Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997. 25  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for some g. Proof. See appendix A Furthermore one easily checks that completeness on successes remains: g: 5 Observation of suspensions In [23] it is shown that the set of suspensions of an LCC agent can be characterized in the Non commutative Logic of Ruet and Abrusci [22, 21] (or even its intuitionnistic version, closer to what DeGroote introduced in [8] and given in appendix B) but there is a condition on the constraints, which must have no non logical axioms. The reason is that confusing a constraint with its consequences would create fake suspensions. As we now ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

.... one easily checks that completeness on successes remains: O success (C; D:A) a f9Xc 2 CjA z ILL(C 0 ;D) X ; c; z g: 5 Observation of suspensions In [23] it is shown that the set of suspensions of an LCC agent can be characterized in the Non commutative Logic of Ruet and Abrusci [22, 21], but there is a condition on the constraints, which must have no non logical axioms. The reason is that confusing a constraint with its consequences would create fake suspensions. As we now can avoid this phenomenon, it is natural to extend the previous result to a general observation of ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....C Gamma; Delta; Omega L ( t=x]A) Omega R = C 8L Gamma; Delta; Omega L (8x: A) Omega R = C Figure 1: Sequent rules for the uniform fragment of OLL. and its dual. There is no explicit mobility modality as in our system. The intuitionistic version of non commutative logic presented in [16, 7] is an acyclic, single conclusion restriction of NL. Thus our system may be seen as the fragment of intuitionistic NL without the commutative connectives, extended with a mobility modality. 3. UNIFORM DERIVATIONS Now that we have a suitable sequent system for OLL, we begin analyzing proof ....

Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

.... Demaille resembles the multimodal approach: one has at the same time the Lambek connectives and the commutative linear connectives, with structural rules linking the two nevertheless it is less powerful, because it is still decidable, and does not involve structural modalities or postulates [24, 82, 6, 23]. There also exists a classical calculus with a left and a right negation [2] with ways to introduce modalities enabling displacement [3] or more sophisticated behaviors: although these systems have not been used in linguistic applications, they may well do so, since one is often requires ....

Paul Ruet. Logique non-commutative et programmation concurrente. Thse de doctorat, spcialit logique et fondements de l'informatique, Universit Paris 7, 1997.

....may be that a proof net syntax can still be provided for such systems. 2 Proof nets Surveys of proof nets include Lamarche and Retor e (1996) and Retor e (1996) Works on (possibly) non commutative or partially commutative proof nets includes Retor e (1993, 1997) Bellin and van de Wiele (1995) Ruet (1997), Abrusci and Ruet (1998) de Groote (1999) and Moot and Puite (1999) Here we consider the possibility of a systematic correspondence between combined algebraic and relational interpretation and paths in proof nets, for which it is convenient to describe, in the following two subsections, some ....

Ruet, Paul: 1997, Logique Non-commutative et Programmation Concurrente par Contraintes, Ph.D. dissertation, Universit'e Paris 7.

....stores: Theorem 3.8 (Completeness) Let A be an LCC agent. We have O store (C; D:A) a f9Xc 2 Cj9 ; A z IMALL(C 00 ;D) X ; c; z g. 3. 3 Suspensions In [23] it is shown that the set of suspensions of an LCC agent can be characterized in the Non commutative Logic of Ruet and Abrusci [22, 21], but there is a condition on the constraints, which must have no non logical axioms. The reason is that confusing a constraint with its consequences would create fake suspensions. As we now can avoid this phenomenon, it s natural to try and generalize the previous result to a general observation ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....an extension of the published version, to be precise. In this calculus one both have non commutative connectives of the Lambek calculus [14] and the usual commutative connectives of multiplicative linear logic [10] This kind of calculus has then been extended to a classical setting by Paul Ruet [18], and further studied by Michele Abrusci and Paul Ruet [1] Akim Demaille [7] As usual a (partial) marking with n(P ) tokens in the place P will be denoted by the formula N P P n(P ) where Omega is the commutative product and P k a short hand for P Omega Delta Delta Delta Omega P ....

