Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Place, Thomas; Zeitoun, Marc

doi:10.1145/3303991

Cited by 22 publications

(22 citation statements)

References 111 publications

(168 reference statements)

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“…An interesting consequence of our results is that since we proved the decidability of separation for the level two in the Straubing-Thérien hierarchy, the main theorem of [PZ19] is an immediate corollary: membership for this level is decidable. However, the algorithm of [PZ19] was actually based on a characterization theorem: languages of level two in the Straubing-Thérien hierarchy are characterized by a syntactic property of a canonical recognizer (i.e., their syntactic monoid). It turns out that one can also deduce this characterization theorem from our results (this does require some combinatorial work however).…”

Section: :32mentioning

confidence: 70%

Separation for dot-depth two

Place¹,

Zeitoun²

2021

Logical Methods in Computer Science

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The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level? Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation is that: (1) while more difficult, separation is more rewarding (2) it provides a more convenient framework (3) all recent membership algorithms are reductions to separation for lower levels. We present a separation algorithm for dot-depth two. While this is our most prominent application, our result is more general. We consider a family of hierarchies that includes the dot-depth: concatenation hierarchies. They are built via a generic construction process. One first chooses an initial class, the basis, which is the lowest level in the hierarchy. Further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis is finite, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results.

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Section: :32mentioning

confidence: 70%

Separation for dot-depth two

Place¹,

Zeitoun²

2021

Logical Methods in Computer Science

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“…In particular, it seems unlikely that BP ol(C)-membership boils down to C-separation in the general case. Indeed, a specialized characterization for the class BP ol(BP ol(ST)) is known [14]. Yet, deciding it involves looking at more general question than BP ol(ST)-separation.…”

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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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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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“…Finding membership algorithms has been an important quest for a long time in formal languages theory. The solutions that were obtained for important classes are milestones in the theory of regular languages [13,22,33,36,38,40]. In the paper, we prove two of them: Schützenberger's theorem [36] and Simon's theorem [38].…”

Section: Regularmentioning

confidence: 88%

“…An algorithm for dot-depth one was published by Knast in 1983 [13]. Despite a lot of partial results along the way, it took thirty more years to solve the next level: the decidability of dot-depth two was shown in 2014 [26,33]. This situation is easily explained: in practice, getting new membership results always required new conceptual ideas and techniques.…”

Section: Introductionmentioning

confidence: 99%

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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Going Higher in First-Order Quantifier Alternation Hierarchies on Words

Cited by 22 publications

References 111 publications

Separation for dot-depth two

Separation for dot-depth two

Separation and covering for group based concatenation hierarchies

Deciding Classes of Regular Languages: The Covering Approach

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