Section: New Results

First efforts at designing proof certificates

Participants : Hichem Chihani, Quentin Heath, Dale Miller, Fabien Renaud.

Work on the ERC Advance Grant ProofCert has progressed along two lines.

Given earlier work within the team [6] , [7] , there now exists a flexible and well understood concept of focused proof for classical and intuitionistic first-order logics. Chihani, Miller, and Renaud have been working to use that notion of proof as a means of providing flexible definition of proof evidence for those two logics. Initial results along those directions have been reported in the [19] and [20] . In those papers, several examples definitions of the semantics of proof certificates (formal documents providing the details of some proof evidence) are provided in such a way that a single, simple proof checker can formally elaborate that evidence into a focused sequent calculus. Such an elaboration thus guarantees the soundness of that proof. These papers also describe a “reference proof checker” that has been built with the expectation that its formal correctness can be established. That checker is also able to do bounded proof reconstruction as well as allow both deterministic and non-deterministic computation to be mixed with deduction.

Our understanding of focused proofs in the presence of both induction and co-induction (inference rules found in model checkers and most theorem provers) is less well developed. As a result, Miller and Tiu have been studied a simple approach of proof certificate in the setting of model checking in the hope of identifying the relevant proof theory designs that need to be developed. In [33] , they showed how tabled deduction in model checking can be used to provide a formal proof certificate for a range of co-inductively defined predicates.