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Profinite Methods in Semigroup Theory
 Int. J. Algebra Comput
, 2000
"... this paper. The extended bibliography given below shows other important contributions by Azevedo, Costa, Delgado, Pin, Teixeira, Volkov, Weil and Zeitoun. ..."
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Cited by 19 (2 self)
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this paper. The extended bibliography given below shows other important contributions by Azevedo, Costa, Delgado, Pin, Teixeira, Volkov, Weil and Zeitoun.
On the lattice of subpseudovarieties of DA
"... Abstract. The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of subpseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hi ..."
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Abstract. The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of subpseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable subpseudovarieties of DA in terms of iterated Mal’cev products with the pseudovarieties of definite and reverse definite semigroups. The complete elucidation of the structure of the lattice LB of band varieties is one of the jewels of semigroup theory: this lattice turns out to be countable, with a simple structure (Birjukov [2], Fennemore [3], Gerhard [5], see Section 2.2 below for the main features of this structure). Moreover, each of its elements can be defined by a small number of identities (at most 3), and we can efficiently solve the membership problem in each variety of bands, as well as the word problem in its free object [6]. As bands are locally finite, the lattice L(B) of pseudovarieties of finite
On FO 2 quantifier alternation over words
 In Proceedings of the 34th Symposium on Mathematical Foundations of Computer Science (MFCS 2009), number 5734 in LNCS
, 2009
"... Abstract. We show that each level of the quantifier alternation hierarchy within FO 2 [<] on words is a variety of languages. We use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the q ..."
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Abstract. We show that each level of the quantifier alternation hierarchy within FO 2 [<] on words is a variety of languages. We use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier alternation hierarchy – and conjecturally equal to it. It follows that the latter hierarchy is decidable within one unit, a much more precise result than what is known about the quantifier alternation hierarchy within FO[<], where no decidability result is known beyond the very first levels. Firstorder logic is an important object of study in connection with computer science and language theory, not least because many important and natural problems are firstorder definable: our understanding of the expressive power of this logic and the efficiency of the solution of related algorithmic problems are of direct interest in such fields as verification. Here, by firstorder logic, we mean the firstorder logic of the linear order, FO[<], interpreted on finite words.
The FO 2 alternation hierarchy is decidable
"... We consider the twovariable fragment FO 2 [<] of firstorder logic over finite words. Numerous characterizations of this class are known. Thérien and Wilke have shown that it is decidable whether a given regular language is definable in FO 2 [<]. From a practical point of view, as shown by Weis, FO ..."
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We consider the twovariable fragment FO 2 [<] of firstorder logic over finite words. Numerous characterizations of this class are known. Thérien and Wilke have shown that it is decidable whether a given regular language is definable in FO 2 [<]. From a practical point of view, as shown by Weis, FO 2 [<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO 2 [<]definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO 2 [<]quantifier alternation hierarchy: deterministic and codeterministic products, Mal’cev products with definite and reverse definite semigroups, iterated block products with Jtrivial monoids, and some inductively defined omegaterm identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Thérien, and Vollmer.
POINTLIKE SETS WITH RESPECT TO R AND J
"... Abstract. We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety R of all finite Rtrivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperi ..."
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Abstract. We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety R of all finite Rtrivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute Jpointlike sets, where J denotes the pseudovariety of all finite Jtrivial semigroups. We finally show that, in contrast with the situation for R, the natural adaptation of Henckell’s algorithm to J computes pointlike sets, but not all of them. 1.
On FO² quantifier alternation over words
"... We show that each level of the quantifier alternation hierarchy within FO²[<] on words is a variety of languages. We use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier al ..."
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We show that each level of the quantifier alternation hierarchy within FO²[<] on words is a variety of languages. We use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier alternation hierarchy – and conjecturally equal to it. It follows that the latter hierarchy is decidable within one unit, a much more precise result than what is known about the quantifier alternation hierarchy within FO[<], where no decidability result is known beyond the very first levels.
Quantifier Alternation in TwoVariable FirstOrder Logic with Successor Is Decidable ∗
"... We consider the quantifier alternation hierarchy within twovariable firstorder logic FO 2 [<, suc] over finite words with linear order and binary successor predicate. We give a single identity of omegaterms for each level of this hierarchy. This shows that for a given regular language and a nonn ..."
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We consider the quantifier alternation hierarchy within twovariable firstorder logic FO 2 [<, suc] over finite words with linear order and binary successor predicate. We give a single identity of omegaterms for each level of this hierarchy. This shows that for a given regular language and a nonnegative integer m it is decidable whether the language is definable by a formula in FO 2 [<, suc] which has at most m quantifier alternations. We also consider the alternation hierarchy of unary temporal logic TL[X, F, Y, P] defined by the maximal number of nested negations. This hierarchy coincides with the FO 2 [<, suc] quantifier alternation hierarchy.