Safety Verification for Two-Way Finite Automata with Monotonic Counters

We look at a model of a two-way nondeterministic finite automaton augmented with monotonic counters operating on inputs of the form a1i1 ...anin for some fixed n and distinct symbols a1, ..., an, where i1, ..., in are nonnegative integers. Our results concern the following Presburger safety verification problem: Given a machine M, a state q, and a Presburger relation E over counter values, is there (i1, ..., in) such that M, when started in its initial state on the left end of the input a1i1 ...anin with all counters initially zero, reaches some configuration where the state is q and the counter values satisfy E? We give positive and negative results for different variations and generalizations of the model (e.g., augmenting the model with reversal-bounded counters, discrete clocks, etc.). In particular, we settle an open problem in [10].