Pomset logic: a logical and grammatical alternative to the Lambek calculus

Thirty years ago, I introduced a non commutative variant of classical linear logic, called POMSET LOGIC, issued from a particular denotational semantics or categorical interpretation of linear logic known as coherence spaces. In addition to the multiplicative connectives of linear logic, pomset logic includes a non-commutative connective, "<" called BEFORE, which is associative and self-dual: (A<B)^=A^ < B^ (observe that there is no swapping), and pomset logic handles Partially Ordered MultiSETs of formulas. This classical calculus enjoys a proof net calculus, cut-elimination, denotational semantics, but had no sequent calculus, despite my many attempts and the study of closely related deductive systems like the calculus of structures. At the same period, Alain Lecomte introduced me to Lambek calculus and grammars. We defined a grammatical formalism based on pomset logic, with partial proof nets as the deductive systems for parsing-as-deduction, with a lexicon mapping words to partial proof nets. The study of pomset logic and of its grammatical applications has been out of the limelight for several years, in part because computational linguists were not too keen on proof nets. However, recently Sergey Slavnov found a sequent calculus for pomset logic, and reopened the study of pomset logic. In this paper we shall present pomset logic including both published and unpublished material. Just as for Lambek calculus, Pomset logic also is a non commutative variant of linear logic — although Lambek calculus appeared 30 years before linear logic ! — and as in Lambek calculus it may be used as a grammar. Apart from this the two calculi are quite different, but perhaps the algebraic presentation we give here, with terms and the semantic correctness criterion, is closer to Lambek's view.

research
06/06/2016

Proof nets for the Displacement calculus

We present a proof net calculus for the Displacement calculus and show i...
research
08/10/2018

Proof Nets and the Linear Substitution Calculus

Since the very beginning of the theory of linear logic it is known how t...
research
03/01/2018

Sequentialization for full N-Graphs via sub-N-Graphs

Since proof-nets for MLL- were introduced by Girard (1987), several stud...
research
01/10/2021

The Logic for a Mildly Context-Sensitive Fragment of the Lambek-Grishin Calculus

While context-free grammars are characterized by a simple proof-theoreti...
research
12/31/2021

First order linear logic and tensor type calculus for categorial grammars

We study relationship between first order multiplicative linear logic (M...
research
12/16/2019

On the Unity of Logic: a Sequential, Unpolarized Approach

The present work aims to give a unity of logic via standard sequential, ...
research
01/21/2013

Computational Aspects of the Calculus of Structure

Logic is the science of correct inferences and a logical system is a too...