L-Convex Polyominoes are Recognizable in Real Time by 2D Cellular Automata

TL;DR

This paper demonstrates that L-convex polyominoes, a specific class of geometric shapes, can be recognized in real time by 2D cellular automata, highlighting their computational efficiency and advancing understanding of shape recognition.

Contribution

It proves that the recognition of L-convex polyominoes by tiling systems can be achieved by 2D cellular automata in real time, introducing techniques specific to cellular automata.

Findings

L-convex polyominoes are recognizable in real time by 2D cellular automata.

The recognition uses a characterization similar to tiling systems but adapted for cellular automata.

Real time recognition imposes unique constraints requiring specialized techniques.

Abstract

A polyomino is said to be L-convex if any two of its cells are connected by a 4-connected inner path that changes direction at most once. The 2-dimensional language representing such polyominoes has been recently proved to be recognizable by tiling systems by S. Brocchi, A. Frosini, R. Pinzani and S. Rinaldi. In an attempt to compare recognition power of tiling systems and cellular automata, we have proved that this language can be recognized by 2-dimensional cellular automata working on the von Neumann neighborhood in real time. Although the construction uses a characterization of L-convex polyominoes that is similar to the one used for tiling systems, the real time constraint which has no equivalent in terms of tilings requires the use of techniques that are specific to cellular automata.