Rational Series and Automata Theory
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Automata and their languages are best understood in terms of their  
generating functions, which are formal power series in noncommuting  
variables with coefficients in the Boolean semiring { True, False }.  
Via this dictionary, regular languages correspond to rational series.  
By a seminal result of Schutzenberger, rational series are precisely  
those series that have linear representations; the corresponding  
matrices here recover the usual finite state machine associated to a  
regular language. For anyone who knows a bit of algebra, this point of  
view makes quick work of the introductory computer science curriculum  
on automata theory.
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This point of view also facilitates generalizations to other domains.  
If we replace the Boolean semiring by the real numbers, we can study  
probabilistic automata, of the sort used e.g. in speech recognition  
and finance. Here, polyhedral geometry comes into play.