Recursively Enumerable Languages Research Articles

Regular frequency computations

An ( m , n ) -computation of a function f is given by a deterministic Turing machine which on n pairwise different inputs produces n output values where at least m of the n values are in accordance with f. In such a case, we say that the Turing machine computes f with frequency ⩾ m / n . The most prominent result for frequency computations is due to Trakhtenbrot: The class of ( m , n ) -computable functions equals the class of computable functions if and only if 2 m > n . Via characteristic functions the definition of ( m , n ) -computability carries over to sets. Here Trakhtenbrot's result reads as: The class of ( m , n ) -computable sets equals the class of recursive sets if and only if 2 m > n . The notion of frequency computation can be extended to other models of computation. For resource bounded computations, the behavior is completely different: for e.g., whenever n ′ - m ′ > n - m , it is known that under any reasonable resource bound there are sets ( m ′ , n ′ ) -computable, but not ( m , n ) -computable. However, scaling down to finite automata, the analogue of Trakhtenbrot's result holds again: We show here that the class of languages ( m , n ) -recognizable by deterministic finite automata equals the class of regular languages if and only if 2 m > n . This was originally stated by Kinber, but his proof has a flaw, as pointed out by Tantau. Conversely, for 2 m ⩽ n , the class of languages ( m , n ) -recognizable by deterministic finite automata is uncountable for a two-letter alphabet. When restricted to a one-letter alphabet, then every ( m , n ) -recognizable language is regular. This was also shown by Kinber. We give a new and more direct proof for this result.

AN APPLICATION OF FIRST-ORDER LOGIC TO THE STUDY OF RECOGNIZABLE LANGUAGES

A variation of first-order logic with variables for exponents is developed to solve some problems in the setting of recognizable languages on the free monoid, accommodating operators such as product, bounded shuffle and reversion. Restricting the operators to powers and product, analogous results are obtained for recognizable languages of an arbitrary finitely generated monoid M, in particular for a free inverse monoid of finite rank. As a consequence, it is shown to be decidable whether or not a recognizable subset of M is pure or p-pure.

Macro forest transducers

Xml documents conceptually are not trees, but forests. Therefore, we extend the concept of macro tree transducers (mtt's) to a transformation formalism of forests, macro forest transducers (mft's). We show that mft's form a strict superset of mtt's operating on forests (represented as binary trees). On the other hand, the transformation of every mft can be simulated by the composition of two mtt's. Although macro forest transducers are more powerful, the complexity of inverse type inference, i.e., computing the pre-image of a recognizable language, is almost the same as for tree transducers.

An algebraic approach to data languages and timed languages

Algebra offers an elegant and powerful approach to understand regular languages and finite automata. Such framework has been notoriously lacking for timed languages and timed automata. We introduce the notion of monoid recognizability for data languages, which includes timed languages as special case, in a way that respects the spirit of the classical situation. We study closure properties and hierarchies in this model and prove that emptiness is decidable under natural hypotheses. Our class of recognizable languages properly includes many families of deterministic timed languages that have been proposed until now, and the same holds for non-deterministic versions.

Translating the Middle Ages: Modernism and the Ideal of the Common Language

Robert Crawford recently delivered an impassioned address on the monolingual assumptions that underpin ‘English’ literature. ‘English’, as a concept and as a discipline, has been historically constructed to appropriate within its canon other literatures written in English but not produced in England or even in Great Britain. At the same time, this ‘English’ excludes the literature produced in England or Great Britain in other languages: Anglo-Norman writings in French, early modern Latin poetry, Scottish and Welsh writing. Today’s pluralism does not extend to the past of English literature, from which ‘plurilingualism’ is erased in what Crawford called ‘English’s Alzheimer disease’. And this, Crawford argued, is symptomatic of a persistent imperialist construction of ‘English’.1 Crawford’s contention might well put us in mind of Lawrence Venuti’s argument on the translator’s invisibility. For Venuti, the historical privileging of a scientific, transparent version of the English language has worked to discourage ‘foreignness’: the best kind of translation is ‘fluent’, and reads as if we were experiencing the work in the original, without any mediation. The translation is expected to be transparent, the translator invisible, and English, implicitly, to remain solidly monolithic and easily recognizable.2 Crawford’s and Venuti’s concerns enable us to return to a question frequently discussed by commentators on modernism: its claimed elitism. The accusation ultimately stems from a traditional, and indeed canonical, requirement of ‘accessibility’ predicated on the use of transparent, plain, recognizable language and forms. This demand can be traced back to the ideal of linguistic simplicity and anti-metaphorical

