On the Existential Theories of Büchi Arithmetic and Linear p-adic Fields

We consider the complexity of the satisfiability problems for the existential fragment of Büchi arithmetic and for the existential fragment of linear arithmetic over p-adic fields. Our main results are that both problems are NP-complete. The NP upper bound for existential linear arithmetic over p-adic fields resolves an open question posed by Weispfenning [J. Symb. Comput., 5(1/2) (1988)] and holds despite the fact that satisfying assignments in both theories may have bit-size super-polynomial in the description of the formula. A key technical contribution is to show that the existence of a path between two states of a finite-state automaton whose language encodes the set of solutions of a given system of linear Diophantine equations can be witnessed in NP.