Showing papers by "Jean-Yves Girard published in 1995"

Linear logic: its syntax and semantics

Abstract: This is perfect in mathematics, but wrong in real life, since real implication is causal. A causal implication cannot be iterated since the conditions are modified after its use ; this process of modification of the premises (conditions) is known in physics as reaction. For instance, if A is to spend $1 on a pack of cigarettes and B is to get them, you lose $1 in this process, and you cannot do it a second time. The reaction here was that $1 went out of your pocket. The first objection to that view is that there are in mathematics, in real life, cases where reaction does not exist or can be neglected : think of a lemma which is forever true, or of a Mr. Soros, who has almost an infinite amount of dollars.

Advances in Linear Logic

Abstract: Linear logic: its syntax and semantics J. Y. Girard Part I. Categories and Semantics: 1. Bilinear logic in algebra and linguistics J. Lambek 2. A category arising in linear logic, complexity theory and set theory A. Blass 3. Hypercoherences: a strongly stable model of linear logic T. Erhard Part II. Complexity and Expressivity: 4. Deciding provability of linear logic formulas P. D. Lincoln 5. The direct simulation of Minsky machines in linear logic M. I. Kanovich 6. Stochastic interaction and linear logic P. D. Lincoln, J. Mitchell and A. Scedrov 7. Inheritance with exceptions C. Fouquere and J. Vauzeilles Part III. Proof Theory: 8. On the fine structure of the exponential rule S. Martini and A. Masini 9. Sequent calculi for second order logic V. Danos, J. B. Joinet and H. Schellinx Part IV. Proff Nets: 10. From proof nets to interaction nets Y. Lafont 11. Empires and kingdoms in MLL G. Bellin and J. Van De Wiele 12. Noncommutative proof nets V. M. Abrusci 13. Volume of multiplicative formulas and provability F. Metayer Part V. Geometry of Interaction: 14. Proof nets and Hilbert space V. Danos and L. Regnier 15. Geometry of interacion III: accomodating the additives J. Y. Girard.

Geometry of interaction III: accommodating the additives

Abstract: The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C∗-algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponential-free conclusions.

On Geometry of Interaction

Abstract: The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C*-algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponential-free conclusions.