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First-order and counting theories of ω-automatic structures

Published online by Cambridge University Press:  12 March 2014

Dietrich Kuske  and
Markus Lohrey
Dietrich Kuske
Affiliation:
Universität Leipzig, Institut Für Informatik, Postfach 100920, D-04009 Leipzig, Germany, E-mail: kuske@informatik.uni-leipzig.de
Markus Lohrey
Affiliation:
Universität Leipzig, Institut Für Informatik, Postfach 100920, D-04009 Leipzig, Germany, E-mail: lohrey@informatik.uni-leipzig.de
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Abstract

The logic 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 many elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 1 , March 2008 , pp. 129 - 150
Copyright
Copyright © Association for Symbolic Logic 2008

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