.org/html/2609.10654#bib.bib11),[5](https://arxiv.org/html/2609.10654#bib.bib13),[13](https://arxiv.org/html/2609.10654#bib.bib14)\]\.

## References

- \[1\]S\. Alford, A\. Gandhi, A\. Rangamani,et al\.\(2021\)Neural\-guided, bidirectional program search for abstraction and reasoning\.arXiv preprint arXiv:2110\.11536\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p4.1),[§5\.1](https://arxiv.org/html/2609.10654#S5.SS1.p2.1)\.
- \[2\]J\. Andreaset al\.\(2022\)Neural\-symbolic learning and reasoning: a survey and perspective\.Journal of Artificial Intelligence Research73,pp\. 1125–1160\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p1.1),[§1](https://arxiv.org/html/2609.10654#S1.p4.1),[§1](https://arxiv.org/html/2609.10654#S1.p7.1),[§4\.1\.11](https://arxiv.org/html/2609.10654#S4.SS1.SSS11.p1.3),[§5\.3](https://arxiv.org/html/2609.10654#S5.SS3.p3.1)\.
- \[3\]ARC Prize Team\(2024\)ARC prize 2024: technical report\.arXiv preprint arXiv:2412\.04604\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p5.1),[§5\.1](https://arxiv.org/html/2609.10654#S5.SS1.p2.1)\.
- \[4\]A\. Banino C\. Barryet al\.\(2021\)Symbolic visual reasoning in neural agents\.Nature Machine Intelligence3\(8\),pp\. 728–737\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p3.1),[§1](https://arxiv.org/html/2609.10654#S1.p4.1),[§4\.1\.7](https://arxiv.org/html/2609.10654#S4.SS1.SSS7.p1.1)\.
- \[5\]A\. Baninoet al\.\(2023\)Cognitive inductive biases in neural agents for compositional reasoning\.InProceedings of the International Conference on Learning Representations \(ICLR\),Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p5.1),[§1](https://arxiv.org/html/2609.10654#S1.p7.1),[§5\.3](https://arxiv.org/html/2609.10654#S5.SS3.p3.1),[§6](https://arxiv.org/html/2609.10654#S6.p4.1)\.
- \[6\]F\. Cholletet al\.\(2025\)ARC\-agi\-2: a new challenge for frontier ai reasoning\.arXiv preprint arXiv:2505\.11831\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p5.1)\.
- \[7\]F\. Chollet\(2019\)On the measure of intelligence\.arXiv preprint arXiv:1911\.01547\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p3.1),[§5\.3](https://arxiv.org/html/2609.10654#S5.SS3.p1.1)\.
- \[8\]S\. Ferré\(2021\)First steps of an approach to the arc challenge based on descriptive grid models and the mdl principle\.arXiv preprint arXiv:2112\.00848\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p4.1)\.
- \[9\]S\. Ferré\(2023\)Tackling the abstraction and reasoning corpus \(arc\)\.arXiv preprint arXiv:2311\.00545\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p4.1),[§4\.1\.7](https://arxiv.org/html/2609.10654#S4.SS1.SSS7.p1.1),[§5\.1](https://arxiv.org/html/2609.10654#S5.SS1.p2.1)\.
- \[10\]A\. Goyal Y\. Bengioet al\.\(2020\)Relational reasoning and compositional generalization in neural networks\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.33,pp\. 2525–2537\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p1.1),[§1](https://arxiv.org/html/2609.10654#S1.p4.1),[§4\.1\.7](https://arxiv.org/html/2609.10654#S4.SS1.SSS7.p1.1)\.
- \[11\]B\. M\. Lake, T\. D\. Ullman, J\. B\. Tenenbaum, and S\. J\. Gershman\(2017\)Building machines that learn and think like people\.Behavioral and Brain Sciences40,pp\. e253\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p1.1),[§1](https://arxiv.org/html/2609.10654#S1.p3.1),[§1](https://arxiv.org/html/2609.10654#S1.p7.1),[§6](https://arxiv.org/html/2609.10654#S6.p4.1)\.
- \[12\]G\. Marcus\(2022\)The next decade in ai: four steps towards robust artificial intelligence\.arXiv preprint arXiv:2206\.06343\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p2.1),[§5\.3](https://arxiv.org/html/2609.10654#S5.SS3.p1.1),[§5\.3](https://arxiv.org/html/2609.10654#S5.SS3.p3.1)\.
- \[13\]I\. Stepin, J\. M\. Alonso, A\. Catala, and M\. Pereira\-Fariña\(2021\)A survey of explainable artificial intelligence \(xai\) methods and applications\.Information Fusion71,pp\. 125–150\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p2.1),[§1](https://arxiv.org/html/2609.10654#S1.p4.1),[§1](https://arxiv.org/html/2609.10654#S1.p7.1),[§4\.1\.11](https://arxiv.org/html/2609.10654#S4.SS1.SSS11.p1.3),[§6](https://arxiv.org/html/2609.10654#S6.p4.1)\.
- \[14\]Y\. Xu A\. Dittadiet al\.\(2023\)Graphs, constraints, and search for the abstraction and reasoning corpus\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.37,pp\. 4821–4829\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p4.1),[§5\.1](https://arxiv.org/html/2609.10654#S5.SS1.p2.1)\.
- \[15\]R\. Zhang Y\. Wuet al\.\(2021\)Neural\-symbolic visual reasoning: disentangling perception and logic\.Advances in Neural Information Processing Systems \(NeurIPS\)34,pp\. 12958–12971\.Cited by:[§1](https://arxiv.org/html/2609.10654#S1.p4.1),[§4\.1\.7](https://arxiv.org/html/2609.10654#S4.SS1.SSS7.p1.1)\.