## Section: Partnerships and Cooperations

### European Initiatives

#### FP7 Projects

##### Proofcert

Abstract: The ProofCert proposal aims at building a foundation that will allow a broad spectrum of formal methods—ranging from automatic model checkers to interactive theorem provers—to work together to establish formal properties of computer systems. This project starts with a wonderful gift to us from decades of work by logicians and proof theorist: their efforts on logic and proof has given us a universally accepted means of communicating proofs between people and computer systems. Logic can be used to state desirable security and correctness properties of software and hardware systems and proofs are uncontroversial evidence that statements are, in fact, true. The current state-of-the-art of formal methods used in academics and industry shows, however, that the notion of logic and proof is severely fractured: there is little or no communication between any two such systems. Thus any efforts on computer system correctness is needlessly repeated many times in the many different systems: sometimes this work is even redone when a given prover is upgraded. In ProofCert, we will build on the bedrock of decades of research into logic and proof theory the notion of proof certificates. Such certificates will allow for a complete reshaping of the way that formal methods are employed.

#### Collaborations in European Programs, except FP7

##### STRUCTURAL: ANR blanc International

Participants : Kaustuv Chaudhuri, Nicolas Guenot, Willem Heijltjes, François Lamarche, Dale Miller, Lutz Straßburger.

Total funding by the ANR: 242 390,00 EUR (including 12 000 EUR pôle de compétivité: SYSTEMTIC Paris région)

This project is a consortium of four partners, two French and two Austrian, who are all internationally recognized for their work on structural proof theory, but each coming from a different tradition. One of the objective of the project is build a bridge between these traditions and develop new proof-theoretic tools and techniques of structural proof theory having a strong potential of applications in computer science, in particular at the level of the models of computation and the extraction of programs and effective bounds from proofs.

On one side, there is the tradition coming from mathematics, which is mainly concerned with first-order logic, and studies, e.g., Herbrand's theorem, Hilbert's epsilon-calculus, and Goedel's Dialectica interpretation. On the other side, there is the tradition coming from computer science, which is mainly concerned with propositional systems, and studies, e.g., Curry-Howard isomorphism, algebraic semantics, linear logic, proof nets, and deep inference. A common ground of both traditions is the paramount role played by analytic proofs and the notion of cut elimination. We will study the inter-connections of these different traditions, in particular we focus on different aspects and developments in deep inference, the Curry-Howard correspondence, term-rewriting, and Hilbert's epsilon calculus. As a byproduct this project will yield a mutual exchange between the two communities starting from this common ground, and investigate, for example, the relationship between Herbrand expansions and the computational interpretations of proofs, or the impact of the epsilon calculus on proof complexity.

Besides the old, but not fully exploited, tools of proof theory, like the epsilon-calculus or Dialectica interpretation, the main tool for our research will be deep inference. Deep inference means that inference rules are allowed to modify formulas deep inside an arbitrary context. This change in the application of inference rules has drastic effects on the most basic proof theoretical properties of the systems, like cut elimination. Thus, much of the early research on deep inference went into reestablishing these fundamental results of logical systems. Now, deep inference is a mature paradigm, and enough theoretical tools are available to think to applications. Deep inference provides new properties, not available in shallow deduction systems, namely full symmetry and atomicity, which open new possibilities at the computing level that we intend to investigate in this project. We intend to investigate the precise relation between deep inference and term rewriting, and hope to develop a general theory of analytic calculi in deep inference. In this way, this project is a natural continuation of the ANR project INFER which ended in May 2010.

##### PHC Procope: From Proofs to Counterexamples for Programming

Participants : Kaustuv Chaudhuri, Nicolas Guenot, Willem Heijltjes, Lutz Straßburger.

Finding counterexamples is an endeavor which is as important as proving theorems. But while the latter has seen a huge amount of research effort—we have nowadays a large quantity of tools for automated and interactive theorem proving—the former has mainly been neglegted by proof theorists. One of the reasons is that finding counterexamples or countermodels has been considered a model theoretical activity, rather than a proof theoretical one. Only recently, researchers have begun to explore the well-known duality between "proof search" and "search for countermodels" in a purely proof theoretical way. The main objective of this collaboration is to develop the necessary proof theory for automatically generating such counterexamples in a more general setting.

##### PHC Germaine de Staël: Extending the Realm of the Curry-Howard-Correspondence

Participants : Nicolas Guenot, Willem Heijltjes, Lutz Straßburger.

Title: Extending the Realm of the Curry-Howard-Correspondence

Swiss Partner: University of Bern, Institut für Informatik und angewandte Mathematik (IAM)

The Curry-Howard correspondence between proofs and programs is probably the most interesting and surprising connection between mathematics and computer science. It was discovered in the 1960s, but its main development started in the 1980s. The basis of the correspondence is a correspondence between intuitionistic proofs and typed functional programs (written as terms of lambda-calculus).

Our goal is to develop such a correspondence for new formalisms, like hypersequents, nested sequents and deep inference, in order to better understand their proofs and, we hope, either to discover new programming constructs or to give a new logical interpretation to existing ones.