Journal of Logic and Computation - current issue

Lambda Calculus, Type Theory, and Natural Language II

Fox, C., Fernandez, M., Lappin, S. — 2008-03-13

Sequentially Indexed Grammars

van Eijck, J. — 2008-03-13

This article defines the grammar class of sequentially indexed grammars (SIGs) that results of a change in the index stack handling mechanism of indexed grammars (Aho, 1968, Journal of the ACM, 15, 647–671; 1969, Journal of the ACM, 16, 383–406). SIGs are different from linear indexed grammars (Gazdar, 1988, Natural Language, Parsing and Linguistic, Theories, pp. 69–94) (the rule format is simpler) and they generate a strictly larger language class. We give a polynomial algorithm for parsing with SIGs that is a rather straightforward extension of the Earley algorithm for parsing with context-free grammars. SIGs are attractive because of the simple rule format, the natural correspondence between indices and traces, and the perspicuity of the parsing scheme.

M. H. Newman's Typability Algorithm for Lambda-calculus

Hindley, J. R. — 2008-03-13

This article is essentially an extended review with historical comments. It looks at an algorithm published in 1943 by M. H. A. Newman, which decides whether a lambda-calculus term is typable without actually computing its principal type. Newman's algorithm seems to have been completely neglected by the type-theorists who invented their own rather different typability algorithms over 15 years later.

Proof Nets for Basic Discontinuous Lambek Calculus

Morrill, G., Fadda, M. — 2008-03-13

The theory of proof nets for continuity based on the Lambek calculus is well-developed, but we need a compatible extension to include discontinuity. Earlier work set out ingredients: hypersequent calculus and proof nets expanded with parameter edges. This article completes a preliminary line by finalizing a version of proof nets for the basic discontinuous Lambek calculus BDLC (the minimal system with one point of discontinuity) and proving correctness with respect to the hypersequent calculus.

Hyperintensions

Pollard, C. — 2008-03-13

Standard possible worlds semantics has been known from the start to have a problem with granularity: for a wide range of natural-language (NL) entailment patterns, not enough meaning distinctions are available to make predictions consistent with robust intuitions. Though numerous solutions have been proposed, often of great ingenuity and technical sophistication, none of these has gained widespread acceptance. As a result, most semanticists have made a practical decision to work in a framework known to have dubious foundations and leave the foundational problems to mathematical logicians. Here, a new approach is proposed which may be simple enough and conservative enough to be practical for working empirical and computational semanticists. More specifically, I show how the use of a higher-order logic with definable subtypes leads to a novel and surprisingly straightforward solution of the granularity problem. I also call attention to a hitherto unnoticed problem in standard approaches to NL semantics having to do with non-principal ultrafilters and show why it does not arise under my proposal. The two main technical innovations that drive the proposal are (i) axiomatizing NL entailment as a preorder (as opposed to an order) on the set of (primitive) propositions, and (ii) defining worlds as certain sets of propositions (viz. ultrafilters). These innovations provide just the tools we need to develop a formally explicit theory of hyperintensions, ¹ mathematical models of Fregean senses of a finer granularity than the familiar intensions (functions to extensions from worlds, where the worlds in turn are theoretical primitives) of mainstream Kripke/Montague-inspired NL semantics.

Computable Models

Turner, R. — 2008-03-13

We investigate mathematical modelling with theories of data types. We provide a formal setting for the formulation of such theories (TPL) and use it to introduce the notion of a computational model. We explore the notion and provide several case studies.