Unambiguous recognizable two-dimensional languages

Abstract

We consider the family UREC of unambiguous recognizable two-dimensional languages. We prove that there are recognizable languages that are inherently ambiguous, that is UREC family is a proper subclass of REC family. The result is obtained by showing a necessary condition for unambiguous recognizable languages. Further UREC family coincides with the class of picture languages defined by unambiguous 2OTA and it strictly contains its deterministic counterpart. Some closure and non-closure properties of UREC are presented. Finally we show that it is undecidable whether a given tiling system is unambiguous

Snake-Deterministic Tiling Systems

“…UREC is the class of all unambiguous languages. It is known that UREC ⊂ REC and that it is undecidable whether a tiling system is unambiguous [3].…”

Characterizations of recognizable picture series

“…There, Giammarresi and Restivo also posed the conjecture that unambiguous languages are properly included in the family of recognizable languages. This conjecture was solved very recently in [1], where the authors showed that unlike with words, there exist recognizable picture languages that are inherently ambiguous, i.e. not computable by unambiguous 2OTA; moreover, the problem whether a tiling system is unambiguous is undecidable.…”

“…Tiles are pictures of size (2, 2) and dominoes have size (1,2) or (2, 1). For a picture p, we denote by T ( p) (respectively D( p)) the set of all sub-tiles (respectively sub-dominoes) of p. A language L ⊆ Γ ++ is local (respectively hv-local) if there exists a set Θ of tiles (respectively dominoes) over Γ .…”

“…not computable by unambiguous 2OTA; moreover, the problem whether a tiling system is unambiguous is undecidable. In [25][26][27] we proved an equivalence theorem for unambiguous picture languages in terms of unambiguous 2OTA and unambiguous tiling systems, unambiguously rational operations and also of unambiguous (quadrapolic) picture automata (the equivalence of unambiguous 2OTA and unambiguous tiling systems was independently derived in [1]). …”

Picture Languages: From Wang Tiles to 2D Grammars

“…For p ∈ Σ +,+ let p be the set of subpictures of size (2,2) of p. 5 In the sequel the concepts of tile, and local language are central.…”