One quantifier alternation in first-order logic with modular predicates

Kufleitner, Manfred; Walter, Tobias

doi:10.1051/ita/2014024

Cited by 10 publications

(13 citation statements)

References 27 publications

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“…It is simple to show that this basis consists of group languages and has decidable separation. Hence, our generic theorem applies: it strengthens and unifies results of [6,12], which dealt with membership at levels up to 3/2, and [34], which proved separation at level 1 with a combinatorial proof leading to a brute force algorithm, orthogonal to our techniques. Finally, when the basis consists of languages recognized by commutative groups, we obtain by [27,8] the quantifier alternation hierarchy of first-order logic endowed with predicates counting the number of occurrences of a letter before a position, modulo some integer.…”

Section: Introductionsupporting

confidence: 64%

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Separation and covering for group based concatenation hierarchies

Place

Zeitoun

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Concatenation hierarchies are natural classifications of regular languages. All such hierarchies are built through the same construction process: one starts from an initial, specific class of languages (the basis) and builds new levels using two generic operations. Concatenation hierarchies have gathered a lot of interest since the early 70s, notably thanks to an alternate logical definition: each concatenation hierarchy can be defined as the quantification alternation hierarchy within a variant of first-order logic over words (while the hierarchies differ by their bases, the variants differ by their set of available predicates).Our goal is to understand these hierarchies. A typical approach is to look at two decision problems: membership and separation. In the paper we are interested in the latter, which is more general. For a class of languages C, C-separation takes two regular languages as input and asks whether there exists a third one in C including the first one and disjoint from the second one. Settling whether separation is decidable for the levels within a given concatenation hierarchy is among the most fundamental and challenging questions in formal language theory. In all prominent cases, it is open, or answered positively for low levels only. Recently, a breakthrough was made using a generic approach for a specific kind of hierarchy: those with a finite basis. In this case, separation is always decidable for levels 1/2, 1 and 3/2. Our main theorem is similar but independent: we consider hierarchies with possibly infinite bases, but that may only contain group languages. An example is the group hierarchy of Pin and Margolis: its basis consists of all group languages. Another example is the quantifier alternation hierarchy of first-order logic with modular predicates (FO(<, MOD)): its basis consists of the languages that count the length of words modulo some number. Using a generic approach, we show that for any such hierarchy, if separation is decidable for the basis, then it is decidable as well for levels 1/2, 1 and 3/2 (we actually solve a more general problem called covering). This complements the aforementioned result nicely: all bases considered in the literature are either finite or made of group languages. Thus, one may handle the lower levels of any prominent hierarchy in a generic way.Both authors acknowledge support from the DeLTA project (ANR-16-CE40-0007).

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Section: Introductionsupporting

confidence: 64%

“…This hierarchy has been widely investigated in the literature. For membership, it was known that the problem is decidable for BΣ 1 (<, MOD) (see [6]) as well as Σ 1 (<, MOD) and Σ 2 (<, MOD) (see [12]). Furthermore, it was recently shown that separation is decidable for BΣ 1 (<, MOD) [34].…”

Section: Factmentioning

confidence: 99%

Separation and covering for group based concatenation hierarchies

Place

Zeitoun

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…Another interesting class of predicates are modular predicates. In [7] the authors have studied Σ 2 [<, MOD] over finite words. The results of [7] can be generalised to infinite words by adapting the alphabetic topology to the modular setting.…”

Section: Summary and Open Problemsmentioning

confidence: 99%

Level Two of the Quantifier Alternation Hierarchy over Infinite Words

Kufleitner

Walter

2016

Computer Science – Theory and Applications

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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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“…In [7] the authors have studied Σ 2 [<, MOD] over finite words. The results of [7] can be generalised to infinite words by adapting the alphabetic topology to the modular setting.…”

Section: Summary and Open Problemsmentioning

confidence: 99%

“…In [7] the authors have studied Σ 2 [<, MOD] over finite words. The results of [7] can be generalised to infinite words by adapting the alphabetic topology to the modular setting. As for successor predicates, we believe that an appropriate effective characterization of this topology might help in deciding BΣ 2 [<, MOD] over infinite words.…”

Section: Summary and Open Problemsmentioning

confidence: 99%