What's Decidable about Weighted Automata?

Weighted automata map input words to numerical values. Applications of weighted automata include formal verification of quantitative properties, as well as text, speech, and image processing. In the 90's, Krob studied the decidability of problems on rational series, which strongly relate to weighted automata. In particular, it follows from Krob's results that the universality problem (that is, deciding whether the values of all words are below some threshold) is decidable for weighted automata with weights in N boolean OR {infinity}, and that the equality problem is undecidable when the weights are in Z boolean OR {infinity}. In this paper we continue the study of the borders of decidability in weighted automata, describe alternative and direct proofs of the above results, and tighten them further. Unlike the proofs of Krob, which are algebraic in their nature, our proofs stay in the terrain of state machines, and the reduction is from the halting problem of a two-counter machine. This enables us to significantly simplify Krob's reasoning and strengthen the results to apply already to a very simple class of automata: all the states are accepting, there are no initial nor final weights, and all the weights are from the set {-1, 0, 1}. The fact we work directly with automata enables us to tighten also the decidability results and to show that the universality problem for weighted automata with weights in N boolean OR {infinity}, and in fact even with weights in Q(>= 0) boolean OR {infinity}, is PSPACE-complete. Our results thus draw a sharper picture about the decidability of decision problems for weighted automata, in both the front of equality vs. universality and the front of the N boolean OR {infinity} vs. the Z boolean OR {infinity} domains.