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On the Exact Algorithmic Extraction of Finite Tesselations Through Prime Extraction of Minimal Representative Forms

Sushish Baral, Paulo Garcia, Warisa Sritriratanarak

TL;DR

This paper proposes a hierarchical algorithm that discovers exact tessellations in finite planar grids, addressing the problem where multiple independent patterns may coexist within a hierarchical structure and provides deterministic behavior for exact, axis-aligned, rectangular tessellations.

Abstract

The identification of repeating patterns in discrete grids is rudimentary within symbolic reasoning, algorithm synthesis and structural optimization across diverse computational domains. Although statistical approaches targeting noisy data can approximately recognize patterns, symbolic analysis utilizing deterministic extraction of periodic structures is underdeveloped. This paper aims to fill this gap by employing a hierarchical algorithm that discovers exact tessellations in finite planar grids, addressing the problem where multiple independent patterns may coexist within a hierarchical structure. The proposed method utilizes composite discovery (dual inspection and breadth-first pruning) for identifying rectangular regions with internal repetition, normalization to a minimal representative form, and prime extraction (selective duplication and hierarchical memoization) to account for irregular dimensions and to achieve efficient computation time. We evaluate scalability on grid sizes from 2x2 to 32x32, showing overlap detection on simple repeating tiles exhibits processing time under 1ms, while complex patterns which require exhaustive search and systematic exploration shows exponential growth. This algorithm provides deterministic behavior for exact, axis-aligned, rectangular tessellations, addressing a critical gap in symbolic grid analysis techniques, applicable to puzzle solving reasoning tasks and identification of exact repeating structures in discrete symbolic domains.

On the Exact Algorithmic Extraction of Finite Tesselations Through Prime Extraction of Minimal Representative Forms

TL;DR

This paper proposes a hierarchical algorithm that discovers exact tessellations in finite planar grids, addressing the problem where multiple independent patterns may coexist within a hierarchical structure and provides deterministic behavior for exact, axis-aligned, rectangular tessellations.

Abstract

The identification of repeating patterns in discrete grids is rudimentary within symbolic reasoning, algorithm synthesis and structural optimization across diverse computational domains. Although statistical approaches targeting noisy data can approximately recognize patterns, symbolic analysis utilizing deterministic extraction of periodic structures is underdeveloped. This paper aims to fill this gap by employing a hierarchical algorithm that discovers exact tessellations in finite planar grids, addressing the problem where multiple independent patterns may coexist within a hierarchical structure. The proposed method utilizes composite discovery (dual inspection and breadth-first pruning) for identifying rectangular regions with internal repetition, normalization to a minimal representative form, and prime extraction (selective duplication and hierarchical memoization) to account for irregular dimensions and to achieve efficient computation time. We evaluate scalability on grid sizes from 2x2 to 32x32, showing overlap detection on simple repeating tiles exhibits processing time under 1ms, while complex patterns which require exhaustive search and systematic exploration shows exponential growth. This algorithm provides deterministic behavior for exact, axis-aligned, rectangular tessellations, addressing a critical gap in symbolic grid analysis techniques, applicable to puzzle solving reasoning tasks and identification of exact repeating structures in discrete symbolic domains.
Paper Structure (15 sections, 2 theorems, 8 figures, 4 tables, 4 algorithms)

This paper contains 15 sections, 2 theorems, 8 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Motivating Example
  4. Preliminaries
  5. Hierarchical Composite and Prime Discovery
  6. Definitions
  7. High-Level Methodology
  8. Algorithm Description
  9. Experimental Results
  10. Test Cases
  11. Decomposition Results
  12. Scalability Analysis
  13. Discussion
  14. Related Work
  15. Conclusion

Key Result

Lemma 2.2

If $\mathbf{G}$ is of size $n\times m$, there are $\frac{nm((n+1)(m+1)-4)}{4}$ possible rectangular bounding boxes in $\mathbf{G}$.

Figures (8)

  • Figure 1: ARC tasks; reproduced from https://arcprize.org/play?task=0a938d79
  • Figure 2: Example grid and identified tesselations, with connections to respective tiles.
  • Figure 3: BFS pruning tree for composite discovery showing node status by color: green (continue), pink/red (tessellation found), orange (duplicate), light blue (near minimum), dark red (empty), purple (prime). Three composites discovered at bottom.
  • Figure 4: Normalization process reducing three composites (Tc, pink) to minimal forms (Tc', purple) through iterative row/column halving with duplication.
  • Figure 5: Prime extraction BFS tree with overlap detection and normalization. Colors indicate: purple (primes), orange (duplicates), dark red (empty). Bottom panel shows level-wise organization (Levels 1-4).
  • ...and 3 more figures

Theorems & Definitions (6)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Corollary 2.4
  • Definition 2.5