Towards a language theory for infinite N-free pomsets

N-free or series–parallel pomsets are a model for the behavior of modularly constructed concurrent systems. The investigation of recognizable languages of finite N-free pomsets was initiated by Lodaya and Weil who extended the theorems by Kleene and by Myhill and Nerode on recognizable word languages to this setting. In this paper, we extend Lodaya and Weil's results in several aspects: (a) We consider the relation of recognizable sets to monadic second order logic in order to generalize Büchi's theorem. (b) We prove our results (and extensions of results by Lodaya and Weil) for sets of infinite N-free pomsets. And (c), we investigate first-order axiomatizable, starfree, and aperiodic sets of infinite N-free pomsets and prove results in the spirit of McNaughton and Papert's and Schützenberger's theorems for finite words.

A structural property of regular frequency computations

The notion of an (m,n)-computation was already introduced in 1960 by Rose and further investigated by Trakhtenbrot in 1963. It has been extended to finite automata by Kinber in 1976 and he has shown an analogue of Trakhtenbrot's result: The class of languages (m,n)-recognizable by deterministic finite automata is equal to the class of regular languages if and only if 2m>n. Furthermore, for a unary alphabet, the class of (m,n)-recognizable languages coincides with the class of regular languages for all m and n. In this paper, we will present the first structural property of (m,n)-recognizable languages which is valid for all 1⩽m⩽n and for all alphabets. Kinber's result for unary alphabets becomes a corollary. This property is also used to show that certain non-unary languages are not (m,n)-regular and that the class of all (m,n)-recognizable languages is not closed under the reversal operation. However, this class forms a Boolean algebra.

Commutative images of rational languages and the Abelian kernel of a monoid

Natural algorithms to compute rational expressions for recognizable languages, even those which work well in practice, may produce very long expressions. So, aiming towards the computation of the commutative image of a recognizable language, one should avoid passing through an expression produced this way. We modify here one of those algorithms in order to compute directly a semilinear expression for the commutative image of a recognizable language. We also give a second modification of the algorithm which allows the direct computation of the closure in the profinite topology of the commutative image. As an application, we give a modification of an algorithm for computing the Abelian kernel of a finite monoid obtained by the author in 1998 which is much more efficient in practice.

Decidability Equivalence between the Star Problem and the Finite Power Problem in Trace Monoids

In the last decade, some researches on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the interest of the finite power property to achieve partial solutions of this problem. We prove that the star problem is decidable in some trace monoid if and only if in the same monoid, it is decidable whether a recognizable language has the finite power property. Intermediary results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without C4-submonoid. We also deal with some earlier ideas, conjectures, and questions which have been raised in the research on the star problem and the finite power property, e.g., we show the decidability of these problems for recognizable languages which contain at most one non-connected trace.

The Kleene–Schützenberger Theorem for Formal Power Series in Partially Commuting Variables

Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schützenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmański to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semiring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schützenberger's and Ochmański's theorems.

RECOGNIZABLE AND LOGICALLY DEFINABLE LANGUAGES OF INFINITE COMPUTATIONS IN CONCURRENT AUTOMATA

Automata with concurrency relations [Formula: see text], which occurred in formal verification methods of concurrent programs, are labeled transition systems with a collection of binary relations describing when two actions in a given state of the automaton can happen independently of each other. The concurrency monoid M($\mathcal{A}$) comprises all finite computation sequences of [Formula: see text], modulo a canonical congruence induced by the concurrency relations, with composition as monoid operation. Then M∞($\mathcal{A}$) denotes the set of all infinite products in M($\mathcal{A}$); its elements can be represented by labeled partially ordered sets. Under suitable assumptions on [Formula: see text], we show that a language L in M∞($\mathcal{A}$) is recognizable iff it is definable by a formula of monadic second order logic, and that it is recognizable iff it can be constructed from recognizable languages in M($\mathcal{A}$) using co-rational expressions. This generalizes various recent results in trace theory.

Free Profinite ℛ-Trivial Monoids

This article is concerned with the structure of semigroups of implicit operations on R, the pseudovariety of all ℛ-trivial semigroups. We give two complementary descriptions for these semigroups: first by means of labeled ordinals, and second by means of labeled infinite trees of finite depth. In the first representation, the product of implicit operations has a simple characterization. In the second one, the topology on implicit operations is best visible. A similar study is conducted for the implicit operations on DRG, the non-aperiodic analogue of R. This leads to a description of the corresponding variety of recognizable languages and to the computation of certain joins of pseudovarieties.

