Universal First-Order Quantification over Automata

Deciding formulas that mix arithmetic and uninterpreted predicates is of practical interest, notably for applications in verification. One approach for obtaining a decision procedure consists in building by structural induction an automaton that recognizes the set of models of the formula under analysis, and then testing whether this automaton accepts a non-empty language. A drawback is that universal quantification is usually handled by a reduction to existential quantification and complementation. For logical formalisms in which models are encoded as infinite words, this hinders the practical use of this method due to the difficulty of complementing infinite-word automata. The contribution of this paper is to introduce an algorithm for directly computing the effect of universal first-order quantifiers on automata recognizing sets of models, for formulas involving natural numbers encoded in unary notation. This paves the way to implementable decision procedures for various arithmetic theories.