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GRASP-an efficient SAT solver

GRASP-an efficient SAT solver. Pankaj Chauhan. What is SAT?. Given a propositional formula in CNF, find an assignment to boolean variables that makes the formula true! E.g.  1 = (x 2  x 3 )  2 = (  x 1  x 4 )  3 = (  x 2  x 4 ) A = {x 1 =0, x 2 =1, x 3 =0, x 4 =1}.

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GRASP-an efficient SAT solver

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  1. GRASP-an efficient SAT solver Pankaj Chauhan

  2. What is SAT? • Given a propositional formula in CNF, find an assignment to boolean variables that makes the formula true! • E.g. 1 = (x2  x3) 2 = (x1  x4) 3 = (x2  x4) A = {x1=0, x2=1, x3=0, x4=1} SATisfying assignment! 15-398: GRASP and Chaff

  3. What is SAT? • Solution 1: Search through all assignments! • n variables 2n possible assignments, explosion! • SAT is a classic NP-Complete problem, solve SAT and P=NP! 15-398: GRASP and Chaff

  4. Why SAT? • Fundamental problem from theoretical point of view • Numerous applications • CAD, VLSI • Optimization • Model Checking and other type of formal verification • AI, planning, automated deduction 15-398: GRASP and Chaff

  5. Outline • Terminology • Basic Backtracking Search • GRASP • Pointers to future work Please interrupt me if anything is not clear! 15-398: GRASP and Chaff

  6. Terminology • CNF formula  • x1,…, xn: n variables • 1,…, m: m clauses • Assignment A • Set of (x,v(x)) pairs • |A| < n  partial assignment {(x1,0), (x2,1), (x4,1)} • |A| = n  complete assignment {(x1,0), (x2,1), (x3,0), (x4,1)} • |A= 0  unsatisfyingassignment {(x1,1), (x4,1)} • |A= 1  satisfying assignment {(x1,0), (x2,1), (x4,1)} 1 = (x2  x3) 2 = (x1  x4) 3 = (x2  x4) A = {x1=0, x2=1, x3=0, x4=1} 15-398: GRASP and Chaff

  7. Terminology • Assignment A (contd.) • |A= X  unresolved {(x1,0), (x2,0), (x4,1)} • An assignment partitions the clause databaseinto three classes • Satisfied, unsatisfied, unresolved • Free literals: unassigned literals of a clause • Unitclause: #free literals = 1 15-398: GRASP and Chaff

  8. Basic Backtracking Search • Organize the search in the form of a decision tree • Each node is an assignment, called decision assignment • Depth of the node in the decision tree decision level (x) • x=v@d  xis assigned tovat decision leveld 15-398: GRASP and Chaff

  9. Basic Backtracking Search • Iterate through: • Make new decision assignments to explore new regions of search space • Infer implied assignments by a deduction process. May lead to unsatisfied clauses, conflict! The assignment is called conflicting assignment. • Conflicting assignments leads to backtrack to discard the search subspace 15-398: GRASP and Chaff

  10. x1 x1 = 0@1 x2 x2 = 0@2 Backtracking Search in Action 1 = (x2  x3) 2 = (x1  x4) 3 = (x2  x4) x1 = 1@1  x4 = 0@1  x2 = 0@1  x3 = 1@1  x3 = 1@2 {(x1,1), (x2,0), (x3,1) , (x4,0)} {(x1,0), (x2,0), (x3,1)} No backtrack in this example! 15-398: GRASP and Chaff

  11. x1 x1 = 0@1 x1 = 1@1 x2 x2 = 0@2 Backtracking Search in Action Add a clause 1 = (x2  x3) 2 = (x1  x4) 3 = (x2  x4) 4 = (x1  x2  x3)  x4 = 0@1  x2 = 0@1  x3 = 1@1  x3 = 1@2 conflict {(x1,0), (x2,0), (x3,1)} 15-398: GRASP and Chaff

  12. Davis-Putnam revisited Deduction Decision Backtrack 15-398: GRASP and Chaff

  13. GRASP • GRASP is Generalized seaRch Algorithm for the Satisfiability Problem (Silva, Sakallah, ’96) • Features: • Implication graphs for BCP and conflict analysis • Learning of new clauses • Non-chronological backtracking! 15-398: GRASP and Chaff

  14. GRASP search template 15-398: GRASP and Chaff

  15. GRASP Decision Heuristics • Procedure decide() • Choose the variable that satisfies the most #clauses == max occurences as unit clauses at current decision level • Other possibilities exist 15-398: GRASP and Chaff

