Section: Research Program

Theoretical background

The proof assistants that we consider provide both a programming language, where users can describe algorithms performing tasks in their domain of interest, and a logical language to reason about the programs, thus making it possible to ensure that the algorithms do solve the problems for which they were designed. trustability is gained because algorithms and logical statements provide multiple views of the same topic, thus making it possible to detect errors coming from mismatch between expected and established properties. The verification process is itself a logical process, where the computer can bring rigor in aligning expectations and guarantees.

The foundations of proof assistants rest on the very foundations of mathematics. As a consequence, all aspects of reasoning must be made completely explicit in the process of formally verifying an algorithm. All aspects of the formal verification of an algorithm are expressed in a discourse whose consistency is verified by the computer, so that unclear or intuitive arguments need to be replaced by precise logical inferences.

One of the foundational features on which we rely extensively is Type Theory. In this approach a very simple programming language is equiped with a powerful discipline to check the consistency of usage: types represent sets of data with similar behavior, functions represent algorithms mapping types to other types, and the consistency can be verified by a simple computer program, a type-checker. Although they can be verified by a simple program, types can express arbitrary complex objects or properties, so that the verification work lives in an interesting realm, where verifying proofs is decidable, but finding the proofs is undecidable.

This process for producing new algorithms and theorems is a novelty in the development of mathematical knowledge or algorithms, and new working methods must be devised for it to become a productive approach to high quality software development. Questions that arise are numerous. How do we avoid requiring human assistance to work on mundane aspects of proofs? How do we take advantage of all the progress made in automatic theorem proving? How do we organize the maintenance of ambitious corpora of formally verified knowledge in the long term?

To acquire hands-on expertise, we concentrate our activity on three aspects. The first one is foundational: we develop and maintain a library of mathematical facts that covers many aspects of algebra. In the past, we applied this library to proofs in group theory, but it is increasingly used for many different areas of mathematics and by other teams around the world, from combinatorics to elliptic cryptography, for instance. The second aspect is applicative: we develop a specific tool for proofs in cryptography, where we need to reason on the probability that opponents manage to access information we wish to protect. For this activity, we develop a specific proof system, relying on a wider set of automatic tools, with the objective of finding the tools that are well adapted to this domain and to attract users that are initially specialists in cryptography but not in formal verification. The third domain is robotics, as we believe that the current trend towards more an more autonomous robots and vehicles will raise questions of safety and trustability where formal verification can bring significant added value.