Laurent Regnier. Lambda-Calcul et Reseaux. These de doctorat, Universite Paris VII, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... w and co out of the boxes, one can associate with every cut free proof net of MELL a unique standard proof net. Notice, by the way, that (except for the expansions) a standard proof net is nothing but a proof net of the nouvelle syntaxe de ned by V. Danos and L. Regnier (for example in [Reg92]) We now have to be precise on what we mean by R and R 0 are semantically equivalent (in a given semantics) where R and R 0 are two proof nets with the same conclusions. Indeed, we cannot simply say that this means [jRj] jR 0 j] referring to de nition 1.1.4) because this equality ....

Laurent Regnier. Lambda-Calcul et Reseaux. These de doctorat, Universite Paris VII, 1992.

....of the net. A translation of terms into nets allows to lift the interpretation to the calculus. Decidability of Regularity. The term rewriting system RL is obtained by orientating from left to right all the equations in L (including the dual commutation equations) RL is con uent [13]. A stable form is 1 or any term a (m)b in L where a and b are positive at terms, i.e. they contain no applications of ( or ( and m is stable. Stable forms are normal with respect to RL . Every stable form is equal to some term AB with A and B positive but not ....

.... in L where a and b are positive at terms, i.e. they contain no applications of ( or ( and m is stable. Stable forms are normal with respect to RL . Every stable form is equal to some term AB with A and B positive but not necessarily at. Proposition 1 (AB property [13, 3]) If is a straight path in some net then its weight w( can be rewritten either to a stable form or to 0. Let be a path with w( rewritable to a stable form ab . Since for a positive monomial x, L x x = 1, then L a ab b = 1, and L a w( b = 1. Thus L ....

Laurent Regnier. Lambda-Calcul et Reseaux. PhD thesis, Universite Paris VII, January 1992.

....guess from A itself the presence of the future redex. That is, looking at A itself, we see that is matched with and y is matched with x. This has been noted by many researchers and hence rules like ( x :N)PQ ( x :NQ)P have been introduced in many articles with di erent purposes [1, 5, 8, 9, 11, 12, 14, 16, 17, 19 21, 23]. Such rules enable one to rewrite A so that both redexes become visible in A. Note that: A ( y : f :fy) x ( y : f :fy)x) B. These transformations are rather powerful in that they can group together terms with equal reductional behavior. Let us give here this example: ....

.... x ; P . e.g. it is ne to write ( x: y:x :y)z)u ( x: y:x :y)u)z but not to write ( x: y:x :y)z)u C ( y:x : x: y)z)u. Hence, we study like rules in this paper (which imply g like rules) Now, we discuss where generalised reduction has been used (cf. 13, 9] [19] introduces the notion of a premier redex which is similar to the redex based on y and Q in the rule ( above (which we call generalised redex) 20] uses and (and calls the combination ) to show that the perpetual reduction strategy nds the longest reduction path when the term is ....

L. Regnier. Lambda calcul et reseaux. PhD thesis, University Paris 7, 1992.

....push the context formul of the nal sequent of the transported subderivation back down along the tree. At branchings originally due to explicit contractions on A, the contractions now are inherited by these context formul . cf. the complete exponential reduction of proof nets, as described in [Reg92]. 1.4 The constraint In [AP91] a crucial property of proofs was pointed out by the authors: focalization. In [Gir91a] this property (the stoup) found a semantical counterpart: central cliques of Girard s correlation semantics. In [DJS97] the authors show that focalization ( constraint) is ....

Laurent Regnier. Lambda-Calcul et Reseaux. These de doctorat, Universite Paris VII, 1992.

....translations of calculus into linear logic. Furthermore it has been recently given a nice categorical semantics, the control categories of P. Selinger [14] which as a by product, extends the language of types with a new disjunctive connective. As in the calculus where the equivalence [12, 13] identi es terms that di er only in their sequential structure (e.g. x 1 : x 2 :u)v 1 v 2 and ( x 2 : x 1 :u)v 2 v 1 ) calculus terms contain pieces of information, which are unnecessary from the operational viewpoint. Indeed the control category semantics identi es distinct normal ....

....show that the translation of calculus preserves reductions and that it is surjective. Interestingly enough, it translates A B as A ( B just like the usual encoding of intuitionistic logic 1 . As a consequence the translation is a straightforward extension of the calculus translation [1, 12]. Furthermore, since there is very little di erence between LLP and LC (the two systems are equivalent in the categorical sense) our framework conciliates LC and the calculus, allowing the use of LC as a typing system for terms and endowing calculus with LC structure (e.g. its ....

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Laurent Regnier. Lambda-Calcul et Reseaux. These de doctorat, Universite Paris VII, 1992.

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