Paul Ruet. Non-commutative linear logic with mobilities. In Bulletin of Symbolic Logic, volume 3-2, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....substructural logics do not directly allow one to state temporal properties. Kanovitch and Ito have recently described their approach to adding temporal logic to Light Linear Logic [14] Retor e [22] proposes the use of ordered sequents, and a before connective which models sequentiality. Ruet [23] introduces another variation of non commutative linear logic characterized by the addition of new non commutative variants of multiplicative connectives, and new modalities called mobilities. All these are attempts at building substructural logics that model sequentiality as well as concurrency. ....

Paul Ruet. Non-commutative linear logic with mobilities. In Bulletin of Symbolic Logic, volume 3-2, 1997.

.... In this paper we show that the intuitionistic fragment of a mixed noncommutative version of linear logic (NLI) copes with this difficulty (Section 4) this logic combines both commutative and non commutative connectives, and is based on previous proposals of de Groote [5] and of the first author [27]. Its classical version, which extends commutative linear logic on one hand and the cyclic non commutative linear logic of Girard and Yetter [10, 33] on the other hand, is presented in [28] Here we just consider an intuitionistic fragment : NLI. We show that the stores, the successes and the ....

....and leads to non commutative logic. 4 Intuitionistic non commutative logic We just present here an intuitionistic fragment of this logic, which we call NLI. Also not all the connectives are considered here. The complete presentation of this non commutative logic, based on previous proposals [5, 27], is the topic of [28] The set F of formulas is built from atoms p; q; the constant 1, the existential quantifier 9 and connectives : a multiplicative commutative conjunction tensor Omega , a left non commutative implication Gamma ffl , the additive conjunction with , and the ....

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P. Ruet. Non-commutative linear logic with mobilities. Presented at the Logic Colloquium'96, San Sebastian, Spain, Bulletin of Symbolic Logic 3-2:274--275, Jan. 1997.

....c and d be constraints: c d iff c ffl ffl d ffl . Let A and B be cc agents: A j B iff A ffl j ffl B ffl , A Gamma B iff A ffl Gamma ffl B ffl . 3 Non Commutative Linear Logic The complete presentation of that non commutative linear logic is the topic of another paper [27], and is partially recalled in Appendix. We just present here the intuitionistic fragment of interest for the present paper. The formulas are built from atoms p; q; the constant 1, the existential quantifier 9 and connectives: a (multiplicative) commutative conjunction Omega , a ....

....1 [1 1] Gamma C [i] A i B) 2 ( Gamma; C) Gamma; 1 C[if(A 1 B A) A B)g] Table 2: Sequent calculus for an intuitionistic fragment of INLL. Theorem 1 The sequent calculus given in Table 2 enjoys cut elimination. This is a direct consequence of the cut elimination theorem given in [27] for the sequent calculus of classical non commutative linear logic, as the cut elimination procedure preserves the intuitionistic nature of sequents (i.e. at most one formula on the right of a sequent) Comments: I The graph associated to a sequent enables to express sequentiality constraints ....

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P. Ruet. Non-commutative linear logic with mobilities. Presented at the Logic Colloquium'96, San Sebastian, Spain, LIENS Technical Report in preparation, 1996.

....is represented by a series parallel order) and a phase semantics. The present logic is based on two previous works: the intuitionistic multiplicative version of de Groote (with phase semantics) and a proposal of the author (a classical system, with modalities, but not extending commutative LL) [3, 10]. The present work extends the version of de Groote to the classical case, with all the connectives (and thus Lambek s calculus [6] It differs from other proposals made by Retor e [8, 9] to combine both kinds of connectives. This mixed logic admits a syntax in terms of proof nets as well ....

P. Ruet. Non-commutative linear logic with mobilities. Abstract submitted to the Bulletin of Symbolic Logic, Presented at the Logic Colloquium'96, San Sebastian, Spain (http://www.dmi.ens.fr/sruet/PAPIERS/nll4.ps), 1996.

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