Jean-Eric Pin

Polynomial closure and unambiguous product

This article is a contribution to the algebraic theory of automata, but it also contains an application to Büchi’s sequential calculus. The polynomial closure of a class of languagesC is the set of languages that are finite unions of languages of the formL0a1L1 ···anLn, where theai’s are letters and theLi’s are elements ofC. Our main result is an algebraic characterization, via the syntactic monoid, of the polynomial closure of a variety of languages. We show that the algebraic operation corresponding to the polynomial closure is a certain Mal’cev product of varieties. This result has several consequences. We first study the concatenation hierarchies similar to the dot-depth hierarchy, obtained by counting the number of alternations between boolean operations and concatenation. For instance, we show that level 3/2 of the Straubing hierarchy is decidable and we give a simplified proof of the partial result of Cowan on level 2. We propose a general conjecture for these hierarchies. We also show that if a language and its complement are in the polynomial closure of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages of the variety. This allows us to extend the results of Thomas on quantifier hierarchies of first-order logic.

On two Combinatorial Problems Arising from Automata Theory

We present some partial results on the following conjectures arising from automata theory. The first conjecture is the triangle conjecture due to Perrin and Schiitzenberger. Let A ={a, b } be a two-letter alphabet, d a positive integer and let B d ={ a i ba j |0 ≤ i + j ≤ d }. If X C B d is a code, then | X ≤ d + 1. The second conjecture is due to Cerny and the author. Let be an automaton with n states. If there exists a word of rank n - k in , there exists such a word of length ≤ k 2 .

First-order logic and star-free sets

formulas are built up in the usual way by means of the connectives -I, v , A and the quantifiers 3 and V bounding up both types of variables. Now, we say that a word w on the alphabet A satisfies such a sentence 4 if 6 is true when variables are interpreted as integers, set-variables are interpreted as set of integers and the formula X,X is interpreted as “the letter in position x in w is an a.” McNaughton [3] was the first to consider the case where the set of formulas is restricted to first-order, that is, when set-variables are ignored. He proved that the languages defined in this way are precisely the star-free languages, that is, all languages obtained from finite languages by boolean operations and concatenation product. Later on, star-free languages have been considerably studied. First, a fundamental result of Schiitzenberger shows that star-free languages are exactly the languages recognized by an aperiodic finite monoid (i.e., a monoid all of whose groups are trivial). Further on, a great number of subclasses of star-free languages have been studied [6]. Among the most famous, let us quote the locally testable languages studied by McNaughton and Brzozowski and Simon and the piecewise testable languages, introduced by Simon. Star-free languages are defined by two types of operations: boolean operations on one hand and concatenation product on the other hand. This naturally defines a hierarchy based on the alternative use of these operations. The hierarchy was originally introduced by Brzozowski who showed with Knast [ 11 that the inclusion was proper on each level. Furthermore the class of locally testable languages 393 0022~0000/86

ASH'S TYPE II THEOREM, PROFINITE TOPOLOGY AND MALCEV PRODUCTS: PART I

3.00

Duality and Equational Theory of Regular Languages

This paper is concerned with the many deep and far reaching consequences of Ashs positive solution of the type II conjecture for finite monoids. After reviewing the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture — also verified by Ash — it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ashs theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH1H2…Hn, where g ∈ G and each Hi is a finitely generated subgroup of G. This significantly extends classical results of M. Hall. Finally, we return to the roots of this problem and give connections with the complexity theory of finite semigroups. We show that the largest local complexity function in the sense of Rhodes and Tilson is computable.

On the expressive power of temporal logic

This paper presents a new result in the equational theory of regular languages, which emerged from lively discussions between the authors about Stone and Priestley duality. Let us call lattice of languagesa class of regular languages closed under finite intersection and finite union. The main results of this paper (Theorems 5.2 and 6.1) can be summarized in a nutshell as follows: A set of regular languages is a lattice of languages if and only if it can be defined by a set of profinite equations. The product on profinite words is the dual of the residuation operations on regular languages. In their more general form, our equations are of the form ui¾?v, where uand vare profinite words. The first result not only subsumes Eilenberg-Reitermans theory of varieties and their subsequent extensions, but it shows for instance that any class of regular languages defined by a fragment of logic closed under conjunctions and disjunctions (first order, monadic second order, temporal, etc.) admits an equational description. In particular, the celebrated McNaughton-Schutzenberger characterisation of first order definable languages by the aperiodicity condition xi¾?= xi¾?+ 1, far from being an isolated statement, now appears as an elegant instance of a very general result.

On Reversible Automata

Abstract We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the “restricted” temporal logic (RTL), obtained by discarding the “until” operator. A formal language is RTL-expressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads to a polynomial time algorithm to check whether the formal language accepted by an n -state deterministic automaton is RTL-expressible.

Polynomial Closure and Unambiguous Product

A reversible automaton is a finite (possibly incomplete) automaton in which each letter induces a partial one-to-one map from the set of states into itself. We give four non-trivial characterizations of the languages accepted by a reversible automaton equipped with a set of initial and final states and we show that one can effectively decide whether a given rational (or regular) language can be accepted by a reversible automaton. The first characterization gives a description of the subsets of the free group accepted by a reversible automaton that is somewhat reminiscent of Kleenes theorem. The second characterization is more combinatorial in nature. The decidability follows from the third — algebraic -characterization. The last characterization relates reversible automata to the profinite group topology of the free monoid.

Locally trivial categories and unambiguous concatenation

This article is a contribution to the algebraic theory of automata, but it also contains an application to Büchi’s sequential calculus. The polynomial closure of a class of languagesC is the set of languages that are finite unions of languages of the formL0a1L1 ···anLn, where theai’s are letters and theLi’s are elements ofC. Our main result is an algebraic characterization, via the syntactic monoid, of the polynomial closure of a variety of languages. We show that the algebraic operation corresponding to the polynomial closure is a certain Mal’cev product of varieties. This result has several consequences. We first study the concatenation hierarchies similar to the dot-depth hierarchy, obtained by counting the number of alternations between boolean operations and concatenation. For instance, we show that level 3/2 of the Straubing hierarchy is decidable and we give a simplified proof of the partial result of Cowan on level 2. We propose a general conjecture for these hierarchies. We also show that if a language and its complement are in the polynomial closure of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages of the variety. This allows us to extend the results of Thomas on quantifier hierarchies of first-order logic.

THE WREATH PRODUCT PRINCIPLE FOR ORDERED SEMIGROUPS

Abstract 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.