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A survey on small fragments of firstorder logic over finite words
 International Journal of Foundations of Computer Science
, 2008
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FRAGMENTS OF FIRSTORDER LOGIC OVER INFINITE WORDS (EXTENDED ABSTRACT)
 SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
, 2008
"... We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of firstorder logic FO[<] for ωlanguages: Σ2, ∆2, FO 2 ∩ Σ2 (and by duality FO 2 ∩ Π2), and FO 2. These descriptions extend the respective results for finite words. In part ..."
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We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of firstorder logic FO[<] for ωlanguages: Σ2, ∆2, FO 2 ∩ Σ2 (and by duality FO 2 ∩ Π2), and FO 2. These descriptions extend the respective results for finite words. In particular, we relate the above fragments to language classes of certain (unambiguous) polynomials. An immediate consequence is the decidability of the membership problem of these classes, but this was shown before by Wilke [20] and Bojańczyk [2] and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.
EFFICIENT ALGORITHMS FOR MEMBERSHIP IN BOOLEAN HIERARCHIES OF REGULAR LANGUAGES
, 2008
"... 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 forbiddenchain characterizations for the single levels of these hierarchies and obtain ..."
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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 forbiddenchain characterizations for the single levels of these hierarchies and obtain the following results: • The classes of the Boolean hierarchy over level Σ1 of the dotdepth 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 twoletter alphabet, the classes of the Boolean hierarchy over level Σ2 of the StraubingThérien hierarchy are decidable in NL. This is the first decidability result for this hierarchy. • The membership problems for all mentioned Booleanhierarchy classes are logspace manyone hard for NL. • The membership problems for quasiaperiodic languages and for dquasiaperiodic languages are logspace manyone complete for PSPACE.
LOGIC MEETS ALGEBRA: THE CASE OF REGULAR LANGUAGES PASCAL TESSON AND DENIS THÉRIEN
"... Abstract. The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point ..."
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Abstract. The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and blockproducts of pseudovarieties of monoid. We also explain the impact of these connections to circuit complexity theory. 1.
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"... erstellt und ausschließlich die angegebenen Quellen und Hilfsmittel benutzt zu haben. ..."
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erstellt und ausschließlich die angegebenen Quellen und Hilfsmittel benutzt zu haben.
Kleene theorems for product systems
"... We prove Kleene theorems for two subclasses of labelled product systems which are inspired from wellstudied subclasses of 1bounded Petri nets. For product Tsystems we define a corresponding class of expressions. The algorithms from systems to expressions and in the reverse direction are both pol ..."
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We prove Kleene theorems for two subclasses of labelled product systems which are inspired from wellstudied subclasses of 1bounded Petri nets. For product Tsystems we define a corresponding class of expressions. The algorithms from systems to expressions and in the reverse direction are both polynomial time. For product free choice systems with a restriction of structural cyclicity, that is, the initial global state is a feedback vertex set, going from systems to expressions is still polynomial time; in the reverse direction it is polynomial time with access to an NP oracle for finding deadlocks.