Fragments of First-Order Logic over Infinite Words

Diekert, Volker; Kufleitner, Manfred

doi:10.1007/s00224-010-9266-7

Cited by 19 publications

(63 citation statements)

References 35 publications

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“…The subsequent two topologies are derived from Diekert's and Kufleitner's [3] alphabetic topology which is useful for investigations in restricted first-order theories for infinite words.…”

Section: Topologies Related To Finite Automatamentioning

confidence: 99%

“…We start with the alphabetic topology which was introduced in [3]. Then we consider a variant of the alphabetic topology.…”

Section: The Alphabetic Topologiesmentioning

confidence: 99%

“…For the next definition we fix the following notation (cf. [3]). For A ⊆ X the ω-language A im is the set of all ω-words ξ ∈ X ω where exactly the letters in A occur infinitely often.…”

Section: Definition 3 the Alphabetic Topology Is Defined By The Basementioning

confidence: 99%

“…for investigations in first-order logic [3], to characterise the set of random infinite words [1] or the set of disjunctive infinite words [15] and to describe the converging behaviour of not necessarily hyperbolic iterative function systems [5,14].…”

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confidence: 99%

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Topologies Refining the Cantor Topology on X ω

Schwarz¹,

Staiger²

2010

IFIP Advances in Information and Communication Technology

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Abstract. The space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science. Usually, it is convenient to regard this space as a metric space, the Cantor-space. It turned out that for several purposes topologies other than the one of the Cantor-space are useful, e.g. for studying fragments of first-order logic over infinite words or for a topological characterisation of random infinite words.It is shown that both of these topologies refine the topology of the Cantor-space. Moreover, from common features of these topologies we extract properties which characterise a large class of topologies. It turns out that, for this general class of topologies, the corresponding closure and interior operators respect the shift operations and also, to some respect, the definability of sets of infinite words by finite automata.The space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science (see [7,20] and the surveys [6,13,17,18]). Usually, it is convenient to regard this space as a topological space provided with the Cantor topology. This topology can be also considered as the natural continuation of the left topology of the prefix relation on the space of finite words; for a survey see [2].It turned out that for several purposes other topologies on the space of infinite words are also useful [9,12], e.g. for investigations in first-order logic [3], to characterise the set of random infinite words [1] or the set of disjunctive infinite words [15] and to describe the converging behaviour of not necessarily hyperbolic iterative function systems [5,14].Most of these papers use topologies on the space of infinite words which are certain refinements of the Cantor topology showing a certain kind of shift invariance. The aim of this paper is to give a unified treatment of those topologies and to investigate their relations to Cantor topology.Special attention is paid to subsets of the space of infinite words definable by finite automata. It turns out that several of the refinements of the Cantor topology under consideration behave well with respect to finite automata, that is, the corresponding closure and interior operators preserve at least one of the classes of finite-state or regular ω-languages.

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“…The subsequent two topologies are derived from Diekert's and Kufleitner's [3] alphabetic topology which is useful for investigations in restricted first-order theories for infinite words.…”

Section: Topologies Related To Finite Automatamentioning

confidence: 99%

“…We start with the alphabetic topology which was introduced in [3]. Then we consider a variant of the alphabetic topology.…”

Section: The Alphabetic Topologiesmentioning

confidence: 99%

Section: Definition 3 the Alphabetic Topology Is Defined By The Basementioning

confidence: 99%

mentioning

confidence: 99%

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Topologies Refining the Cantor Topology on X ω

Schwarz¹,

Staiger²

2010

IFIP Advances in Information and Communication Technology

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“…It turns out that topology is a very useful tool for restricting the infinite behaviour of the algebraic approach accordingly, see e.g. [4,6,10,21]. In particular, the combination of algebra and topology is convenient for the study of languages in Γ ∞ , the set of finite and infinite words over the alphabet Γ.…”

Section: Introductionmentioning

confidence: 99%

Level Two of the Quantifier Alternation Hierarchy Over Infinite Words

Kufleitner

Walter

2017

Theory Comput Syst

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The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragment Σ 2 over infinite words. Here, Σ 2 consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is BΣ 2 . Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an ω-regular language L, it is decidable whether L is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun's recent decidability result for BΣ 2 from finite to infinite words.

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Well-Quasi Orders and Hierarchy Theory

Selivanov

2020

Trends in Logic

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We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.

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Fragments of First-Order Logic over Infinite Words

Cited by 19 publications

References 35 publications

Topologies Refining the Cantor Topology on X ω

Topologies Refining the Cantor Topology on X ω

Level Two of the Quantifier Alternation Hierarchy Over Infinite Words

Well-Quasi Orders and Hierarchy Theory

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