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How a CDCL SAT solver works

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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Overview of how a Conflict-Driven Clause Learning (CDCL) SAT solver operates with focus on key features such as 1-UIP learning.

Presentation of a sample problem with clauses (ω1 to ω7) for illustration of the SAT solver in action.Progressive decision-making steps showing selected variables (x1 to x4) and the implications of these decisions in the graph.

Demonstration of deduction process from decisions made, represented in the implication graph with various clauses.

Ongoing deduction moves visualized step-by-step, depicting the role of clauses and their implications as the solver progresses.

Identifying conflicts arising in the decision-making process, including visual representation of graphs indicating reasons and conflicts.

Advanced diagnosis of conflicts in each step when variables lead to contradictions, emphasizing the implications derived from existing clauses.

Explanation of the backtracking process to previous decision levels in an attempt to resolve conflicts, showing effects on implication graphs.

Final deductions and decisions leading to the identification of a satisfiable model for the provided clause database, concluding the SAT solving process.

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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 CDCLSAT Solver works Masahiro Sakai Twitter: @masahiro_sakai
  • 2.
    Abstract • Demonstrating howa 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 amodel 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.
  • 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