Straubing's wreath product principle provides a description of the languages recognized by the wreath product of two monoids. A similar principle for ordered semigroups is given in this paper. Applications to language theory extend standard results of the theory of varieties to positive varieties. They include a characterization of positive locally testable languages and syntactic descriptions of the operations L ! La and L ! LaA . Next we turn to concatenation hierarchies. It was shown by Straubing that the n-th level Bn of the dot-depth hierarchy is the variety V n LI, where LI is the variety of locally trivial semigroups and V n is the n-th level of the Straubing-Therien hierarchy. We prove that a similar result holds for the half levels. It follows in particular that a level or a half level of the dot-depth hierarchy is decidable if and only if the corresponding level of the Straubing-Therien hierarchy is decidable. 1 Introduction All semigroups and monoids considered in t...

Q: I see. A: Now an old result of Buchi [8, 9] states that a language is regular if and only if it can be de ned by a monadic second order sentence. The result also holds for in nite words: just change regular into !-regular in the statement. You remember that a set of in nite words is !-regular if and only if it is accepted by a Buchi automaton. Q: Or, equivalently, if it is a nite union of sets of the form XY , where X and Y are regular languages. But wait a minute! Why did you consider suddenly monadic second order logic? What about full second order and rst order logic? A: You're right. Both classes are very interesting too. But monadic second order is an important border: if you consider weaker logics, such as rst order, you are sure to deal with regular languages and you can hope to have easy solutions for your problems on logic by converting them into problems on nite automata. This will be the topic of our conversation today and you will see that reality is qu

A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fi xed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent in the sense that they possess the same sets of satisfying assignments. We prove a Dichotomy Theorem by showing that for all sets C of allowed constraints, this problem is either polynomial-time solvable or coNP-complete, and we give a simple criterion to determine which case holds. Another equivalence problem...

We introduce a strict hierarchy fL B n g of language classes which exhausts the class of starfree regular languages. It is shown for all n 0 that the classes L B n have decidable membership problems. As the main result, we prove that our hierarchy is levelwise comparable by inclusion to the dot-depth hierarchy, more precisely, L B n contains all languages having dot-depth n+1=2. This yields a lower bound algorithm for the dot-depth of a given language. The same results hold for a hierarchy fL L n g and the Straubing-Thérien hierarchy.

By definition, the class B1 of dot-depth one languages is the Boolean closure of the class B 1=2 of languages that can be written as finite unions of u0A + u1 A + un , where u i 2 A . So dot-depth one languages can be described by Boolean combinations of patterns (u0 , u1 , ..., un ) in words which captures locally testable and piecewise testable properties. From a descriptional complexity point of view, the lengths of the u i reflect sequential aspects, while the Boolean operations measure combinatorial complexity. We prove that the Boolean hierarchy over B 1=2 is decidable and strict, which has consequences in first-order logic and complexity theory. Moreover, we effectively characterize the fine structure of B1 w.r.t. the mentioned sequential and combinatorial measures. This allows the exact location of a given language in this two-dimensional landscape in a computable way.

We make the following progress on the dot--depth problem: (1) We introduce classes C B n and C L n of starfree languages defined via forbidden patterns in finite automata. It is shown for all n 0 that C B n (C L n ) contains level n + 1=2 of the dot--depth hierarchy (Straubing--Therien hierarchy, resp.). Since we prove that C B n and C L n have decidable membership problems, this yields a lower bound algorithm for the dot--depth of a given language. (2) We prove many structural similarities between our hierarchies fC B n g and fC L n g, and the mentioned concatenation hierarchies. Both show the same inclusion structure and can be separated by the same languages. Moreover, we see that our pattern classes are not too large, since C L n does not capture level n + 1=2 of the dot--depth hierarchy. We establish an effective conjecture for the dot-- depth problem, namely that C B n and C L n , and the respective levels of concatenation hierarchies in fact coincide. I...

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The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: • The classes of the Boolean hierarchy over level Σ1 of the dot-depth hierarchy are decidable in NL (previously only the decidability was known). The same remains true if predicates mod d for fixed d are allowed. • If modular predicates for arbitrary d are allowed, then the classes of the Boolean hierarchy over level Σ1 are decidable. • For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level Σ2 of the Straubing-Thérien hierarchy are decidable in NL. This is the first decidability result for this hierarchy. • The membership problems for all mentioned Boolean-hierarchy classes are logspace many-one hard for NL. • The membership problems for quasi-aperiodic languages and for d-quasi-aperiodic languages are logspace many-one complete for PSPACE.