A Purely Logical Account of Sequentiality in Proof Search

Bruscoli, Paola

doi:10.1007/3-540-45619-8_21

Cited by 34 publications

(51 citation statements)

References 11 publications

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“…The addition of seq opens new syntactic possibilities, for example in dealing with process algebras. It has been used already, in a purely multiplicative setting, to model CCS's prefixing [5]. Furthermore, we show a class of equivalent extensions of NEL, which all enjoy the subformula property.…”

Section: Introductionmentioning

confidence: 88%

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: Introductionmentioning

confidence: 88%

A Non-commutative Extension of MELL

Guglielmi

Straßburger

2002

Logic for Programming, Artificial Intelligence, and Reasoning

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“…As a matter of fact, BV's self-dual non-commutativity captures very precisely the sequentiality notion of CCS [Mil89] (and so of other process algebras), as Bruscoli shows in [Bru02].…”

Section: Introductionmentioning

confidence: 99%

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

Guglielmi¹,

Straβburger²

2011

Math. Struct. Comp. Sci.

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System NEL is the mixed commutative/non-commutative linear logic BV augmented with linear logic's exponentials, or, equivalently, it is MELL augmented with the non-commutative self-dual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential composition and it faithfully models causal quantum evolution. In this paper, we show cut elimination for NEL, based on a property that we call splitting. NEL is presented in the calculus of structures, which is a deep-inference formalism, because no Gentzen formalism can express it analytically. The splitting theorem shows how and to what extent we can recover a sequent-like structure in NEL proofs. Together with the decomposition theorem, proved in the previous paper of the series, this immediately leads to a cut-elimination theorem for NEL.

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“…Candidate rules for non-deterministic choice to extend SBVr could be 1 : A further reason to extend SBVr with non-deterministic choice is to keep developing the programme started in [4], aiming at a purely logical characterization of full CCS.…”

Section: Discussionmentioning

confidence: 99%

“…The point about BV is that, thanks to its "deepness", it naturally extends multiplicative linear logic (MLL) [5] with the non commutative binary operator Seq, denoted " ". Second, from [4] we recall that Seq soundly and completely models the sequential communication of CCS concurrent and communicating systems [11], while Par ( ) models parallel composition. Third, we recall that λ-calculus β-reduction incorporates a sequential behavior.…”

Section: Introductionmentioning

confidence: 99%

Linear Lambda Calculus and Deep Inference

Roversi

2011

Lecture Notes in Computer Science

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Abstract. We introduce a deep inference logical system SBVr which extends SBV [6] with Rename, a self-dual atom-renaming operator. We prove that the cut free subsystem BVr of SBVr exists. We embed the terms of linear λ-calculus with explicit substitutions into formulas of SBVr. Our embedding recalls the one of full λ-calculus into π-calculus. The proof-search inside SBVr and BVr is complete with respect to the evaluation of linear λ-calculus with explicit substitutions. Instead, only soundness of proof-search in SBVr holds. Rename is crucial to let proof-search simulate the substitution of a linear λ-term for a variable in the course of linear β-reduction. Despite SBVr is a minimal extension of SBV its proof-search can compute all boolean functions, exactly like linear λ-calculus with explicit substitutions can do.

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A Purely Logical Account of Sequentiality in Proof Search

Cited by 34 publications

References 11 publications

A Non-commutative Extension of MELL

A Non-commutative Extension of MELL

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

Linear Lambda Calculus and Deep Inference

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