Quantitative Languages Defined by Functional Automata

TL;DR

This paper studies the properties of functional weighted automata across various measures, establishing decidability results for functionality and related problems, and exploring limitations like undecidability of realizability.

Contribution

It provides the first decidability results for functionality in weighted automata under multiple measures and extends the framework to functional group automata.

Findings

Functionality is decidable for sum, mean, discounted sum, and ratio measures.

Threshold, inclusion, and equivalence problems are decidable for functional automata.

Realizability becomes undecidable for sum, mean, and ratio measures.

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.