March 2008 First-order and counting theories of ω-automatic structures
Dietrich Kuske, Markus Lohrey
J. Symbolic Logic 73(1): 129-150 (March 2008). DOI: 10.2178/jsl/1208358745

Abstract

The logic ℒ(𝒬u) extends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the set C”). This logic is investigated for structures with an injectively ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there are κ many elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of ℒ(𝒬u) that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.

Citation

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Dietrich Kuske. Markus Lohrey. "First-order and counting theories of ω-automatic structures." J. Symbolic Logic 73 (1) 129 - 150, March 2008. https://doi.org/10.2178/jsl/1208358745

Information

Published: March 2008
First available in Project Euclid: 16 April 2008

zbMATH: 1141.03015
MathSciNet: MR2387935
Digital Object Identifier: 10.2178/jsl/1208358745

Rights: Copyright © 2008 Association for Symbolic Logic

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Vol.73 • No. 1 • March 2008
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