Technical Report TR-2005-07

BibliographyLohrey, Markus; Kuske, Dietrich: First-order and counting theories of omega-automatic structures.
University of Stuttgart, Faculty of Computer Science, Electrical Engineering, and Information Technology, Technical Report Computer Science No. 2005/07.
22 pages, english.
CR-SchemaF.4.1 (Mathematical Logic)
Keywordsautomatic structures; logic in computer science
Abstract

The logic L(Q_u) 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 injective omega-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable (Blumensath, Grädel 2004). It is shown that, as in the case of automatic structures (Khoussainov, Rubin, Stephan 2004) also modulo-counting quantifiers as well as infinite cardinality quantifiers ("there are c many elements satisfying ...") lead to decidable theories. For a structure of bounded degree with injective omega-automatic presentation, the fragment of L(Q_u) that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (omega-automaticity and bounded degree) are necessary for this result to hold.

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Contactlohrey@informatik.uni-stuttgart.de
Department(s)University of Stuttgart, Institute of Formal Methods in Computer Science, Theoretical Computer Science
Entry dateSeptember 28, 2005
   Publ. Computer Science