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How a CDCL SAT solver works

Masahiro Sakai, profile picture
Masahiro Sakai

1. The document demonstrates how a CDCL SAT solver works on a small example problem with 9 variables and 7 clauses. 2. It shows the step-by-step deductions made by the solver as it decides variable assignments, deduces implications, and detects a conflict. 3. When a conflict is found, the solver performs conflict analysis to determine the reason for the inconsistency and learns a new clause, which is added to its clause database.

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How a CDCL SAT Solver works Masahiro Sakai Twitter: @masahiro_sakai
Abstract • Demonstrating how a modern SAT solver works on a small example. • Target Algorithm • Conflict-Driven Clause Learning (CDCL) • 1-UIP learning + far-backtracking • details are omitted efficient data (decision heuristics, structure, etc.,)
An Example Problem • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x1@1 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x2 x2@2 x1@1 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω1 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω1 x4@4 x4 ω2 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω3 ω1 x4@4 x7@4 x4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω3 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω6 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 ω6 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω6 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 ω6 x6@4 x8@4 Clause DB Conflict • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph x3@3 ω1 x2@2 ω5 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: x3@3 Side ¬x8 ¬x9 ω1 x2@2 ω5 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ¬x2 ¬x7 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ¬x2 ¬x7 ω1 x4@4 x7@4 It contains one literal in the current ω3 ω6 ω2 ω4 decision level (i.e. 4). x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ¬x2 ¬x7 ω1 x4@4 x7@4 It contains one literal in the current ω3 ω6 ω2 ω4 decision level (i.e. 4). x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9 New clause is learnt !
Backtrack Implication Graph Learnt clause: ¬x2 ¬x7 x3@3 ω1 x2@2 ω5 x2 was set at level 2 x1@1 x7 was set at level 4 x5@4 x9@4 ω3 ω5 ω6 Assertion level of this clause (which is the ω1 second highest level) x4@4 x7@4 is 2. ω6 ω2 ω3 ω4 Backtrack to level 2! x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω6 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 ω6 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω3 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω3 ω1 x4@4 x7@4 x4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω1 x4@4 x4 ω2 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω1 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x2 x2@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x2 x2@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4@4 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4@4 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω1 x4@4 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω1 x4@4 x4 x4 ω2 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω3 ω3 ω1 x4@4 x4 x4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω3 ω3 ω1 x4@4 x4 x4 ω2 ω3 x6@4 Clause DB Conflict • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph x3@3 ω8 ω1 x2@2 ¬x7@2 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: x3@3 Side ¬x5 ¬x6 x7 ω8 ω1 x2@2 ¬x7@2 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ¬x1 ¬x4 x7 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ¬x1 ¬x4 x7 ω1 x4@4 It contains one literal in the current ω3 decision level (i.e. 4). ω2 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ¬x1 ¬x4 x7 ω1 x4@4 It contains one literal in the current ω3 decision level (i.e. 4). ω2 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 New clause is learnt !
Backtrack Implication Graph Learnt clause: ¬x1 ¬x4 x7 x3@3 ω8 ω1 x2@2 ¬x7@2 x1 was set at level 1 x1@1 x4 was set at level 4 x5@4 x7 was set at level 2 ω3 ω3 ω1 Assertion level of this x4@4 clause is 2. ω2 ω3 Backtrack to level 2! x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω3 ω3 ω1 x4@4 x4 x4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Backtrack Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 ¬x4@2 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 ¬x4@2 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 x5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 x5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 ω3 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 ω3 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x8@5 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x8@5 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ω6 ¬x6@4 x4 x4 x5 x8@5 ¬x9@5 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ω6 ¬x6@4 x4 x4 x5 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ω6 ¬x6@4 x4 x4 x5 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 x8 Clause DB Conflict • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph x3@3 ω8 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph ω7: x3@3 ¬x8 x9 ω8 x2@2 ¬x7@2 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 ¬x8 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 ¬x8 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 It contains one literal ω6 ¬x6@4 in the current x8@5 ¬x9@5 ω3 decision level (i.e. 5). ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 ¬x8 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 It contains one literal ω6 ¬x6@4 in the current x8@5 ¬x9@5 ω3 decision level (i.e. 5). ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 •New ¬x2 ¬x7 learnt ! ω8: clause is • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Backtrack Implication Graph Learnt clause: ¬x8 x3@3 ω8 x2@2 ¬x7@2 x8 was set at level 5 x1@1 ω9 ω9 ω3 Assertion level of this ¬x4@2 clause (which is the second highest level) ω6 ¬x6@4 is 0 by convention. x8@5 ¬x9@5 ω3 Backtrack to level 0 ω7 ω7 x5@4 (root level) Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Backtrack Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ω6 ¬x6@4 x4 x4 x5 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Backtrack Implication Graph x1 x2 x3 x3 x3 x4 x4 x5 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Deduce Implication Graph x1 x2 x3 x3 x3 x4 x4 x5 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Deduce Implication Graph x1 x2 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Decide Implication Graph x1 x2 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Decide Implication Graph x1 x1 x2 x1@1 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Decide Implication Graph x1 x1 x2 x2 x2@2 x1@1 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Deduce Implication Graph x1 x1 x2 x2 x2@2 x1@1 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Deduce Implication Graph x1 x1 ω8 x2 x2 x2@2 ¬x7@2 x1@1 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Deduce Implication Graph x1 x1 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Decide Implication Graph x1 x1 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Decide Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 x3 ¬x4@2 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Decide Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 x3 ¬x4@2 x4 x4 x5 x5 ¬x8@0 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Deduce Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 x3 ¬x4@2 x4 x4 x5 x5 ¬x8@0 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Deduce Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Decide Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Decide Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 All variables are ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 assigned without Clause DB • ω1: •¬x1 ¬x4 x5 ω6: ¬x8 ¬x9 • • conflict ω2: ω3: • • ¬x4 ¬x5 x6 ¬x6 x7 ω7: ¬x8 ω8: ¬x2 x9 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB Satisfiable • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
Implication Graph Found a model x3@3 ω8 M= x2@2 ¬x7@2 x1@1 ω9 { x1, x2, x3, ω9 ω3 ¬x4@2 ¬x4, x5, ¬x6 ¬x6@4 ¬x8@0 ω3 ¬x7, ¬x8, x9 } x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
fin.
References • J. P. Marques-Silva and K. A. Sakallah, "GRASP: a search algorithm for propositional satisfiability," Computers, IEEE Transactions on, vol. 48, no. 5, pp. 506-521, May 1999. • Handbook of Satisfiability, A. Biere, M. Heule, H. Van Maaren, and T. Walsh, Eds. IOS Press, Feb. 2009.

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How a CDCL SAT solver works

  • 1. How a CDCL SAT Solver works Masahiro Sakai Twitter: @masahiro_sakai
  • 2. Abstract • Demonstrating how a modern SAT solver works on a small example. • Target Algorithm • Conflict-Driven Clause Learning (CDCL) • 1-UIP learning + far-backtracking • details are omitted efficient data (decision heuristics, structure, etc.,)
  • 3. An Example Problem • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 4. Decide Implication Graph Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 5. Decide Implication Graph x1 x1@1 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 6. Decide Implication Graph x1 x2 x2@2 x1@1 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 7. Decide Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 8. Decide Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 9. Deduce Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 10. Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω1 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 11. Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω1 x4@4 x4 ω2 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 12. Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω3 ω1 x4@4 x7@4 x4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 13. Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω3 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 14. Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 15. Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω6 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 ω6 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 16. Deduce Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω6 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 ω6 x6@4 x8@4 Clause DB Conflict • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 17. Diagnose Implication Graph x3@3 ω1 x2@2 ω5 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 18. Diagnose Implication Graph Reason ω6: x3@3 Side ¬x8 ¬x9 ω1 x2@2 ω5 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 19. Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 20. Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 21. Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 x1@1 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 22. Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 23. Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 24. Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ¬x2 ¬x7 ω1 x4@4 x7@4 ω2 ω3 ω4 ω6 x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 25. Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ¬x2 ¬x7 ω1 x4@4 x7@4 It contains one literal in the current ω3 ω6 ω2 ω4 decision level (i.e. 4). x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 26. Diagnose Implication Graph Reason ω6: ω4: x3@3 Side ¬x8 ¬x9 ¬x7 x8 ω1 x2@2 ω5 ¬x7 ¬x9 ω5: x1@1 ¬x2 ¬x7 x9 x5@4 x9@4 ω3 ω5 ω6 ¬x2 ¬x7 ω1 x4@4 x7@4 It contains one literal in the current ω3 ω6 ω2 ω4 decision level (i.e. 4). x6@4 x8@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9 New clause is learnt !
