A homomorphism theorem for weighted context-free grammars

Abstract

Productions of a context-free grammar can be given coefficients from semirings, inducing weights for both derivations in the grammar and strings over the terminal alphabet. For a weighted context-free grammar in Greibach normal form, the weight of any string, as well as the set of derivations of the string, may be determined from the image under a homomorphism which maps each terminal symbol to a polynomial. The definition of the homomorphism is a straightforward function of the productions. Some examples of interesting semiring structures are included.