Context-Free Fuzzy Languages
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In an earlier paper (Santos, 1973b), the author introduced two distinct but equivalent formulations of max-product grammars. One corresponds to the maximal interpretation (Salomaa, 1969) of probabilistic grammars (Santos, 1972), while the other provides a grammar for generating fuzzy languages (Santos, 1973a). Various types of max-product grammars were given. Among them was the context-free max-product grammar (CMG). In the present paper, we shall study in detail the properties of CMG and languages generated by CMG. The main contents of the paper are contained in the next two sections. In Section II, the definitions of CMG and fuzzy languages generated by CMG are given. It is shown that every CMG may be put in Chomsky normal form and Greibach normal form (Hopcroft and Ullman, 1969) provided the fuzzy languages generated by the CMG are finitary. This forms the basis of all subsequent discussions. In Section III, various properties of languages generated by CMG with cut points are established. It is shown that the family A ° of languages generated by CMG with cut points contains the family of context-free languages as a proper subfamily. Moreover, it is shown that there exists a context-sensitive language which is not in Lf.
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