Chapter 1 Bounded and Ordered Satisfiability : Connecting Recognition with Lambek-style Calculi to Classical Satisfiability Testing

It is well known that the Lambek Grammars are weakly equivalent to the Context-Free Grammars (CFGs, Pentus 1993, 1997), and that testing string membership with a CFG is in (Earley 1970). Nevertheless, Pentus (2003) has recently proven that sequent derivability in the Lambek Calculus with product is NPcomplete. The complexity of the corresponding problem for the product-free fragment remains unknown. This fragment is significant, given the at best limited apparent motivation for products in linguistic applications of the calculus. In this paper, when we mention the Lambek Calculus (LC) or Lambek Grammars (LG), we are referring to the product-free fragment. Pentus (1997) has presented an algorithm that transforms a product-free Lambek Grammar to a weakly equivalent CFG, but the transformed grammar is exponentially larger in the worst case. Since the grammar is considered a part of the instance for the decision problem of string membership, this does not resolve the open complexity problem for the product-free calculus. This paper studies the connection between the LG string membership problem and the SAT problem, which was the first problem shown to be NP-complete (Cook 1971). Much of the previous work on parsing with Lambek grammars has derived its inspiration from recognition algorithms for rewriting systems (Hepple 1992), string algebras (Morrill 1996) or graph theory (Moot and Puite 1999; Penn 2002), but fundamentally, LC is a logical framework, like the classical propositional logic upon which SAT is based. The crucial difference is the sensitivity to resources and order that LC incorporates. What we argue here is: (1) that a sense of order can be imposed on classical SAT using the polarity that propositional variables already possess (unlike LC), (2) that the corresponding ordered SAT problem is still NP-complete, (3) that this new version of SAT leads to a new and simpler proof of the NP-completeness of the product-free Lambek Calculus with permutation (LP) by an implementation of “locks” and “keys” somewhat reminiscent of the proposed modal extensions of categorial logics (Kurtonina and Moortgat 1996), (4) that the problem can be further restricted bounded-distance SAT in order to fit into LC, but (5) that bounded-distance SAT