The logic ℒ(אu) 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 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 [4], It is shown that, as in the case of automatic structures [13], also modulo-counting quantifiers as well as infinite cardinality quantifiers ("there are x many elements satisfying ...") lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of ℒ (אU) 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. © Springer-Verlag Berlin Heidelberg 2006.
CITATION STYLE
Kuske, D., & Lohrey, M. (2006). First-order and counting theories of ω-automatic structures. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 3921 LNCS, pp. 322–336). https://doi.org/10.1007/11690634_22
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