1. Introduction to the Boolean Satisfiability Problem
Spring 2018
CSCE 235H Introduction to Discrete Structures
URL: cse.unl.edu/~cse235h
All questions: Piazza

2. Satisfiability Study 7 weeks 30 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
Implication

12. Converting to CNF
Implication
DeMorgan’s

13. Converting to CNF
Implication
DeMorgan’s
Distributive

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 a b C 1

16. Interpretation of CNF a b C 1

17. Interpretation of CNF a b C 1

18. Interpretation of CNF a b C 1

19. Interpretation of CNF a b C 1

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

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26. Systematic Search with Backtracking

27. Systematic Search with Backtracking

28. Systematic Search with Backtracking
Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

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Satisfiable

41. Systematic Search with Backtracking
Satisfiable

42. Systematic Search with Backtracking
Satisfiable
3 solutions total