Subshifts defined by nondeterministic and alternating plane-walking automata

2024-09-12

AI Summary

This paper investigates the recognition power of two-dimensional wandering automata for infinite two-dimensional word languages (i.e., subshifts), focusing on expressive boundaries under directional nondeterminism and rule alternation. We introduce, for the first time, a formal model incorporating both existential and universal alternation—yielding a strictly increasing alternating hierarchy. We prove that the existential and universal nondeterministic classes are incomparable and each strictly subsumes the deterministic class; moreover, the recognized subshifts form a proper subclass of sofic subshifts. Our primary contributions are: (i) the first two-dimensional automaton model with quantifier alternation; (ii) a precise characterization of a strict three-level hierarchy among deterministic, existential-nondeterministic, and universal-nondeterministic recognition; and (iii) the key conjecture that the alternating level induces an infinite strictly ascending hierarchy of subshift classes.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Intelligent Robots: Other Foundations and Applications
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Uncertain Direction Automata
• Subshift Recognition
• Automata Uncertainty Types

Innovation

• Augmented Planar Walking Automata
• Uncertainty Handling
• Subshift Classification