A logical approach to locality in pictures languages

This paper deals with descriptive complexity 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 is the simplest and most natural model of parallel computation and linear time is their minimal time class allowing synchronization. We uniformly generalize to any dimension the characterization by Giammarresi et al. (“Monadic SecondOrder Logic over Rectangular Pictures and Recognizability by Tiling Systems”, Inf. and Comput. 125(1): 32–45, 1996) 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 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. These results show that in some sense the logics involved can be made “local” with respect to the underlying regular structures. Finally, we describe some hierarchy results that show the optimality of our logical characterizations and delineate their limits.