Towards a Dynamic Type Theory

Universitt at des Saarlandes Over the past few years, there have been a series of attempts Zee89, GS90, EK95, Mus94, KKP95] to combine the Montagovian type theoretic framework Mon74] with dynamic approaches, such as DRT Kam81]. The motivation for these developments is to obtain a general logical framework for discourse semantics that combines compositionality and dynamic binding. Let us look at an example of compositional semantics construction in-DRT which is one of the above formalisms KKP95]. By the use of-reduction we arrive at a rst-order DRT representation of the sentence Every man sleeps. For the purposes of the logical analysis in this paper-DRT employs a linearised variant of this representation, fg:(fXg:man(X)) ! fg:sleep(X)) where discourse referents are introduced by a dynamic binding operator. Note that here-reduction may lead to free variables to be captured, which in pure-calculus is the thing impossible. For example, if we want to construct sleeps today, as is done below, the free event variable E in the functor is captured by the discourse referent of the argument. In-DRT, is the conjunction operator for DRSes that intuitively merges two DRSes by uniting the sets of discourse referents and that of conditions. Unfortunately, the above mentioned uniied formalisms have failed so far to duplicate a key aspect of type theory that has lead to interesting linguistic analyses in computational linguistics. Type theory (or higher-order logic) is a two-layered formalism, where the algebraic content (the behaviour of higher-order functions) is neatly packaged into a formalism of its own, namely simply typed-calculus; while the logical content (the speciic semantics of connectives and quantiiers) is built on top of it. Thus the use of type theory allows us to deal with the complexities of natural language semantics on two distinct levels: the simply typed-calculus provides the theory of-reduction (which is the motor of compositionality) whereas the logical side of semantics is dealt with by a system that is rather like predicate logic. By focussing on known mechanisms for dealing with each of the two subsystems, it has proved possible to use type theoretic techniques for natural language processing systems. Higher-order uniication Hue75] solves equations in the simply typed-calculus and leads to analyses of Ellipsis DSP91], and Focus GK96, Pul94]. Note that these accounts are inadequate for the treatment of the logical structure, so they make insuucient predictions about quantiiers and connectives. First-order automated theorem proving Fit90] is used to reason about …