Descriptive complexity for pictures languages (extended abstract)

TL;DR

This paper explores the descriptive complexity of picture languages across any dimension using logical fragments, providing uniform characterizations of recognizable languages and cellular automata complexity classes.

Contribution

It generalizes existing characterizations to any dimension and offers the first machine-independent logical characterizations of cellular automata complexity classes.

Findings

Logical characterizations of recognizable picture languages

First machine-independent characterizations of cellular automata complexity

Hierarchy results showing the limits of logical characterizations

Abstract

This paper deals with descriptive complexity of picture languages of any dimension by syntactical fragments of existential second-order logic. - We uniformly generalize to any dimension the characterization by Giammarresi et al. \cite{GRST96} of the class of \emph{recognizable} picture languages in existential monadic second-order logic. - We state several logical characterizations of the class of picture languages recognized in linear time on nondeterministic cellular automata of any dimension. They are the first machine-independent characterizations of complexity classes of cellular automata. Our characterizations are essentially deduced from normalization results we prove for first-order and existential second-order logics over pictures. They are obtained in a general and uniform framework that allows to extend them to other "regular" structures. Finally, we describe some hierarchy results that show the optimality of our logical characterizations and delineate their limits.