Laplace, Turing et la géométrie impossible du "jeu de l 'imitation " : aléas, déterminisme et programmes dans le test de Turing

Abstract

Laplace, Turing and the "imitation game" impossible geometry : randomness, determinism and programs in Turing's test. From the physico-mathematical viewpoint, the imitation game between a man and a machine, proposed by Turing in 1950, is a game between a discrete and a continuous system. Turing stresses several times the laplacian nature of his discrete-state machine, yet he tries to show the undetectability of a functional imitation, by his machine, of a system (the brain) that, in his words, is not a discrete-state machine, as it is sensitive to limit conditions. We shortly compare this tentative imitation with Turing's mathematical modeling of morphogenesis (his 1952 paper, focusing on continuous systems which are sensitive to initial conditions). On the grounds of recent advances about dynamical systems, we show the detectability of a Turing Machine in many dynamical processes. Turing's hinted distinction between imitation and modeling is developed, as well as a discussion on the repeatability of computational processes. Most references are physico-mathematical in nature, but the analysis is purely conceptual.