Denis Bechet, Philippe de Groote, and Christian Retore. A complete axiomatisation of the inclusion of series-parallel partial orders. In H. Common, editor, Rewriting Techniques and Applications, RTA 1997,Lecture Notes in Computer Science, pages 230--240. Springer-Verlag, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....connective, derived from coherence semantics. He shows a rewriting system that yields cut elimination for certain proof nets. That rewriting system is essentially equivalent to system SBVc. Retore got his rewriting rules from a study of the inclusion relation for series parallel orders, in [4]. Since series parallel orders are at the basis of my definition of structures, Retore s result and mine are similar because they come from the same underlying combinatorial phenomena. The di#erence between his work and mine is in the use we make of the rewriting system: he applies its rules in ....

Denis Bechet, Philippe de Groote, and Christian Retore. A complete axiomatisation of the inclusion of series-parallel orders. In H. Comon, editor, Rewriting Techniques and Applications, RTA '97, volume 1232 of Lecture Notes in Computer Science, pages 230--240. Springer-Verlag, 1997.

....orders on the same set such that : 1 k 2 ] P [ 1 2 ] Proof. By lemma 3.23, the above rule is sound. Conversely, let and be two seriesparallel orders on the same set such that P . We use the axiomatisation of the inclusion between series parallel orders given by (Bechet, de Groote and Retor e 1997) ; the result stated in propositions 3.2 and 4.1 of (Bechet, de Groote and Retor e 1997) is the following : i is obtained from by means of the following re exive and transitive congruence : Paul Ruet 18 (1) k (2) 1 k 2 ) 1 ) k 2 (3) 1 k 2 ) ....

....By lemma 3.23, the above rule is sound. Conversely, let and be two seriesparallel orders on the same set such that P . We use the axiomatisation of the inclusion between series parallel orders given by (Bechet, de Groote and Retor e 1997) the result stated in propositions 3.2 and 4. 1 of (Bechet, de Groote and Retor e 1997) is the following : i is obtained from by means of the following re exive and transitive congruence : Paul Ruet 18 (1) k (2) 1 k 2 ) 1 ) k 2 (3) 1 k 2 ) 1 k ( 2 ) 4) 1 k 2 ) 1 k 2 ) 1 1 ) k ( 2 2 ) Now P ....

D. Bechet, Ph. de Groote, and Ch. Retore. A complete axiomatisation for the inclusion of series-parallel orders. In Proc. Int. Conf. on Rewriting Techniques and Applications. Springer LNCS, 1997.

....connective, derived from coherence semantics. He shows a rewriting system that yields cut elimination for certain proof nets. That rewriting system is essentially equivalent to system EV . Retor e got his rewriting rules from a study of the inclusion relation for series parallel orders, in [5]. Since series parallel orders are at the basis of my de nition of structures, Retor e s result and mine are similar because they come from the same underlying combinatorial phenomena. The di erence between his work and mine is in the use we make of the rewriting system: he applies its rules in ....

Denis Bechet, Philippe de Groote, and Christian Retore. A complete axiomatisation of the inclusion of series-parallel orders. In H. Comon, editor, Rewriting Techniques and Applications, RTA '97, volume 1232 of Lecture Notes in Computer Science, pages 230-240. Springer-Verlag, 1997.

....sequent calculus In this section we present the calculus P.D.G. Actually our version slightly extends the published version, 6] but not of the author s project. Indeed, when this paper was printed he did not yet know the rules axiomatizing the inclusion of seriesparallel partial orders [3] but this calculus was designed to incorporate such rules. 3.1 Formulae Formulae are defined by: F : P j F Omega F j F fi F j F Gammaffi F j F n F j F = F So this calculus contains the following connectives: two multiplicative conjunctions: fi the non commutative conjunction of the ....

....OE 0 = OE (X 0 Theta X 0 ) is an SP order as well. If t is an SP term denoting OE one obtains a term denoting OE 0 from t by replacing each x 2 (X Gamma X 0 ) by and reducing the term by applying the following equalities: t; t) ft; g: 5 Finally we have found in [3] a complete axiomatization for the inclusion of SP orders as a rewriting system over SP terms: Proposition 15 Let OE and OE 0 be two SP orders with the same domain, and let t and t 0 two SP terms denoting them. One has OE oe OE 0 if and only if t Gamma t 0 where Gamma is the ....

Denis Bechet, Philippe de Groote, and Christian Retor. A complete axiomatisation of the inclusion of series-parallel partial orders. In H. Comon, editor, Rewriting Techniques and Applications, RTA`97, volume 1232 of LNCS, pages 230--240. Springer Verlag, 1997.

....step of this process, a proof structure consists in a proof structure with links plus a directed cograph on its conclusions. This transformation and its inverse are shown to preserve correctness, that is the absence of chordless circuit. Next, in section 5, we show that the rewriting system of [2] which axiomatizes the inclusion of (directed) cographs preserves the correctness of directed R B cographs except one rewriting rule concerning the compositions of directed cographs corresponding to and Omega . From these results we obtain another look at cut elimination for pomset logic, ....

....(z; x) z; t)g R 6= This looks exactly as being N free but here the relation is symmetric, so (u; v) 2 R , v; u) 2 R] What about the smallest class of relations containing 1 , and closed under parallel, series, and symmetric series compositions Let us call them directed cographs. In [2] we characterized them as follows: R is antireflexive 8x 2 E (x; x) 62 R. R , the directed part of R is N free R l , undirected part of R is P 4 free R is weakly transitive, i.e. x; y) 2 R (y; z) 2 R ) x; z) 2 R and (x; y) 2 R (y; z) 2 R ) x; z) 2 R In ....

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Denis Bechet, Philippe de Groote, and Christian Retore. A complete axiomatisation of the inclusion of series-parallel partial orders. In H. Comon, editor, Rewriting Techniques and Applications, RTA`97, volume 1232 of LNCS, pages 230--240. Springer Verlag, 1997.

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Denis Bechet, Philippe de Groote, and Christian Retore. A complete axiomatisation of the inclusion of series-parallel partial orders. In H. Common, editor, Rewriting Techniques and Applications, RTA 1997,Lecture Notes in Computer Science, pages 230--240. Springer-Verlag, 1997.

No context found.

Denis Bechet, Philippe De Groote, and Christian Retore. A complete axiomatisation of the inclusion of series-parallel orders. In H. Comon, editor, Rewriting Techniques and Applications, RTA '97, volume 1232 of Lecture Notes in Computer Science, pages 230--240. Springer-Verlag, 1997.

No context found.

Denis Bechet, Philippe De Groote, and Christian Retore. A complete axiomatisation of the inclusion of series-parallel orders. In H. Comon, editor, Rewriting Techniques and Applications, RTA '97, volume 1232 of Lecture Notes in Computer Science, pages 230--240. Springer-Verlag, 1997.

No context found.

Denis Bechet, Philippe de Groote, and Christian Retore. A complete axiomatisation of the inclusion of series-parallel partial orders. In H. Common, editor, Rewriting Techniques and Applications, RTA 1997,Lecture Notes in Computer Science, pages 230--240. Springer-Verlag, 1997.

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