Degrees of Restriction for Two-Dimensional Automata

TL;DR

This paper introduces and studies hierarchical models of two-dimensional automata with restricted head movements, establishing recognition hierarchies and comparing deterministic and nondeterministic variants.

Contribution

It develops the concept of degrees of restriction for 2D automata, creating extended models with bounded restricted moves and analyzing their recognition power.

Findings

Recognition hierarchies for extended three-way automata

Recognition hierarchies for extended two-way automata

Incomparability between deterministic and nondeterministic models

Abstract

A three-way (resp., two-way) two-dimensional automaton has a read-only input head that moves in three (resp., two) directions on a finite array of cells labelled by symbols of the input alphabet. Restricting the input head movement of a two-dimensional automaton results in a model that is weaker in terms of recognition power. In this paper, we introduce the notion of "degrees of restriction" for two-dimensional automata, and we develop sets of extended two-dimensional automaton models that allow for some bounded number of restricted moves. We establish recognition hierarchies for both deterministic and nondeterministic extended three-way two-dimensional automata, and we find similar hierarchies for both deterministic and nondeterministic extended two-way two-dimensional automata. We also prove incomparability results between nondeterministic and deterministic extended three-way two-dimensional automata. Lastly, we consider closure properties for some operations on languages recognized by extended three-way two-dimensional automata.