....or the relation to other system are not discussed. One is referred to, for instance [11, 19, 17] Regarding the commutative and non commutative calculus P.D.G. it is defined in [6] as PNC IMLL (Partially Non Commutative Intuitionistic Multiplicative Linear Logic) the reader is also referred to [18, 1] for further work and to [7] for the study and comparison of various versions. INRIA Petri nets and partially commutative linear logic 7 Regarding series parallel partial orders, which play an important rle, the reader is referred to the excellent survey [15] Acknowledgments Thanks to Philippe ....

Paul Ruet. Logique non-commutative et programmation concurrente. Thse de doctorat, spcialit logique et fondements de l'informatique, Universit Paris 7, 1997.

....explore uses of an order aware logic in computer science, we would like it to conservatively extend both traditional and linear logic. Recently, following early explorations [Yet90, Abr90, BG91] two such proposals have been made: one by Abrusci and Ruet rooted in a non commutative conjunction [Rue97, AR98], and one by the authors based on ordered hypotheses [PP99a] Our system of natural deduction provides a foundation for applications in functional programming, which we investigated via the Curry Howard isomorphism and properties of an ordered calculus. Subsequently, we exhibited a related ....

....context constructors in the sequent calculus presentation. The logic is essentially all of linear logic with the addition of a non commutative tensor and and its dual. There is no explicit mobility modality as in our system. The intuitionistic version of non commutative logic presented in [Rue97, Gro96] is an acyclic, single conclusion restriction of NL. Thus our system may be seen as the fragment of intuitionistic NL without the commutative connectives, extended with a mobility modality. 3 Uniform Derivations Now that we have a suitable sequent system for OLL we begin analyzing proof structure ....

Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....rules on modal formulas. As an alternative we propose a system which distinguishes among unrestricted, linear, and ordered hypotheses. Our presentation of INCLL is in the form of natural deduction with proof terms, thereby departing from previous formulations based on the sequent calculus [BG91,Abr90,Rue97]. This establishes the connection to functional computation by an extension of the Curry Howard isomorphism. INCLL is a conservative Partially supported by the National Science Foundation under grant CCR 9804014. Partially supported by the National Science Foundation under grant ....

Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....The notion of commutativity usually concerns binary operators rather than individual properties. However, as we have previously pointed out, INCLL captures non commutativity at the level of hypotheses rather than connectives. Both Ruet s intuitionistic non commutative linear logic [21] and Ruet and Abrusci s classical non commutative linear logic [2] or cyclic linear logic) have two context constructors, one of which is commutative. Additionally neither of these systems directly admits the concept of a mobile hypothesis (although the latter system may be able to capture it by ....

Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....To fully explore uses of an order aware logic in computer science, we would like it to conservatively extend both traditional and linear logic. Recently, following early explorations [Abr90, BG91] two such proposals have been made: one by Abrusci and Ruet rooted in a non commutative conjunction [Rue97, AR98], and one by the authors based on ordered hypotheses [PP99] Our system of natural deduction provides a foundation for applications in functional programming, which we investigated via the Curry Howard isomorphism and properties of an ordered calculus. Subsequently, we exhibited a related sequent ....

....and warrants further investigation from all angles. Besides further work on language design, implementation, and programming methodology, we also plan to investigate the potential applications of our particular approach to order in concurrent constraint programming as proposed by Ruet [Rue97]. Finally, the existence of canonical forms for the natural deduction system of INCLL [PP99] means that it might be possible to add it to the linear logical framework [CP98] in a conservative manner. However, we have not yet considered such issues as type reconstruction or unification. ....

Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....To fully explore uses of an order aware logic in computer science, we would like it to conservatively extend both traditional and linear logic. Recently, following early explorations [Abr90, BG91] two such proposals have been made: one by Abrusci and Ruet rooted in a non commutative conjunction [Rue97, AR98], and one by the authors based on ordered hypotheses [PP99] Our system of natural deduction provides a foundation for applications in functional programming, which we investigated via the Curry Howard isomorphism and properties of an ordered calculus. Subsequently, we exhibited a related sequent ....

....computation and warrants further investigation from all angles. Besides further work on language design, implementation, and programming methodology, we also plan to investigate the potential applications of our particular approach to order in concurrent constraint programming as proposed by Ruet [Rue97]. Finally, the existence of canonical forms for the natural deduction system of INCLL [PP99] means that it might be possible to add it to the linear logical framework [CP98] in a conservative manner. However, we have not yet considered such issues as type reconstruction or unification. ....

Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....over reals, and then programming one (of many possible) constraint solvers faithful to the cc ask tell paradigm as an lcc program on top of that constraint system. The rest of the paper is organized as follows. Section 2 gives the necessary background on the lcc framework as developed in [13, 4], and describes the tight correspondance between lcc computations and proofs in linear logic. Section 3 then introduces the constraint system LC designed to serve as a base for (nonmonotonic) constraint computing over reals. Section 4 illustrates the use of LC by expressing a constraint solver ....

....solver that is complete with respect to satisfiability and entailment of linear equations and inequations as a couple of lcc agents, effectively defining a cc(R) language. 2 The lcc Framework We recall below the main features of the linear concurrent constraint framework as defined in [4, 13] 2.1 Syntax Definition 21 (Linear constraint system) A linear constraint system is a pair (C; C ) where: C is a set of formulas of intuitionistic linear logic (ILL) see for example [5] called the linear constraints, built from a set V of variables, a set Sigma of function and relation ....

[Article contains additional citation context not shown here]

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universit'e Denis Diderot, Paris 7, 1997.

....useful for reactive systems. Standard CC programs can however be recovered by the usual translation of intuitionistic logic into linear logic [10] Section 3 settles the basic soundness and completeness results of CC and LCC operational semantics w.r.t. intuitionistic linear logic, relying on [30] and preliminary results from [29] Results similar to those of this section are part of the folklore on CC languages [19, 34] but have not been published. Here we prove that 1) the stores of CC computations can be characterized in intuitionistic logic, and 2) both the stores and the successes of ....

....to model checking. Preliminary results on counter phase model generation methods can be found in [26] The method can be generalized to handle more safety properties of LCC programs. In particular the characterization of LCC suspensions [31] in the non commutative logic of the 30 second author [30], can be used to prove deadlock properties using non commutative phase spaces. The extension induced by the logic of CC languages to linear constraint systems is also interesting to study in its own right as it reconciles declarative programming with some form of imperative programming. We ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universit'e Denis Diderot, Paris 7, 1997.

....useful for reactive systems. Standard CC programs can however be recovered by the usual translation of intuitionistic logic into linear logic [10] Section 3 settles the basic soundness and completeness results of CC and LCC operational semantics w.r.t. intuitionistic linear logic, relying on [30] and preliminary results from [29] Results similar to those of this section are part of the folklore on CC languages [19, 34] but have not been published. Here we prove that 1) the stores of CC computations can be characterized in intuitionistic logic, and 2) both the stores and the successes of ....

....way to model checking. Preliminary results on counter phase model generation methods can be found in [26] The method can be generalized to handle more safety properties of LCC programs. In particular the characterization of LCC suspensions [31] in the non commutative logic of the second author [30], can be used to prove deadlock properties using non commutative phase spaces. The extension induced by the logic of CC languages to linear constraint systems is also interesting to study in its own right as it reconciles declarative programming with some form of imperative programming. We ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....a sequent, etc. The number of negations however, remains open : there could be a commutative one and a non commutative one : Order varieties. The solution is based on : a syntactic idea : the seesaw rule, and its semantic counterpart : the structure of order variety. Order varieties see (Ruet 1997; Abrusci and Ruet 1999) and section 3 are structures that can be presented by partial orders in several ways, a good analogy being the oriented circle which becomes a total order as soon as an origin is xed. An essential property of order varieties (proposition 2.5) is that in a sequent ....

....there are several possible choices : 1. ed formulas do not commute in a non commutative situation : from [ A;B] we may not infer [B; A] even if B is itself a ed formula. This has been considered by Demaille in (Demaille 1999) 2. Bags of ed formulas commute. This has been considered in (Ruet 1997), and it is consistent with the intuition that there is basically a single par and a single tensor and the isomorphisms : A; B) A B) A; B) 3. ed formulas are central : they commute with everyone. This is the choice we make here, as it is simpler than 2, while preserving ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

....for r links. Then there is a simple definition of proof nets by a trip condition, which can be generalized in presence of commutative connectives. The second author introduced a mixed non commutative commutative sequent calculus enjoying cut elimination and a corresponding phase semantics [17], starting from the intuitionistic version of De Groote [6] and questions arising in the theory of concurrency [19] The main technical ingredient is the structure of order varieties, which enable to express symmetry constraints in a sequent. An order variety is a structure which, provided a point ....

