Abstract
The logic \({\mathcal L}({\mathcal Q}_u)\) 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 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 \(\varkappa\) many elements satisfying ...”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of \({\mathcal L}({\mathcal Q}_u)\) 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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Kuske, D., Lohrey, M. (2006). First-Order and Counting Theories of ω-Automatic Structures. In: Aceto, L., Ingólfsdóttir, A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2006. Lecture Notes in Computer Science, vol 3921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11690634_22
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DOI: https://doi.org/10.1007/11690634_22
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