Weighted grammars and Kleene's theorem
Information Processing Letters
The Kleene-Schützenberger theorem for formal power series in partially commuting variables
Information and Computation
Automata, Languages, and Machines
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Theoretical Computer Science - Latin American theoretical informatics
Branching automata with costs: a way of reflecting parallelism in costs
Theoretical Computer Science - Implementation and application of automata
Derivatives of rational expressions with multiplicity
Theoretical Computer Science
Characterizations of recognizable picture series
Theoretical Computer Science
Skew and infinitary formal power series
Theoretical Computer Science
On Aperiodic and Star-Free Formal Power Series in Partially Commuting Variables
Theory of Computing Systems
Note: Schützenberger's theorem on formal power series follows from Kleene's theorem
Theoretical Computer Science
Weighted automata with discounting
Information Processing Letters
CSL '08 Proceedings of the 22nd international workshop on Computer Science Logic
Expressiveness and Closure Properties for Quantitative Languages
LICS '09 Proceedings of the 2009 24th Annual IEEE Symposium on Logic In Computer Science
Probabilistic Weighted Automata
CONCUR 2009 Proceedings of the 20th International Conference on Concurrency Theory
Weighted automata and weighted logics with discounting
Theoretical Computer Science
Weighted finite automata over strong bimonoids
Information Sciences: an International Journal
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Fuzzy Sets and Systems
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
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Describing average- and longtime-behavior by weighted MSO logics
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
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CIAA'10 Proceedings of the 15th international conference on Implementation and application of automata
Weighted tree automata over valuation monoids and their characterization by weighted logics
Algebraic Foundations in Computer Science
Weighted automata and weighted MSO logics for average and long-time behaviors
Information and Computation
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We study Kleene's theorem about the equivalence of automata and expressions in a quantitative setting both for finite and infinite words. The quantities originate from valuation monoids and ω-indexed valuation monoids which cover not only semirings but also cost models like average cost, long-run peaks of resource consumption, or discounting sums of rewards. For finite words we deduce the characterization of weighted automata by regular weighted expressions directly from Kleene's theorem. For infinite words we define three different behaviors of weighted Büchi automata depending on the way runs are evaluated. Depending on the properties of the underlying ω-indexed valuation monoid, we explore the connections between the different behaviors of weighted Büchi automata and ω-regular weighted expressions. Again, we use classical results on ω-languages to derive results in the quantitative setting.