Quantitative Languages Defined by Functional Automata

Abstract

Abstract. A weighted automaton is functional if any two accepting runs on the same finite word have the same value. In this paper, we investigate functional weighted automata for four different measures: the sum, the mean, the discounted sum of weights along edges and the ratio between rewards and costs. On the positive side, we show that functionality is decidable for the four measures. Furthermore, the existential and universal threshold problems, the language inclusion problem and the equivalence problem are all decidable when the weighted automata are functional. On the negative side, we also study the quantitative extension of the realizability problem and show that it is undecidable for sum, mean and ratio. We finally show how to decide whether the language associated with a given functional automaton can be defined with a deterministic one, for sum, mean and discounted sum. The results on functionality and determinizability are expressed for the more general class of functional group automata. This allows one to formulate within the same framework new results related to discounted sum automata and known results on sum and mean automata. Ratio automata do not fit within this general scheme and different techniques are required to decide functionality.

The Target Discounted-Sum Problem

“…In the case of functional automata, it is shown in [16] that the non-strict versions of the inclusion and universality problems are decidable in PTIME. They leave the strict versions of these problems as an open question.…”

“…Proof: The ≤-inclusion and ≤-universality problems are shown in [16] to be in PTIME. A solution to the <-universality problem then directly follows from Theorem 26: Given a functional DSA A and a threshold value t, for every finite word w A(w) < t iff for every word w we have A(w) ≤ t, and there is no word w such that A(w) = t.…”

“…A DSA A over finite words is said to be functional if for every word w, all accepting runs of A on w have the same value [16].…”

“…Discounting the influence of future events takes place in many natural processes, such as temperature change, capacitor charge, and effective interest rate, for which reason it is a key paradigm in economics and it is widely studied in game theory, Markov decision processes, and automata theory [1], [4], [5], [8], [10], [11], [16], [17], [20]. Yet, the decidability question of basic problems with regard to these models are still open.…”

“…This problem is a natural milestone for open problems that involve discounting, such as problems on discounted-sum automata [5], [7], [16], [17], discounted-sum two-player games [6], [23], and multi-objective discounted-sum reachability [8].…”