Partial Proof Trees as Building Blocks for a Categorial Grammar

We describe a categorial system (PPTS) based on partial proof trees (PPTs) as the building blocks of the system. The PPTs are obtained by unfolding the arguments of the type that would be associated with a lexical item in a simple categorial grammar. The PPTs are the basic types in the system and a derivation proceeds by combining PPTs together. We describe the construction of the finite set of basic PPTs and the operations for combining them. PPTS can be viewed as a categorial system incorporating some of the key insights of lexicalized tree adjoining grammar, namely the notion of an extended domain of locality and the consequent factoring of recursion from the domain of dependencies. PPTS therefore inherits the linguistic and computational properties of that system, and so can be viewed as a 'middle ground' between a categorial grammar and a phrase structure grammar. We also discuss the relationship between PPTS, natural deduction, and linear logic proof-nets, and argue that natural deduction rather than a proof-net system is more appropriate for the construction of the PPTs. We also discuss how the use of PPTs allows us to 'localize' the management of resources, thereby freeing us from this management as the PPTs are combined.