Ambiguity and Complementation in Recognizable Two-dimensional Languages

Abstract

The theory of one-dimensional (word) languages is well founded and investigated since fifties. From several years, the increasing interest for pattern recognition and image processing motivated the research on two-dimensional or picture languages, and nowadays this is a research field of great interest. A first attempt to formalize the concept of finite state recognizability for two-dimensional languages can be attributed to Blum and Hewitt ([7]) who started in 1967 the study of finite state devices that can define two-dimensional languages, with the aim to finding a counterpart of what regular languages are in one dimension. Since then, many approaches have been presented in the literature following all classical ways to define regular languages: finite automata, grammars, logics and regular expressions. In 1991, a unifying point of view was presented in [13] where the family of tiling recognizable picture languages is defined (see also [14]). The definition of recognizable picture language takes as starting point a well known characterization of recognizable word languages in terms of local languages and projection. Namely, any recognizable word language can be obtained as projection of a local word language defined over a larger alphabet. Such notion can be extended in a natural way to the two-dimensional case: more precisely, local picture languages are defined by means of a set of square arrays of side-length two (called tiles) that represents the only allowed blocks of that size in the pictures of the language (with special treatment for border symbols). Then, we say that a picture language is tiling recognizable if it can be obtained as a projection of a local picture language. The family of all tiling recognizable picture languages is called REC. Remark that, when we consider words as particular pictures (that is pictures in which one side has length one), this definition of recognizability coincides with the one for the words, i.e. the definition given in terms of finite automata. The family REC can be characterized by several formalisms such as different variants of tiling systems, on-line tessellation automata, Wang systems, existential monadic second order logic, ”special” regular expressions, etc. (see