Deterministic recognizability of picture languages by Wang automata

Picture languages are a generalization of string languages to two dimensions: a picture is a twodimensional array of elements from a finite alphabet. Several classes of picture languages have been considered in the literature [5,7,3,10]. In particular, here we refer to class REC introduced in [5] with the aim to generalize to 2D the class of regular string languages. REC is a robust class that has various characterizations; in particular, it is the class of picture languages that can be generated by tiling systems, a model introduced in [4], or equivalently by Wang systems [?]. A central notion in string regular language theory is determinism, whereas the concept of determinism for picture languages is far from being well understood. Tiling systems are implicitly nondeterministic: REC is not closed under complement, and the membership problem is NP-complete [8]. Clearly, this latter fact severely hinders the potential applicability of the notation. The identification of a reasonably “rich” deterministic subset of REC would spur its application, since it would allow linear parsing w.r.t. the number of pixels of the input picture. In past and more recent years, several different deterministic subclasses of REC have been studied, e.g. the classes defined by deterministic 4-way automata [7] or deterministic online tessellation acceptors [6]. This latter model inspired the notion of determinism of [1], that relies on four diagonalbased scanning strategies, each starting from one of the four corners of the picture. Here will call the corresponding deterministic class Diag-DREC1. In [9] we introduced the class Snake-DREC, based on a different kind of determinism for tiles, using a boustrophedonic scanning strategy. Snake-DREC properly extends Diag-DREC and is closed under complement, rotation and symmetries. However, like Diag-DREC, it is not closed under intersection and union. When pictures of only one row (or column) are considered, this model reduces to deterministic finite state automata. Quite surprisingly, such notion of determinism coincides with line unambiguity of Row-UREC (or Col-UREC) introduced in [1] to have backtracking at most linear in one dimension of the input picture. The notion of determinism for tiling systems is quite different than the same notion for 4-way automata [7,5]. For instance, both Snake-DREC and Diag-DREC are incomparable to the class of languages recognized by 4-way deterministic automata. Moreover, any notion of determinism in tiling systems (and online tessellation acceptors) seems to require some pre-established strategy used for scanning the picture. Indeed, both Diag-DREC and Snake-DREC are based on some fixed kind of strategy. In [2], a first generalization of the concept of scanning strategy is presented; with the same goal, here we propose an alternative framework, where a scanning strategy is defined as a method to sort all cells of a picture,