Girard, J.-Y., A. Scedrov & P. Scott : "Normal forms and cut-free proofs as natural transformations", in "Logic from Computer Science", MSRI publications, Y. Moschovakis (editor), 21, Springer-Verlag, (1992), 217--241.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....L. Pinto hereditary Harrop logic [22] It may also be used as a basis for some inductive proofs about derivations in LJ or natural deductions in NJ (e.g. those in [12] where the strong eliminability of cut, rather than just the admissibility, is required; and, we anticipate, those in [5] and [15]) Herbelin called (following [6] his calculus LJT , a name we avoid in case of confusion with that in the first author s [9] we call its cut free fragment MJ because it is intermediate between [13] Gentzen s cut free LJ and NJ. We apologise to Herbelin for not adopting his nomenclature. We ....

....sentence of [17] of including the (in our notation) cut reduction rule cut 4 (P, M , y.cut 3 (Q,lx.M,Ms) cut 3 (Q,cut 4 (P, M , y.lx.M) cut 2 (P, M , y. Ms) is unresolved. Elsewhere we show (or plan to show) how the MJ calculus is well suited both for applications to inductive arguments [5, 12, 15] about other sequent calculi and for proof search. The same methodology, of finding a calculus between a sequent calculus (admitting lots of permutations) and a natural deduction system (without the immediate subformula property) should also be applied to substructural logics. As suggested by a ....

Girard, J.-Y., A. Scedrov & P. Scott : "Normal forms and cut-free proofs as natural transformations", in "Logic from Computer Science", MSRI publications, Y. Moschovakis (editor), 21, Springer-Verlag, (1992), 217--241.

.... search in hereditary Harrop logic [22] It may also be used as a basis for some inductive proofs about derivations in LJ or natural deductions in NJ (e.g. those in [12] where the strong eliminability of cut, rather than just the admissibility, is required; and, we anticipate, those in [5] and [15]) Herbelin called (following [6] his calculus LJT , a name we avoid in case of confusion with that in the first author s [9] we call its cut free fragment MJ because it is intermediate between [13] Gentzen s cut free LJ and NJ. We apologise to Herbelin for not adopting his nomenclature. We ....

....sentence of [17] of including the (in our notation) cut reduction rule cut 4 (P, M , y.cut 3 (Q,lx.M,Ms) cut 3 (Q,cut 4 (P, M , y.lx.M) cut 2 (P, M , y. Ms) is unresolved. Elsewhere we show (or plan to show) how the MJ calculus is well suited both for applications to inductive arguments [5, 12, 15] about other sequent calculi and for proof search. The same methodology, of finding a calculus between a sequent calculus (admitting lots of permutations) and a natural deduction system (without the immediate subformula property) should also be applied to substructural logics. As suggested by a ....

Girard, J.-Y., A. Scedrov & P. Scott : "Normal forms and cut-free proofs as natural transformations", in "Logic from Computer Science", MSRI publications, Y. Moschovakis (editor), 21, Springer-Verlag, (1992), 217--241.

.... logical relations to formalise the notion of parametricity has been proposed several times, and with varying degrees of abstraction, most notably in the work of Reynolds and his student Ma, Rey83, MR91] The use of dinaturality has been proposed by Bainbridge, Freyd, Girard, Scedrov, and Scott, [BFSS90, GSS91]. On an abstract level, the idea behind the use of logical relations is that given an intepretation of the second order lambda calculus, it is possible to form a category whose objects are (say) n ary relations, and in which the morphisms come form morphisms between the types on which the ....

J-Y. Girard, A. Scedrov, and P.J. Scott. Normal forms and cut-free proofs as natural transformations. In Y.N. Moschovakis, editor, Logic From Computer Science, pages 217--241. Mathematical Sciences Research Institute Publications (V.21), Springer Verlag, 1991.

....in (a variant of) LJ that are generated by the standard interpretation of cut free LJ derivations as normal natural deductions are precisely those generated by certain basic permutations. We believe it can also be used to prove permutation invariant properties of sequent derivations, as in [19] and to give simpler inductive proofs about normal natural deductions, such as the theorem in [5] Proof search in MJ avoids the inefficiencies associated with the permutations in LJ; but it does not avoid the problems of contraction. It remains to be seen whether one can avoid permutations and ....

Girard, J.-Y., A. Scedrov & P. Scott : "Normal forms and cut-free proofs as natural transformations", in "Logic from Computer Science", MSRI publications, Y. Moschovakis (editor), 21, Springer-Verlag, (1992), 217--241.

....be formulas in multiplicative linear logic, interpreted as definable multivariant functors on RT VEC. Then the vector space of diadditive dinatural transformations has as basis the denotations of cut free proofs in the theory MLL MIX. We obtain the following corollary by the methods outlined in [19, 11]. Corollary 2.15 Diadditive dinatural transformations compose. Thus we obtain an (indexed) autonomous category by taking as objects formulas, interpreted as multivariant functors. Morphisms will be diadditive dinatural transformations. 7 Remark 2.16 (Fully Faithful Representation Theorem) ....

J. Y. Girard, A. Scedrov, P. Scott, Normal Forms and Cut-free Proofs as Natural Transformations, in : Logic From Computer Science, Mathematical Science Research Institute Publications 21, (1991), pp. 217-241. (Also available by anonymous ftp from: theory.doc.ic.ac.uk, in: papers/Scott).

....For a detailed discussion, see Section 4 below. As such, we believe that the work presented in this paper may be viewed as the beginnings of a theory of logical relations for linear logic and concurrency. In this paper, we present a semantics based upon an extension of functorial polymorphism [5, 9, 22] to the linear setting. In this setting, types are definable multivariant functors on a category of topological vector spaces. We then interpret terms, i.e. deductions in the theory MLL MIX as certain dinatural transformations between such functors. The key property is that these transformations ....

....and a reflective subcategory of T MOD(G) via the functor ( Furthermore the forgetful functor to RT VEC preserves the autonomous structure. 7 Functorial Polymorphism We shall give a further development of the theory of Functorial Polymorphism applied to linear logic, following [5, 9, 22]. For other developments of the general theory, cf [16, 17] Recall that in functorial polymorphism, the types of a calculus are interpreted directly as certain multivariant functors, while terms are an appropriate multivariant version of natural transformation known as a dinatural ....

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J. Y. Girard, A. Scedrov, P. Scott, Normal Forms and Cut-free Proofs as Natural Transformations, in : Logic From Computer Science, Mathematical Science Research Institute Publications 21, (1991), pp. 217-241. (Also available by anonymous ftp from: theory.doc.ic.ac.uk, in: papers/Scott).

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