Recursively Enumerable Languages Research Articles

Logical definability on infinite traces

The main results of the present paper are the equivalence of definability by monadic second-order logic and recognizability for real trace languages, and that first-order definable, star-free, and aperiodic real trace languages form the same class of languages. This generalizes results on infinite words and on finite traces to infinite traces. It closes an important gap in the different characterizations of recognizable languages of infinite traces.

TWO-DIMENSIONAL FINITE STATE RECOGNIZABILITY

The purpose of this paper is to investigate about a new notion of finite state recognizability for two-dimensional (picture) languages. This notion takes as starting point the characterization of one-dimensional recognizable languages in terms of local languages and projections. Such notion can be extended in a natural way to the two-dimensional case. We first introduce a notion of local picture language and then we define,a recognizable picture language as a projection of a local picture language. The family of recognizable picture languages is denoted by REC. We study some combinatorial and language-theoretic properties of family REC. In particular we prove some closure properties with respect to different kinds of operations. From this, we derive that some natural families of two-dimensional languages (finite languages, regular languages, locally testable languages) are recognizable. Further we give some necessary conditions for recognizability which provides tools to show that certain languages are not recognizable. Although REC shares several properties of recognizable string languages, however, differently from the case of words, we prove here that REC is not closed under complementation and that the emptyness problem is undecidable for this family of languages. Finally, we report some characterizations of family REC by means of machine-based models and logic-based formalisms.

Recognizable languages in concurrency monoids

Automata with concurrency relations A are labelled transition systems with a collection of binary relations indicating when two actions, in a given state of the automaton, are concurrent. We investigate concurrency monoids M( A ) comprising all finite computation sequences of A , modulo a canonical congruence induced by the concurrency relations, with composition as monoid operation. Under suitable assumptions on A we obtain a Kleene-type characterization of the recognizable languages of M( A ). This generalizes results of Cori, Métivier, Perrin and Ochmanski in trace theory.

Rational and Recognizable Complex Trace Languages

Mazurkiewicz defined traces as an algebraic model of finite concurrent processes. In order to model non-terminating processes a good notion of infinite traces was needed, which finally led to the notion of complex traces. For complex traces an associative concatenation and an ω-iteration are defined. This paper defines and investigates rational and recognizable complex trace languages. We prove various closure results such as the closure under boolean operations (for recognizable languages), concatenation, and left and right quotients by recognizable sets. Then we study sufficient conditions ensuring the recognizability of the finite and infinite iterations of complex trace languages. We introduce a generalization of the notion of concurrent iteration which leads to the main result of the paper: the generalization of Kleene′s and Ochmański′s theorems to complex trace languages.

LEARNING OF RECOGNIZABLE PICTURE LANGUAGES

Learning of certain classes of two-dimensional picture languages is considered in this paper. Linear time algorithms that learn in the limit, from positive data the classes of local picture languages and locally testable picture languages are presented. A crucial step for obtaining the learning algorithm for local picture languages is an explicit construction of a two-dimensional on-line tessellation acceptor for a given local picture language. A polynomial time algorithm that learns the class of recognizable picture languages from positive data and restricted subset queries, is presented in contrast to the fact that this class is not learnable in the limit from positive data alone.

A CHARACTERIZATION OF RECOGNIZABLE PICTURE LANGUAGES

This paper first shows that REC, the family of recognizable picture languages in Giammarresi and Restivo,3 is equal to the family of picture languages accepted by two-dimensional on-line tessellation acceptors in Inoue and Nakamura.5 By using this result, we then solve open problems in Giammarresi and Restivo,3 and show that (i) REC is not closed under complementation, and (ii) REC properly contains the family of picture languages accepted by two-dimensional nondeterministic finite automata even over a one letter alphabet.

Deterministic asynchronous automata for infinite traces

This paper shows the equivalence between the family of recognizable languages over infinite traces and the family of languages which are recognized by deterministic asynchronous cellular Muller automata. We thus give a proper generalization of McNaughton's Theorem from infinite words to infinite traces. Thereby we solve one of the main open problems in this field. As a special case we obtain that every closed (w.r.t. the independence relation) word language is accepted by someI-diamond deterministic Muller automaton.

Recognizable and Rational Subsets of Algebras

In this paper recognizable and rational subsets of general algebras of finite type are studied. These are natural generalizations of the familiar notions originally defined for monoids, but they also include as special cases recognizable and rational tree languages, for example. Some closure properties of the families of recognizable and rational subsets are studied. The operations considered seem to be intimately connected with the recognizability and rationality properties of subsets, and they can also be defined for subsets of abstract algebras, regardless of the nature of the elements. The last part of the paper is devoted to generalized forms of Kleene’s theorem and related results.

Recognizable trace languages, distributed automata and the distribution problem

We introduce new finite state "parallel machines," the (?-)distributed automata, for trace languages. We prove that these machines give a new characterization of recognizable trace languages: a trace language is recognizable if and only if it is recognized by a (?-)distributed automaton. At last, we show how the classical problem of distribution of uninterpreted tasks on several processors can be straightforward modelized by recognizable trace languages and solved using (?-)distributed.

