Subshifts defined by nondeterministic and alternating plane-walking automata

TL;DR

This paper extends plane-walking automata to include nondeterminism and alternation, showing these models recognize a strict subclass of sofic subshifts and establishing a hierarchy of recognized subshifts.

Contribution

It introduces nondeterministic and alternating variants of plane-walking automata and analyzes their expressive power in recognizing subshifts, establishing new hierarchies and incomparability results.

Findings

Recognized subshifts form a strict subclass of sofic subshifts.

Existential and universal nondeterminism classes are incomparable.

Deterministic class is strictly contained within nondeterministic classes.

Abstract

Plane-walking automata were introduced by Salo & T\"orma to recognise languages of two-dimensional infinite words (subshifts), the counterpart of $4$-way finite automata for two-dimensional finite words. We extend the model to allow for nondeterminism and alternation of quantifiers. We prove that the recognised subshifts form a strict subclass of sofic subshifts, and that the classes corresponding to existential and universal nondeterminism are incomparable and both larger that the deterministic class. We define a hierarchy of subshifts recognised by plane-walking automata with alternating quantifiers, which we conjecture to be strict.