## Section: New Results

### Models of Computation

Alejandro Díaz-Caro and Gilles Dowek have shown how to provide a structure of probability space to the set of execution traces on a non-confluent abstract rewrite system, by defining a variant of a Lebesgue measure on the space of traces. Then, they showed how to use this probability space to transform a non-deterministic calculus into a probabilistic one. As an example, they applied this technique to the previously introduced non-deterministic calculus. [25]

Ali Assaf and Alejandro Díaz-Caro, together with Simon Perdrix (Nancy), Christine Tasson (PPS) and Benoît Valiron (PPS) have determined the relationship between the algebraic $\lambda $-calculus, a fragment of the differential $\lambda $-calculus and the linear-algebraic $\lambda $-calculus, a candidate $\lambda $-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and their set of terms is closed under linear combinations. However, the two languages were built using different approaches: the former is a call-by-name language whereas the latter is call-by-value; the former considers algebraic equalities whereas the latter approaches them through rewrite rules. They have analysed how these different approaches relate to one another, proposing four canonical languages based on each of the possible choices: call-by-name versus call-by-value, algebraic equality versus algebraic rewriting. They have shown that the various languages simulate one another. Preliminary versions of this work where published in [47] and [41] . Now they are working on a journal version filling the gaps between these previous works.

Together with Pablo Arrighi (Grenoble) and Benoît Valiron (PPS), Alejandro Díaz-Caro has described a type system for the linear-algebraic lambda-calculus. The type system accounts for the linear-algebraic aspects of this extension of lambda-calculus: It is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. They have proven that the resulting typed lambda-calculus is strongly normalising and features a weak subject reduction. In addition, they have shown how to naturally encode matrices and vectors in this typed calculus [34] .

Gilles Dowek has investigated a new definition of the notion of a chaotic system that can be applied to discrete systems and that is compatible with the principle of a finite density of information.

The paper Call-by-value non-determinism in a linear logic type discipline by Alejandro Díaz-Caro, Giulio Manzonetto and Michele Pagani has been published [26] .

The paper Universality in two dimensions of Gilles Dowek and Nachum Dershowitz has been published.

The paper Linear-algebraic lambda-calculus: higher-order, encodings and confluence of Pablo Arrighi and Gilles Dowek has been published.

The book *Lambda Calculus with Types*
, written by
Henk Barendregt, Wil Dekkers, Richard Statman, and 11 contributors,
including Gilles Dowek, has been published.