Antonio Restivo

Two-dimensional languages

The aim of this chapter is to generalize concepts and techniques of formal language theory to two dimensions. Informally, a two-dimensional string is called a picture and is defined as a rectangular array of symbols taken from a finite alphabet. A two-dimensional language (or picture language) is a set of pictures.

Recognizable picture languages

The purpose of this paper is to propose a new notion of recognizability for picture (two-dimensional) languages extending the characterization of one-dimensional recognizable languages in terms of local languages and alphabetic mappings. We first introduce the family of local picture languages (denoted by LOC) and, in particular, prove the undecidability of the emptiness problem. Then we define the new family of recognizable picture languages (denoted by REC). We study some combinatorial and language theoretic properties of REC such as ambiguity, closure properties or undecidability results. Finally we compare the family REC with the classical families of languages recognized by four-way automata.

Automata and forbidden words

Abstract Let L ( M ) be the (factorial) language avoiding a given anti-factorial language M . We design an automaton accepting L ( M ) and built from the language M . The construction is effective if M is finite. If M is the set of minimal forbidden words of a single word ν, the automaton turns out to be the factor automaton of ν (the minimal automaton accepting the set of factors of ν). We also give an algorithm that builds the trie of M from the factor automaton of a single word. It yields a nontrivial upper bound on the number of minimal forbidden words of a word.

Monadic second-order logic over rectangular pictures and recognizability by tiling systems

Abstract It is shown that a set of pictures (rectangular arrays of symbols) is recognized by a finite tiling system iff it is definable in existential monadic second-order logic. As a consequence, finite tiling systems constitute a notion of recognizability over two-dimensional inputs which at the same time generalizes finite-state recognizability over strings and also matches a natural logic. The proof is based on the Ehrenfeucht–Fraisse technique for first-order logic and an implementation of “threshold counting” within tiling systems.

Burrows--Wheeler transform and Sturmian words

right Michael Burrows and David Wheeler introduced in 1994 (cf. [1]) a reversible transformation on strings (BWT from now on) that arouses considerable interest and curiosity in the field of Data Compression. Such a transformation produces a permutation BWT(w) of an input stringw that is easier to compress than the original one, in the sense that there exist very simple universal data compression algorithms providing surprisingly good performances when the original string is preprocessed by applying BWT . Actually they achieve compression rates that are very close to the best known compression rate. Moreover, due to their simplicity, such algorithms can be implemented with relatively low complexity. The underlying idea consists of grouping together symbols that appear in similar contexts. The output of such a transformation gives a sequence where it is very likely to have many instances of a same character close one another. A theoretical analysis of some BWT-based algorithms is available in [7].

Fine and Wilf's theorem for three periods and a generalization of Sturmian words

We extend the theorem of Fine and Wilf to words having three periods. We then define the set 3-PER of words of maximal length for which such result does not apply. We prove that the set 3-PER and the sequences of complexity 2n + 1, introduced by Arnoux and Rauzy to generalize Sturmian words, have the same set of factors.

An extension of the Burrows–Wheeler Transform

We describe and highlight a generalization of the Burrows-Wheeler Transform (bwt) to a multiset of words. The extended transformation, denoted by ebwt, is reversible. Moreover, it allows to define a bijection between the words over a finite alphabet A and the finite multisets of conjugacy classes of primitive words in A^*. Besides its mathematical interest, the extended transform can be useful for applications in the context of string processing. In the last part of this paper we illustrate one such application, providing a similarity measure between sequences based on ebwt.

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.

Reconstruction of L-convex Polyominoes

Abstract We introduce the family of L-convex polyominoes, a subset of convex polyominoes whose elements satisfy a special convexity property. We develop an algorithm that reconstructs an L-convex polyomino from the set of its maximal L-polyominoes.

Minimal Forbidden Words and Symbolic Dynamics

We introduce a new complexity measure of a factorial formal language L: the growth rate of the set of minimal forbidden words. We prove some combinatorial properties of minimal forbidden words. As main result we prove that the growth rate of the set of minimal forbidden words for L is a topological invariant of the dynamical system defined by L.