We describe how the Davis-Putnam-Logemann-Loveland procedure DPLL is bisimilar to the goal-directed proof-search mechanism described by a standard but carefully chosen sequent calculus. We thus relate a procedure described as a transition system on states to the gradual completion of incomplete proof-trees.For this we use a focused sequent calculus for polarised classical logic, for which we allow analytic cuts. The focusing mechanisms, together with an appropriate management of polarities, then allows the bisimulation to hold: The class of sequent calculus proofs that are the images of the DPLL runs finishing on UNSAT, is identified with a simple criterion involving polarities.We actually provide those results for a version DPLL(T ) of the procedure that is parameterised by a background theory T for which we can decide whether conjunctions of literals are consistent. This procedure is used for Satisfiability Modulo Theories (SMT) generalising propositional SAT. For this, we extend the standard focused sequent calculus for propositional logic in the same way DPLL(T ) extends DPLL: with the ability to call the decision procedure for T . DPLL(T ) is implemented as a plugin for PSYCHE, a proofsearch engine for this sequent calculus, to provide a sequentcalculus based SMT-solver.

Goal-directed proof search in first-order logic uses metavariables 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. This paper investigates a generalisation of such mechanisms whereby theory-specific constraints are produced instead of substitutions. In order to design modular proofsearch 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 axiomatisation 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.

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In this paper, we introduce two focussed sequent calculi, LK p (T ) and LK + (T ), that are based on Miller-Liang's LKF system [LM09] for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T , and the possibility to polarise literals "on the fly" during proof-search.These features are used in other works [FLM12, FGLM13] to simulate the DPLL(T ) procedure [NOT06] as proof-search in the extension of LK p (T ) with a cut-rule.In this report we therefore prove cut-elimination in LK p (T ).Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T . On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs.In order to prove this, we introduce a second sequent calculus, LK + (T ) that extends LK p (T ) with a relaxed focussing discipline, but we then show an encoding of LK + (T ) back into the more restrictive system LK(T ).We then prove completeness of LK p (T ) (and therefore of LK + (T )) with respect to firstorder reasoning modulo the ground propositional lemmas of the background theory T .

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