On the structure of recognizable languages of dependence graphs

Dans le cadre de la theorie des traces, un graphe de dependance represente le comportement d'un systeme distribue (par exemple un reseau de Petri) par analogie avec la relation mot-automate dans le cas sequentiel. Un langage reconnaissable de graphes de dependances represente ainsi l'ensemble de tous les comportements d'un systeme distribue satisfaisant des conditions de regularite. Dans cet article nous caracterisons les languages de graphes obtenus a partir des precedents par suppression des etiquettes sur les sommets

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.

Local varieties of completely regular monoids

The aim of this paper is to apply recent deep rcsults on completely regular semigrotips to questions about categories, arising in connection with the theory of recognizable languages. A variety [pseudovariety] V of monoids is local if any [finite] category whose local monoids belong to V divides a member of V, in the sense of B. Tilson 1-21] (see below). Our Main Theorem (7.1) essentially states that a monoid variety which satisfies an identity x + l = x, n >/0, that is, which consists of completely regular semigroups, is local whenever all the labels on its Pol~ik ladder are local. (See below for the definitions.) As an immediate consequence, every nontrivial monoid variety consisting of orthogroups is local (An orthogroup is a completely regular semigroup whose idempotents form a subsemigroup.) Further, it quickly follows that any nontrivial pseudovariety of monoids consisting of orthogro,ps is also local At the core of the paper is a study of congruences on locally completely regular categories (those for which each local monoid is completely regular). Pseudovarieties of monoids arise naturally via the well-known correspondence between regular languages and finite monoids (e.g., 1-13]). It has lately been realized that, for some purposes, they are best studied within the framework of pseudovarieties of categories; locality of a (pseudo-)

Varieties of finite monoids and Büchi-McNaughton theorem

Perrin (1982) has proved that a part of Büchi-McNaughton theorem in the form formulated by Eilenberg (1974) can be generalized to a large class of varieties of recognizable languages. We show here that Perrin's assertion can be extended to the whole theorem as well as to the partial improvements of it given by Arnold (1983), Beauquier and Perrin (1984). To do that, suitable classes of automata associated with a variety of finite monoids are introduced and examined.

Context free closed families of languages

We study context free closed families of languages, that is families L verifying the equation CF o L = L where CF denotes the family of context free languages and o is the substitution operation. Call a family L a context free AFL if it is both context free closed and a rational cone. The families CF and RE (of recursively enumerable languages) have this property. We show that the class of context free AFL's is properly contained into that of full AFL's. Moreover the context free AFL generated by a family L, is given by the formula L Δ = CF o LΓ where LΓ denotes the rational cone generated by L. Using this result we establish the remarkable inclusion (L o M)Δ ⊃ LΔ o MΔ holding for all families L and M as well as the equality LΔ = CF o LΓ LΓ being the full AFL generated by L.

The complexity of computing the number of strings of given length in context-free languages

Computing the number of strings of given length contained in a language is related to classical problems of combinatorics, formal languages and computational complexity. Here we study the complexity of this problem in the case of context-free languages. It is shown that, for unambiguous context-free languages such a computation is “easy” and can be carried out by efficient parallel algorithms. On the contrary, for some context-free languages of ambiguity degree two, the problem becomes intractable. These results are related to other classical subjects concerning counting problems, exponential time recognizable languages and sparse sets.

An undecidable problem about rational sets and contour words of polyominoes

The contour of a polyomino can be represented as a word called a contour word written on a four letters alphabet A = { x, x′, y, y′ } corresponding to the four directions right, left, up and down. We prove that it is undecidable whether a recognizable language R ⊂ A ∗ contains a contour word.

On the varieties of languages associated with some varieties of finite monoids with commuting idempotents

Eilenberg has shown that there is a one-to-one correspondence between varieties of finite monoids and varieties of recognizable languages. In this paper, we give a description of a variety of languages close to the class of piecewise testable languages considered by I. Simon. The corresponding variety of monoids is the variety of j -trivial monoids with commuting idempotents. This result is then generalized to the case of finite monoids with commuting idempotents whose regular D -classes are groups from a given variety of groups.

On supremal languages of classes of sublanguages that arise in supervisor synthesis problems with partial observation

This paper characterizes the class of closed and (M, N)-recognizable languages in terms of certain structural aspects of relevant automata. This characterization leads to algorithms that effectively compute the supremal (M, N)-recognizable sublanguage of a given language. One of these algorithms is used, in an alternating manner with an algorithm which yields the supremal (∑u, N)-invariant resulting algorithm is proved. An example illustrates the use of these algorithms.

Partially ordered finite monoids and a theorem of I. Simon

We use semigroup expansions to show that every finite J-trivial monoid is a quotient of a finite monoid admitting a partial order that is compatible with multiplication. As a result we obtain a radically new proof of the theorem of I. Simon characterizing the recognizable languages whose syntactic monoids are J-trivial.

Locally trivial categories and unambiguous concatenation

We use the recently developed theory of finite categories and the two-sided kernel to study the effect of the unambiguous concatenation product of recognizable languages on the syntactic monoids of the languages involved. As a result of this study we obtain an algebraic characterization (originally due to Pin) of the closure of a variety of languages under boolean operations and unambiguous concatenation, and a new proof of a theorem of Straubing characterizing the closure of a variety of languages under boolean operations and concatenation. We also note some connections to the study of the dot-depth hierarchy.