Focalization and phase models for classical extensions of non-associative Lambek calculus

TL;DR

This paper proves a focalization property for classical non-associative Lambek calculus extensions, demonstrating normalization of Cut-free derivations using syntactic phase models.

Contribution

It establishes Andreoli's focalization property for CNL and LG, providing a normalization result and a unified sequent presentation for these classical extensions.

Findings

Focalization property proven for CNL and LG

Normalization of Cut-free derivations demonstrated

Construction of syntactic phase models for focused proofs

Abstract

Lambek's non-associative syntactic calculus (NL) excels in its resource consciousness: the usual structural rules for weakening, contraction, exchange and even associativity are all dropped. Recently, there have been proposals for conservative extensions dispensing with NL's intuitionistic bias towards sequents with single conclusions: De Groote and Lamarche's classical non-associative Lambek calculus (CNL) and the Lambek-Grishin calculus (LG) of Moortgat and associates. We demonstrate Andreoli's focalization property for said proposals: a normalization result for Cut-free sequent derivations identifying to a large extent those differing only by trivial rule permutations. In doing so, we proceed from a `uniform' sequent presentation, deriving CNL from LG through the addition of structural rules. The normalization proof proceeds by the construction of syntactic phase models wherein every `truth' has a focused proof, similar to work of Okada and of Herbelin and Lee.