## Section: Research Program

### Research Program

#### Panorama of Deductive Verification

There are two main families of approaches for deductive
verification. Methods in the first family build on top of mathematical
proof assistants (e.g., Coq, Isabelle) in which both the model and the
program are encoded; the proof that the program meets its
specification is typically conducted in an interactive way using the
underlying proof construction engine. Methods from the second family
proceed by the design of standalone tools taking as input a program in
a particular programming language (e.g., C, Java) specified with a
dedicated annotation language (e.g., ACSL [49],
JML [56]) and automatically producing a set of
mathematical formulas (the *verification conditions*) which are
typically proved using automatic provers (e.g., Z3 [70],
Alt-Ergo [58], CVC4 [48]).

The first family of approaches usually offers a higher level of assurance than the second, but also demands more work to perform the proofs (because of their interactive nature) and makes them less easy to adopt by industry. Moreover, they generally do not allow to directly analyze a program written in a mainstream programming language like Java or C. The second kind of approaches has benefited in the past years from the tremendous progress made in SAT and SMT solving techniques, allowing more impact on industrial practices, but suffers from a lower level of trust: in all parts of the proof chain (the model of the input programming language, the VC generator, the back-end automatic prover), potential errors may appear, compromising the guarantee offered. Moreover, while these approaches are applied to mainstream languages, they usually support only a subset of their features.

#### Overall Goals of the Toccata Project

One of our original skills is the ability to conduct proofs by using automatic provers and proof assistants at the same time, depending on the difficulty of the program, and specifically the difficulty of each particular verification condition. We thus believe that we are in a good position to propose a bridge between the two families of approaches of deductive verification presented above. Establishing this bridge is one of the goals of the Toccata project: we want to provide methods and tools for deductive program verification that can offer both a high amount of proof automation and a high guarantee of validity. Indeed, an axis of research of Toccata is the development of languages, methods and tools that are themselves formally proved correct.

In industrial applications, numerical calculations are very common (e.g. control software in transportation). Typically they involve floating-point numbers. Some of the members of Toccata have an internationally recognized expertise on deductive program verification involving floating-point computations. Our past work includes a new approach for proving behavioral properties of numerical C programs using Frama-C/Jessie [47], various examples of applications of that approach [54], the use of the Gappa solver for proving numerical algorithms [68], an approach to take architectures and compilers into account when dealing with floating-point programs [55], [66]. We also contributed to the Handbook of Floating-Point Arithmetic [65]. A representative case study is the analysis and the proof of both the method error and the rounding error of a numerical analysis program solving the one-dimension acoustic wave equation [52] [51]. Our experience led us to a conclusion that verification of numerical programs can benefit a lot from combining automatic and interactive theorem proving [53], [54], [59]. Verification of numerical programs is another main axis of Toccata.

Our scientific programme detailed below is structured into four axes:

Let us conclude with more general considerations about our agenda of the next four years: we want to keep on