Section: New Results

Study of Formalisms

Towards an implementation of the Implicit Calculus of Constructions

Participants : Bruno Barras, Bruno Bernardo.

Bruno Bernardo and Bruno Barras are working on an Implicit Calculus of Constructions with dependent sums (also known as -types) and with decidable type inference. In this calculus, all the static information (types and proof objects), though it appears explicitly, is transparent and does not affect the computational behavior. Bruno Bernardo has already defined and studied an Implicit Calculus of Constructions with decidable typing and is now working on extending it with - types, in order to have a more expressive language.

A problem in the metatheory of this extension has been uncovered: as in first-order logic with implicit existential quantifier, the subject-reduction property does not hold. However, this negative result does not affect more explicit versions of ICC, where introduction and elimination rules of the implicit quantifier are used explicitly in the proof derivation, but are considered implicit in the conversion rule. Next steps are to extend Alexandre Miquel's models based on coherence spaces in order to prove the consistency and the strong normalization property of the system.

Normalization and deduction modulo

Participants : Mathieu Boespflug, Denis Cousineau.

Mathieu Boespflug has been working on generalizing normalization by evaluation to term reduction modulo arbitrary rewrite rules in addition to beta-reduction. This allows for a cheap yet efficient implementation of a normalizer, as required in many proof assistants and also finds applications in partial evaluation. The implementation is cheap because most of the work is offloaded to an existing evaluator. His work has focused in particular on minimizing any overhead on beta-reduction caused by normalization by evaluation, showing that normalization by evaluation can reduce terms at nearly the same speed as the underlying evaluator.

He also wrote a full implementation of a type checker for proofs written in the -modulo calculus, called Dedukti . Version 1.0 of Dedukti was released in September. At this stage, the type checker works by translating the input terms into a functional program that can be compiled by the GHC Haskell compiler. Type checking the input terms is a side-effect of executing the obtained program. The translation makes essential use of untyped normalization by evaluation, a method for finding normal forms  [5] , [7] .

Previously, Denis Cousineau worked on the property of strong normalization in logical frameworks where theories are expressed with rewrite rules. He defined, in particular, a semantic criterion not only correct but also complete for the property of strong normalization for theories expressed in Minimal Deduction modulo (a minimal version of Deduction modulo with the only two connectors and ) [18] . In 2009, Denis Cousineau has extended his results on a sound and complete semantic criterion for proof normalization to dependent types ( -calculus modulo) [1] , [8] .

Resolution and resolution modulo

Participants : Gilles Dowek, Guillaume Burel.

Gilles Dowek has given a new formulation of Resolution modulo, called Polarized resolution modulo, that is both more general than Resolution modulo, because it includes polarized rewrite rules, and more restrictive, because all rewrite rules must be clausal [21]

Gilles Dowek has shown that simple type theory can be expressed as such a clausal rewrite system [22] .

Polarized resolution modulo can be seen as a restriction of Equational resolution, that mixes clause restrictions and literal restrictions. Guillaume Burel and Gilles Dowek have compared this method with other restrictions of resolution and shown that as a consequence of Gödel second incompleteness theorem, Polarized resolution modulo is not an instance of Ordered resolution [20] .

The text of the invited conference at LSFA 2007 of Gilles Dowek has been published. This text analyzes the history of the convergence of reduction-based methods and model-based methods in proof theory [2] .

Formalisation in deduction modulo

Participant : Gilles Dowek.

Gilles Dowek has given a formulation in Deduction modulo of the system FA2 of J.-L. Krivine and M. Parigot. In particular, he has shown that one originality of this system is to express the specifications not as propositions, but in the congruence of the theory [23] .

In a volume published in honor of Peter Andrews' birthday, Gilles Dowek has analyzed the history of higher-order Skolem theorem showing that this history illustrates a confrontation of two points of view on simple type theory: the logical and the theoretical points of view [3] .

Constructive mathematics

Participant : Arnaud Spiwack.

Arnaud Spiwack (together with Thierry Coquand) has studied the notion of “finite set” in the setting of constructive mathematics. They describe how it naturally splits into several different properties. All of which are equivalent in ZFC. From a constructive point of view they have, however, quite different behaviours, and correspond algorithmically to different finite structures.

Encoding rewriting strategies in lambda-calculi with patterns

Participant : Germain Faure.

Germain Faure proposed an improvement to the pure pattern calculus: we claim that it is strictly more powerful to define the application of the match fail as the pure -term defining the boolean false instead of the identity function as it is done in the original version of the pure pattern calculus of Jay and Kesner. We show that using non algebraic patterns we are able to encode in a natural way any rewriting strategies as well as the branching construct “|” used in functional programming languages. We close the open question (raised by Cirstea and Kirchner) whether rewriting strategies can be directly encoded in -calculi with patterns. This work leads to a research report [14] .

Semantics

Participants : Bruno Barras, Benjamin Werner.

Benjamin Werner and Bruno Barras work on various aspects of the set-theoretical models of Coq's type theory.