V. Danos (1990) "La logique lin'eaire appliqu'ee `a l"etude de divers processus de normalization et principalement du -calcul." Ph.D. dissertation. 24  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... autonomous category. Thus, our logics LF and LF I provide an alternate sequent calculus presentation for these fragments, and the functor nets of [15] provide a net theoretic syntax. In the exponential case, these will be a variant of the familiar exponential boxes introduced by Danos and Regnier [16] and studied from a categorical perspective in [10, 15] We have also noted that our logic LF can provides an alternative sequent calculus for the analysis of Retor e s noncommutative connective on coherence spaces. It should be of interest to compare our sequent calculus with that introduced by ....

V. Danos (1990) "La logique lin'eaire appliqu'ee `a l"etude de divers processus de normalization et principalement du -calcul." Ph.D. dissertation. 24

....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 ....

....if there is no ambiguity, and refer to it as the D R graph of R. For further information, cf. 12, 6] Occasionally (e.g. in Section 5. 3 below) we will consider the system of sequent calculus for multiplicative linear logic, with the additional structural rule of Mix, also called direct logic DL [7, 11, 31, 4, 10]: mix : Gamma Delta Gamma; Delta Definition. A proof structure R is a proof net for Direct Logic DL if for every switching s of R, the graph G s (R) is acyclic (but not necessarily connected. The following fundamental result (Girard [13] relates sequent calculus and proof nets for ....

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V. Danos. La logique lin'eaire appliqu'ee `a l"etude de divers processus de normalisation et principalement du -calcul, Th`ese de doctorat, U. Paris VII, April 1990.

....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 ....

....if there is no ambiguity, and refer to it as the D R graph of R. For further information, cf. 12, 6] Occasionally (e.g. in Section 5. 3 below) we will consider the system of sequent calculus for multiplicative linear logic, with the additional structural rule of Mix, also called direct logic DL [7, 11, 31, 4, 10]: mix : 0 1 0; 1 Definition. A proof structure R is a proof net for Direct Logic DL if for every switching s of R, the graph G s (R) is acyclic (but not necessarily connected. The following fundamental result (Girard [13] relates sequent calculus and proof nets for MLL. 19 Theorem 4 ....

[Article contains additional citation context not shown here]

V. Danos. La logique lin'eaire appliqu'ee `a l"etude de divers processus de normalisation et principalement du -calcul, Th`ese de doctorat, U. Paris VII, April 1990.

....and faithful embedding between two logical system. The key is, computational properties are preserved between the original system and the decorated system. From this property, one can see that confluency of cut elimination procedure for LKT and LKQ is an immediate corollary to confluency for MELL[2]. In this paper, we focus on the intuitionistic version of decoration which is analogue of linear one. Our method can be seen as yet another proof of SN and CR property of cut elimination for LKT and LKQ, since typed calculus is also proven to be SN and CR [11] Our contribution is to the ....

....properties: Cut elimination step for LKT and LKQ is one to one to the cut elimination for its linear decoration. Proposition 3 (DJS) Cut elimination procedure for LKT LKQ is Strongly Normalizing (SN) and Church Rosser (CR) Because Cut elimination procedure of MELL is proven to be SN and CR [2]. All properties above for linear decoration also holds for intuitionistic decoration. Proposition 4 (DJS) LKT and LKQ is sound and complete w.r.t. classical provability. Soundness is easy, as ignoring semicolon in LKT and LKQ derivation induce LK derivation. For completeness, DJS shows a ....

Vincent Danos. La Logique Lin'eaire Appliqu'ee `a l"etude de Divers Processus de Normalisation (Principalement du -Calcul). PhD thesis, University of Paris VII, June 1990.

....5 q Propositon 2.2 (DJS) Cut elimination procedure of LKT LKQ is isomorphic to the one that of its linear decoration. Propositon 2. 3 (DJS) Cut elimination procedure of LKT LKQ is Strongly Normalizing (SN) and Church Rosser (CR) Because Cut elimination procedure of MELL is proven to be SN and CR [3]. Above properties for linear decoration is also true for intuitionistic decoration. Thus intuitionistic decoration can be seen as a part of yet another proof of SN and CR of LKT LKQ, since typed calculus is also proven to be SN and CR [12] 2.4 Constrictive Morphisms Both LKT LKQ can be ....

.... B (CBV) respectively. This shows that computational interpretation on this small fragment of MELL proof net is already popular in computer science. Thus the approach of Curry Howard isomorphism on MELL PN as programs and cut elimination as computation seems to be promising. It is SN and CR [3] and can be extended to the second order (system F polymorphism [9] It seems interesting to explore the categorical proof theory (categorical logic) for MELL proof net(PN) in which reflective transitive closure of cut elimination procedure are the equalities between PNs(as morphisms of ....

