Motivation. CS389L: Automated Logical Reasoning. Lecture 17: SMT Solvers and the DPPL(T ) Framework. SMT solvers. The Basic Idea.

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1 Motivation Lecture 17: SMT rs and the DPPL(T ) Framework şıl Dillig n previous lectures, we looked at decision procedures for conjunctive formulas in various first-order theories This lecture: How to handle boolean structure when deciding satisfiability modulo theories n practice, cannot convert to DNF because causes exponential blow-up in formula size SMT (satisfiability modulo theory) solvers use clever techniques to handle boolean structure şıl Dillig, 1/28 şıl Dillig, 2/28 SMT solvers The Basic dea To use solver, we construct a propositional formula, called boolean abstraction, that overapproximates satisfiability Key idea underlying SMT solvers: Combine theory solvers with solvers solver: Decision procedure for checking satisfiability in conjunctive fragment solver handles boolean structure, and theory solver handles theory-specific reasoning f boolean abstraction is UN, we are done also unsat modulo theory f boolean abstraction is, doesn t necessarily mean original formula is Use theory solver to check if assignment returned by solver is satisfiable modulo theory f not, add additional boolean constraints (called theory s) to guide the search for an assignment that is satisfiable modulo theory şıl Dillig, 3/28 şıl Dillig, 4/28 Boolean Abstraction SMT formula in theory T formed according to CFG: F := a T F 1 F 2 F 1 F 2 F What is the boolean abstraction of this formula? For each SMT formula, define a bijective function B, called boolean abstraction function, that maps SMT formula to overapproximate formula F : x = z ((y = z x < z) (x = z)) Function B defined inductively as follows: B(a T ) = b (b fresh) B(F 1 F 2 ) = B(F 1 ) B(F 2 ) B(F 1 F 2 ) = B(F 1 ) B(F 2 ) B( F ) = B(F 1 ) Boolean abstraction is also called boolean skeleton Since B is a bijective function, B 1 also exists What is B 1 (b 2 b 1 )? şıl Dillig, 5/28 şıl Dillig, 6/28 1

2 Boolean Abstraction as Overapproximation Observe: The boolean abstraction constructed this way overapproximates satisfiability of the formula solver may yield assignments that are not sat modulo T because boolean abstraction is an over-approximation n this case, we need to learn theory s Two different approaches for learning theory s s this formula satisfiable? F : x = z ((y = z x < z ) (x = z )) Boolean abstraction: B(F ) = b1 ((b2 b3 ) b1 ) s this satisfiable? What is a sat assignment? What is B 1 (A)? s B 1 (A) satisfiable? s ıl Dillig, Need for Clauses SMT Solving Off-line Version 9/28 Consider example from before: B(F ) : b1 ((b2 b3 ) b1 ) Sat assignment to B(F ) A : b1 b2 b3 B 1 (A) is unsat What is new boolean abstraction? s ıl Dillig, Unfortunately, s obtained this way are too weak Suppose A is a conjunction of 100 literals such that B 1 (A) = x = y x < y a1 a2... a98 A prevents exact same assignment But it doesn t prevent many other bad assignments involving x = y x 6= y such as: B 1 (A) = x = y x < y a1 a2... a98 s ıl Dillig, n fact, there are 298 unsat assignments containing x = y x 6= y but A prevents only one of them! 8/28 F : x = z ((y = z x < z ) (x = z )) s this formula? 10/28 mprovement to Off-line SMT So far, we just add negation of current assignment as theory (b1 ((b2 b3 ) b1 )) (b1 b2 b3 ) Shortcoming of This Approach On-line (lazy): ntegrate theory solver into the CDCL loop OfflineSMT(φ){ ψ := B(φ) while(true){ A := CDCL(φ) if(a = ) return UN; res := (B 1 (A)); if(res) return ; ψ := ψ A s ıl Dillig, Off-line (eager): Use solver as black-box First look into off-line version because it s easier s ıl Dillig, 7/28 11/28 s ıl Dillig, 2 Rather than adding A as a, better idea is to find an unsatisfiable core of B 1 (A) Given a set S of clauses, an unsat core of S 0 is a subset S 0 such that S 0 is also unsat deally, we would like to find the minimal unsatisfiable core Minimal unsatisfiable core C has property that if you drop any single atom of C, result is satisfiable What is a minimal unsat core of x = y x < y a1 a2... a98? x = y x < y 12/28

