Deterministic and Unambiguous Families within Recognizable Two-dimensional Languages

Abstract

Recognizable two-dimensional languages (REC) are defined by tiling systems that generalize to two dimensions non-deterministic finite automata for strings. We introduce the notion of deterministic tiling system and the corresponding family of languages (DREC) and study its structural and closure properties. Furthermore we show that, in contrast with the one-dimensional case, there exist other classes between deterministic and non-deterministic families that we separate by means of examples and decidability properties.

Structure and properties of strong prefix codes of pictures

“…The following diagram summarizes the results. Languages X 1 , X 2 , X 3 that separate the classes can be chosen as the following: As future research, we will try to remove the finiteness condition and consider prefix sets in sub-classes of REC (the family of tiling system recognizable languages), such as deterministic ones (Anselmo et al 2010;Anselmo and Madonia 2009).…”

“…In this paper we refer to the definition of a code given in Anselmo et al (2013a) where two-dimensional codes are introduced in the setting of the theory of recognizable two-dimensional languages and coherently to the notion of language unambiguity as in Anselmo et al (2006Anselmo et al ( , 2010.…”

Infinite Two-Dimensional Strong Prefix Codes: Characterization and Properties

“…Any picture p ∈ X ++ can be decomposed starting at top-left-corner and checking the subpicture p[(1, 1), (2, 2)]; it can be univocally decomposed in X. Then, proceed similarly for the next contiguous subpictures of size (2,2).…”

“…[10,15,17,19,[22][23][24]). A crucial difference with the string language theory is that in two dimensions many problems become undecidable and even for finite-state recognizability we loose the equivalence between determinism and non-determinism ( [2,6,17]).…”

Non-overlapping matrices

“…The notion of non-overlapping strings can be naturally extended to the two dimensional case by means of matrices (or pictures): any two of them A, B (the case A = B is allowed) do not overlap if it is not possible to move A (B) over B (A) in a way such that the corresponding entries match. Also in this case, it is possible to find different kinds of definitions of non-overlapping sets of matrices, as in [6], [7], [8], or several deep and interesting investigations about some their properties ( [1,2,3,4,5] and all the references therein). In particular, in [1] the role of the frame of the matrices is deeply analysed, showing that it is possible to generate a corss-bifix-free set of matrices simply framing with a suitable frame any matrix of a given set.…”