Hilbert's tenth problem
New Decidability Results Concerning Two-Way Counter Machines
SIAM Journal on Computing
Reversal-Bounded Multicounter Machines and Their Decision Problems
Journal of the ACM (JACM)
Two-Way Counter Machines and Diophantine Equations
Journal of the ACM (JACM)
On the Emptiness Problem for Two-Way NFA with One Reversal-Bounded Counter
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
On two-way nondeterministic finite automata with one reversal-bounded counter
Theoretical Computer Science - Insightful theory
On the solvability of a class of diophantine equations and applications
Theoretical Computer Science
Reachability in Succinct and Parametric One-Counter Automata
CONCUR 2009 Proceedings of the 20th International Conference on Concurrency Theory
Particle swarm optimisation based Diophantine equation solver
International Journal of Bio-Inspired Computation
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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.