9 research outputs found
Polynomials, fragments of temporal logic and the variety DA over traces
We show that some language theoretic and logical characterizations of recognizable word languages whose syntactic monoid is in the variety DA also hold over traces. To this aim we give algebraic characterizations for the language operations of generating the polynomial closure and generating the unambiguous polynomial closure over traces.
We also show that there exist natural fragments of local temporal logic that describe this class of languages corresponding to DA. All characterizations are known to hold for words
Fragments of first-order logic over infinite words
We give topological and algebraic characterizations as well as language
theoretic descriptions of the following subclasses of first-order logic FO[<]
for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and
Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_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 and Bojanczyk and
is therefore not our main focus. The paper is about the interplay of algebraic,
topological, and language theoretic properties.Comment: Conference version presented at 26th International Symposium on
Theoretical Aspects of Computer Science, STACS 200
The expressive power of simple logical fragments over traces
We compare the expressive power of some first-order fragments and of two simple temporal logics over Mazurkiewicz traces. Over words, most of these fragments have the same expressive power whereas over traces we show that the ability of formulating concurrency increases the expressive power.
We also show that over so-called dependence structures it is impossible to formulate concurrency with the first-order fragments under consideration. Although the first-order fragments and over partial orders both can express concurrency of two actions, we show that in general they are incomparable over traces. For we give a characterization in terms of temporal logic by allowing an operator for parallelism
On logical hierarchies within FO^2-definable languages
We consider the class of languages defined in the 2-variable fragment of the
first-order logic of the linear order. Many interesting characterizations of
this class are known, as well as the fact that restricting the number of
quantifier alternations yields an infinite hierarchy whose levels are varieties
of languages (and hence admit an algebraic characterization). Using this
algebraic approach, we show that the quantifier alternation hierarchy inside
FO^{2}[<] is decidable within one unit. For this purpose, we relate each level
of the hierarchy with decidable varieties of languages, which can be defined in
terms of iterated deterministic and co-deterministic products. A crucial notion
in this process is that of condensed rankers, a refinement of the rankers of
Weis and Immerman and the turtle languages of Schwentick, Th\'erien and
Vollmer.Comment: arXiv admin note: text overlap with arXiv:0904.289
Rankers over infinite words
We consider the fragments FO2, the intersection of Sigma2 and FO2, the intersection of Pi2 and FO2, and Delta2 of first-order logic FO[<] over finite and infinite words. For all four fragments, we give characterizations in terms of rankers. In particular, we generalize the notion of a ranker to infinite words in two possible ways. Both extensions are natural in the sense that over finite words, they coincide with classical rankers and over infinite words, they both have the full expressive power of FO2. Moreover, the first extension of rankers admits a characterization of the intersection of Sigma2 and FO2 while the other leads to a characterization of the intersection of Pi2 and FO2. Both versions of rankers yield characterizations of the fragment Delta2. As a byproduct, we also obtain characterizations based on unambiguous temporal logic and unambiguous interval temporal logic
Fragments of first-order logic over infinite words
We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic for omega-languages: Sigma2, FO2, the intersection of FO2 and Sigma2, and Delta2 (and by duality Pi2 and the intersection of FO2 and Pi2). 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 and Bojanczyk and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties
Polynomials, Fragments of Temporal Logic and the Variety DA over Traces
Abstract We show that some language theoretic and logical characterizations of recognizable word languages whose syntactic monoid is in the variety DA also hold over traces. To this end we give algebraic characterizations for the language operations of generating the polynomial closure and generating the unambiguous polynomial closure over traces. We also show that there exist natural fragments of local temporal logic that describe this class of languages corresponding to DA. All characterizations are known to hold for words
Polynomials, Fragments of Temporal Logic and the Variety DA over Traces
Abstract. We show that some language theoretic and logical characterizations of recognizable word languages whose syntactic monoid is in the variety DA also hold over traces. To this aim we give algebraic characterizations for the language operations of generating the polynomial closure and generating the unambiguous polynomial closure over traces. We also show that there exist natural fragments of local temporal logic that describe this class of languages corresponding to DA. All characterizations are known to hold for words. 1 Introduction Traces were introduced by Mazurkiewicz as a generalization of words to de-scribe the behavior of concurrent processes [4]. Since then traces have become a rather popular setting to study concurrency. A lot of aspects of traces and tracelanguages have been researched, see [1] for an overview. Over words it has turned out that finite semigroups are a powerful tech-nique to refine the class of recognizable languages [2]. Two natural operations on classes of languages are the polynomial closure and the unambiguous polyno-mial closure. For particular classes of languages, so called language varieties, it has been shown that there exist algebraic counterparts in terms of the so-calledMal'cev product [10]. In Section 3 (resp. Section 4) we will show that this correspondence between the Mal'cev product and the polynomial closure (resp. theunambiguous polynomial closure) for restricted varieties also holds over traces