Descriptive complexity for pictures languages

This paper deals with logical characterizations of picture languages of any dimension by syntactical fragments of existential second-order logic. Two classical classes of picture languages are studied: the class of recognizable picture languages, i.e. projections of languages defined by local constraints (or tilings): it is known as the most robust class extending the class of regular languages to any dimension; the class of picture languages recognized on nondeterministic cellular automata in linear time : cellular automata are the simplest and most natural model of parallel computation and linear time is the minimal time-bounded class allowing synchronization of nondeterministic cellular automata. We uniformly generalize to any dimension the characterization by Giammarresi et al. [7] of the class of 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. They are the first machine-independent characterizations of complexity classes of cellular automata. Our characterizations are essentially deduced from normalization results we prove for firstorder 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. 1998 ACM Subject Classification F.1.1 Models of Computation, F.1.3 Complexity Measures and Classes, F.4.3 Formal Languages