3 Computing Minimal Unsat Core How can we compute minimal unsat core of conjunctive T formula without modifying theory solver? Let φ be original unsatisfiable conjunct Let s compute minimal unsat core of φ : x = y f (x ) + z = 5 f (x ) 6= f (y) y 3 Drop x = y from φ. s result unsat? Drop one atom from φ, call this φ0 Drop f (x ) + z = 5. s result unsat? f φ0 is still unsat, φ := φ0 New formula: φ : x = y f (x ) 6= f (y) y 3 Repeat this for every atom in φ Drop f (x ) 6= f (y). s result unsat? Clearly, resulting φ is minimal unsat core of original formula Finally, drop y 3. s result unsat? So, minimal unsat core is x = y f (x ) 6= f (y) s ıl Dillig, 13/28 s ıl Dillig, SMT mproved Off-line Version 14/28 Motivation for On-line SMT OfflineSMT(φ){ ψ := B(φ) while(true){ A := CDCL(φ) if(a = ) return UN; res := (B 1 (A)); if(res) return ; χ := UnsatCore(B 1 (A)) ψ := ψ B(χ) s ıl Dillig, This strategy is much better than simple strategy where we add A as theory. But still need to wait for full assignment from the solver, which can be problematic Consider very large formula F containing x = y and x < y with corresponding boolean variables b1 and b2 As soon as sat solver makes assignment b1 = >, b2 = >, we are doomed because this is unsatisfiable in theory Thus, no need to continue with solving after this bad partial assignment s ıl Dillig, 15/28 On-line SMT 16/28 DPLL-Based r Architecture dea: Don t use solver as blackbox ntegrate theory solver right into the CDCL n other words, theory s become another kind of that solvers already learn... no conflict conflict Conflict UN s ıl Dillig, 17/28 s ıl Dillig, 3 dea: ntegrate theory solver right into this solving loop! 18/28

4 Suppose solver has made assignment in step and performed f no detected, immediately invoke theory solver UN Combination of DPLL-based solver and decision procedure for conjunctive T formula called DPLL(T ) framework Specifically, suppose A is current partial assignment to boolean abstraction Use theory solver to decide if B 1 (A) is unsat f B 1 (A) unsat, add theory A to clause database Or better, add negation of unsat core of A to clause database şıl Dillig, 19/28 şıl Dillig, 20/28 Propagation What we described so far is sufficient to solve SMT formula, but we can be even more clever! Suppose original formula contains literals x = y, y = z, x < z with corresponding boolean variables b 1, b 2, b 3 Suppose solver makes partial assignment b 1 :, b 2 : UN Add theory and continue doing, which will detect n next step, free to assign b 3 : or b 3 : But assignment b 3 : is stupid b/c will lead to in T As before, decides what level to to şıl Dillig, 21/28 şıl Dillig, 22/28 Propagation Lemma, cont dea: solver can communicate which literals are implied by current partial assignment n our example, x < z implied by current partial assignment x = y y = z Thus, can safely add b 1 b 2 b 3 to clause database These kinds of clauses implied by theory are called theory propagation lemmas UN Adding theory propagation lemmas prevents bad assignments to boolean abstraction şıl Dillig, 23/28 şıl Dillig, 24/28 4

5 nferring Propagation Lemmas How do we obtain theory propagation lemmas? Option #1: Treat theory solver as blackbox, query whether particular literal a is implied by current partial assisgnment? Option #2: Modify theory solver so that it can figure out implied literals Second option is considered more efficient, but have to figure out how to do this for each different theory Which Propagation Lemmas to Add Which theory propagation lemmas do we add? Option #1: Figure out and add all literals implied by current partial assignment; called exhaustive theory propagation Option #2: Only figure out literals obviously implied by current partial assignment Exhaustive theory propagation can be very expensive; second option considered preferable There isn t much of a science behind which literals are obviously implied rs use different strategies to obtain simple-to-find implications şıl Dillig, 25/28 şıl Dillig, 26/28 SMT rs Today Summary All competitive SMT solvers today are based on the on-line version Many existing off-the-shelf SMT solvers: Z3, CVC3, Yices, Math, etc. Lots of on-going research on SMT, esp. related to quantifier support Annual competition SMT-COMP between solvers; tools ranked in various categories SMT solvers decide satisfiability in boolean combinations of different theories nstead of converting to DNF, they handle boolean structure using solving technqiues Competitive solvers are based on DPLL(T ) framework şıl Dillig, 27/28 şıl Dillig, 28/28 5