One quantifier alternation in first-order logic with modular predicates

Kufleitner, Manfred; Walter, Tobias

doi:10.1051/ita/2014024

Kufleitner, Manfred ¹ ; Walter, Tobias ¹

¹ University of Stuttgart, FMI, Germany.

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 49 (2015) no. 1, pp. 1-22.

Résumé

Adding modular predicates yields a generalization of first-order logic $F O$ over words. The expressive power of $F O [<, M O D]$ with order comparison $x < y$ and predicates for $x \equiv i m o d n$ has been investigated by Barrington et al. The study of $F O [<, M O D]$ -fragments was initiated by Chaubard et al. More recently, Dartois and Paperman showed that definability in the two-variable fragment $F O^{2} [<, M O D]$ is decidable. In this paper we continue this line of work. We give an effective algebraic characterization of the word languages in $Σ_{2} [<, M O D]$ . The fragment $Σ_{2}$ consists of first-order formulas in prenex normal form with two blocks of quantifiers starting with an existential block. In addition we show that $Δ_{2} [<, M O D]$ , the largest subclass of $Σ_{2} [<, M O D]$ which is closed under negation, has the same expressive power as two-variable logic $F O^{2} [<, M O D]$ . This generalizes the result $F O^{2} [<] = Δ_{2} [<]$ of Thérien and Wilke to modular predicates. As a byproduct, we obtain another decidable characterization of $F O^{2} [<, M O D]$ .

Reçu le : 2013-10-17
Accepté le : 2014-06-18

MR Zbl

DOI : 10.1051/ita/2014024

Classification : 68Q70, 03D05, 20M35, 68Q45
Mots clés : Finite monoid, syntactic homomorphism, logical fragment, first-order logic, modular predicate

Affiliations des auteurs :

Kufleitner, Manfred ¹ ; Walter, Tobias ¹

¹ University of Stuttgart, FMI, Germany.

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Kufleitner, Manfred; Walter, Tobias. One quantifier alternation in first-order logic with modular predicates. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 49 (2015) no. 1, pp. 1-22. doi : 10.1051/ita/2014024. http://www.numdam.org/articles/10.1051/ita/2014024/

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