### First-order and counting theories of ω-automatic structures

Dietrich Kuske and Markus Lohrey
Source: J. Symbolic Logic Volume 73, Issue 1 (2008), 129-150.

#### 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.

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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1208358745
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Digital Object Identifier: doi:10.2178/jsl/1208358745

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