Relational Semantics for the Lambek-Grishin Calculus

We study ternary relational semantics for symmetric versions of the Lambek calculus with interaction principles due to Grishin (1983). We obtain completeness on the basis of a Henkin-style weak filter construction. 1 Background, motivation The categorial calculi proposed by Lambek and their current typelogical extensions respect an “intuitionistic” restriction: in a Gentzen presentation, Lambek sequents are of the form Γ⇒ B, where B is a single formula, and Γ is a tree structure with formulas A1, . . . , An at the yield. Depending on the particular calculus one works with, the antecedent structure can degenerate into a list or a multiset of formulas. The intuitionistic restriction is a serious expressive limitation when it comes to using the Lambek framework in the analysis of natural language syntax and semantics. Core phenomena such as displacement or scope construal are beyond the reach of the basic Lambek calculus; to deal with such phenomena, various extensions have been proposed based on structural rules, which can be introduced implicitly or explicitly, and with global or modally-controlled application regimes. The price one pays for such extensions is high: whereas the basic Lambek calculus has a polynomial recognition problem [3], already the simplest extension with an associative regime for ⊗ is known to be NP complete as shown in [8]. In a remarkable paper written in 1983, V.N. Grishin [4] has proposed a different strategy for generalizing the Lambek calculi. The starting point for Grishin’s approach is a symmetric extension of the Lambek calculus: in addition to the familiar operators ⊗, \, / (product, left and right division), one also considers a dual family ⊕, ,;: coproduct, right and left difference. The resulting vocabulary is given in (1). A,B ::= p | atoms: s sentence, np noun phrases, . . . A⊗B | B\A | A/B | product, left vs right division A⊕B | A B | B ;A coproduct, right vs left difference (1) Algebraically, the Lambek operators form a residuated triple; likewise, the ⊕ family forms a dual residuated triple. The minimal symmetric categorial grammar, which we will refer to as LG∅, 1We thank Anna Chernilovskaya and the anonymous MOL’07 referees for helpful comments on an earlier version of this paper. 2A little pronunciation dictionary: read B\A as ‘B under A’, A/B as ‘A over B’, B ; A as ‘B from A’ and A B as ‘A less B’. We follow [6] in using the notation ⊕ for the coproduct, which is a multiplicative operation.