An Abstract Framework for Satisfiability Modulo Theories

DPLL Modulo Theories Works with any DPLL engine and T -solver but is best with 1. an on-line DPLL engine and 2. an incremental T -solver Tableaux 2007 – p.29/40 Abstract DPLL Modulo TheoriesDPLL Modulo Theories Works with any DPLL engine and T -solver but is best with 1. an on-line DPLL engine and 2. an incremental T -solver It consists of the following rules: Propagate, Decide, Fail, Restart (as in the propositional case) and T -Backjump, T -Learn, T -Forget (theory versions of Backjump, Learn, Forget, resp.) Tableaux 2007 – p.29/40 Theory Rules T -Backjump B1 l •B2 || F,C → B1 k || F,C if    1. B1 l •B2 |= ¬C, 2. for some clause D ∨ k F,C |=T D ∨ k, B1 |= ¬D, k is undefined in M, k or k occurs in B1 l •B2 || F,C Not.: F |=T G iff every model of T that satisfies F satisfies G Tableaux 2007 – p.30/40 Theory Rules T -Backjump B1 l •B2 || F,C → B1 k || F,C if    1. B1 l •B2 |= ¬C, 2. for some clause D ∨ k F,C |=T D ∨ k, B1 |= ¬D, k is undefined in M, k or k occurs in B1 l •B2 || F,C T -Learn B || F → B || F, C if { all atoms of C occur in B || F, F |=T C T -Forget B || F, C → B || F if F |=T C Tableaux 2007 – p.30/40 Correctness of Abstract DPLL Modulo Theories Proposition For a rule application strategy to be fair it suffices to apply T -Learn/T -Forget only finitely many times, apply Restart only with increased periodicity, and stop with a state B || F only if B is T -consistent and B |= F or F is irreducible by Propagate, Decide and T -Backjump Tableaux 2007 – p.31/40 From Complete to Incomplete Theory Solvers Recall: On reaching a state B || G with B |= G, the T -solver must determine whether B |=T ⊥ Tableaux 2007 – p.32/40 From Complete to Incomplete Theory Solvers Recall: On reaching a state B || G with B |= G, the T -solver must determine whether B |=T ⊥ At the very least, the T -solver must be refutationally sound: never calling a T -satisfiable set B of literals T -unsatisfiable, Tableaux 2007 – p.32/40 From Complete to Incomplete Theory Solvers Recall: On reaching a state B || G with B |= G, the T -solver must determine whether B |=T ⊥ At the very least, the T -solver must be refutationally sound: never calling a T -satisfiable set B of literals T -unsatisfiable, Ideally, it should also be refutationally complete: Tableaux 2007 – p.32/40 From Complete to Incomplete Theory Solvers Recall: On reaching a state B || G with B |= G, the T -solver must determine whether B |=T ⊥ At the very least, the T -solver must be refutationally sound: never calling a T -satisfiable set B of literals T -unsatisfiable, Ideally, it should also be refutationally complete: always able to recognize a T -unsatisfiable set B of literals as such. Tableaux 2007 – p.32/40 From Complete to Incomplete Theory Solvers Recall: On reaching a state B || G with B |= G, the T -solver must determine whether B |=T ⊥ At the very least, the T -solver must be refutationally sound: never calling a T -satisfiable set B of literals T -unsatisfiable, Ideally, it should also be refutationally complete: always able to recognize a T -unsatisfiable set B of literals as such. For certain theories, it is advantageous to relax the refutational completeness requirement. Tableaux 2007 – p.32/40