Semirings and Formal Power Series

This chapter presents basic foundations for the theory of weighted automata: semirings and formal power series. A fundamental question is how to extend the star operation (Kleene iteration) from languages to series. For this, we investigate ordered, complete and continuous semirings and the related concepts of star semirings and Conway semirings. We derive natural properties for the Kleene star of cycle-free series and also of matrices often used to analyze the behavior of weighted automata. Finally, we investigate cycle-free linear equations which provide a useful tool for proving identities for formal power series.

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