Introduction The DPLL and DPLL( T ) algorithms Architecture & Implementation Experimental Evaluation Demo Conclusion & Future Work Pluggable SAT-Solvers for SMT-Solvers Bas Schaafsma DISI, University of Trento & Fondazione Bruno Kessler May 29, 2013 http://mathsat.fbk.eu/ 1
Introduction The DPLL and DPLL( T ) algorithms The SAT/SMT problem Architecture & Implementation Applications of SAT/SMT-Solvers Experimental Evaluation Motivation Demo Conclusion & Future Work The SMT problem The SAT problem Given a Boolean formula F , is there an assignment for which F evaluates to true ? The SMT problem SAT extended with a set of theories T 1 ∪ T 2 · · · ∪ T n Example ( EUF ∪ LA ( Z )): ( x + 2 y = 6 ∨ y = 9) ∧ ¬ ( f ( x ) = f ( y )) ∧ x = 2 http://mathsat.fbk.eu/ 2
Introduction The DPLL and DPLL( T ) algorithms The SAT/SMT problem Architecture & Implementation Applications of SAT/SMT-Solvers Experimental Evaluation Motivation Demo Conclusion & Future Work Some Useful Theories Theory of Linear Arithmetic ( LA ) F LA = L ( x = ) ∧ L ( x< ) Theory of BitVectors ( BV ) F BV = L ( zext � � ( x � � )) > u � � ) Theory of Arrays ( ARR ) F ARR∪LA = L ( read ( a, )= ) ∧ ¬ L ( read ( write ( a,, ) ,i )= read ( a,i )) http://mathsat.fbk.eu/ 3
Introduction The DPLL and DPLL( T ) algorithms The SAT/SMT problem Architecture & Implementation Applications of SAT/SMT-Solvers Experimental Evaluation Motivation Demo Conclusion & Future Work Applications of SMT-Solvers • Bounded Model Checking • Equivalence Testing [GPB01] • Property Driven Reachability Testing [CNR12] • Scheduling [ABP + 11] • Test Case Generation [GLM12] • Software model checking through Predicate Abstraction [FQ02] • Program Synthesis [SGCF11] • ... http://mathsat.fbk.eu/ 4
Introduction The DPLL and DPLL( T ) algorithms The SAT/SMT problem Architecture & Implementation Applications of SAT/SMT-Solvers Experimental Evaluation Motivation Demo Conclusion & Future Work Pluggable SAT solvers: Motivation • Developing a new (allround) SMT solver entails more than a new SAT solver. → MathSAT5 ∼ 150kloc vs MiniSAT ∼ 6kloc • Success of SAT solvers highly dependent on heuristics. • Tuning SAT solvers requires investment of time and money. • SAT-Solver is a deciding factor for BV & BV ∪ ARR instances. • We want to combine state-of-the-art SAT solvers & SMT solvers. • This is NOT a straight forward bitblasting approach. http://mathsat.fbk.eu/ 5
Introduction The DPLL and DPLL( T ) algorithms The SAT/SMT problem Architecture & Implementation Applications of SAT/SMT-Solvers Experimental Evaluation Motivation Demo Conclusion & Future Work Introduction 1 The DPLL and DPLL( T ) algorithms 2 Architecture & Implementation 3 Experimental Evaluation 4 Demo 5 Conclusion & Future Work 6 http://mathsat.fbk.eu/ 6
Introduction The DPLL and DPLL( T ) algorithms Architecture & Implementation The DPLL algorithm Experimental Evaluation The DPLL( T ) algorithm Demo Conclusion & Future Work The DPLL algorithm 1: Preprocess( F ) 2: while true do BCP( F ) 3: if not conflict then 4: if all variables assigned then 5: return SAT 6: end if 7: decide() 8: else 9: C conflict ← analyze() 10: if top level conflict found then 11: return UNSAT 12: end if 13: backtrack(C conflict ) 14: end if 15: 16: end while http://mathsat.fbk.eu/ 7
Introduction The DPLL and DPLL( T ) algorithms Architecture & Implementation The DPLL algorithm Experimental Evaluation The DPLL( T ) algorithm Demo Conclusion & Future Work DPLL( T ) = DPLL + .. • For correctness: Theory consistency checks. Case splitting. • For optimization: Early pruning. Theory deductions. • (Incrementality) For specific details check [ST09]. http://mathsat.fbk.eu/ 8
Introduction The DPLL and DPLL( T ) algorithms Architecture & Implementation The DPLL algorithm Experimental Evaluation The DPLL( T ) algorithm Demo Conclusion & Future Work (Simplified) DPLL( T ) algorithm 1: Preprocess( F ) 2: while true do BCP( F ) 3: if not conflict and theories consistent then 4: if all variables assigned and no case splitting needed. 5: then return SAT 6: end if 7: decide() 8: else 9: C conflict ← analyze() 10: if top level conflict found then 11: return UNSAT 12: end if 13: backtrack(C conflict ) 14: end if 15: 16: end while http://mathsat.fbk.eu/ 9