  • 27. Backtrack Implication Graph Learnt clause: ¬x2 ¬x7 x3@3 ω1 x2@2 ω5 x2 was set at level 2 x1@1 x7 was set at level 4 x5@4 x9@4 ω3 ω5 ω6 Assertion level of this clause (which is the ω1 second highest level) x4@4 x7@4 is 2. ω6 ω2 ω3 ω4 Backtrack to level 2! x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 28. Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω6 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 ω6 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 29. Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 ω5 x1@1 x5@4 x9@4 x3 ω3 ω5 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 30. Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω3 ω1 x4@4 x7@4 x4 ω2 ω3 ω4 x6@4 x8@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 31. Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω3 ω1 x4@4 x7@4 x4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 32. Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω1 x4@4 x4 ω2 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 33. Backtrack Implication Graph x1 x3@3 x2 ω1 x2@2 x1@1 x5@4 x3 ω1 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 34. Backtrack Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 x4@4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 35. Backtrack Implication Graph x1 x3@3 x2 x2@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 36. Backtrack Implication Graph x1 x2 x2@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 37. Deduce Implication Graph x1 x2 x2@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 38. Deduce Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 39. Decide Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 40. Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 41. Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4@4 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 42. Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4@4 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 43. Deduce Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω1 x4@4 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 44. Deduce Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω1 x4@4 x4 x4 ω2 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 45. Deduce Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω3 ω3 ω1 x4@4 x4 x4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 46. Deduce Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω3 ω3 ω1 x4@4 x4 x4 ω2 ω3 x6@4 Clause DB Conflict • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 47. Diagnose Implication Graph x3@3 ω8 ω1 x2@2 ¬x7@2 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 48. Diagnose Implication Graph Reason ω3: x3@3 Side ¬x5 ¬x6 x7 ω8 ω1 x2@2 ¬x7@2 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 49. Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 50. Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 51. Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 x1@1 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 52. Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 53. Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 54. Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ¬x1 ¬x4 x7 ω1 x4@4 ω2 ω3 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 55. Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ¬x1 ¬x4 x7 ω1 x4@4 It contains one literal in the current ω3 decision level (i.e. 4). ω2 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω5: ¬x2 ¬x7 x9
  • 56. Diagnose Implication Graph Reason ω3: ω2: x3@3 Side ¬x5 ¬x6 x7 ¬x4 x6 ω8 ω1 x2@2 ¬x7@2 ¬x4 ¬x5 x7 ω1: x1@1 ¬x1 ¬x4 x5 x5@4 ω3 ω3 ¬x1 ¬x4 x7 ω1 x4@4 It contains one literal in the current ω3 decision level (i.e. 4). ω2 Conflict x6@4 Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 New clause is learnt !
  • 57. Backtrack Implication Graph Learnt clause: ¬x1 ¬x4 x7 x3@3 ω8 ω1 x2@2 ¬x7@2 x1 was set at level 1 x1@1 x4 was set at level 4 x5@4 x7 was set at level 2 ω3 ω3 ω1 Assertion level of this x4@4 clause is 2. ω2 ω3 Backtrack to level 2! x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 58. Backtrack Implication Graph x1 x3@3 ω8 x2 ω1 x2@2 ¬x7@2 x1@1 x5@4 x3 x3 ω3 ω3 ω1 x4@4 x4 x4 ω2 ω3 x6@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 59. Backtrack Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 60. Deduce Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 x3 x3 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 61. Deduce Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 ¬x4@2 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 62. Decide Implication Graph x1 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 ¬x4@2 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 63. Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 64. Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 x5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 65. Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 x5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 66. Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 ω3 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 67. Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 ω3 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 68. Decide Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x8@5 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 69. Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x8@5 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 70. Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ω6 ¬x6@4 x4 x4 x5 x8@5 ¬x9@5 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 71. Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ω6 ¬x6@4 x4 x4 x5 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 72. Deduce Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ω6 ¬x6@4 x4 x4 x5 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 x8 Clause DB Conflict • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 73. Diagnose Implication Graph x3@3 ω8 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 74. Diagnose Implication Graph ω7: x3@3 ¬x8 x9 ω8 x2@2 ¬x7@2 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 75. Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 76. Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 77. Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 ¬x8 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 ω6 ¬x6@4 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 78. Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 ¬x8 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 It contains one literal ω6 ¬x6@4 in the current x8@5 ¬x9@5 ω3 decision level (i.e. 5). ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9
  • 79. Diagnose Implication Graph ω7: ω6: x3@3 ¬x8 x9 ¬x8 ¬x9 ω8 x2@2 ¬x7@2 ¬x8 x1@1 ω9 Reason ω9 ω3 Side ¬x4@2 It contains one literal ω6 ¬x6@4 in the current x8@5 ¬x9@5 ω3 decision level (i.e. 5). ω7 ω7 x5@4 Conflict Side Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 •New ¬x2 ¬x7 learnt ! ω8: clause is • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 80. Backtrack Implication Graph Learnt clause: ¬x8 x3@3 ω8 x2@2 ¬x7@2 x8 was set at level 5 x1@1 ω9 ω9 ω3 Assertion level of this ¬x4@2 clause (which is the second highest level) ω6 ¬x6@4 is 0 by convention. x8@5 ¬x9@5 ω3 Backtrack to level 0 ω7 ω7 x5@4 (root level) Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 81. Backtrack Implication Graph x1 x3@3 ω8 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 ¬x4@2 ω6 ¬x6@4 x4 x4 x5 x8@5 ¬x9@5 ω3 ω7 ω7 x5@4 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 82. Backtrack Implication Graph x1 x2 x3 x3 x3 x4 x4 x5 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 83. Deduce Implication Graph x1 x2 x3 x3 x3 x4 x4 x5 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 84. Deduce Implication Graph x1 x2 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 85. Decide Implication Graph x1 x2 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 86. Decide Implication Graph x1 x1 x2 x1@1 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 87. Decide Implication Graph x1 x1 x2 x2 x2@2 x1@1 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 88. Deduce Implication Graph x1 x1 x2 x2 x2@2 x1@1 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 89. Deduce Implication Graph x1 x1 ω8 x2 x2 x2@2 ¬x7@2 x1@1 x3 x3 x3 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 90. Deduce Implication Graph x1 x1 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 91. Decide Implication Graph x1 x1 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 ¬x4@2 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 92. Decide Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 x3 ¬x4@2 x4 x4 x5 ¬x8@0 x8 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 93. Decide Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 x3 ¬x4@2 x4 x4 x5 x5 ¬x8@0 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 94. Deduce Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 x3 x3 x3 x3 ¬x4@2 x4 x4 x5 x5 ¬x8@0 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 95. Deduce Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 96. Decide Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 97. Decide Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 98. Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 99. Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 All variables are ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 assigned without Clause DB • ω1: •¬x1 ¬x4 x5 ω6: ¬x8 ¬x9 • • conflict ω2: ω3: • • ¬x4 ¬x5 x6 ¬x6 x7 ω7: ¬x8 ω8: ¬x2 x9 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 100. Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 101. Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB Satisfiable • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 102. Implication Graph x1 x1 x3@3 ω8 x2 x2 x2@2 ¬x7@2 x1@1 ω9 ω9 ω3 x3 x3 x3 x3 ¬x4@2 ¬x6@4 x4 x4 x5 x5 ¬x8@0 ω3 x8 x9 x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 103. Implication Graph Found a model x3@3 ω8 M= x2@2 ¬x7@2 x1@1 ω9 { x1, x2, x3, ω9 ω3 ¬x4@2 ¬x4, x5, ¬x6 ¬x6@4 ¬x8@0 ω3 ¬x7, ¬x8, x9 } x9@5 x5@4 Clause DB • ω1: ¬x1 ¬x4 x5 • ω6: ¬x8 ¬x9 • ω2: ¬x4 x6 • ω7: ¬x8 x9 • ω3: ¬x5 ¬x6 x7 • ω8: ¬x2 ¬x7 • ω4: ¬x7 x8 • ω9: ¬x1 ¬x4 x7 • ω5: ¬x2 ¬x7 x9 • ω10: ¬x8
  • 104. fin.
  • 105. References • J. P. Marques-Silva and K. A. Sakallah, "GRASP: a search algorithm for propositional satisfiability," Computers, IEEE Transactions on, vol. 48, no. 5, pp. 506-521, May 1999. • Handbook of Satisfiability, A. Biere, M. Heule, H. Van Maaren, and T. Walsh, Eds. IOS Press, Feb. 2009.

Editor's Notes