## Semirings, Automata, Languages |

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AFP's alphabet Assume automata theory behavior Boolean semiring Chomsky normal form claim coefficients component Consider construction context-free grammar context-free languages convergent sequences Corollary corresponding defined definition denote discrete convergence e-free elimination theory equality follows Example family of power formal power series functional composition Furthermore Greibach normal form Hadamard product Hence idempotent iff there exists implies infer integer isomorphic copy Kronecker product language theory Lemma limit function linear system mapping monoid Moreover morphism non-null blocks non-null entries normal form notation notion of convergence Observe obtain partially ordered semiring polynomial principal cone proof of Theorem pushdown automata pushdown tape pushdown transition matrix quasiregular rational power series regulated rational representation regulated rational transduction regulated representation reset type resp restart type right side second equality Section semi-AFP specified standard type strong convergence strong solution symbol system of equations unique solution variables welldefined Z-rational zerosumfree