Level Two of the Quantifier Alternation Hierarchy over Infinite Words

Kufleitner, Manfred; Walter, Tobias

doi:10.1007/978-3-319-34171-2_16

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…On the other hand, combining algebraic properties with topology has proven a fruitful alternative in some cases where algebra alone is not enough; see e.g. [6,12,16]. In particular, this approach was used in yet unpublished work by Boussidan and the second author for the characterization of the join levels of the alternation hierarchies, and by the authors for the characterization of the half-levels [11].…”

Section: Introductionmentioning

confidence: 99%

Deciding FO2 Alternation for Automata over Finite and Infinite Words

Henriksson¹,

Kufleitner²

2021

Preprint

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We consider two-variable first-order logic FO 2 and its quantifier alternation hierarchies over both finite and infinite words. Our main results are forbidden patterns for deterministic automata (finite words) and for Carton-Michel automata (infinite words). In order to give concise patterns, we allow the use of subwords on paths in finite graphs. This concept is formalized as subword patterns. Deciding the presence or absence of such a pattern in a given automaton is in NL. In particular, this leads to NL algorithms for deciding the levels of the FO 2 quantifier alternation hierarchies. This applies to both full and half levels, each over finite and infinite words. Moreover, we show that these problems are NL-hard and, hence, NL-complete.

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Section: Introductionmentioning

confidence: 99%

Deciding FO2 Alternation for Automata over Finite and Infinite Words

Henriksson¹,

Kufleitner²

2021

Preprint

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“…[18,Chapter VIII]. On the other hand, the combination of such algebraic properties with topology can provide characterizations of various language classes; see e.g [7,12,15]. Usually, the topological conditions restrict the behavior at infinity.…”

Section: Introductionmentioning

confidence: 99%

Nesting negations in FO2 over infinite words

Henriksson,

Kufleitner

2020

Preprint

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We consider two-variable first-order logic FO 2 over infinite words. Restricting the number of nested negations defines an infinite hierarchy; its levels are often called the half-levels of the FO 2 quantifier alternation hierarchy. For every level of this hierarchy, we give an effective characterization. For the lower levels, this characterization is a combination of an algebraic and a topological property. For the higher levels, algebraic properties turn out to be sufficient. Within two-variable first-order logic, each algebraic property is a single ordered identity of omega-terms. The topological properties are the same as for the lower half-levels of the quantifier alternation hierarchy without the two-variable restriction (i.e., the Cantor topology and the alphabetic topology).Our result generalizes the corresponding result for finite words. The proof uses novel techniques and is based on a refinement of Mal'cev products for ordered monoids.

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“…On the other hand, new issues specific to infinite words arise, for example, we were not able to generalize the transfer result from Σ iseparation to Σ i+1 -membership (as a consequence, membership for Σ 4 remains open). Note that a different proof for deciding BΣ 2 -membership has been obtained independently in [9]. It gives a decidable characterization based on topology and algebra.…”

Section: Introductionmentioning

confidence: 99%

Quantifier Alternation for Infinite Words

Pierron¹,

Place²,

Zeitoun³

2016

Lecture Notes in Computer Science

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Abstract. We investigate the expressive power of quantifier alternation hierarchy of first-order logic over words. This hierarchy includes the classes Σi (sentences having at most i blocks of quantifiers starting with an ∃) and BΣi (Boolean combinations of Σi sentences). So far, this expressive power has been effectively characterized for the lower levels only. Recently, a breakthrough was made over finite words, and decidable characterizations were obtained for BΣ2 and Σ3, by relying on a decision problem called separation, and solving it for Σ2.The contribution of this paper is a generalization of these results to the setting of infinite words: we solve separation for Σ2 and Σ3, and obtain decidable characterizations of BΣ2 and Σ3 as consequences.

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Level Two of the Quantifier Alternation Hierarchy over Infinite Words

Cited by 4 publications

References 14 publications

Deciding FO2 Alternation for Automata over Finite and Infinite Words

Deciding FO2 Alternation for Automata over Finite and Infinite Words

Nesting negations in FO2 over infinite words

Quantifier Alternation for Infinite Words

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