Weighted Automata and Weighted Logics

In automata theory, Büchi’s and Elgot’s fundamental theorems [6, 7, 25] established the coincidence of regular and ω-regular languages with languages definable in monadic second-order logic. At the same time, Schützenberger [57] investigated finite automata with weights and characterized their behaviours as rational formal power series. Both of these results have inspired a wealth of extensions and further research, cf. [4, 24, 42, 56, 60] for monographs and surveys as well as the chapters [26, 55] of this handbook, and also led to recent practical applications, e.g. in verification of finite-state programs (model checking, [3,43,46]), in digital image compression [11,33,35,36] and in speechto-text processing [8,49,51], cf. also chapters [1,29,38,50] of the present handbook [17]. It is the goal of this chapter to introduce a logic with weights taken from an arbirary semiring and to present conditions under which the behaviours of weighted finite automata are precisely the series definable in our weighted monadic second-order logic. We will deal with both finite and infinite words. In comparison to the essential predecessors [13, 14, 20], our logic will be defined in a purely syntactical way, and the results apply to arbitrary (also noncommutative) semirings. Our motivation for this weighted logic is as follows. First, weighted automata and their behaviour can be viewed as a quantitative extension of classical automata. The latter decide whether a given word is accepted or not, whereas weighted automata also compute e.g. the ressources, time or cost used or the probability of its success when executing the word. We would like to have an extension of Büchi’s and Elgot’s theorems to this setting. Second, classical logic for automata describes whether a certain property (e.g. “there exist three consecutive a’s”) holds for a given word or not. One could be in-