What’s Decidable about Weighted Automata?

Abstract

Abstract. 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. A weighted automaton is defined with respect to a semiring. For the tropical semiring, the weight of a run is the sum of the weights of the transitions taken along the run, and the value of a word is the minimal weight of an accepting run on it. In the 90's, Krob studied the decidability of problems on rational series defined with respect to the tropical semiring. Rational series are strongly related to weighted automata, and Krob's results apply to them. 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 defined with respect to the tropical semiring with domain ∪ {∞}, and that the equality problem is undecidable when the domain is ¡ ∪ {∞}. 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, make the undecidability result accessible to the automata-theoretic community, and strengthen it 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 on the transitions are from the set {−1, 0, 1}. The fact we work directly with the automata enables us to tighten also the decidability results and to show that the universality problem for weighted automata defined with respect to the tropical semiring with domain ∪ {∞}, and in fact even with domain ¢ ≥0 ∪ {∞}, is PSPACE-complete. Our results thus draw a sharper picture about the decidability of decision problems for weighted automata, in both the front of containment vs. universality and the front of the ∪ {∞} vs. the ¡ ∪ {∞} domains.

Quantitative Languages Defined by Functional Automata

“…We also show that the equivalence problem can be decided in polynomial space for Ratio via an easy reduction to functionality. Note that those decidability results are in sharp contrast with the corresponding results for the full class of nondeterministic weighted automata: for that class, only the existential threshold problem is known to be decidable, the language inclusion problem is undecidable for Sum [25,1], and therefore for Avg and Ratio, while the problem is open for Dsum.…”

“…It is known that inclusion is undecidable for non-deterministic Sum-automata [25,1], and therefore also for Avg and Ratio-automata. To the best of our knowledge, it is open whether it is decidable for Dsum-automata.…”

“…We show that all those problems are decidable for the four classes of measure functions that we consider in this paper when the automata are functional. In 1 We do not consider the measure functions Min and Max that map a sequence to the minimal and the maximal value that appear in the sequence as the nondeterministic automata that use those measure functions can be made deterministic and all the decision problems for them have known and simple solutions [11].…”

Parameterized Weighted Containment

“…Weighted finite automaton (WFA, for short) map words to numerical values. Technically, every transition in a weighted automaton A has a cost, and the value that A assigns to a finite word w, denoted val (A, w), is the sum of the costs of the transitions participating in the most expensive accepting run of A on w. 1 Applications of weighted automata include formal verification, where they are used for the verification of quantitative properties [6], as well as text, speech, and image processing, where the weights of the automaton are used in order to account for the variability of the data and to rank alternative hypotheses [8,20].…”

“…The source of the difficulty is the infinite domain of values that the automata may assign to words. In particular, the problem of weighted containment is in general undecidable [1,18]. Given the importance of the problem, researchers have studied decidable fragments and approximations of weighted containment.…”