Showing papers by "Jean-Yves Girard published in 1992"
TL;DR: It is proved that any functional term of appropriate type actually encodes a polynomial-time algorithm and that conversely any polynometric-time function can be obtained in this way.
275 citations
01 Jan 1992
TL;DR: In this article, the usual syntax guarantees that certain naturality equations from category theory are necessarily provable, and a categorical approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cutfree proofs.
Abstract: What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what non-trivial identifications must hold between lambda terms, thought-of as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cut-free proofs. This viewpoint is connected to Reynolds’ relational interpretation of parametricity ([27], [2]), and to the Kelly-Lambek-Mac Lane-Mints approach to coherence problems in category theory.
47 citations
8 citations
TL;DR: The notion of a ptyx (plural, ptykes) is obtained by generalising the ordinals On and the dilators Dil introduced by Girard into a typed hierarchy by using strictly increasing maps f: x -* y as morphisms.
Abstract: ?0. Introduction. The notion of a ptyx (plural, ptykes) is obtained by generalising the ordinals On and the dilators Dil introduced by Girard [1] into a typed hierarchy. The notion is due to Girard, and a detailed treatment will appear in Girard [2]. In this Introduction we will give a summary of the theory of ptykes. It will be an advantage to be familiar with the theory of dilators as introduced in Girard [1] or as presented in Girard and Normann [3]. The class On of ordinals is organised into a category ON by using strictly increasing maps f: x -* y as morphisms. A dilator will be a functor F: ON -* ON commuting with pullbacks and direct limits. Associated with each dilator F there is a denotation system DF, obtained as follows: For each x E On and y E F(x), y can be given a unique denotation (c; x0,.. . , x"1; X)F, where ceF(n) (n= {O .. ., n-1}), y = F(0)(c) (where 0 is defined by 0(i) = xi for i < n), and for no m < n and 4: m -+ n do we have c E im(4). We let the trace Tr(F) be the set of pairs (c, n) occurring in a denotation. Tr(F) has a natural well-ordering: (c, n) < (d, m) when
6 citations