First-order and counting theories of ω-automatic structures

The logic L(Qu) 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 infinityquantifier, the resulting theory of any such structure is known to be decidable [5]. It is shown that, as in the case of automatic structures [18], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there are κ many elements satisfying ...”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of L(Qu) 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.