Maximal Incrementality in Linear Categorial Deduction

Recent work has seen the emergence of a common framework for parsing categorial grammar (CG) formalisms that fall within the 'type-logical' tradition (such as the Lambek calculus and related systems), whereby some method of linear logic theorem proving is used in combination with a system of labelling that ensures only deductions appropriate to the relevant grammatical logic are allowed. The approaches realising this framework, however, have not so far addressed the task of incremental parsing a key issue in earlier work with 'flexible' categorial grammars. In this paper, the approach of (Hepple, 1996) is modified to yield a linear deduction system that does allow flexible deduction and hence incremental processing, but that hence also suffers the problem of 'spurious ambiguity'. This problem is avoided via normalisation. 1 I n t r o d u c t i o n A key attraction of the class of formalisms known as 'flexible' categorial grammars is their compatibility with an incremental style of processing, in allowing sentences to be assigned analyses that are fully or primarily left-branching. Such analyses designate many initial substrings of a sentence as interpretable constituents, allowing its interpretation to be generated 'on-line' as it is presented. Incremental interpretation has been argued to provide for efficient language processing, by allowing early filtering of implausible readings. 1 This paper is concerned with the parsing of categorial formalisms that fall within the 'type-logical' 1Within the categorial field, the significance of incrementality has been emphasised most notably in the work of Steedman, e.g. (Steedman, 1989). tradition, whose most familiar representative is the associative Lambek calculus (Lambek, 1958). Recent work has seen proposals for a range of such systems, differing in their resource sensitivity (and hence, implicitly, their underlying notion of 'linguistic structure'), in some cases combining differing resource sensitivities in one system. 2 Many of these proposals employ a 'labelled deductive system' methodology (Gabbay, 1996), whereby types in proofs are associated with labels which record proof information for use in ensuring correct inferencing. A common framework is emerging for parsing type-logical formalisms, which exploits the labelled deduction idea. Approaches within this framework employ a theorem proving method that is appropriate for use with linear logic, and combine it with a labelling system that restricts admitted deductions to be those of a weaker system. Crucially, linear logic stands above all of the type-logical formalisms proposed in the hierarchy of substructural logics, and hence linear logic deduction methods can provide a common basis for parsing all of these systems. For example, Moortgat (1992) combines a linear proof net method with labelling to provide deduction for several categorial systems. Morrill (1995) shows how types of the associative Lambek calculus may be translated to labelled implicational linear types, with deduction implemented via a version of SLD resolution. Hepple (1996) introduces a linear deduction method, involving compilation to first order formulae, which can be combined with various labelling disciplines. These approaches, however, are not directed toward incremental processing. In what follows, we show how the method of (Hepple, 1996) can be modified to allow processing which has a high degree of incrementality. These modifications, however, give a system which suffers 2See, for example, the formalisms developed in (Moortgat & Morrill, 1991), (Moortgat & Oehrle, 1994), (Morrill, 1994), (Hepple, 1995).