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Introduction to Boolean Satisfiability and CNF Conversion

Dive into the Boolean Satisfiability Problem, learn to convert formulas to CNF, explore SAT solvers like MiniSAT, and discover applications in scheduling, verification, and more. Understand key concepts such as literals, clauses, and systematic search with backtracking. Gain exposure to fundamental research in computer science with practical examples and exercises.

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Introduction to Boolean Satisfiability and CNF Conversion

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  1. Introduction to the Boolean Satisfiability Problem Spring 2017 CSCE 235H Introduction to Discrete Structures URL: cse.unl.edu/~cse235h All questions: Piazza

  2. Satisfiability Study • 7 weeks • 25 min lectures in recitation • ~2 hours of homework per week • Goals: • Exposure to fundamental research in CS • Understand how to model problems • Learn to use SAT solver, MiniSAT

  3. Boolean Satisfiability Problem • Given: • A Boolean formula • Question: • Is there an assignment of truth values to the Boolean variables such that the formula holds true?

  4. Boolean Satisfiability Problem

  5. Boolean Satisfiability Problem SATISFIABLE a=true, b=true

  6. Boolean Satisfiability Problem SATISFIABLE a=true, b=true UNSATISFIABLE Left side of implication is a tautology. Right side of implication is a contradiction. True cannot imply false.

  7. Applications of SAT • Scheduling • Resource allocation • Hardware/software verification • Planning • Cryptography

  8. Conjunctive Normal Form • Variable • Literal • Clause • Formula

  9. Converting to CNF • All Boolean formulas can be converted to CNF • The operators can be rewritten in terms of • can be rearranged using • De Morgan’s Laws • Distributive Laws • Double Negative • May result in exponential size increase of the formula

  10. Converting to CNF

  11. Converting to CNF

  12. Converting to CNF

  13. Converting to CNF

  14. Interpretation of CNF • Every clause must be satisfied by at least one true literal • Total possible number of solutions increases as number of variables increases • Clauses constrain the possible solutions • Smaller clauses are more constraining

  15. Interpretation of CNF

  16. Interpretation of CNF

  17. Interpretation of CNF

  18. Interpretation of CNF

  19. Interpretation of CNF

  20. Determining SAT/UNSAT • All that is required to show satisfiability is to find a valid solution • Many techniques available: • Guessing and checking • Systematic search • Inference

  21. Systematic Search with Backtracking • Construct a binary tree of all combinations • Proceeds in a depth first manner • Each level corresponds to a variable • Each branch corresponds to a truth assignment • Branches of the tree are ‘pruned’ when the assignment cannot be extended in a satisfiable manner

  22. Systematic Search with Backtracking

  23. Systematic Search with Backtracking

  24. Systematic Search with Backtracking

  25. Systematic Search with Backtracking

  26. Systematic Search with Backtracking Satisfiable

  27. Systematic Search with Backtracking Satisfiable

  28. Systematic Search with Backtracking Satisfiable

  29. Systematic Search with Backtracking Satisfiable

  30. Systematic Search with Backtracking Satisfiable

  31. Systematic Search with Backtracking Satisfiable

  32. Systematic Search with Backtracking Satisfiable

  33. Systematic Search with Backtracking Satisfiable 3 solutions total

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