Continuation Semantics for Symmetric Categorial Grammar

Abstract

Categorial grammars in the tradition of Lambek [1,2] are asymmetric: sequent statements are of the form ${\Gamma}\Rightarrow{A}$, where the succedent is a single formula A, the antecedent a structured configuration of formulas A ₁,...,A _n . The absence of structural context in the succedent makes the analysis of a number of phenomena in natural language semantics problematic. A case in point is scope construal: the different possibilities to build an interpretation for sentences containing generalized quantifiers and related expressions. In this paper, we explore a symmetric version of categorial grammar based on work by Grishin [3]. In addition to the Lambek product, left and right division, we consider a dual family of type-forming operations: coproduct, left and right difference. Communication between the two families is established by means of structure-preserving distributivity principles. We call the resulting system LG. We present a Curry-Howard interpretation for derivations. Our starting point is Curien and Herbelin’s sequent system for λμ calculus [4] which capitalizes on the duality between logical implication (i.e. the Lambek divisions under the formulas-as-types perspective) and the difference operation. Importing this system into categorial grammar requires two adaptations: we restrict to the subsystem where linearity conditions are in effect, and we refine the interpretation to take the left-right symmetry and absence of associativity/commutativity into account. We discuss the continuation-passing-style (CPS) translation, comparing the call-by-value and call-by-name evaluation regimes. We show that in the latter (but not in the former) the types of LG are associated with appropriate denotational domains to enable a proper treatment of scope construal.