A Non-commutative Extension of MELL

Abstract

IntroductionNon-commutative logical operators have a long tradition [12,22,2,13,16,3], and their proof theoretical properties have been studied in the sequent calculus [7] and in proof nets [8]. Recent research has shown that the sequent calculus is not adequate to deal with very simple forms of non-commutativity [9,10,21]. On the other hand, proof nets are not ideal for dealing with exponentials and additives, which are desirable for getting good computational power. In this paper we show a logical system that joins a simple form of noncommutativity with commutative multiplicatives and exponentials. This is done in the formalism of the calculus of structures [9,10], which overcomes the difficulties encountered in the sequent calculus and in proof nets. Structures are expressions intermediate between formulae and sequents, and in fact they unify those two latter entities into a single one, thereby allowing more control over mutual dependencies of logical relations.We perform a proof theoretical analysis for cut elimination, with new tools, and we explore some further important properties, which are not available in more traditional settings and which we can collectively regard as 'modularity'. Despite the complexities of the proof theoretical investigation, the system obtained is very simple. This paper contributes the following new results: 1Wedefine a propositional logical system, called NEL (Non-commutative exponential linear logic), which extends MELL (multiplicative exponential linear logic [8]) by a non-commutative, self-dual logical operator called seq. This system, which was first imagined in [10], is conservative over MELL augmented by the mix and nullary mix rules [1,6]. System NEL can be immediately understood by anybody acquainted with the sequent calculus, and is aimed at the same range of applications as MELL. In nearly all computer science languages, sequential composition plays a fundamental role,

A Purely Logical Account of Sequentiality in Proof Search

“…In this paper I showed how to close this gap, through a purely logical restriction of SBV , and I showed how to represent PA BV in SBV . I argued that this process algebra can be extended to a Turing-equivalent one, comprising much of CCS, while still maintaining a perfect correspondence to the purely logical formal system studied in [9]. Further steps, to enhance expressivity, are possible in even more extended formal systems, by way of additives, along the lines of [15].…”

“…These are notoriously extremely effective in reducing exponentially the search space for proofs, provided one knows exactly which structure to use in cuts. As Theorems 2.6 and 3.8 point out, several different systems are equivalent to BV L. Extending our system to SNEL, an extension of SBV with exponentials studied in [9], will bring in an even larger range of possibilities.…”

“…Point 3: Recent work by Guglielmi and Straßburger provides the extension: they designed a Turing-equivalent system, called SNEL, which conservatively extends SBV with exponentials [9]. Since we find there the usual exponential of linear logic, it should be possible to map fixpoint operators by simple, known replication techniques.…”

A system of interaction and structure V: the exponentials and splitting

“…We can also consider NEL as MELL (multiplicative exponential linear logic [Gir87]) plus seq. NEL, which was first presented in [GS02], is conservative over both BV and over MELL augmented by the mix and nullary mix rules [FR94,Ret93,AJ94]. Note that, like BV, NEL cannot be analytically expressed outside of deep inference.…”

“…The main results of this paper have already been presented, without proof, in [GS02]. For several years, the proofs of the statements have been available in a manuscript on the web.…”