Section: Research Program

Logical Frameworks

A thesis, which is at the root of our research effort, is that logical systems should be expressed as theories in a logical framework. As a consequence, proof-checking systems should not be focused on one theory, such as Simple type theory, Martin-Löf's type theory, or the Calculus of constructions, but should be theory independent. On the more theoretical side, the proof search algorithms, or the algorithmic interpretation of proofs should not depend on the theory in which proofs are expressed, but this theory should just be a parameter. This is for instance expressed in the title of our invited talk at ICALP 2012: A theory independent Curry-De Bruijn-Howard correspondence [25].

Various limits of Predicate logic have led to the development of various families of logical frameworks: $\lambda$-prolog and Isabelle have allowed terms containing free variables, the Edinburgh logical framework has allowed proofs to be expressed as $\lambda$-terms, Pure type systems have allowed propositions to be considered as terms, and Deduction modulo theory has allowed theories to be defined not only with axioms, but also with computation rules.

The $\lambda \Pi$-calculus modulo theory, that is implemented in the system Dedukti and that is a synthesis of the Edinburgh logical framework and of Deduction modulo theory, subsumes them all. Part of our research effort is focused on improving the $\lambda \Pi$-calculus modulo theory, for instance allowing to define congruences with associative and commutative rewriting. Another part of our research effort is focused on the automatic analysis of theories to prove their confluence, termination, and consistency either by pencil and paper proofs or automatically [4].