  16. GRASP Deduction • Boolean Constraint Propagation using implication graphs E.g. for the clause  = (x y), ify=1, then we must havex=1 • For a variable xoccuring in a clause , assignment 0 to all other literals is calledantecedent assignmentA(x) • E.g. for  = (x y  z), A(x) = {(y,0), (z,1)}, A(y) = {(x,0),(z,1)}, A(z) = {(x,0), (y,0)} • Variables directly responsible for forcing the value ofx • Antecedent assignment of a decision variable is empty 15-398: GRASP and Chaff

  17. Implication Graphs • Nodes are variable assignments x=v(x) (decision or implied) • Predecessors of x are antecedent assignments A(x) • No predecessors for decision assignments! • Special conflict vertices have A() = assignments to vars in the unsatisfied clause • Decision level for an implied assignment is (x) = max{(y)|(y,v(y))A(x)} 15-398: GRASP and Chaff

  18. 4 x2=1@6 x5=1@6 1 4 3 6 x4=1@6  conflict 6 3 2 5 2 5 x6=1@6 x3=1@6 Example Implication Graph Current truth assignment: {x9=0@1 ,x10=0@3, x11=0@3, x12=1@2, x13=1@2} Current decision assignment: {x1=1@6} x10=0@3 1 = (x1  x2) 2 = (x1  x3  x9) 3 = (x2  x3  x4) 4 = (x4  x5  x10) 5 = (x4  x6  x11) 6 = (x5  x6) 7 = (x1  x7  x12) 8 = (x1 x8) 9 = (x7  x8   x13) x1=1@6 x9=0@1 x11=0@3 15-398: GRASP and Chaff

  19. GRASP Deduction Process 15-398: GRASP and Chaff

  20. GRASP Conflict Analysis • After a conflict arises, analyze the implication graph at current decision level • Add new clauses that would prevent the occurrence of the same conflict in the future  Learning • Determine decision level to backtrack to, might not be the immediate one  Non-chronological backtrack 15-398: GRASP and Chaff

  21. Learning • Determine the assignment that caused the conflict, negation of this assignment is called conflict induced clause C() • The conjunct of this assignment is necessary condition for  • So adding C() will prevent the occurrenceof  again 15-398: GRASP and Chaff

  22. Learning • Find C() by a backward traversal of the IG, find the roots of the IG in the transitive fanin of  • For our example IG, C() = (x1  x9  x10  x11) 15-398: GRASP and Chaff

  23. (x,v(x)) if A(x) =  causesof(x) = (x)  [  causesof(y)]o/w (y,v(y))  (x) Learning (some math) • For any node of an IG x, partition A(x) into (x) = {(y,v(y)) A(x)|(y)<(x)} (x) = {(y,v(y)) A(x)|(y)=(x)} • Conflicting assignment AC() = causesof(), where 15-398: GRASP and Chaff

  24. C() =  xv(x) (x,v(x))  AC() Learning (some math) • Deriving conflicting clause C()from the conflicting assignment AC() is straight forward • For our IG, AC() = {x1=1@6, x9 =0@1, x10 = 0@3, x11 = 0@3} 15-398: GRASP and Chaff

  25. Learning • Unique implication points (UIPs) of an IG also provide conflict clauses • Learning of new clauses increases clause database size • Heuristically delete clauses based on a user parameter • If size of learned clause > parameter, don’t include it 15-398: GRASP and Chaff

  26. Backtracking Failure driven assertions (FDA): • If C() involves current decision variable, then after addition, it becomes unit clause, so different assignment for the current variable is immediately tried. • In our IG, after erasing the assignment at level 6, C() becomes a unit clause x1 • This immediately implies x1=0 15-398: GRASP and Chaff

  27. Continued IG Decision level 3 x8=1@6 x9=0@1 C() 8 9 x10=0@3 5 ’ x13=1@2 x1=0@6 C() 9 9 C() 7 x11=0@3 x7=1@6 x1 7 6 x12=1@2 Due to C()  ’ 15-398: GRASP and Chaff

  28. Backtracking Conflict Directed Backtracking • Now deriving a new IG after setting x1=0 by FDA, we get another conflict ’ • Non-chronological backtrack to decision level 3, because backtracking to any level 5, 4 would generate the same conflict ’ 15-398: GRASP and Chaff

  29. Backtracking Conflict Directed Backtracking contd. • Backtrack level is given by • = d-1chronological backtrack • < d-1non-chronological backtrack AC(’) = {x9 =0@1, x10 = 0@3, x11 = 0@3, x12=1@2, x13=1@2} C() = (x9  x10  x11   x12   x13)  = max{(x)|(x,v(x))AC(’)} 15-398: GRASP and Chaff

  30. Procedure Diagnose() 15-398: GRASP and Chaff

  31. Is that all? • Huge overhead for constraint propagation • Better decision heuristics • Better learning, problem specific • Better engineering! Chaff 15-398: GRASP and Chaff

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