Weighted automata and weighted logics

• Published in: Theoretical computer science Vol. 380; no. 1; pp. 69 - 86
• Main Authors: Droste, Manfred, Gastin, Paul
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 21.06.2007; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Formal power series; General logic; Language theory and syntactical analysis; Logic and foundations; Mathematical logic, foundations, set theory; Mathematics; Memory and file management (including protection and security); Memory organisation. Data processing; Miscellaneous; Sciences and techniques of general use; Software; Theoretical computing; Weighted automata; Weighted logics; Weighted automata; Formal power series; Weighted logics; Compression; Word; Automaton; Computer theory; Formal series; Mapping; Text; Image; Power series; Area; First order logic; First order; Sentence; Speech processing; Power
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2007.02.055

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.