A Local System for Classical Logic

Brünnler, Kai; Tiu, Alwen

doi:10.1007/3-540-45653-8_24

Cited by 83 publications

(142 citation statements)

References 6 publications

(4 reference statements)

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“…It also holds for other systems in the calculus of structures, including systems SBV and SELS [10], Fig. 4 Permuting b↑ up and b↓ down classical logic [4] and full propositional linear logic.…”

Section: Theorem For Every Derivationmentioning

confidence: 95%

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A Non-commutative Extension of MELL

Guglielmi

Straßburger

2002

Logic for Programming, Artificial Intelligence, and Reasoning

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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,

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Section: Theorem For Every Derivationmentioning

confidence: 95%

“…Our calculus has later been used successfully in [19] for defining pure MELL and showing decomposition and cut elimination for it. In [4] a completely local definition of classical logic is shown: in that system, not only the cut rule, but also contraction is atomic.…”

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confidence: 99%

A Non-commutative Extension of MELL

Guglielmi

Straßburger

2002

Logic for Programming, Artificial Intelligence, and Reasoning

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“…Here is another interesting question: Could it be that in the noncommutative case we can find normal representatives for proofs instead of having to rely on equivalence classes? • The relation with the calculus of structures [GS01,BT01] and its use of deep inference. We should mention that the idea behind our approach originates from the new viewpoints that are given by deep inference.…”

Section: 2mentioning

confidence: 99%

“…and get an arrow, that we call switch [GS01,BT01] but is more traditionally known as weak distributivity [HdP93,CS97b] or linear distributivity 12 , and that we denote by τ ∅,B,C,D .…”

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confidence: 99%

From Proof Nets to the Free *-Autonomous Category

Lamarche¹,

Straßburger²

2006

Logical Methods in Computer Science

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Abstract. In the first part of this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. In the second part of the paper we show that the identifications enforced on proofs are such that the class of two-conclusion proof nets defines the free * -autonomous category.

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“…The paper [12] details how the atomic lambda-calculus and its sharing mechanisms are derived from deep inference [6], a proof methodology where inferences apply in context, reminiscent of term rewriting. Sharing in deep inference is by explicit contraction rules, which implement atomic duplication by interacting with individual inferences.…”

Section: Introductionmentioning

confidence: 99%