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    "description": "In automata theory, a fundamental result of B\u00fcchi and Elgot states that the recognizable languages are precisely the ones definable by sentences of monadic second order logic. We will present a generalization of this result to the context of weighted automata. We develop syntax and semantics of a quantitative logic; like the behaviors of weighted automata, the semantics of sentences of our logic are formal power series describing \u2018how often\u2019 the sentence is true for a given word. Our main result shows that if the weights are taken in an arbitrary semiring, then the behaviors of weighted automata are precisely the series definable by sentences of our quantitative logic. We achieve a similar characterization for weighted B\u00fcchi automata acting on infinite words, if the underlying semiring satisfies suitable completeness assumptions. Moreover, if the semiring is additively locally finite or locally finite, then natural extensions of our weighted logic still have the same expressive power as weighted automata.", 
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