Over words, two variables are as powerful as one quantifier alternation

We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that two-sided semidirect products with J as the right-hand factor preserve decidability. , the second author provided an algebraic characterization of the levels of the hierarchy, showing that they correspond to the levels of weakly iterated two-sided semidirect products of the pseudovariety J of finite J -trivial monoids. This still left open the problem of decidability of the hierarchy: effectively determining from a description of a regular language the lowest level of the hierarchy to which the language belongs. This problem was apparently solved in Almeida-Weil [2], from which explicit identities for the iterated product varieties can be extracted. However, an error in that paper called the correctness of these results into question. Here we show that the given identities do indeed characterize these pseudovarieties. In particular, since it is possible to verify effectively whether a given finite monoid satisfies one of these identities, we obtain an effective procedure for exactly determining the alternation depth of a regular language definable in two-variable logic. ACM Subject Classification

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“…In other words, the two-variable definable languages form the variety of languages corresponding to DA(Thérien and Wilke, [19]). …”

Section: Varieties and Identitiesmentioning

confidence: 99%

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

Krebs

Straubing

2017

ACM Trans. Comput. Logic

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“…Our proof is a straightforward adaptation of a proof of Thérien and Wilke in [16], which analyzed the languages recognized by semigroups in DA.…”

Section: Treating Contexts Like Wordsmentioning

confidence: 99%

“…Among them one can find [11,16,14,8]: a) word languages that can be defined in the temporal logic with operators F and F −1 ; b) word languages that can be defined by a first-order formula with two variables, but with unlimited quantifier alternations; c) word languages whose syntactic semigroup belongs to the semigroup variety DA; d) word languages recognized by two-way ordered deterministic automata; e) a certain form of "unambiguous" regular expressions.…”

Section: Tree Languages Defined In First-order Logic With One Quantifmentioning

confidence: 99%

Tree Languages Defined in First-Order Logic with One Quantifier Alternation

Bojańczyk

Segoufin

Automata, Languages and Programming

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Abstract. We study tree languages that can be defined in ∆2. These are tree languages definable by a first-order formula whose quantifier prefix is ∃ * ∀ * , and simultaneously by a first-order formula whose quantifier prefix is ∀ * ∃ * . For the quantifier free part we consider two signatures, either the descendant relation alone or together with the lexicographical order relation on nodes. We provide an effective characterization of tree and forest languages definable in ∆2. This characterization is in terms of algebraic equations. Over words, the class of word languages definable in ∆2 forms a robust class, which was given an effective algebraic characterization by Pin and Weil [11].

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“…Using the results of [17], one can show that this language is inexpressible in UTL (let alone UTL\X). For example, one can compute the syntactic monoid associated with L and invoke the characterisation of syntactic monoids of UTL-definable languages from [17] to obtain the desired result. We omit the details.…”

Section: Lemma 10 Utl\x ⊆ CL : Every Utl\x Formula Can Be Encoded Asmentioning

confidence: 99%

Linear Completeness Thresholds for Bounded Model Checking

Kroening

Ouaknine

Strichman

et al. 2011

Computer Aided Verification

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Abstract.Bounded model checking is a symbolic bug-finding method that examines paths of bounded length for violations of a given LTL formula. Its rapid adoption in industry owes much to advances in SAT technology over the past 10-15 years. More recently, there have been increasing efforts to apply SAT-based methods to unbounded model checking. One such approach is based on computing a completeness threshold : a bound k such that, if no counterexample of length k or less to a given LTL formula is found, then the formula in fact holds over all infinite paths in the model. The key challenge lies in determining sufficiently small completeness thresholds. In this paper, we show that if the Büchi automaton associated with an LTL formula is cliquey, i.e., can be decomposed into clique-shaped strongly connected components, then the associated completeness threshold is linear in the recurrence diameter of the Kripke model under consideration. We moreover establish that all unary temporal logic formulas give rise to cliquey automata, and observe that this group includes a vast range of specifications used in practice, considerably strengthening earlier results, which report manageable thresholds only for elementary formulas of the form F p and G q.

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Over words, two variables are as powerful as one quantifier alternation

Cited by 105 publications

References 13 publications

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

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

Tree Languages Defined in First-Order Logic with One Quantifier Alternation

Linear Completeness Thresholds for Bounded Model Checking

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