Accessibility setting

Section: New Results

Models of Computation

Together with Pablo Arrighi (Grenoble), Gilles Dowek has reformulated Gandy's proof of the physical Church-Thesis in the quantum case [11] . Gilles Dowek has proposed the idea that the Galileo thesis could be seen as a consequence of the physical Church-Turing thesis and therefore as a consequence of Gandy's principles [15] . Gilles Dowek has proposed a definition of a notion of non deterministic computation over the real numbers [14] that could be used as a language to describe continuous non deterministic physical phenomena. All this work has then been presented in a tutorial at the conference Language and Automata Theory and Applications [28] .

Together with Pablo Arrighi, Gilles Dowek has investigated further the principle of a finite density of information [38] and in particular the impact of this definition on the notion of a chaotic dynamical system [37] .

Together with Pablo Arrighi, Gilles Dowek has investigated a generalization of the notion of cellular automaton where the principle of a bounded density of information is formulated independently of the geometry of space. This led to the notion of a Causal graph dynamic [12] .

Nachum Dershowitz and Gilles Dowek have shown that extending Turing machines with a two-dimensional tape, made this formalism usable in practice to implement classical algorithms [45] .

Alejandro Díaz-Caro and Gilles Dowek have proposed to take a fresh look at non deterministic $\lambda$-calculi—such as quantum $\lambda$-calculi—and derive non determinism from type isomorphism [30] .

Together with Giulio Manzonetto (Paris 13) and Michele Pagani (Paris 13), Alejandro Díaz-Caro has considered an extension of the call-by-value $\lambda$-calculus with a may-convergent non-deterministic choice and a must-convergent parallel composition, endowed with a type system. They have proved that a term is typable if and only if it is converging, and that its typing tree carries enough information to give a bound on the length of its lazy call-by-value reduction. Moreover, when the typing tree is minimal, such a bound becomes the exact length of the reduction [31] .

Together with Barbara Petit (Sardes), Alejandro Díaz-Caro has considered the non-deterministic extension of the call-by-value lambda calculus, which corresponds to the additive fragment of the linear-algebraic lambda-calculus. They have defined a fine-grained type system, capturing the right linearity present in such formalisms. After proving the subject reduction and the strong normalisation properties, they have proposed a translation of this calculus into the System F with pairs, which corresponds to a non linear fragment of linear logic. The translation provides a deeper understanding of the linearity in this setting [32] .

Together with Pablo Arrighi, Barbara Petit, Pablo Burias (Rosario), Mauro Jaskelioff (Rosario), and Benoît Valiron (Penn), Alejandro Díaz-Caro has studied possible typing systems for the full linear-algebraic $\lambda$-calculus in which the non-deterministic calculus can be seen as a particular case. They have proposed a type system that keeps track of “the amount of a type” that is present in each term [13] . As an example of its use, they have shown that it can serve as a guarantee that the normal form of a term is barycentric, that is that its scalars are summing to one. They also proposed a type system similar to the one presented in [32] , but for the full calculus, ensuring confluence and convergence [23] . Finally, they provided a full type system that is able to statically describe the linear combinations of terms resulting from the reduction of programs, also ensuring convergence [19] .