Malacaria, P., Regnier, L.: Some results on the interpretation of #-calculus in operator algebras. In: Proceedings of LICS. (1991)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....mathematical tool inside the model used to interpret the cut elimination process. This operation is not always defined and sufficient conditions have been given which ensure termination of the computation : in the case of second order Linear Logic ( Gir88a, Gir95a] and of untyped lambda calculus [MR91]) operators coming from the syntax do satisfy such conditions (a nilpotency condition for instance in the case of LL) Various frameworks have been used to describe goi models: bounded operators on Hilbert spaces ( Gir88a, DR95] partial applications ( Dan90, Reg92] and clauses ( Gir95a] This ....

P. Malacaria and L. Regnier. Some results on the interpretation of -calculus in operator algebras. In Proceedings of the Sixth Annual Symposium on Logic in Computer Science, pages 63--72. IEEE Computer Society Press, 1991.

....for various logical proof reduction (normalization) strategies. A second theme of this paper is that information flow in cut elimination is related to sending receiving protocols in calculus. In Section 5, this intuition is made precise in a detailed analysis of information flow in pure nets [11, 22, 8] which is a theory of graphical networks for representing untyped lambda terms in the style of proof nets for linear logic. We shall show how to dynamically orient a proof net (assigning the symbols I (for input) and O (for output) in a manner coherent with respect to introduction and elimination ....

....of information flow for pure nets, thus obviating the need for bidirectional buffers 3 . Terms of the calculus have a natural direction, namely from the inputs to the output. This is obvious from the intended functional interpretation, as well as the dynamics of evaluation. V. Danos, L. Regnier [11, 30, 22], J. van de Wiele and others have studied pure nets, a formalism of proof nets for untyped calculus: pure nets are nets built from formulas I and O (representing input and output, resp. using links of the forms I O O I cut I O O . ....

[Article contains additional citation context not shown here]

P. Malacaria and L. Regnier. Some Results on the Interpretation of -calculus in Operator Algebras, Logics in Computer Science (LICS), 1991, Amsterdam, IEEE Computer Press, pp. 63-72.

....nilpotency of oeU , i.e. that some power (oeU) n is zero, a condition which is experimentally true for all pairs (U; oe) arising from logical proofs 4 . In this case the central part of the execution 4 For non logical systems like pure calculus there is still a notion of weak nilpotency, see [G88A, MR91] Geometry of interaction III 11 formula can be written as a finite sum : U(1 Gamma oeU) Gamma1 = U U oeU U oeU oeU : the length of the sum being equal to the order of nilpotency n of oeU . Observe that n can be extremely big : the basic way to get a high order of nilpotency n is ....

Malacaria, P. & Regnier, L. : Some results on the interpretation of -calculus in operator algebras, Proc. of LICS '91, IEEE Computer Society Press, 1991, pp. 63-72. 62 Jean-Yves Girard

.... into dynamic, operational semantics of linear logic proof reduction (cut elimination) is provided by geometry of interaction, proposed by Girard [47, 48] an exposition is in Danos [36] Geometry of interaction is further investigated in the work of Danos [37] Regnier [83] Malacaria and Regnier [78], Abramsky and Jagadeesan [3] and Gonthier et al. 55, 56] see below) Linear logic continues to be a very active field of research. Up to date developments are discussed on the electronic forum Linear . One subscribes by sending an email message to linear request cs.stanford.edu. An ....

P. Malacaria and L. Regnier. Some results on the interpretation of - calculus in operator algebras. In Proc. 6-th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pages 63--72. IEEE Computer Society Press, Los Alamitos, California, July 1991.

....(straightforward) generalization, we have pursued an alternative approach in [Asperti and Laneve 1994] which is more useful in the perspective of optimal implementations. We have implemented any IS into pure nets, a type free variant of Girard s Proof Nets, that have been studied in [Danos 1990, Malacaria and Regnier 1991, Regnier 1991] with the aim of encoding pure calculus. The implementation, on one hand, confirms the logical foundation of IS s and, on the other hand, provides the intermediate step towards the optimal implementation in [Asperti and Laneve 1993a, Laneve 1993] We refer the interested reader to ....

P. Malacaria and L. Regnier (1991) Some results on the interpretation of -calculus in operator algebras. In Proceedings 6 th Annual Symposium on Logic in Computer Science, Amsterdam, pages 63 -- 72.

