Parsing And Derivational Equivalence

It is a tacit assumption of much linguistic inquiry that all distinct derivations of a string should assign distinct meanings. But despite the tidiness of such derivational uniqueness, there seems to be no a priori reason to assume that a gramma r must have this property. If a grammar exhibits derivational equivalence, whereby distinct derivations of a string assign the same meanings, naive exhaustive search for all derivations will be redundant, and quite possibly intractable. In this paper we show how notions of derivation-reduction and normal form can be used to avoid unnecessary work while parsing with grammars exhibiting derivational equivalence. With grammar regarded as analogous to logic, derivations are proofs; what we are advocating is proof-reduction, and normal form proof; the invocation of these logical techniques adds a further paragraph to the story of parsing-as-deduction. I n t r o d u c t i o n The phenomenon of derivational equivalence is most evident in work on generalised categorial grammars, where it has been referred to as ~spurious ambiguity'. It has been argued that the capacity to assign left-branching, and therefore incrementally interpretable, analyses makes these grammars of particular psychological interest. We will illustrate our methodology by reference to generalised categorial grammars using a combinatory logic (as opposed to say, lambda-calculus) semantics. In particular we consider combinatory (categorial) grammars with rules and generalised rules *We thank Mike Reape for criticism and suggestions in relation to this material, and Inge Bethke and Henk Zeevat for reading a late draft. All errors are our own. The work was carried out by the alphabetically first author under ESRC Postgraduate Award C00428722003 and by the second under ESRC Postgraduate Award C00428522008 and an SERC Postdoctoral Fellowship in IT. of the kind of Steedman (1987), and with metarules (Morri~ 19ss). Although the problem of derivational equivalence is most apparent in generalised categorial grammars, the problem is likely to recur in many grammars characterising a full complement of constructions. For example, suppose that a grammar is capable of characterising right extraposit ion of an object 's adjunct to clause-final position. Then sentences such as Joha met a m an yesterday who swims will be generated. But it is probable that the same g r ammar will assign Joha met a maa who swims a right extraposi t ion derivation in which the relative clause happens to occupy its normal position in the string; the normal and right extraposition derivations generate the same strings with the same meanings, so there is derivational equivalence. Note tha t a single equivalence of this kind in a g r ammar undermines a methodological assumption of derivational uniqueness. Combinatory Logic and Combinatory G r a m m a r Combina tory logic (CL; Curry and Feys, 1958; Curry, Hindley and Seldin, 1972; Hindley and Seldin, 1986) refers to systems which are applicative, like the lambda-calculi, but which formalise functional abst ract ion through a small number of basic 'combinators ' , ra ther than through a variable-binding opera tor like A. We will define a typed combinatory logic. Assume a set of basic types, say e and t. Then the set of types is defined as follows: (1) a. If A is a basic type then A is a type b. If A and B are types then A-*B is a type A convention of r ight-associativity will be used for types, so tha t e.g. (e-, t)-*(e-, t) may be writ-