Class 6: SAT

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class 6: Solving SAT David Evans University of Virginia cs2102: Discrete Mathematics
2. Plan • Converting to CNF: ⟹ • 3SAT Problem • PS2 Questions
3. Converting to CNF ⟹ T T T T F F F T T F F T
4. Converting to CNF ⟹ T T T T F F F T T F F T The result is T iff the inputs do not match any row where the output is F. The inputs do not match a row, if any one of them does not match the input on that row.
5. Converting to CNF ⟹ T T T T F F F T T F F T ¬ ∨
6. Converting to CNF ⊕ T T F T F T F T T F F F
7. Satisfiability Definition. A formula is in 3SAT if it is in 3CNF form and it is satisfiable.
8. Satisfiability Definition. A formula is in 3SAT if it is in 3CNF form and it is satisfiable. Is this formula in 3SAT?
9. Satisfiability Definition. A formula is in 3SAT if it is in 3CNF form and it is satisfiable. Is this formula in 3SAT?
10. Magic 3SAT Box 3SAT
11. Magic 3SAT Box 3SAT Input Formula T if ∈ 3SAT, F otherwise Note: “magic box” ≈ “algorithm”
12. What’s a “Problem”?
13. Defining a Problem A problem is defined by: – A precise description of a set of inputs – A statement of the property the output must have A machine solves a problem if for every input it eventually produces a satisfactory output.
14. Problems and Algorithms problem: set of possible inputs and desired property of output. procedure: precise description of an information process. algorithm: a procedure that solves a problem. To solve a problem, an algorithm must (eventually) produce the correct output for any problem input. program: description of a procedure that can be executed by a computer.
15. 3SAT Decision Problem Input: logical formula in 3CNF form (in some representation) Output: True iff the formula has some satisfying assignment, otherwise False.
16. SAT Assignment Problem Input: logical formula in 3CNF form (in some representation)
17. SAT Assignment Problem Input: logical formula in 3CNF form (in some representation) Output: satisfying variable assignments that make the formula True if the formula has some satisfying assignment, otherwise “Not Satisfiable”.
18. Finding the Satisfying Assignment 3SAT Input Formula T if ∈ 3SAT, F otherwise Can we use the 3SAT box to find a satisfying assignment?
19. Finding the Satisfying Assignment 3SAT 3SAT
20. def solve(f): vars = variables in f if vars.empty(): done else: = vars.choose() # choose a variable in f if 3SAT( ∧ ): = True solve(f with all occurrences of replaced by T) else: = False solve(f with all occurrences of replaced by F)
21. Question Break
31. Charge • PS2 Due Friday (6:29pm) • Next week: – Building SAT Solvers – (Mathematical) Data Types • Sets, Functions – Read MCS Chapter 4