## Section: New Results

### Verification of scientific algorithms

#### Termination of Delaunay triangulation by edge flipping

Participants : Jean-François Dufourd [ Université de Strasbourg ] , Yves Bertot.

An algorithm to produce planar Delaunay triangulations has been formally described in the framework of the ANR Galapagos project. In particular, we showed that the termination of the algorithm could be proved with the help of a systematic description of finite sets.

#### Co-recursion and real numbers

Participants : Yves Bertot, Nicolas Julien, Ioana Paşca.

The traditional understanding that real numbers are fractional numbers with an infinite sequence of digits after the decimal point can be modeled using infinite streams of digits, a special case of co-inductive data-types. The main work of this year was to optimize a generic approach to computations based on Newton's algorithm, for instance for the square-root function. Part of this work was published in [17] .

#### Regularity of interval matrices

Participants : Yves Bertot, Ioana Paşca.

We are studying an article by Rex and Rohn that gives efficient sufficient conditions for regularity of matrices with interval coefficients. Although it is in a preliminary stage, we can foresee that it will rely on our previous work on formalizing linear algebra.

Like our previous work on Kantorovitch's theorem that lead to our formalization of Newton's method, this topic was proposed to us by colleagues from the COPRIN team, who are more involved in robotics. In the long run we expect that a formal description of the convergence theorems makes it possible to propose new tools for the verification of controling software in this domain.

#### Certificate translation for optimizing compiler

Participants : Gilles Barthe [ IMDEA Madrid ] , Benjamin Grégoire, Sylvain Heraud, Cesar Kunz [ IMDEA Madrid ] , Anne Pacalet.

In a Proof Carrying Code environment, certificate generation remains an open problem. Certifying compilers can automatically produce certificates but are mostly restricted to basic safety properties. Certificate translation is an alternative method that transforms certificates of source programs into certificates of compiled programs. In an earlier work we have developed the theory of certificate translation. This years we have developed an implementation. It was the subject of a paper at International Conference on Formal Engineering Methods [7] , in particular we study the impact of certificate translation on the size of certificates.

#### Certifying code generation

Participants : Benjamin Grégoire, Jan-Olaf Blech [ Verimag Grenoble ] .

In [12] , we have presented an approach to guarantee the correctness of compiler transformations with respect to a formal notion of correctness. We certify the results of each compilation run. With the help of a compiler generated certificate and a certificate checker, we verify the results of each compilation run automatically. Thus, we ensure the correctness of the compilation run without having to look at concrete compilation algorithms. A journal version has been submitted to Formal Methods in System Design.

#### Formally verified structural abstract interpretation

Participant : Yves Bertot.

A paper describing a formal study of abstract interpretation has been published in the lecture notes of a summer school [10] .

#### Properties of Gene networks

Participants : Yves Bertot, Johan Segura, Adrien Richard [ Laboratoire CNRS I3S, Université de Nice-Sophia Antipolis ] .

Biologists often try to predict the dynamic behavior of complex
biological systems composed of several genes by studying interactions
between some of these genes. For instance, a gene may be known to
activate or inhibit the expression of another one. An abstract
*interaction graph* , with positive and negative edges, is then
used as a tool to model activatory and inhibitory effects between
genes. Another abstract model relies on notions of finite-state
automata. We studied the relations between some aspects of the two
models. This year we concentrated on an article by Shih and Dong
on sufficient conditions for the existence of a single fixed point,
in the case where each gene can only have two states.

An article on the work in the same domain from previous years has been submitted to an international conference.