....for various logical proof reduction (normalization) strategies. A second theme of this paper is that information flow in cut elimination is related to sending receiving protocols in calculus. In Section 5, this intuition is made precise in a detailed analysis of information flow in pure nets [11, 22, 8] which is a theory of graphical networks for representing untyped lambda terms in the style of proof nets for linear logic. We shall show how to dynamically orient a proof net (assigning the symbols I (for input) and O (for output) in a manner coherent with respect to introduction and elimination ....

....information flow for pure nets, thus obviating the need for bidirectional buffers 3 . Terms of the calculus have a natural direction, namely from the inputs to the output. This is obvious from the intended functional interpretation, as well as the dynamics of evaluation. V. Danos, L. Regnier [11, 30, 22], J. van de Wiele and others have studied pure nets, a formalism of proof nets for untyped calculus: pure nets are nets built from formulas I and O (representing input and output, resp. using links of the forms I O O I cut I O O . ....

[Article contains additional citation context not shown here]

P. Malacaria and L. Regnier. Some Results on the Interpretation of -calculus in Operator Algebras, Logics in Computer Science (LICS), 1991, Amsterdam, IEEE Computer Press, pp. 63-72.

....(straightforward) generalization, we have pursued an alternative approach in (Asperti and Laneve 1994) which is more useful in the perspective of optimal implementations. We have implemented any IS into pure nets, a type free variant of Girard s Proof Nets, that have been studied in (Danos 1990; Malacaria and Regnier 1991; Regnier 1991) with the aim of encoding pure calculus. The implementation, on one hand, confirms the logical foundation of IS s and, on the other hand, provides the intermediate step towards the optimal implementation in (Asperti and Laneve 1993a; Laneve 1993) We refer the interested reader to ....

P. Malacaria and L. Regnier (1991) Some results on the interpretation of -calculus in operator algebras. In Proceedings 6 th Annual Symposium on Logic in Computer Science, Amsterdam, pages 63 -- 72.

....the basis. Moreover the issue of head linear reduction is mentioned in [Gir91] An improvement of proof net extraction technique should be possible in a fragment of LU. ffl Linear logic approach to pure calculus is given by the calculus of pure nets developed in [Dan89] Dan90] Reg90] [MaRe91], DuVdW91] and [Reg92] An algebraic characterization of the proof extraction in the framework of dynamic algebras should be interesting. ffl The Bohm out Technique and Bohm Theorem, first introduced in [Boh68] is treated in full generality in [Bar84] and [Kri90] The issue of equations in ....

P.Malacaria, L.Regnier, Some Results on the Interpretation of -calculus in Operator Algebras, LICS, 1991.

..... q p p q p q (q ) q) p ) p ) x 12 x 01 (x 14 ) x 03 ) Theorem 8 For any term T , the virtual structure T v is split and square free; in other words it is a virtual net. Square freedom for calculus was proved in [6] and splitness is easily proved by induction on T . ....

Pasquale Malacaria and Laurent Regnier. Some results on the interpretation of -calculus in operator algebras. In Proceedings L.I.C.S., 1991.

.... (e n ) The big model of L R is the set of bounded operators on H, denoted by B(H) The interest of this new model is that we are going to use our remark about the linearity of the commutation relation by superposing small solutions; another interest may be understood by reading [Gir88b] or [MR91]. The basic structure. B(H) is clearly an involutive monoid (w.r.t. composition and adjunction) wherein there is a zero and a one. It even is a C algebra. The shift morphism. Take : H Omega H Gamma H to be the isomorphism defined by (e i Omega e j ) e di;je , where H Omega H means ....

Pasquale Malacaria and Laurent Regnier. Some results on the interpretation of -calculus in operator algebra. In Proceedings of the Sixth Annual Symposium on Logic in Computer Science, Amsterdam, 1991. IEEE Computer Society Press. V. Danos & L. Regnier

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Malacaria, P., Regnier, L.: Some results on the interpretation of #-calculus in operator algebras. In: Proceedings of LICS. (1991)

No context found.

Malacaria, P., Regnier, L.: Some results on the interpretation of #-calculus in operator algebras. In: Proceedings of LICS. (1991)

No context found.

P. Malacaria and L. Regnier. Some results on the interpretation of -calculus in Operator Algebras. In Proceedings of the Sixth International Symposium on Logic in Computer Science, IEEE Press, 63-72, 1991.

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