First-order and counting theories ofω-automatic structures

Abstract

AbstractThe logicextends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the setC”). This logic is investigated for structures with an injectivelyω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there aremany elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injectiveω-automatic presentation, the fragment ofthat 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.