.... (adequacy: theorem 4.3) and ff is induced by the proof net via definition 3.23. 5. In practice, in the sequent calculus, one may prefer to work with presentations, i.e. partial orders, rather than order varieties. It is possible to define a sequent calculus on presentations: this is done in [17,18]. In that case, one needs to add structural rules, seesaw and entropy: Gamma k Delta seesaw Gamma Delta Gamma[ Delta Sigma] entropy Gamma[ Delta k Sigma] which enable to change the presentation. In the present paper, we simply consider the sequent calculus on order ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universit'e Denis Diderot, Paris 7, 1997.

....for r links. Then there is a simple definition of proof nets by a trip condition, which can be generalized in presence of commutative connectives. The second author introduced a mixed non commutative commutative sequent calculus enjoying cut elimination and a corresponding phase semantics [17], starting from the intuitionistic version of De Groote [6] and questions arising in the theory of concurrency [18] The main technical ingredient is the structure of order varieties, which enable to express symmetry constraints in a sequent. An order variety is a structure which, provided a point ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universit'e Denis Diderot, Paris 7, 1997.

....linear logic [6] Section 3 settles the basic soundness and completeness results of LCC operational semantics w.r.t. intuitionistic linear logic. Results similar to those of this section are part of the folklore on CC languages [15, 24] but have not been published, we refer to the appendix B and [5, 20] for complete proofs. Completeness results show that ILL can be used to prove liveness properties of LCC programs, i.e. properties expressing that something good will eventually happen. Then we show in section 4 how safety properties of CC and LCC programs (i.e. that some derivations never ....

....proofs in a somewhat similar way to model checking. These issues should serve further investigation. The method can be generalized to handle more safety properties of LCC programs. In particular the characterization of LCC suspensions [21] in the non commutative logic of the second author [20], can be used to prove deadlock properties using noncommutative phase spaces. The extension induced by the logic of CC languages to linear constraint systems is also interesting to study in its own right as it reconciles declarative programming with some form of imperative programming. We ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universit'e Denis Diderot, Paris 7, 1997.

....in a sequent, etc. The number of negations however, remains open: there could be a commutative one and a non commutative one : Order varieties. The solution is based on: a syntactic idea: the seesaw rule, and its semantic counterpart: the structure of order variety. Order varieties (see [13, 1] and section 3) are structures that can be presented by partial orders in several ways, a good analogy being the oriented circle which becomes a total order as soon as an origin is fixed. An essential property of order varieties (proposition 2.5) is that in a sequent Gamma structured by an ....

....there are several possible choices: 1. ed formulas do not commute in a non commutative situation: from Gamma[ A; B] we may not infer Gamma[B; A] even if B is itself a ed formula. This has been considered by Demaille in [3] 2. Bags of ed formulas commute. This has been considered in [13], and it is consistent with the intuition that there is basically a single par and a single tensor and the isomorphisms: A; B) A Phi B) A; B) 3. ed formulas are central: they commute with everyone. This is the choice we make here, as it is simpler than 2, while ....

[Article contains additional citation context not shown here]

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universit'e Denis Diderot, Paris 7, 1997.

....proofs in a somewhat similar way to model checking. These issues should serve further investigation. The method can be generalized to handle more safety properties of LCC programs. In particular the characterization of LCC suspensions [28] in the non commutative logic of the second author [27], can be used to prove deadlock properties using non commutative phase spaces. The extension induced by the logic of CC languages to linear constraint systems is also interesting to study in its own right as it reconciles declarative programming with some form of imperative programming. We ....

P. Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universit'e Denis Diderot, Paris 7, 1997.

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Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997. 25

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Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997. 18

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Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

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Paul Ruet. Logique non-commutative et programmation concurrente par contraintes. PhD thesis, Universite Denis Diderot, Paris 7, 1997.

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