Section: Application Domains

Trustworthy implementations of theorem proving techniques

Instead of integrating different provers by exchanging proof evidence and relying on a backend proof-checker, another approach to integration consists in re-implementing the theorem proving techniques as proof-search strategies, on an architecture that guarantees correctness. Focused systems can serve as the basis of such an architecture, identifying points for choice and backtracking, and providing primitives for the exploration of the search space. These form a trusted Application Programming Interface that can be used to program and experiment various proof-search heuristics without worrying about correctness. No proof-checking is needed if one trusts the implementation of the API.

This approach has led to the development of the Psyche engine.

Two major research directions are currently being explored, based on the above:

• The first one is about understanding how to deal with quantifiers in presence of one or more theories: On the one hand, traditional techniques for quantified problems, such as unification  [47] or quantifier elimination are usually designed for either the empty theory or very specific theories. On the other hand, the industrial techniques for combining theories (Nelson-Oppen, Shostak) are designed for quantifier-free problems, and quantifiers are there dealt with incomplete clause instantiation methods or trigger-based techniques  [61] . We are working on making the two approaches compatible.

• The above architecture's modular approach raises the question of how its different modules can safely cooperate (in terms of guaranteed correctness), while some of them are trusted and others are not. The issue is particularly acute if some of the techniques are run concurrently and exchange data at unpredictable times. For this we explore new solutions based on Milner's LCF  [79] . In [30] , we argued that our solutions in particular provide a way to fulfil the “Strategy Challenge for SMT-solving” set by De Moura and Passmore  [88] .