Memoisation for Glue Language Deduction and Categorial Parsing

The multiplicative fragment of linear logic has found a number of applications in computational linguistics: in the "glue language" approach to LFG semantics, and in the formulation and parsing of various categorial grammars. These applications call for efficient deduction methods. Although a number of deduction methods for multiplicative linear logic are known, none of them are tabular methods, which bring a substantial efficiency gain by avoiding redundant computation (c.f. chart methods in CFG parsing): this paper presents such a method, and discusses its use in relation to the above applications. 1 I n t r o d u c t i o n The multiplicative fragment of linear logic, which includes just the linear implication (o-) and multiplicative (®) operators, has found a number of applications within linguistics and computational linguistics. Firstly, it can be used in combination with some system of labelling (after the 'labelled deduction' methodology of (Gabbay, 1996)) as a general method for formulating various categorial grammar systems. Linear deduction methods provide a common basis for parsing categorial systems formulated in this way. Secondly, the multiplicative fragment forms the core of the system used in work by Dalrymple and colleagues for handling the semantics of LFG derivations, providing a 'glue language' for assembling the meanings of sentences from those of words and phrases. Although there are a number of deduction methods for multiplicative linear logic, there is a notable absence of tabular methods, which, like chart parsing for CFGs, avoid redundant computation. Hepple (1996) presents a compilation method which allows for tabular deduction for implicational linear logic (i.e. the fragment with only o--). This paper develops that method to cover the fragment that includes the multiplicative. The use of this method for the applications mentioned above is discussed. 2 Multiplicative Linear Logic Linear logic is a 'resource-sensitive' logic: in any deduction, each assumption ('resource') is used precisely once• The formulae of the multiplicative fragment of (intuitionistic) linear logic are defined by ~" ::= A I ~'o-~" J 9 v ® ~ (A a nonempty set of atomic types). The following rules provide a natural deduction formulation: Ao--B : a B : b o -E A : (ab) [B : v] A : a o--I A o B : ),v.a [B: x],[C : y] B ® C : b A : a A : a B : b ®E ®I A" @ • E.,~(b, a) A ® B : (a ® b) The elimination (E) and introduction (I) rules for o-correspond to steps of functional application and abstraction, respectively, as the term labelling reveals. The o--I rule discharges precisely one assumption (B) within the proof to which it applies. The ®I rule pairs together the premise terms, whereas ®E has a substitution like meaning. 1 Proofs that Wo--(Xo--Z), Xo--Y, Yo--Z =~ W and that Xo-Yo-Z, Y@Z =v X follow: W o ( X o Z ) : w X o Y : x Y o Z : y [Z:z]