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

Deblina Kar

arXiv:2609.10654v1cs.AIcs.LG

TL;DR

ARC asks whether systems can infer abstract rules from limited examples and transfer them to unseen grid tasks. This paper proposes a three-stage rule-chaining framework that combines symbolic induction, perceptual pattern synthesis, and structural abstraction, achieving 95.4% average accuracy while solving 230 of 240 test cases. The authors present the architecture as an interpretable route toward human-aligned abstraction without task-specific tuning.

  • Problem

    ARC probes the challenge of generalizing abstract and compositional rules from few examples, a capability that remains difficult for systems relying on statistical fitting.

  • Method

    The framework sequentially combines deterministic rule discovery, compositional pattern synthesis, and context-aware structural abstraction, reusing reasoning traces across stages.

  • Results

    95.4% average accuracy was reported, with 230 of 240 test cases solved across the complete framework.

  • Takeaways & Limitations

    The results support explicit rule composition and perceptually guided hybrid reasoning as a path toward interpretable, traceable, and reusable artificial reasoning.

Abstract

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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 percent. 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.

1. Introduction

ARC evaluates whether systems can infer abstract transformations from few examples and apply them to unseen grids. This paper proposes a three-stage rule-chaining framework combining deterministic induction, compositional reasoning, and structural abstraction.

  • 1. Introduction: ARC tests cognitive generalization by requiring rules inferred from input–output examples to be applied to unseen test inputs.Its grid puzzles emphasize symmetry, color transformation, shape manipulation, and structural composition while minimizing statistical cues.
  • 1. Introduction: The benchmark remains largely unsolved because symbolic systems face combinatorial and grounding challenges while neural systems lack interpretability and systematic generalization.
  • 1. Introduction: ARC-AGI-2 expands task complexity to evaluate structured reasoning, symbolic interpretability, adaptive generalization, and multi-level rule chaining.
  • 1. Introduction: The framework integrates deterministic rule discovery, compositional pattern reconstruction, and structural abstraction for progressively higher levels of reasoning.The stages identify geometric, color, and spatial mappings; recombine grid sub-blocks; and capture nested, hierarchical, and relational patterns.
  • 1. Introduction: Evaluated on more than 230 ARC and ARC-AGI-2 tasks, the framework reports robust generalization across deterministic, compositional, and abstract reasoning categories.The system operates through a progressive fallback hierarchy that reuses earlier insights and partial rules.

2. Data Organization and Feature Extraction

The study organizes ARC tasks into standardized partitions and tabular representations containing raw grids and derived geometric, shape, and color attributes. These representations support downstream strategy selection and rule induction.

  • 2. Data Organization and Feature Extraction: The dataset is divided into 1000 training tasks, 120 evaluation tasks, and 240 test tasks.
  • 2. Data Organization and Feature Extraction: Each task stores training input–output pairs and test inputs in JSON, with the model predicting missing test-output grids.
  • 2. Data Organization and Feature Extraction: ARC grids are represented as two-dimensional arrays of integers from 0 to 9, with each integer denoting a color index.
  • 2. Data Organization and Feature Extraction: Figures 1 and 2 respectively visualize processed task features and representative training–test input–output structures.
  • 2. Data Organization and Feature Extraction: Derived features encode dimensions, area, binary shape correspondence, and color consistency between input and output grids.
  • 2. Data Organization and Feature Extraction: These structured attributes provide the foundation for rule induction and solver selection based on visual and symbolic similarity.

3. Framework Overview

The framework uses three progressively intelligent solvers to move from deterministic rule discovery through compositional reasoning to context-aware abstraction. This hierarchy is designed to combine interpretability with generalization.

  • 3. Framework Overview: The framework transitions from deterministic rule discovery to compositional reasoning and finally to context-aware abstraction across three solvers.Each solver addresses a different level of abstraction in complex transformation-based visual reasoning tasks.

Solver 1 – Deterministic Rule Discovery and Logic Reasoning

Solver 1 learns direct input–output transformations through explicit rule extraction, feature mapping, template extraction, and adaptive logic detection. It is intended for deterministic or geometric relationships.

