## Section: New Results

### Axiomatic constraint systems for proof search modulo theories

Goal-directed proof search in first-order logic uses meta-variables to delay the choice of witnesses; substitutions for such variables are produced when closing proof-tree branches, using first-order unification or a theory-specific background reasoner. We have investigated a generalization of such mechanisms whereby theory-specific constraints are produced instead of substitutions. In order to design modular proof-search procedures over such mechanisms, we provide a sequent calculus with meta-variables, which manipulates such constraints abstractly. Proving soundness and completeness of the calculus leads to an acclimatization that identifies the conditions under which abstract constraints can be generated and propagated in the same way unifiers usually are. We then extract from our abstract framework a component interface and a specification for concrete implementations of background reasoners. This is a common work with Damien Rouhling (ENS Lyon), Stéphane Lengrand (CNRS, LIX) and Jean-Marc Notin (CNRS, LIX), based on the PhD contributions of Mahfuza Farooque (unaffiliated). It is described in [8] .