Abstract
In this chapter we develop the theory of finite automata starting from ideas based on linear algebra over semirings. Many results in the theory of automata and languages depend only on a few equational axioms. For example, Conway has shown that Kleene’s fundamental theorem equating the recognizable languages with the regular ones follows from a few simple identities defining Conway semirings. Such semirings are equipped with a star operation subject to the sum star identity and product star identity.
We define finite automata over power series semirings and over Conway semirings, and prove theorems of the Kleene–Schützenberger type. Moreover, we introduce finite linear systems and show the coincidence of the set of components of the solutions of such finite linear systems with the set of behaviors of finite automata.
Then we generalize the Büchi theory on languages over infinite words. We define the algebraic structures needed for this generalization: semiring–semimodule pairs and quemirings. Then we define finite automata over quemirings and prove theorems of the Kleene–Büchi type. Moreover, we consider linear systems over quemirings as a generalization of regular grammars with finite and infinite derivations and show the coincidence of the set of components of the solutions of such linear systems with the set of behaviors of finite automata over quemirings.
The first author was partially supported by grant no. MTM2007-63422 from the Ministry of Education and Science of Spain.
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Ésik, Z., Kuich, W. (2009). Finite Automata. In: Droste, M., Kuich, W., Vogler, H. (eds) Handbook of Weighted Automata. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01492-5_3
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DOI: https://doi.org/10.1007/978-3-642-01492-5_3
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