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We introduce and investigate weighted context-free grammars over an arbitrary bimonoid K. Thus, we do not assume that the operations of K are commutative or idempotent or they distribute over each other. We prove a Chomsky-Schützenberger type theorem for the series generated by our grammars. Moreover, we show that the class of series generated by weighted right-linear grammars over a linearly ordered alphabet Σ and K coincides with that of recognizable series over Σ and K.

Weighted Context-Free Grammars Over Bimonoids