Abstract
A multioperator monoid \(\mathcal{A}\) is a commutative monoid with additional operations on its carrier set. A weighted tree automaton over \(\mathcal{A}\) is a finite state tree automaton of which each transition is equipped with an operation of \(\mathcal{A}\). We define M-expressions over \(\mathcal{A}\) in the spirit of formulas of weighted monadic second-order logics and, as our main result, we prove that if \(\mathcal{A}\) is absorptive, then the class of tree series recognizable by weighted tree automata over \(\mathcal{A}\) coincides with the class of tree series definable by M-expressions over \(\mathcal{A}\). This result implies the known fact that for the series over semirings recognizability by weighted tree automata is equivalent with definability in syntactically restricted weighted monadic second-order logic. We prove this implication by providing two purely syntactical transformations, from M-expressions into formulas of syntactically restricted weighted monadic second-order logic, and vice versa.
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The work of Z. Fülöp was partially supported by the TÁMOP-4.2.2/08/1/2008-0008 program of the Hungarian National Development Agency.
The work of T. Stüber was partially supported by Deutsche Forschungsgemeinschaft, project DFG VO 1011/4-1.
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Fülöp, Z., Stüber, T. & Vogler, H. A Büchi-Like Theorem for Weighted Tree Automata over Multioperator Monoids. Theory Comput Syst 50, 241–278 (2012). https://doi.org/10.1007/s00224-010-9296-1
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DOI: https://doi.org/10.1007/s00224-010-9296-1