Maria Madonia

Unambiguous recognizable two-dimensional languages

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.

Deterministic and Unambiguous Families within Recognizable Two-dimensional Languages

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.

Deterministic and unambiguous two-dimensional languages over one-letter alphabet

The paper focuses on deterministic and unambiguous recognizable two-dimensional languages with particular attention to the case of a one-letter alphabet. The family DREC(1) of deterministic languages over a one-letter alphabet is characterized as both L(DOTA)(1), the class of languages accepted by deterministic on-line tessellation acceptors, and L(2AFA)(1), the class of languages recognized by 2-way alternating finite automata. We show that there are inherently ambiguous languages and unambiguously recognizable languages that cannot be deterministically recognized even in the case of a one-letter alphabet. In particular we show that on-line tessellation acceptors are more powerful than their deterministic counterpart, even in the case of a one-letter alphabet. Finally we show that DREC(1) is complex enough not to be characterized in terms of classical operations.

A computational model for tiling recognizable two-dimensional languages

Tiling systems are a well accepted model to define recognizable two-dimensional languages but they are not an effective device for recognition unless a scanning strategy for the pictures is fixed. We define a tiling automaton as a tiling system equipped with a scanning strategy and a suitable data structure. The class of languages accepted by tiling automata coincides with the REC family. In this framework it is possible to define determinism, non-determinism and unambiguity. Then (deterministic) tiling automata are compared with the other known (deterministic) automata models for two-dimensional languages.

New operations and regular expressions for two-dimensional languages over one-letter alphabet

We consider the problem of defining regular expressions to characterize the class of recognizable picture languages in the case of a one-letter alphabet. We define a diagonal concatenation and its star and consider two different families, L(D) and L(CRD), of languages denoted by regular expressions involving such operations plus classical operations. L(D) is characterized both in terms of rational relations and in terms of two-dimensional automata moving only right and down. L(CRD) is included in REC and contains languages defined by three-way automata while languages in L(CRD) necessarily satisfy some regularity conditions. Finally, we introduce new definitions of advanced stars expressing the necessity of conceptually different definitions for iteration.

Tiling automaton: a computational model for recognizable two-dimensional languages

Two-dimensional languages can be recognized by tiling systems. A tiling system becomes an effective device for recognition when a scanning strategy on pictures is fixed. We define a Tiling Automaton as a tiling system together with a scanning strategy and a suitable data structure. In this framework it is possible to define determinism, nondeterminism and unambiguity. The class of languages accepted by tiling automata coincides with REC family. Tiling automata are able to simulate on-line tessellation automata. Then (deterministic) tiling automata are compared with the other known (deterministic) automata models for recognition of two-dimensional languages.

Prefix picture codes: A decidable class of two-dimensional codes

A two-dimensional code of pictures is defined as a set X ⊆ Σ** such that any picture over Σ is tilable in at most one way with pictures in X. It is proved that in general it is undecidable whether a finite set of picture is a code. The subclass of prefix codes is introduced and it is proved that it is decidable whether a finite set of pictures is a prefix code. Further a polynomial time decoding algorithm for finite prefix codes is given. Maximality and completeness of finite prefix codes are studied.

Two Dimensional Prefix Codes of Pictures

A two-dimensional code is defined as a set X ⊆ Σ** such that any picture over Σ is tilable in at most one way with pictures in X. The codicity problem is undecidable. The subclass of prefix codes is introduced and it is proved that it is decidable whether a finite set of pictures is a prefix code. Further a polynomial time decoding algorithm for finite prefix codes is given. Maximality and completeness of finite prefix codes are studied: differently from the one-dimensional case, they are not equivalent notions. Completeness of finite prefix codes is characterized.

Deterministic two-dimensional languages over one-letter alphabet

We study the family DREC(1) of deterministic tiling recognizable two-dimensional languages in the case of a one-letter alphabet. The family coincides with both the class of languages accepted by deterministic on-line tessellation acceptors (L(DOTA)(1)) and the one of languages recognized by 2-way alternating finite automata (L(2AFA)(1)). We show that DREC(1) is complex enough to contain languages that cannot be realized by classical operations, while other languages constructed using classical operations cannot be deterministically recognized. Furthermore we prove that there are unambiguously recognizable languages that cannot be deterministically recognized even in the case of one-letter alphabet. In particular L(DOTA)(1) is different from L(OTA)(1) (its non-deterministic counterpart).

Strong Prefix Codes of Pictures

A set X ⊆ Σ** of pictures is a code if every picture over Σ is tilable in at most one way with pictures in X. The definition of strong prefix code is introduced and it is proved that the corresponding family of finite strong prefix codes is decidable and it has a polynomial time decoding algorithm. Maximality for finite strong prefix codes is also considered. Given a strong prefix code, it is proved that there exists a unique maximal strong prefix code that contains it and that has a minimal size. The notion of completeness is also investigated in relation to maximality.