A logical approach to locality in pictures languages

This paper deals with descriptive complexity of picture languages of any dimension by fragments of existential second-order logic: 1) We generalize to any dimension the characterization by Giammarresi et al. (1996) of the class of recognizable picture languages in existential monadic second-order logic. 2) We state natural logical characterizations of the class of picture languages of any dimension d≥1 recognized in linear time on nondeterministic cellular automata, a robust complexity class that contains, for d=1, all the natural NP-complete problems. Our characterizations are essentially deduced from normalization results for first-order and existential second-order logics over pictures. •The class of recognizable picture languages is characterized by existential monadic second-order logic.•Logical characterizations of the class of picture languages recognized in linear time on nondeterministic cellular automata.•This shows the logics involved can be made “local” with respect to the underlying regular structures.•Hierarchy results among some logical frameworks are deduced from the above.