On two-way FA with monotonic counters and quadratic Diophantine equations

We show an interesting connection between two-way deterministic finite automata with monotonic counters and quadratic Diophantine equations. The automaton M operates on inputs of the form a1i1...anin for some fixed n and distinct symbols a1,...,an, where i1,...,in are nonnegative integers. We consider the following reachability 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? In particular, we look at the case when the relation E is an equality relation, i.e., a conjunction of relations of the form Ci = Cj. We show that this case and variations of it are equivalent to the solvability of some special classes of systems of quadratic Diophantine equations. We also study the nondeterministic version of two-way finite automata augmented with monotonic counters with respect to the reachability problem. Finally, we introduce a technique which uses decidability and undecidability results to show "separation" between language classes.