Games, equations and dot-depth two monoids

Given any finite alphabet A and positive integers m1, …, mk, congruences on A*, denoted by ~(m1, …, mk) and related to a version of the Ehrenfeucht-Fraisse game, are defined. Level k of the Straubing hierarchy of aperiodic monoids can be characterized in terms of the monoids A*/~(m1, … mk). A natural subhierarchy of level 2 and equation systems satisfied in the corresponding varieties of monoids are defined. For A = 2, a necessary and sufficient condition is given for A*/~(m1, … , mk) to be of dot-depth exactly 2. Upper and lower bounds on the dot-depth of the A*/~(m1, … mk) are discussed

Languages of Dot-depth One over Infinite Words

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.Comment: Presented at LICS 201

On a Product of Finite Monoids

In this paper, for each positive integer m, we associate with a finite monoid S0 and m finite commutative monoids S1,…, Sm, a product &#x25CAm(Sm,…, S1, S0). We give a representation of the free objects in the pseudovariety &#x25CAm(Wm,…, W1, W0) generated by these (m + 1)-ary products where Si &#x2208 Wi for all 0 &#x2264 i &#x2264 m. We then give, in particular, a criterion to determine when an identity holds in &#x25CAm(J1,…, J1, J1) with the help of a version of the Ehrenfeucht-Fraïssé game (J1 denotes the pseudovariety of all semilattice monoids). The union &#x222Am>0&#x25CAm (J1,…, J1, J1) turns out to be the second level of the Straubing’s dot-depth hierarchy of aperiodic monoids

On dot-depth two

Etant donnés des entiers positifs m1, …, mk, on définit des congruences ~(m1, …, mk) en relation avec une version du jeu de Ehrenfeucht-Fraissé, et qui correspondent au niveau k de la hiérarchie de concaténation de Straubing. Etant donné un alphabet fini A, une condition nécessaire et suffisante est donnée pour que les monoïdes définis par ces congruences soient de dot-delpth exactement

Equations and Dot-Depth One

This paper studies the fine structure of the Straubing hierarchy of star-free languages. Sequences of equations are defined and are shown to be sufficiently strong to characterize completely the monoid varieties of a natural subhierarchy of level one. In a few cases, it is also shown that those sequences of equations are equivalent to finite ones. Extensions to a natural sublevel of level two are discussed

On a complete set of generators for dot-depth two

AbstractA complete set of generators for Straubing's dot-depth-two monoids has been characterized as a set of quotients of the form A∗/∼(n,m), where n and m denote positive integers, A∗ denotes the free monoid generated by a finite alphabet A, and ∼(n,m) denote congruences related to a version of the Ehrenfeucht—Fraïssé game. This paper studies combinatorial properties of the ∼(n,m)'s and in particular the inclusion relations between them. Several decidability and inclusion consequences are discussed

Some Logical Characterizations of the Dot-Depth Hierarchy and Applications

A logical characterization of natural subhierarchies of the dot-depth hierarchy refining a theorem of Thomas and a congruence characterization related to a version of the Ehrenfeucht—Fraïssé game generalizing a theorem of Simon are given. For a sequence ¯ = (ml , …, mk) of positive integers, subclasses (m1, ...,mk) of languages of level k are defined. (ml, …, mk) are shown to be decidable. Some properties of the characterizing congruences are studied, among them, a condition which insures (m1, mk) to be included in ( , …, ). A conjecture of Pin concerning tree hierarchies of monoids (the dot-depth being a particular case) is shown to be false

Going higher in the First-order Quantifier Alternation Hierarchy on Words

We investigate the quantifier alternation hierarchy in first-order logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language to the levels $\mathcal{B}\Sigma_2$ (boolean combination of formulas having only 1 alternation) and $\Sigma_3$ (formulas having only 2 alternations beginning with an existential block). Our proof works by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels