Section: New Results

Sequent Calculus with Calls to a Decision Procedure

Participants : Mahfuza Farooque, Stéphane Graham-Lengrand.

In the PSI project, a version of the focused sequent calculus (for first-order classical logic) has been designed, which can call external decision procedures. Several results were achieved in 2013 since the last Activity Report:

Firstly, a bug was discovered in the proof of cut-elimination, which was used to prove the logical completeness of the calculus. Fixing the bug required minor changes in the definition of the system, but incurred a major re-development of the meta-theory. Out of this technical work, one idea emerged: in presence of a non-trivial theory, changing the polarity of literals may change the provability of formulas. This was quite unexpected, but it led to interesting issues, such as finding sufficient conditions on polarities to guarantee cut-elimination and logical completeness. An substantial achievement in this research topic was to successfully address such issues, which gave rise to a new version of the report [30] .

Secondly, more techniques from automated reasoning were captured as proof-search in this sequent calculus (the incremental construction of proof-trees): besides the SMT-solving algorithm DPLL(T) treated successfully in 2012 (which was written down and published this year in [21] ), the techniques of clause tableaux and connection tableaux were captured this year. This includes in particular a notion of clause tableaux modulo theories that C. Tinelli introduced in 2007  [60] . This new range of captured techniques is interesting as clause tableaux are designed to handle quantifiers, which DPLL(T) does not. This gives a new hope to combine the efficiency of SAT-solvers for propositional reasoning with the handling of quantifiers.