A Non-Commutative Extension of MELL

We extend multiplicative exponential linear logic (MELL) by a non-commutative, self-dual logical operator. The extended system, called NEL, is defined in the formalism of the calculus of structures, which is a generalisation of the sequent calculus and provides a more refined analysis of proofs. We should then be able to extend the range of applications of MELL, by modelling a broad notion of sequentiality and providing new properties of proofs. We show some proof theoretical results: decomposition and cut elimination. The new operator represents a significant challenge: to get our results we use here for the first time some novel techniques, which constitute a uniform and modular approach to cut elimination, contrary to what is possible in the sequent calculus

Hilbert e i fondamenti della matematica: il metodo assiomatico

Il ruolo esercitato da David Hilbert nel campo dei fondamenti della matematica è insigne

Non-commutative logic I: the multiplicative fragment

INTRODUCTION Unrestricted exchange rules of Girard's linear logic [8] force the commutativity of the multiplicative connectives\Omega (times, conjunction) and & (par, disjunction) , and henceforth the commutativity of all logic. This a priori commutativity is not always desirable --- it is quite problematic in applications like linguistics or computer science ---, and actually the desire of a non-commutative logic goes back to the very beginning of LL [9]. Previous works on non-commutativity deal essentially with non-commutative fragments of LL, obtained by removing the exchange rule at all. At that point, a simple remark on the status of exchange in the sequent calculus is necessary to be clear: there are two presentations of exchange in commutative LL, either sequents are finite sets of occurrences of formulas and exchange is obviously implicit, or sequents are fin

Tree Adjoining Grammars in Noncommutative Linear Logic (Extended Abstract)

V. Michele Abrusci 1 , Christophe Fouquer'e 2 , and Jacqueline Vauzeilles 2 1 CILA, Universit'a di Bari, 70121 Bari, Italy 2 LIPN-CNRS URA 1507, Universit'e Paris-Nord, 93430 Villetaneuse Abstract. This paper 1 presents a logical formalization of Tree-Adjoining Grammar (TAG). TAG deals with lexicalized trees and two operations are available: substitution and adjunction. Adjunction is generally presented as an insertion of a tree inside another, surrounding the subtree at the adjunction node. This seems to be contradictory with standard logical ability. We prove that some logic, namely a fragment of noncommutative intuitionistic linear logic (N-ILL), can serve this purpose. Briefly speaking, linear logic is a logic considering facts as resources. NILL can then be considered either as an extension of Lambek calculus, or as a restriction of linear logic. We model the TAG formalism in four steps: trees (initial or derived) and the way they are constituted, the operations (substit..