We show that the emptiness problem for two-way nondeterministic finite automata augmented with one reversal-bounded counter (i.e., the counter alternates between nondecreasing and nonincreasing modes a fixed number of times) operating on bounded languages (i.e., subsets of w_1~*…w_k~* for some nonnull words W_1, …, w_k) is decidable, settling an open problem in [11, 12]. The proof is a rather involved reduction to the solution of a special class of Diophantine systems of degree 2 via a class of programs called two-phase programs. The result has applications to verification of infinite state systems.
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