An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

Krebs, Andreas; Straubing, Howard

doi:10.1145/3149822

Cited by 13 publications

(16 citation statements)

References 24 publications

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“…To conclude a decomposition result, we then prove that this local property entails a global property. These steps are remarkably similar to results from the theory of logic on words and finite monoids which also reduce the problem of decidability to separation [13] and Local-Global theorems [9].…”

Section: Resultssupporting

confidence: 67%

Wreath Products of Distributive Forest Algebras

Hahn

Krebs

Straubing

2018

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

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It is an open problem whether definability in Propositional Dynamic Logic (PDL) on forests is decidable. Based on an algebraic characterization by in terms of forest algebras, Straubing (2013) described an approach to PDL based on a k-fold iterated distributive law. A proof that all languages satisfying such a k-fold iterated distributive law are in PDL would settle decidability of PDL. We solve this problem in the case k = 2: All languages recognized by forest algebras satisfying a 2-fold iterated distributive law are in PDL. Furthermore, we show that this class is decidable. This provides a novel nontrivial decidable subclass of PDL, and demonstrates the viability of the proposed approach to deciding PDL in general.

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Section: Resultssupporting

confidence: 67%

Wreath Products of Distributive Forest Algebras

Hahn

Krebs

Straubing

2018

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

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“…It would be interesting to see whether the proof scheme in [12] could also be used for proving our characterization of Σ 2 m .…”

Section: Resultsmentioning

confidence: 99%

“…. , } yields the following chain of inequalities: [12,35] to our recursion scheme without any major changes in the proofs. The converse also works: One could use the Almeida-Weil recursion scheme in our setting; however, one would need to add some arguments in the proof of Proposition 12 (essentially, in the proof of the Claim one would have to additionally use the assumption M ∈ DA).…”

Section: From Identities To Logicmentioning

confidence: 99%

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The Half-Levels of the FO2 Alternation Hierarchy

2016

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Abstract. The alternation hierarchy in two-variable first-order logic FO 2 [<] over words was shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. We consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment Σ 2 m of FO 2 is defined by disallowing universal quantifiers and having at most m − 1 nested negations. The Boolean closure of Σ 2 m yields the m th level of the FO 2 -alternation hierarchy. We give an effective characterization of Σ 2 m , i.e., for every integer m one can decide whether a given regular language is definable in Σ 2 m . Among other techniques, the proof relies on an extension of block products to ordered monoids.

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“…Finally, several subhierarchies of ∆ 2 = FO 2 were considered in the recent years [30,48,49,50,51,53,52,106,119].…”

Section: Theorem 71 a Language Is Locally Testable If Its Syntactic mentioning

confidence: 99%

The Dot-Depth Hierarchy, 45 Years Later

2017

The Role of Theory in Computer Science

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In 1970, R. S. Cohen and Janusz A. Brzozowski introduced a hierarchy of star-free languages called the dot-depth hierarchy. This hierarchy and its generalisations, together with the problems attached to them, had a long-lasting influence on the development of automata theory. This survey article reports on the numerous results and conjectures attached to this hierarchy.This paper is a follow-up of the survey article Open problems about regular languages, 35 years later [57]. The dot-depth hierarchy, also known as Brzozowski hierarchy, is a hierarchy of star-free languages first introduced by Cohen and Brzozowski [25] in 1971. It immediately gave rise to many interesting questions and an account of the early results can be found in Brzozowski's survey [20] from 1976. Terminology, notation and backgroundMost of the terminology used in this paper was introduced in [57]. We just complete these definitions by giving the ordered versions of the notions of syntactic monoid and variety of finite monoids. Syntactic order and positive varietiesAn ordered monoid is a monoid equipped with an order compatible with the multiplication: x y implies zx zy and xz yz.The syntactic preorder 1 of a language L of A * is the relation L defined on A * by u L v if and only if, for every x, y ∈ A * , xuy ∈ L ⇒ xvy ∈ L. * The author was funded from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 670624).1 Unfortunately, the author used the opposite order in earlier papers (from 1995 to 2011). 1The syntactic congruence of L is the associated equivalence relation ∼ L , defined by u ∼ L v if and only if u L v and v L u.The syntactic monoid of L is the quotient M (L) of A * by ∼ L and the natural morphism η : A * → A * /∼ L is called the syntactic morphism of L. The syntactic preorder L induces an order on the quotient monoid M (L). The resulting ordered monoid is called the syntactic ordered monoid of L.For instance, the syntactic monoid of the language {a, aba} is the monoid M = {1, a, b, ab, ba, aba, 0} presented by the relations a 2 = b 2 = bab = 0. Its syntactic order is given by the relations 0 < ab < 1, 0 < ba < 1, 0 < aba < a, 0 < b.The syntactic ordered monoid of a language was first introduced by Schützenberger [86] in 1956, but thereafter, he apparently only used the syntactic monoid.A positive variety of languages is a class of languages closed under finite unions, finite intersections, left and right quotients and inverses of morphisms. A variety of languages is a positive variety closed under complementation.Similarly, a variety of finite ordered monoids is a class of finite ordered monoids closed under taking ordered submonoids, quotients and finite products. Varieties of finite (ordered) semigroups are defined analogously. If V is a variety of ordered monoids, let V d denote the dual variety, consisting of all ordered monoids (M, ) such that (M, ) ∈ V. We refer the reader to the books [2, 28, 62] for more details.Eilenberg's variety theor...

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An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

Cited by 13 publications

References 24 publications

Wreath Products of Distributive Forest Algebras

Wreath Products of Distributive Forest Algebras

The Half-Levels of the FO2 Alternation Hierarchy

The Dot-Depth Hierarchy, 45 Years Later

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