Generic Results for Concatenation Hierarchies

Place, Thomas; Zeitoun, Marc

doi:10.1007/s00224-018-9867-0

Cited by 19 publications

(64 citation statements)

References 45 publications

(90 reference statements)

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“…Finally, we have the following property which holds because F ⊆ Pol(co-D). This is a corollary of a generic property of polynomial closure presented in [27].…”

Section: Omitted Proofs In Sectionmentioning

confidence: 57%

“…Additionally, one may combine Theorem 8 with a result of [27] to get information on level 5/2 of group based hierarchies. It is shown in [27] that given an arbitrary hierarchy, if separation is decidable for some half level, then so is the membership problem for the next half level. Hence, we get the following corollary of Theorem 8.…”

Section: Concatenation Hierarchies and Main Theoremmentioning

confidence: 99%

“…We may now handle the last two operations in the definition of PBPol (C)-complete subsets. First, we use K to construct two auxiliary classes of languages D and F that we shall need for the proof (the argument reuses some standard results which are detailed in [27]).…”

Section: Omitted Proofs In Sectionmentioning

confidence: 99%

“…Consequently, there are finitely many languages in Pol (C) such that every K ∈ K is built by applying unions and of marked concatenations to complements of these languages. It then follows from [27,Lemma 17] that we may build a finite quotient-closed lattice D ⊆ Pol (C) which contains all these languages as well as L ∈ C ⊆ Pol (C). Thus, by definition, we have K ∈ Pol (co-D) for every K ∈ K. Finally, since K is finite, we may reuse [27,Lemma 17] to build another finite quotient-closed lattice F such that F ⊆ Pol(co-D) and K ∈ F for every K ∈ K. This concludes the proof.…”

Section: Omitted Proofs In Sectionmentioning

confidence: 99%

“…Therefore, our generic statement applies to the group hierarchy of Pin and Margolis. The other examples have a nice logical interpretation. Indeed, the result of Thomas [33] was generalized in [27]: every concatenation hierarchy corresponds, level by level, to 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.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Finally, we have the following property which holds because F ⊆ Pol(co-D). This is a corollary of a generic property of polynomial closure presented in [27].…”

Section: Omitted Proofs In Sectionmentioning

confidence: 57%

Section: Concatenation Hierarchies and Main Theoremmentioning

confidence: 99%

Section: Omitted Proofs In Sectionmentioning

confidence: 99%

Section: Omitted Proofs In Sectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Geometrical Closure of Binary $$V_{3/2}$$ Languages

Dubernard

Guaiana

Mignot

2019

Language and Automata Theory and Applications

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Deciding Classes of Regular Languages: The Covering Approach

Place

2020

Language and Automata Theory and Applications

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We investigate the membership problem that one may associate to every class of languages C. The problem takes a regular language as input and asks whether it belongs to C. In practice, finding an algorithm provides a deep insight on the class C. While this problem has a long history, many famous open questions in automata theory are tied to membership. Recently, a breakthrough was made on several of these open questions. This was achieved by considering a more general decision problem than membership: covering. In the paper, we investigate how the new ideas and techniques brought about by the introduction of this problem can be applied to get new insight on earlier results. In particular, we use them to give new proofs for two of the most famous membership results: Schützenberger's theorem and Simon's theorem.

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Generic Results for Concatenation Hierarchies

Cited by 19 publications

References 45 publications

Separation and covering for group based concatenation hierarchies

Separation and covering for group based concatenation hierarchies

Geometrical Closure of Binary $$V_{3/2}$$ Languages

Deciding Classes of Regular Languages: The Covering Approach

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