## Section: New Results

### Certified provers

#### Calculus of Congruent Constructions

Participants : Frédéric Blanqui, Jean-Pierre Jouannaud, Pierre-Yves Strub, Qian Wang.

In [26] , [51] , we described a modification of the Calculus of Inductive Constructions allowing the use of decision procedures in the computation mechanism. In [21] , we gave a new definition of the calculus without most of the restrictions made in [26] , [51] , and proved its core logic in Coq. This development has been the basis of CoqMT, our new version of Coq. As a paradigmatic example, we developed the basic theory of dependent lists with CoqMT. Compared with the same development for non-dependent lists, very few modifications were necessary to carry out the proofs.

We also started several generalisations of the previous work. Two are especially important: the ability to consider polymorphic first-order theories, and the extraction of equations from pattern matching.

#### Certification of SAT solvers

Participants : Frédéric Blanqui, Jean-Pierre Jouannaud, Pierre-Yves Strub, Bow-Yaw Wang, Lianyi Zhang.

We started to work on the certification of unsatisfiability proofs given by a set of regular input resolution proofs as provided by the PicoSAT solver(http://fmv.jku.at/picosat/ ) and described in http://fmv.jku.at/tracecheck/README.tracecheck . In order to make experiments, we also developped a new version of MiniSAT in OCaml, outputting a trace in the PicoSAT format.

#### Maxterm covering for satisfiability

Participants : Ming Gu, Fei He, Liangze Yin.

Boolean satisfiability (SAT) is to find if there is a true interpretation for a Boolean formula. Many real-world problems can be transformed into SAT problems and many of these problem instances can be effectively solved via satisfiability, such as testing, formal verification, synthesis, various routing problems, etc. In [18] , we present a novel efficient SAT algorithm based on maxterm covering. The satisfiability of a clause set is determined in terms of the number of relative maxterms of the empty clause with respect to the clause set. If the number of relative maxterms is zero, it is unsatisfiable, otherwise satisfiable. A set of synergic heuristic strategies are presented and elaborated. We conduct a number of experiments on 3-SAT problems at the phase transition region of density 4.3, which have been cited as the hardest group of SAT problems. Our experimental results on public benchmarks attest to the fact that, by incorporating our proposed heuristic strategies, our enhanced algorithm can handle 3-SAT problems with 400 variables. The approach runs 3 to 40 times faster than zChaff does for both satisfiable and unsatisfiable problems.