  • Solver 1 – Deterministic Rule Discovery and Logic Reasoning: Solver 1 resolves deterministic or geometric tasks by extracting explicit rules and mapping features between input and output grids.It combines template extraction, symbolic rule generation, and a CNN-based matching layer for spatial and color transformations.

Solver 2 – Structural and Compositional Generalization

Solver 2 extends Solver 1’s rule set to structural decomposition, block composition, pattern alignment, and spatial correspondence for more complex compositional tasks.

  • Solver 2 reconstructs outputs through block composition, pattern alignment, and spatial correspondence inference.
  • It extends Solver 1 to multi-block and symmetry-driven problems, enabling broader compositional generalization.

Solver 3 – Abstract and Context-Aware Reasoning

Solver 3 completes the architecture by using abstraction-driven reasoning to infer semantic and relational dependencies in unseen or irregular grid transformations.

  • Solver 3 uses pattern abstraction, context embedding, and meta-rule induction to reason about unseen transformations and irregular compositions.
  • Each solver follows feature extraction, symbolic rule generation and evaluation, and adaptive logic selection to produce an output grid.

4. Methodology

The methodology combines perceptual encoding, symbolic rule induction, structural composition, sequential meta-rule chaining, fallback recovery, and reasoning traces within a hybrid architecture.

  • Solver 1 combines perception, symbolic logic, and composition to infer deterministic, interpretable transformations through multiple reasoning phases.
  • It extracts canonical templates from connected regions, geometric boundaries, and textures, then compares input–output regions as structured entities.
  • Candidate reflection, rotation, duplication, and substitution rules are evaluated using pixel overlap, geometric alignment, and perceptual similarity.
  • A CNN guides symbolic search through region-level correspondence rather than directly predicting outputs, reducing combinatorial search complexity.
  • Object-level transformations and spatial composition reconstruct translated, rotated, mirrored, tiled, or symmetric arrangements from discrete grid objects.
  • Meta-rule induction composes transformations such as “rotate →fill →mirror” or “crop →duplicate →recolor” into coherent multi-step pipelines.
  • A hierarchical fallback system applies visual correction, spatial reconstruction, and partial-solution fusion when perfect rule matches fail.
  • Detailed reasoning traces record which transformations were applied, their order, and why they were selected, supporting transparency and reproducibility.

4.2. Solver 2: Compositional and Pattern-Driven Generalization

Solver 2 performs higher-order compositional reasoning by merging blocks, generating geometric continuations, aligning patterns, and adapting fallback strategies through symbolic–perceptual integration.

  • It detects nested or overlapping sub-blocks and merges them into larger outputs while preserving spatial and color coherence.
  • Dynamic geometry operators generate spiral, ring, rotational, and concentric continuations guided by neighborhood continuity and grid topology.
  • Pattern overlays and contextual blending superimpose related substructures, using color dominance and contextual priority to resolve overlaps.
  • Expansion and rotation routines transfer extracted concepts such as symmetry and diagonal continuity across grids of varying scale.
  • Spatial realignment and repeated block classification generate and evaluate hypotheses for reconstructing cohesive global patterns.
  • Adaptive fallback trims padding, reconstructs local blocks, and fills missing components when direct reasoning paths fail.
  • Solver 2 combines symbolic logic and perceptual pattern synthesis to reconstruct unseen grid configurations from structure and pattern rather than memorized mappings.
  • Successful tasks update a reward ledger that reprioritizes effective patterns for future reasoning paths.

4.3. Solver 3: Abstract, Context-Aware and Meta-Reasoning Layer

Solver 3 performs the framework’s highest-level reasoning by inferring latent, hierarchical, and contextual relationships rather than relying solely on visible pixel correspondences. It combines structural perception, analogical transformations, contextual completion, and symbolic feedback to reconstruct abstract or incomplete grid patterns.

