A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus

Abstract

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.

Psyche: A Proof-Search Engine Based on Sequent Calculus with an LCF-Style Architecture

“…For this we use a focused sequent calculus LK p (T ) [4,5], which extends the logic programming paradigm to polarised classical logic modulo a background theory T . Polarities and Focusing [6,1] are tools that can be used to describe effective proof-search strategies in Sequent Calculus, hugely narrowing the search-space offered by Gentzen's original rules.…”

“…For this, our sequent calculus LK p (T ) [4,5] describing classical logic modulo a theory T uses polarities and focusing (see e.g. [1]).…”

“…Polarities do not affect the (classical) provability of formulae, but only the shape of proofs and therefore the structure of the proof-search space. Focusing is the phenomenon whereby the inference rules decomposing the connectives of the same polarity can be chained without losing completeness (thus narrowing the search-space), see [5] for a full description. But in brief, focusing organises the proof-search process as an alternance between two kinds of phases: synchronous and asynchronous.…”

“…For this we describe in [5] how to see DPLL(T ), canonically expressed as a transition system [9], as a simple bottom-up proof-construction mechanism in LK p (T ). More practically, every rule of DPLL(T ) can be seen as the insertion of a particular coin in Psyche's slot machine.…”

“…We implemented this as two different plugins for Psyche: DPLL_Pat and DPLL_WL. These remain toy plugins, because, although it is now clear from [5] how to perform each rule of DPLL(T ) in Psyche, we still have to decide which rule to apply. This is where the two plugins differ: DPLL_Pat looks up the applicability of DPLL(T )'s rules by using Patricia tries to implement sets of clauses, while DPLL_WL looks it up using the technique of watched literals.…”