Picture Recognizability with Automata Based on Wang Tiles

Abstract

Abstract. We introduce a model of automaton for picture language recognition which is based on tiles and is called Wang automaton, since its description relies on the notation of Wang systems. Wang automata combine features of both online tessellation acceptors and 4-ways automata: as in online tessellation acceptors, computation assigns states to each picture position; as in 4-way automata, the input head visits the picture moving from one pixel to an adjacent one, according to some scanning strategy. We prove that Wang automata recognize the class REC, i.e. they are equivalent to tiling systems or online tessellation acceptors, and hence strictly more powerful than 4-way automata. We also consider a very natural notion of determinism for Wang automata, and study the resulting class, comparing it with other deterministic classes considered in the literature, like DREC and Snake-DREC.

Strategies to scan pictures with automata based on Wang tiles

“…Here we recall the notion of 2D scanning strategies as introduced in [18] and in particular the definition of blind scanning strategy.…”

“…The notion of blindness of a scanning strategy was introduced in [18] to this aim. Basically, our idea of blind strategy is based on local properties on the "shape" of visited positions of the input picture.…”

“…For nondeterministic Wang automata the choice of the scanning strategy (as long as it is polite) is not relevant from the point of view of the recognizing power of the device: for every polite scanning strategy µ, the class of picture languages recognized by µ-WA equals REC [18,Theorem 1]. This is no longer true when determinism is concerned.…”

“…Wang automata is a more recent kind of device characterizing REC, introduced by us in [18], and based on the labeled variant of Wang tiles used in Wang systems [9]. Our model in some sense combine features of online tessellation acceptors, and traditional 4-ways automata [14].…”

Two-dimensional jumping finite automata

“…An interesting model of automaton to recognize REC class of picture languages is the two-dimensional online tessellation acceptor [6]. Recently, in [9], Wang automaton, a model of automaton for rectangular picture language recognition, has been introduced, which combines the features of both online tessellation acceptors and 4-way automata but differs in the aspect of having many scanning strategies to visit the input picture.…”