Vincent Danos. La Logique Lin'eaire Appliqu'ee `a l"etude de Divers Processus de Normalisation (Principalement du -Calcul). PhD thesis, University of Paris VII, June 1990.

....logic, and some of those differences we mention here. First, we replace the net criterion (as defined by Danos and Regnier [DR89] for instance) with a more local algorithm for determining the sequentiality of a proof structure. This algorithm is essentially the polynomial time algorithm of Danos [D90]. One reason for doing this is that the usual net criterion is insufficient in the noncommutative case with units we plan to discuss the various issues dealing with the non commutative case in a sequel. Furthermore, our proof of sequentialization applies to the case, essential to the categorical ....

....number of components and m the number of wires of the circuit. To find an application of a rewriting, however, may also require a traversal of the structure. Thus, even from this naive view, to determine whether a circuit is sequential is an O( n m) 2 ) algorithm (a more detailed analysis is in [D90]) 2.7 Sequentialization, the net condition, and empires The traditional criterion for being a proof net, it is worth recalling, uses an apparently unrelated aspect of the circuits. Notice that we have drawn arcs between the ( Omega E) output ports and the ( PhiI ) input ports: these are to ....

Danos, V. "La logique lin'eaire appliqu'ee `a l"etude de divers processus de normalisation et principalement du -calcul", Doctoral dissertation, Universit'e de Paris, 1990.

....is OE Theorem 2.3 (Simulation) Cut elimination step of the LKQ is one to one to the one that of its linear decoration. Corollary 2. 4 Cut elimination procedure for the LKQ is Strongly Normalizing (SN) and ChurchRosser (CR) Because Cut elimination procedure for the MELL is proven to be SN and CR [2]. Remark 2.5 We choose q decoration A q ( B q ) instead of DJS s ( B q ) A q ) 4] This only affects the order of abstraction (such as k 0 :x:t ) and application (such as z(y:u)v) In fact, this decoration induces slightly different calculus known as Fischer s CBV CPS calculus ....

Vincent Danos. La Logique Lin'eaire Appliqu'ee `a l"etude de Divers Processus de Normalisation (Principalement du -Calcul). PhD thesis, University of Paris VII, June 1990.

....properties. Lemma 7. For any (t,m)cut elimination step ( reduction) in LKQ= precisely corresponds to cutelimination step in CLL. It means that LKQ= is essentially a subset of (multiplicative, exponential part of) CLL and its Proof Net. It is strongly normalizable (SN) 9] and ChurchRosser (CR)[3]. We can also extend LKQ= to second order [9] with tamed additives [4] Proposition 8. The reduction system LKQ= is confluent and strongly normalizing. A x t = s with 2] A 2 (replace all occurrence of x A with s A ) 3 t = 3 ; A u1 : A un t = s with 2] A ....

Vincent Danos. La Logique Lin'eaire Appliqu'ee `a l"etude de Divers Processus de Normalisation (Principalement du -Calcul). PhD thesis, University of Paris VII, June 1990.

....q Propositon 2.2 (DJS) Cut elimination procedure of LKT LKQ is one to one to the one that of its linear decoration. Propositon 2. 3 (DJS) Cut elimination procedure of LKT LKQ is Strongly Normalizing (SN) and Church Rosser (CR) Because Cut elimination procedure of MELL is proven to be SN and CR [2]. Above properties for linear decoration is also true for intuitionistic decoration. Thus intuitionistic decoration can be seen as yet another proof of SN and CR of LKT LKQ, since typed calculus is also proven to be SN and CR [11] 2.3 Constrictive Morphisms Both LKT LKQ can be seen as ....

Vincent Danos. La Logique Lin'eaire Appliqu'ee `a l"etude de Divers Processus de Normalisation (Principalement du -Calcul). PhD thesis, University of Paris VII, June 1990.

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V. Danos, "La Logique Lin'eaire Appliqu'ee `a l"etude de Divers Processus de Normalisation (Principalement du -Calcul)," Ph. D. Thesis, University of Paris VII, Jun. 1990.

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Vincent Danos. La Logique Lin'eaire Appliqu'ee `a l"etude de Divers Processus de Normalisation (Principalement du -Calcul). PhD thesis, University of Paris VII, June 1990.

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Vincent Danos. La Logique Lin'eaire Appliqu 'ee `a l"etude de Divers Processus de Normalisation (Principalement du -Calcul). PhD thesis, University of Paris VII, June 1990.

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