Geometry of Lexico-Syntactic Interaction

Interaction of lexical and derivational semantics---for example substitution and lambda conversion--is typically a part of the on-line interpretation process. Proof-nets are to categorial grammar what phrase markers are to phrase structure grammar: unique graphical structures underlying equivalence classes of sequential syntactic derivations; but the role of proof-nets is deeper since they integrate also semantics. In this paper we show how interaction of lexical and derivational semantics at the lexico-syntactic interface can be precomputed as a process of off-line lexical compilation comprising Cut elimination in partial proof-nets. Introduction Consider the paraphrase: following examples of (1) a. b. C. Frodo lives in Bag End. Frodo inhabits Bag End. ((in b) (live])) (2) a. b. C. John tries to find Mary. John seeks Mary. ((try (find rn ) ) j) Typically, for at least ( lb) and (2b) the normalised semantic forms result from a process of substitution and lambda conversion subsequent to or simultaneous with syntactic derivation. We show how such interaction of lexical and derivational semantics at the lexicosyntactic interface can be precomputed as a process of off-line lexical compilation comprising Cut elimination in partial proof-nets. For accessibility, we devote in the initial sections a considerable proportion of space to an introduction to categorial grammar oriented towards proof-nets; see also Morrill (1994), Moortgat (1996) and Carpenter (1997). 1 Categorial grammar We consider categorial grammar with category formulas F (categories) defined by the following grammar: (3) a. b. F : : = A IFV r l F / F I F F ..4 ::= S I N I CN I PP I ... The categories in A are referred to as atomic and correspond to the kinds of expressions which are considered to be "complete". Fairly uncontroversially, this class may be taken to include at least sentences S and names N; what the class is exactly is not fixed by the formalism. Left division categories A~B ( 'A under B') are those of expressions (functors) which concatenate with (arguments) in A on the left to yield Bs. Right division categories B/A ( 'B over A') are those o f expressions (functors) which concatenate with (arguments) in A on the right yielding Bs. Product categories A . B are those of expressions which are the result of concatenating an A with a B; products do not play a dominant role here. More precisely, let L be the set of strings (including the empty string e) over a finite vocabulary V and let + be the operation of concatenation (i.e. (L, +, ~) is the free monoid generated by V) 1 . Each category formula A is interpreted as a subset [[A]] of L. When the interpretation of atomic categories has been fixed, that of complex categories is defined by (4). (4) [[AkB]] = {sl Vs'~ [[A]], s'+s~ [[B]] } [[B/A]] = {sl Vs'~ [[A]], s+s'~ [[B]] } [[A.B]] = {Sl+S21Sle [[,4]] & s2~ [[B]] } 1 In fact Lambek (1958) excluded the empty string ---and hence empty antecedents in the calculus of (5)--but it is convenient to include it here.