Deductive Systems and Grammars: Proofs as Grammatical Structures

During the last fifteen years, much of the research of proof theoretical grammars has been focused on their weak generative capacity. This research culminated in Pentus’ theorem, which showed that Lambek grammars generate precisely the context-free languages. However, during the same period of time, research on other grammar formalisms has stressed the importance of “strong generative capacity,” i.e. the derivation or phrase structure trees that grammars assign to strings. The first topic of this thesis is the strong generative capacity of Lambek grammars. The proof theoretic perspective on grammars allows us to consider different notions of what “structure assigned by a Lambek grammar to a string” is taken to mean. For example, we can take any proof tree that establishes that a grammar generates a certain string or only those that are in some normal form. It can be shown that the formal properties of these notions of structure differ. The main result of this part of the thesis is that, although Lambek grammars generate context-free string languages, their derivation trees are more complex than those of context-free grammars. The latter were characterized by Thatcher as coinciding with the local tree languages, while the derivation trees of Lambek grammars include tree languages which are not regular. Even non-associative Lambek grammars, which recently have become more popular variants of categorial grammar, can be used to generate non-local tree languages. However, their normal form tree languages are always regular. Finally, categorial grammars lacking introduction rules have local derivation trees. Thus, there is a genuine hierarchy of proof theoretical grammars with respect to strong generative capacity. Additionally, we consider the semantic aspect of the proof theoretic approach to language, which is given by the correspondence between proof theory and type