  • Abstract contextual inference: Solver 3 infers implicit or nested relationships among colors, shapes, and positions through structural perception, contextual inference, and emergent symbolic learning.It targets transformations that deterministic and compositional logic alone cannot explain.
  • Analogical transformation: Analogical modules treat rotated, mirrored, and adjacent sub-structures as relationally corresponding patterns rather than matching pixels directly.The framework applies transformations invariant under 90°, 180°, or diagonal reflection.
  • Hierarchical pattern reasoning: Hierarchical embedding recursively extracts compact sub-patterns and embeds them into larger constructs, enabling reconstruction of nested geometries across scales.The described mechanisms build “objects within objects” from minimal structures.
  • Contextual completion: Contextual completion fills missing grid regions using inferred color dominance, composition frequency, and adjacency distributions.These priors estimate logically consistent missing parts from local structure.
  • LLM-guided reasoning: An LLM-guided hybrid loop proposes intermediate subgrids, block alignments, and relational hypotheses that symbolic solvers translate into executable operations.The symbolic modules provide grid fragments and descriptors, while the LLM supplies contextual and analogical guidance.
  • Representative reconstruction: Figure 7’s grid pairs and predicted-versus-expected outputs illustrate composite reconstruction through block overlap and color-consistent merging.The visualization concerns ARC task 20818e16 and emphasizes compositional alignment across grids.

5. Evaluation and Results

The evaluation assigns each solver to a distinct reasoning category and reports complementary performance across deterministic, compositional, and abstract tasks. The complete framework reaches 95.4% average accuracy, solving 230 of 240 test cases, while using machine learning primarily as perceptual assistance rather than end-to-end prediction.

  • Evaluation design: Solver 1 targets deterministic transformations, Solver 2 handles compositional and multi-block patterns, and Solver 3 addresses implicit, relational, or hierarchical transformations.The evaluation organizes the hierarchy by the reasoning category each solver is designed to cover.
  • Evaluation setup: The system was tested on 1000 training, 120 evaluation, and 240 test tasks, with experiments run on an Intel i9 CPU and 32 GB of RAM without GPU acceleration.Performance was compared against symbolic and neural baselines.
  • Solver-level results: Solver 3 yielded the highest total coverage by generalizing to the most abstract reasoning categories.Table 1 summarizes solver performance across reasoning categories, with Solver 1 focused on deterministic tasks and Solver 2 on compositional and cyclic tasks.
  • Overall results: 95.4% average accuracy was achieved across the complete framework, which solved 230 of 240 test cases.The results describe the solvers as a progressive hierarchy in which later stages reuse and abstract earlier reasoning.
  • Neural–symbolic integration: Machine learning serves as a perceptual heuristic that guides symbolic search, identifies visual correspondences, and adjusts rule prioritization rather than producing end-to-end predictions.The framework uses a lightweight CNN encoder and learned similarity scores within a reinforcement-style feedback loop.
  • Interpretability rationale: The reported neural–symbolic synergy links perception, symbolic logic, compositional reasoning, and learned feedback while retaining interpretability.This design is presented as an alternative to pure neural approaches that struggle with transparent rules, few-shot data efficiency, and symbolic transformations.

6. Conclusion and Future Work

The paper concludes that a hierarchical rule-chaining framework can combine deterministic, compositional, and abstract reasoning with reusable reasoning traces. It presents explicit rule composition guided by perceptual similarity and adaptive meta-learning as a path toward interpretable abstraction, while future work extends LLM supervision and reinforcement-based refinement.

  • Conclusion: The framework progressively transitions from deterministic rule discovery to compositional and abstract reasoning by combining symbolic logic, perceptual encoding, and meta-rule induction.Its hierarchical design reuses and refines reasoning traces from earlier solver stages.
  • Conclusion: Explicit rule composition guided by perceptual similarity and adaptive meta-learning is reported to approach human-level abstraction without large-scale neural training.The conclusion frames this combination as a route toward reasoning that is accurate, understandable, traceable, and reusable.
  • Future work: Future work will incorporate LLMs as adaptive reasoning supervisors to infer missing spatial relationships, predict intermediate subgrids, and refine compositional alignment.The proposed LLM-guided overlap module is presented as a preliminary step within the rule-chaining hierarchy.
  • Future work: Reinforcement-based feedback is planned to continuously refine LLM guidance, rule prioritization, and compositional inference.This direction aims to unify explainability, abstraction, and adaptive generalization.
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