Tiling-recognizable Two-dimensional Languages: from Non-determinism to Determinism through Unambiguity

Tiling recognizable two-dimensional languages, also known as REC, generalize recognizable string languages to two dimensions and share with them several theoretical properties. Nevertheless REC is not closed under complementation and the membership problem is NP-complete. This implies that this family REC is intrinsically non-deterministic. The natural and immediate definition of unambiguity corresponds to a family UREC of languages that is strictly contained in REC. On the other hand this definition of unambiguity leads to an undecidability result and therefore it cannot correspond to any deterministic notion. We introduce the notion of line-unambiguous tiling recognizable languages and prove that it corresponds or somehow naturally introduces different notions of determinism that define a hierarchy inside REC. A picture (or two-dimensional string) is a two-dimensional arrays of symbols from a finite alphabet. A set of pictures is called two-dimensional language. Basic notations and operations can be extended from string to pictures. The size of a picture p is a pair (m,n) corresponding to the number of its rows and columns, respectively. Moreover there can be defined an operation of column-concatenation between pictures with the same number of rows and of row-concatenation between pictures with the same number of columns. By iteration, there can be also defined the corresponding rowand columnstar operations. The first generalization of finite-state automata to two dimensions can be attributed to M. Blum and C. Hewitt who in 1967 introduced the notion of a four-way automaton moving on a two-dimensional tape as the natural extension of a onedimensional two-way finite automaton (see [7]). They also proved that the deterministic version corresponds to a language class smaller than the corresponding one defined by the non-deterministic model. Four-way automata was not a successful model since the corresponding language class does not satisfies important properties as closure under concatenation and star operations. Since then, many approaches have been presented in the literature in order to find the ”right way” to generalize in 2D what regular languages are in one dimension: finite automata, grammars, logics and regular expressions (see for example [8, 14, 25, 18, 27]). Here we focus