Monadic second-order definable text languages

Atext is a word together with a (additional) linear ordering. Each text has a generic tree representation, called itsshape. Texts are considered in a logical and in an algebraic framework. It is proved that, for texts of bounded primitivity, the class of monadic second-order definable text languages coincides with both the class of recognizable text languages and the class of text languages generated by right-linear text grammars. In particular it is demonstrated that the construction of the shape of a text can be formalized in terms of our monadic second-order logic. This approach can be extended to the case of graphs.

Controlled H Systems of Small Radius

Several characterizations of recursively enumerable languages are given, using H systems with permitting contexts having splicing rules of small radius. Representations of context-free languages ar...

Minimal equational representations of recognizable tree languages

A tree language is congruential if it is the union of finitely many classes of a finitely generated congruence on the term algebra. It is well known that congruential tree languages are the same as recognizable tree languages. An equational representation is an ordered pair (E, P) , where E is either a ground term equation system or a ground term rewriting system, and P is a finite set of ground terms. We say that (E, P) represents the congruential tree language L which is the union of those ?* E -classes containing an element of P, i.e., for which L=⋃{[p]? * E ∣p∈P}. We define two sorts of minimality for equational representations. We introduce the cardinality vector (∣E∣, ∣P∣) of an equational representation (E, P). Let ? l and ? a denote the lexicographic and antilexicographic orders on the set of ordered pairs of nonnegative integers, respectively. Let L be a congruential tree language. An equational representation (E, P) of L with ? l -minimal (? a -minimal) cardinality vector is called ? l -minimal (? a -minimal). We compute, for an L given by a deterministic bottom-up tree automaton, both a ? l -minimal and a ? a -minimal equational representation of L.

On some factorization problems

We consider three notions of factorization arising in dierent frameworks: factorizing languages, factorization of the natural numbers, factorizing codes. A language X A is called factorizing if there exists a language Y A such that XY = A and the product is unambiguous. This is a decidable property for recognizable languages X. If we consider the particular case of unary alphabets, we prove that nite factorizing languages can be constructed by using Krasner factorizations. Moreover, we extend Krasner’s algorithm to factorizations of A n . We introduce a class of languages, the strong factorizing languages, which are related to the factorizing codes, introduced by

Polynomial closure of group languages and open sets of the Hall topology

This main result of this paper states that a recognizable language is open in the Hall topology if and only if it belongs to the polynomial closure C of the group languages. A group language is a recognizable language accepted by a permutation automaton. The polynomial closure of a class of languages L is the set of languages that are finite unions of languages of the form L0a1L1 ... anLn, where the ai's are letters and the Li's are elements of L. The Hall topology is the topology in which the open sets are finite or infinite unions of group languages. We also give a combinatorial description of these languages and a syntactic characterization. Let L be a recognizable set of A*, let M be its syntactic monoid and let P be its syntactic image. Then L belongs to C if and only if, for every s, t in M and for every idempotent e of M, st in P implies set in P. Finally, we give a characterization on the minimal automaton of L that leads to a polynomial time algorithm to check, given a finite deterministic automaton, whether it recognizes of C.

Profinite categories, implicit operations and pseudovarieties of categories

The last decade has seen two methodological advances of particular direct import for the theory of finite monoids and indirect import for that of rational languages. The first has been the use of categories (considered as “algebras over graphs”) as a framework in which to study monoids and their homomorphisms, the second has been the use of implicit operations to study pseudovarieties of monoids. Still more recent work has emphasized the role of profiniteness in finite monoid theory. This paper fuses these three topics by means of a general study of profinite categories, with applications to C-varieties (pseudovarieties of categories) in general, to those C-varieties arising from M-varieties (pseudovarieties of monoids) in particular, to implicit operations on categories and to recognizable languages over graphs.

Local languages and the Berry-Sethi algorithm

One of the basic tasks in compiler construction, document processing, hypertext software and similar projects is the efficient construction of a finite automaton from a given rational (regular) expression. The aim of the present paper is to give an exposition and a formal proof of the background for the algorithm of Berry and Sethi relating the computation involved to a well-known family of recognizable languages, the local languages.

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.