Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speech-to-text processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We generalize Büchi's and Elgot's fundamental theorems to this quantitative setting. We introduce a weighted version of MSO logic and prove that, for commutative semirings, the behaviours of weighted automata are precisely the formal power series definable with particular sentences of our weighted logic. We also consider weighted first-order logic and show that aperiodic series coincide with the first-order definable ones, if the semiring is locally finite, commutative and has some aperiodicity property.

Abstract. Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speech-to-text processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We generalize Büchi's and Elgot's fundamental theorems to this quantitative setting. We introduce a weighted version of MSO logic and prove that, for commutative semirings, the behaviours of weighted automata are precisely the formal power series definable with our weighted logic. We also consider weighted first-order logic and show that aperiodic series coincide with the first-order definable ones, if the semiring is locally finite, commutative and has some aperiodicity property.

Weighted automata model quantitative aspects of systems like memory or power consumption. Recently, Chatterjee, Doyen, and Henzinger introduced a new kind of weighted automata which compute objectives like the average cost or the long-time peak power consumption. In these automata, operations like average, limit superior, limit inferior, limit average, or discounting are used to assign values to finite or infinite words. In general, these weighted automata are not semiring weighted anymore. Here, we establish a connection between such new kinds of weighted automata and weighted logics. We show that suitable weighted MSO logics and these new weighted automata are expressively equivalent, both for finite and infinite words. The constructions employed are effective, leading to decidability results for the weighted logic formulas considered.

No abstract

No abstract