Introduction The DPLL and DPLL( T ) algorithms Architectural Overview Architecture & Implementation Communication Protocols Experimental Evaluation Case Studies Demo Conclusion & Future Work DPLL( T ) Architectural Overview Theory Solvers .. T -Solver 1 T -Solver n Conflicts/ Generated Lemmas Theory Manager Predicate Assignments Internal SAT-Solver SAT/UNSAT UNSAT Core / Interpolant / Model API F input http://mathsat.fbk.eu/ 10
Introduction The DPLL and DPLL( T ) algorithms Architectural Overview Architecture & Implementation Communication Protocols Experimental Evaluation Case Studies Demo Conclusion & Future Work DPLL( T ) + Pluggable Solver Architectural Overview Theory Solvers .. T -Solver 1 T -Solver n Conflicts/ Generated Lemmas Theory Manager Predicate Assignments SAT-Solver Stub Callback functions SAT API calls Pluggable Solver SAT/UNSAT UNSAT Core / Interpolant / Model API F input http://mathsat.fbk.eu/ 11
Introduction The DPLL and DPLL( T ) algorithms Architectural Overview Architecture & Implementation Communication Protocols Experimental Evaluation Case Studies Demo Conclusion & Future Work Pluggable SAT solvers: A quick overview • 3rd Party SAT solvers can be plugged in MathSAT5 by: Implementing a worker interface. Invoking required callback functions during search. • The worker interface allows MathSAT5 to: Specify the problem for the SAT solver to solve. Communicate deduced values. • Callbacks allow the SAT solver to: Communicate found (partial) models to MathSAT5 Invoke T -consistency checks. http://mathsat.fbk.eu/ 12
Introduction The DPLL and DPLL( T ) algorithms Architectural Overview Architecture & Implementation Communication Protocols Experimental Evaluation Case Studies Demo Conclusion & Future Work Pluggable SAT solvers: Requirements • Must be able to act as an enumerator. • Should support, solving under assumptions. • Able to create new variables, add new clauses during search. • Support variable freezing and reintroduction of eliminated variables. • In order to support popping, must be able to delete all clauses containing certain variables. http://mathsat.fbk.eu/ 13
Introduction The DPLL and DPLL( T ) algorithms Architectural Overview Architecture & Implementation Communication Protocols Experimental Evaluation Case Studies Demo Conclusion & Future Work Worker Interface Functions void s o l v e ( std : : vector < int > & assump , std : : vector < int > & c assump ) ; bool add clause ( std : : vector < int > & clause , bool permanent , bool d u r i n g c a l l b a c k ) ; void s e t f r o z e n ( i n t var , bool b ) ; i n t new var ( bool p o l a r i t y , bool dvar ) ; void enqueue assignment ( i n t assignment ) ; void r em ov e cl a u s es con t a i n i n g ( i n t v ) ; http://mathsat.fbk.eu/ 14
Introduction The DPLL and DPLL( T ) algorithms Architectural Overview Architecture & Implementation Communication Protocols Experimental Evaluation Case Studies Demo Conclusion & Future Work Callback functions TCODE n o c o n f l i c t a t e r b c p ( std : : vector < int > & conf ) ; TCODE model found ( std : : vector < int > & c o n f l i c t ) ; void inform hook of assignment ( i n t assignment ) ; void i n f or m h ook of n e w l e v e l ( ) ; void i n f or m h ook of b a c k t r a c k ( i n t l e v e l ) ; void a s k h o o k f o r t r e a s o n ( i n t assignment , std : : vector < int > & r ) ; http://mathsat.fbk.eu/ 15
Introduction The DPLL and DPLL( T ) algorithms Architectural Overview Architecture & Implementation Communication Protocols Experimental Evaluation Case Studies Demo Conclusion & Future Work Pluggable SAT solvers: Two Case Studies • Extending Minisat (& Cleaneling). • Extending Fiver. http://mathsat.fbk.eu/ 16
Introduction The DPLL and DPLL( T ) algorithms Architectural Overview Architecture & Implementation Communication Protocols Experimental Evaluation Case Studies Demo Conclusion & Future Work Case Study A: Extending Minisat • The internal addClause method should be changed such that: Clauses are added at the correct level. For conflicts, jump back to the level, the conflict was introduced. • The analyze method must take into account that assignments can be from deductions, asking the reason if necessary. • After each round of BCP, in search which does not result in, theory propagation should be called -until fixpoint-. http://mathsat.fbk.eu/ 17
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