The String-Meaning Relations Definable by Lambek Grammars and Context-Free Grammars

We show that the class of string-meaning relations definable by the following two types of grammars coincides: (i) Lambek grammars where each lexical item is assigned a (suitably typed) lambda term as a representation of its meaning, and the meaning of a sentence is computed according to the lambda-term corresponding to its derivation; and (ii) cycle-free context-free grammars that do not generate the empty string where each rule is associated with a (suitably typed) lambda term that specifies how the meaning of a phrase is determined by the meanings of its immediate constituents.