Factorization forests for infinite words and applications to countable scattered linear orderings

Colcombet, Thomas

doi:10.1016/j.tcs.2009.10.013

Cited by 20 publications

(22 citation statements)

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“…We state this theorem here. For more details on factorization forests and a proof of the theorem, we refer the reader to [Bojańczyk 2009;Colcombet 2010;Kufleitner 2008;Colcombet 2015]. Let M be a finite monoid and α : A * → M a morphism.…”

Section: Simon's Factorization Forests Theoremmentioning

confidence: 99%

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Place

Zeitoun

2019

J. ACM

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We investigate quantifier alternation hierarchies in first-order logic on finite words. Levels in these hierarchies are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language in the levels BΣ 2 (finite boolean combinations of formulas having only one alternation) and Σ 3 (formulas having only two alternations and beginning with an existential block). Our proofs work by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels.A:3 to be decidable. The class BΣ 1 (<) consists exactly of all piecewise testable languages, i.e., such that membership of a word only depends on its scattered subwords up to a fixed size. These languages were characterized by Simon [1975] as those whose syntactic monoid is J-trivial. A decidable characterization of Σ 2 (<)-hence of ∆ 2 (<) as well-was obtained by Arfi [1987;, a problem revisited and clarified by Pin and Weil [1995;, who also set up a generic algebraic framework to work with. For ∆ 2 (<), the literature is very rich, see the survey by Tesson and Thérien [2002]. For example, the ∆ 2 (<) definable languages are exactly the ones definable in the two-variable restriction of FO(<) [Thérien and Wilke 1998]. These are also the languages whose syntactic monoid belongs to the class DA, as shown again by Pin and Weil [1995; (see also [Schützenberger 1976]). For higher levels in the hierarchy, getting decidable characterizations remained a major open problem. In particular, the case of BΣ 2 (<) has a very abundant history and a series of combinatorial, logical, and algebraic conjectures have been proposed over the years. We refer to Section 3 and to several surveys cited in this section for a bibliography. So far, the only known effective result was partial, working only when the alphabet is of size 2 [Straubing 1988].Contributions.. In this paper, we establish decidable characterizations for the fragments BΣ 2 (<), ∆ 3 (<) and Σ 3 (<) of first-order logic. These new results are based on a deeper decision problem than membership: the separation problem. Fix a class C of languages. The C-separation problem amounts to deciding whether, given two input regular languages, there exists a third language in C containing the first language while being disjoint from the second one. Solving the C-separation problem is more general than obtaining a decidable characterization for the class C. Indeed, since regular languages are effectively closed under complement, testing membership in C can be achieved by testing whether the input is C-separable from its complement. While this reduction immediately transfers decision procedures for one problem to the other, this is not our primary motivation for looking at separation. Although intrinsically more challenging, a solution to the separation problem requires more understanding than just getting a decidable characterization. This understanding for a given fragment can then be exploited in order to ob...

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Section: Simon's Factorization Forests Theoremmentioning

confidence: 99%

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Place

Zeitoun

2019

J. ACM

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“…Here, we briefly recall this theorem. We refer the reader to [Kuf08,Boj09,Col10] for more details and a proof.…”

Section: 2mentioning

confidence: 99%

Separating Regular Languages with Two Quantifiers Alternations

Place

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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We investigate a famous decision problem in automata theory: separation. Given a class of language C, the separation problem for C takes as input two regular languages and asks whether there exists a third one which belongs to C, includes the first one and is disjoint from the second. Typically, obtaining an algorithm for separation yields a deep understanding of the investigated class C. This explains why a lot of effort has been devoted to finding algorithms for the most prominent classes.Here, we are interested in classes within concatenation hierarchies. Such hierarchies are built using a generic construction process: one starts from an initial class called the basis and builds new levels by applying generic operations. The most famous one, the dot-depth hierarchy of Brzozowski and Cohen, classifies the languages definable in first-order logic. Moreover, it was shown by Thomas that it corresponds to the quantifier alternation hierarchy of first-order logic: each level in the dot-depth corresponds to the languages that can be defined with a prescribed number of quantifier blocks. Finding separation algorithms for all levels in this hierarchy is among the most famous open problems in automata theory.Our main theorem is generic: we show that separation is decidable for the level 3 2 of any concatenation hierarchy whose basis is finite. Furthermore, in the special case of the dot-depth, we push this result to the level 5 2 . In logical terms, this solves separation for Σ3: first-order sentences having at most three quantifier blocks starting with an existential one. CC Creative Commons THOMAS PLACEWe use separation as a mathematical tool whose purpose is to investigate classes of languages: given a fixed class C, obtaining a C-separation algorithm usually requires a solid understanding of C. A typical objective when considering separation is to not only get an algorithm that decides it, but also a generic method for computing a separator, if it exists.Remark 2.1. C-separation generalizes another classical decision problem: C-membership which asks whether a single regular language L belongs to C. Indeed, it is simple to verify that asking whether L ∈ C is equivalent to asking whether L is C-separable from its complement (in this case, the only candidate for being a separator is L itself ).Covering. Our second problem is more general and was originally defined in [PZ16,PZ18a]. Given a language L, a cover of L is a finite set of languages K such that, L ⊆ K∈K K Moreover, given a class C, a C-cover of L is a cover K of L such that all K ∈ K belong to C.Covering takes as input a language L and a finite set of languages L. A separating cover for the pair (L, L) is a cover K of L such that for every K ∈ K, there exists L ∈ L which satisfies K ∩ L = ∅. Finally, given a class C, we say that the pair (L, L) is C-coverable when there exists a separating C-cover. The C-covering problem is now defined as follows:INPUT: A regular language L and a finite set of regular languages L. OUTPUT: Is (L, L) C-coverable ?Remark 2.2. Thi...

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“…Its proof, which can be found in [16], follows the same lines of the recent proofs of Simon's theorem, see e.g. [9,10].…”

Section: Theorem 8 (Unambiguous Forest Factorizationmentioning

confidence: 89%

Regular Transducer Expressions for Regular Transformations

Dave

Gastin

Krishna

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Functional MSO transductions, deterministic two-way transducers, as well as streaming string transducers are all equivalent models for regular functions. In this paper, we show that every regular function, either on finite words or on infinite words, captured by a deterministic two-way transducer, can be described with a regular transducer expression (RTE). For infinite words, the transducer uses Muller acceptance and ω-regular look-ahead. RTEs are constructed from constant functions using the combinators if-then-else (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product, the 2-chained Kleene-iteration and the 2chained omega-iteration. Our proof works for transformations of both finite and infinite words, extending the result on finite words of Alur et al. in LICS'14. In order to construct an RTE associated with a deterministic two-way Muller transducer with look-ahead, we introduce the notion of transition monoid for such two-way transducers where the look-ahead is captured by some backward deterministic Büchi automaton. Then, we use an unambiguous version of Imre Simon's famous forest factorization theorem in order to derive a "good" (ω-)regular expression for the domain of the two-way transducer. "Good" expressions are unambiguous and Kleene-plus as well as ω-iterations are only used on subexpressions corresponding to idempotent elements of the transition monoid. The combinator expressions are finally constructed by structural induction on the "good" (ω-)regular expression describing the domain of the transducer.

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Factorization forests for infinite words and applications to countable scattered linear orderings

Cited by 20 publications

References 30 publications

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Separating Regular Languages with Two Quantifiers Alternations

Regular Transducer Expressions for Regular Transformations

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