Stochastic Local Search for Smt: a Preliminary Report

ion of φ and the “invisible” part τp is (the Boolean abstraction of) the set of the T -lemmas providing the obligations induced by the theory T on the atoms of φ. (We use the superscript p to denote the Boolean abstraction of a T -formula.) To this extent, a traditional “lazy” SMT solver can be seen as a DPLL solver which knows φp but not τp: whenever a model μp for φp is found, it is passed to a T -solver which knows τp and hence checks if μp falsifies τp: if this is the case, it returns one clause cp in τp which is falsified by μp, which is then used by DPLL to drive the future search and is optionally added to φp. The above observation inspired to us a procedure integrating a T -solver into a SLS algorithm of the WalkSAT family (WalkSMT hereafter). 3.1 A basic WalkSMT procedure A high-level description of the pseudo-code of WalkSMT is shown in Algorithm 2. (We present first a basic version WalkSMT, in which we temporarily ignore steps 1-3 and 12-13, which we will describe in §3.2, together with other enhancements.) WalkSMT receives in input a SMT(T ) CNF formula and applies a WalkSAT scheme to its Boolean abstraction φp. InitialTruthAssignment, ChooseUnsatisfiedClause and NextTruthAssignment are the functions described in §2.2. (Notice that their underlying heuristic vary with the different variants of WalkSAT adopted.) Ignoring steps 1-3 and 12-13, the only significant difference wrt. Algorithm 1 is in steps 7-14. Whenever a total model μp is found s.t. μp |= φp, it is passed to T -solver. If (the set of T -literals corresponding to) μp is T satisfiable (i.e., μp |= φp∧τp) the procedures ends returning sat. Otherwise, T -solver returns conflict and a T -lemma cp. Notice that this corresponds to say that μp 6|= φp∧τp, and that cp is one of the (possibly-many) clauses in