C. Retor'e. R'eseaux et S'equents Ordonn'es Th`ese de doctorat, U. Paris VII, Math'ematiques, 1993. 74  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

....derivation D of Gamma such that R = D) Gamma . c) If D reduces to D 0 , then D Gamma reduces to (D 0 ) Gamma . d) If D Gamma reduces to R 0 then there is a D 0 such that D reduces to D 0 and R 0 = D 0 ) Gamma . A similar result can be stated for direct logic ([11, 7, 31]) Given an MLL proof structure S A 1 ; A n with distinguished conclusions A 1 ; A n we want to associate to it a basic synchronous calculus term ( S A 1 ; A n ) whose free names will be in bijective correspondence with the conclusions. The key point of this translation is ....

C. Retor'e. R'eseaux et S'equents Ordonn'es Th`ese de doctorat, U. Paris VII, Math'ematiques, 1993. 74

....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 ....

....then there is a sequent calculus derivation D of 0 such that R = D) 0 . c) If D reduces to D 0 , then D 0 reduces to (D 0 ) 0 . d) If D 0 reduces to R 0 then there is a D 0 such that D reduces to D 0 and R 0 = D 0 ) 0 . A similar result can be stated for direct logic ([11, 7, 31]) Given an MLL proof structure S A 1 ; A n with distinguished conclusions A 1 ; A n we want to associate to it a basic synchronous calculus term ( S A 1 ; A n ) whose free names will be in bijective correspondence with the conclusions. The key point of this translation is ....

C. Retor'e. R'eseaux et S'equents Ordonn'es Th`ese de doctorat, U. Paris VII, Math'ematiques, 1993. 74

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