Characterizations of recognizable picture series

The theory of two-dimensional languages as a generalization of formal string languages was motivated by problems arising from image processing and pattern recognition and also concerns models of parallel computing. In the nineties, Restivo and Giammarresi defined the family REC of recognizable picture languages [GR97]. This family is very robust and has been characterized by many different devices, generalizing well-known equivalences of regular word languages. Several authors obtained an equivalence theorem for picture languages describing recognizable languages in terms of types of automata, sets of tiles, rational operations or existential monadic second-order (MSO) logic [BG05, GRST96, IN77, LS97]. Here we investigate picture series. These are functions that map pictures, i.e. arrays of symbols, over a finite alphabet to elements of a commutative semiring and provide an extension of two-dimensional languages to a quantitative setting. The model of a weighted (quadrapolic) picture automaton (WPA) was introduced by Bozapalidis and Grammatikopoulou [BG05]. WPA are automata operating in a natural way on pictures and whose transitions carry weights. The behavior or the computation of a WPA is a picture series. The family of picture series over the alphabet Σ and a commutative semiring K that are computable by WPA will be denoted by K〈〈Σ,WPA〉〉. Bozapalidis and Grammatikopoulou showed that picture series computed by WPA are closed under certain operations and projections on series. These operations on finite languages define rational picture series. The class of series that are projections of rational series are collected in K〈〈Σ〉〉. We will prove that, for commutative semirings, the behaviors of weighted picture automata are precisely alphabetic projections of series defined in terms of rational operations and also coincide with the families of series characterized by weighted tiling or weighted domino systems. We thus obtain a robust definition of a class of recognizable picture series. We get the results for languages by restricting the semiring to the Boolean semiring. Theorem 1. Let K be a commutative semiring and let Σ be an alphabet. Then K〈〈Σ,WPA〉〉 = K〈〈Σ〉〉. In the course of the thesis we will investigate new properties of unambiguous picture languages. We will apply the notion of unambiguity to several devices describing picture languages and prove an equivalence theorem similar to the known basic equivalences in the theory of two-dimensional languages.