On Deciding Linear Arithmetic Constraints Over p-adic Integers for All Primes

Given an existential formula Φ of linear arithmetic over p-adic integers together with valuation constraints, we study the p-universality problem which consists of deciding whether Φ is satisfiable for all primes p, and the analogous problem for the closely related existential theory of Buchi arithmetic. Our main result is a coNEXP upper bound for both problems, together with a matching lower bound for existential Buchi arithmetic. On a technical level, our results are obtained from analysing properties of a certain class of p-automata, finite-state automata whose languages encode sets of tuples of natural numbers.

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