Handbook of theoretical computer science (vol. B)
Word Processing in Groups
Some Results on Automatic Structures
LICS '02 Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science
Automatic Presentations of Structures
LCC '94 Selected Papers from the International Workshop on Logical and Computational Complexity
LICS '03 Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science
A Model-Theoretic Approach to Regular String Relations
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
Cardinality quantifiers in MLO over trees
CSL'09/EACSL'09 Proceedings of the 23rd CSL international conference and 18th EACSL Annual conference on Computer science logic
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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.