A. Asperti. Causal dependencies in multiplicative linear logic with MIX. Math. Struct. in Comp. Sci., 11:1-31, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(which characterizes proof nets in direct logic) seems to express absence of deadlock in a logical computation; clearly, a stronger notion of deadlock than ours is needed to characterize correctness for such nets. We shall not pursue the matter further here (cf. recent work of A. Asperti [5]) Another example of computationally consistent orientation will be considered in Section 5.4 below. 5.4 Orienting Proof Nets and Linear terms The present section is based on joint work by G.Bellin and J. van de Wiele ( 8] The system IMLL of intuitionistic multiplicative linear logic is based ....

A. Asperti. Causal Dependencies in Multiplicative Linear Logic with MIX, manuscript. To appear in: Mathematical Structures in Computer Science.

....automated deduction. We also observe the importance of the definition of labels for such a characterization. It appears from the works of Asperti on logical analysis of distributed systems that MLL plus the MIX rule might be more suitable for the representation of concurrent logical computations [3]. This rule ; Mix added to MLL has been also studied from the proof net point of view by showing that the verification of correctness for proof nets in this logic is equivalent to a successful termination of a concurrent game in the style of Petri nets [3] Refinements and ....

.... logical computations [3] This rule ; Mix added to MLL has been also studied from the proof net point of view by showing that the verification of correctness for proof nets in this logic is equivalent to a successful termination of a concurrent game in the style of Petri nets [3]. Refinements and relationships with term assignments are proposed in [4] where a system of proof nets for MLL (without constants) MIX is given. From such a study of (multiplicative) classical linear logic, 4] has also proposed a proof nets system for (multiplicative) FILL MIX(without ....

A. Asperti. Causal dependencies in multiplicative linear logic with MIX. Math. Struct. in Comp. Science, 5:351--380, 1995.

....(which characterizes proof nets in direct logic) seems to express absence of deadlock in a logical computation; clearly, a stronger notion of deadlock than ours is needed to characterize correctness for such nets. We shall not pursue the matter further here (cf. recent work of A. Asperti [5]) Another example of computationally consistent orientation will be considered in Section 5.4 below. 5.4 Orienting Proof Nets and Linear terms The present section is based on joint work by G.Bellin and J. van de Wiele ( 8] The system IMLL of intuitionistic multiplicative linear logic is ....

A. Asperti. Causal Dependencies in Multiplicative Linear Logic with MIX, manuscript. To appear in: Mathematical Structures in Computer Science.

.... existence of a cyclic Danos Regnier graph [9] in , and interpreted intuitively as a violation of causality: P (a Omega b) receives from Q (b Omega a) a message a that was caused by the sending by P of a message b to Q (this corresponds to a deadlocked proof structure in the sense of [2]) Now, we a 1 6 6 b 1 Omega Omega Omega b 2 5 5 a 2 Omega Omega Omega a Omega b b Omega a (a) b(y) b(u) 1 Psi Psi Psi Psi 8yb(y) C C C C b(u) Omega p p p p p p p p p 1 a(u) E E E E O n n n n n n n n n n n 9x(a(x) Omega (8yb(y) O ....

.... of the operational semantics as given by the labelled transition system (using, for example, a kind of expansion law) but results from a natural analysis of causality in proof nets related to some standard proof theoretic results, like the sequentialisation theorem and the token games of [2]. Moreover, like in L , actions are multisets of terminal formulas, and we are approximating true concurrency by means of a higherdimensional transition system [8] The extension of this results to the full language constitutes work in progress. 8 Related Work and Conclusion In the present ....

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Andrea Asperti. Causal dependencies in multiplicative linear logic with MIX. Mathematical Structures in Computer Science, 5(3):351--380, September 1995.

....cases. The case of multiplicative logic without Mix was studied by the second author in his thesis [8] There are both logical and geometrical reasons to consider the rule of Mix Gamma Delta Gamma; Delta for instance in the study of classical systems and in the theory of concurrency [2]. Notice also that the usual geometric interpretation of the tensor product of morphisms in monoidal categories is the juxtaposition of morphisms. To recover such a construction in the context of Linear Logic we must add the rule of Mix. We are concerned with the issues of correctness, ....

....for abstract graphs [4, 7] But we can do better than searching all possible chains: only some chains have to be tested, and this can be done in linear time, as we show in the next section. 4 A characterization in terms of concurrent processes We consider a variant of Asperti s token game [2]; the original formulation by Asperti is in terms of Petri Nets; we speak informally of trips of tokens in a proof structure and regard this condition as the right generalization of Girard s no short trip condition to the case of MLL Gamma Mix. There are tokens of type and #. Given a ....

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A. Asperti. Causal Dependencies in Multiplicative Linear Logic with MIX. Mathematical Structures in Computer Science, 5, 1995, pp. 351380.

.... of a cyclic Danos Regnier (D R) graph [10] and interpreted intuitively as a violation of causality: P (a Omega b) receives from Q (b Omega a) a message a that was caused by the sending by P of a message b to Q (this corresponds to a deadlocked proof structure in the sense of [2]) Now, we can capture these internal dependencies by a symmetric binary relation between ports such that a b iff there is a path in some D R graph (a causal chain) connecting a and b. In the present example, we should have a 1 b 1 and b 2 a 2 . Therefore, since a 1 6 a 2 ....

.... of the operational semantics as given by the labelled transition system (using, for example, a kind of expansion law) but results from a natural analysis of causality in proof nets related to some standard proof theoretic results, like the sequentialization theorem and the token games of [2]. Moreover, since in L , actions are multisets of terminal formulas, we are approximating true concurrency by means of a higher dimensional transition system [9] Extending this results to the full language constitutes work in progress. 8 Related Work and Conclusion In the present paper, we ....

[Article contains additional citation context not shown here]

Andrea Asperti. Causal dependencies in multiplicative linear logic with MIX. Mathematical Structures in Computer Science, 5(3):351--380, September 1995.

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A. Asperti. Causal dependencies in multiplicative linear logic with MIX. Math. Struct. in Comp. Sci., 11:1-31, 1993.

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A. Asperti. Causal dependencies in multiplicative linear logic with MIX. Mathematical Structures in Computer Science, 5(3):351-380, September 1995.

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A. Asperti. Causal dependencies in multiplicative linear logic with MIX. Math. Struct. in Comp. Sci., 11:1--31, 1993.

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