CS 4700: Foundations of Artificial Intelligence

1. CS 4700:Foundations of Artificial Intelligence Carla P. Gomes gomes@cs.cornell.edu Module: Satisfiability (Reading R&N: Chapter 7)

2. Intro to logic Current module Proof methods • Proof methods divide into (roughly) two kinds: • Application of inference rules • Legitimate (sound) generation of new sentences from old • Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm • Different types of proofs • Model checking • truth table enumeration (always exponential in n) • improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL) (including some inference rules) • heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms Previous module

3. Satisfiability

4. Propositional Satisfiability problem • Satifiability (SAT): Given a formula in propositional calculus, is there a model • (i.e., a satisfying interpretation, an assignment to its variables) making it true? • We consider clausal form, e.g.: • ( ab c ) AND ( b c) AND ( ac) possible assignments SAT: prototypical hard combinatorial search and reasoning problem. Problem is NP-Complete. (Cook 1971) Surprising “power” of SAT for encoding computational problems.

5. Satisfiability as an Encoding Language

6. Encoding Latin Square Problems in Propositional Logic • Variables: • Each variables represents a color assigned to a cell. • Clauses: • Some color must be assigned to each cell • No color is repeated in the same row • No color is repeated in the same column (clause of length n); n2 (sets of negative binary clauses); n(n-1)/2 n n (example Row i Color k) (sets of negative binary clauses); n(n-1)/2n n (example Colum j Color k)

7. 3D Encoding or Full Encoding • This encoding is based on the cubic representation of the quasigroup: each line of the cube contains exactly one true variable; • Variables: • Same as 2D encoding. • Clauses: • Same as the 2 D encoding plus: • Each color must appear at least once in each row; • Each color must appear at least once in each column; • No two colors are assigned to the same cell;

8. Dimacs format • At the top of the file is a simple header. • p cnf <variables> <clauses> • Each variable should be assigned an integer index. Start at 1, as 0 is used to indicate the end of a clause. The positive integer a positive literal, whereas a negative interger represents a negative literal. • Example • -1 7 0  ( x1  x7)

9. A cell gets at most a color No repetition of color in a column No repetition of color in a row A cell gets a color A given color goes in each column A given color goes in each row Extended Latin Square 2x2 order 2 -1 -1 -1 -1 • p cnf 8 24 • -1 -2 0 • -3 -4 0 • -5 -6 0 • -7 -8 0 • -1 -5 0 • -2 -6 0 • -3 -7 0 • -4 -8 0 • -1 -3 0 • -2 -4 0 • -5 -7 0 • -6 -8 0 • 1 2 0 • 3 4 0 • 5 6 0 • 7 8 0 • 1 5 0 • 2 6 0 • 3 7 0 • 4 8 0 • 1 3 0 • 2 4 0 • 5 7 0 • 6 8 0 1/2 3/4 5/6 7/8 1 – cell 11 is red 2 – cell 11 is green 3 – cell 12 is red 4 – cell 12 is green 5 – cell 21 is red 6 – cell 21 is green 7 – cell 22 is red 8 – cell 22 is green

10. Significant progress in Satisfiability Methods Software and hardware verification – complete methods are critical - e.g. for verifying the correctness of chip design, using SAT encodings Applications: Hardware and Software Verification Planning, Protocol Design, etc. Going from 50 variable, 200 constraints to 1,000,000 variables and 5,000,000 constraints in the last 10 years Current methods can verify automatically the correctness of > 1/7 of a Pentium IV.

11. Model Checking

12. Turing Award Source: Slashdot

13. A “real world” example

14. Bounded Model Checking instance: i.e. ((not x1) or x7) and ((not x1) or x6) and … etc.

15. 10 pages later: … (x177 or x169 or x161 or x153 … or x17 or x9 or x1 or (not x185)) clauses / constraints are getting more interesting…

16. 4000 pages later: !!! a 59-cnf clause… …

17. Finally, 15,000 pages later: Note that: … !!! MiniSAT solver solves this instance in less than one minute.

18. Effective propositional inference

19. Effective propositional inference • Two families of algorithms for propositional inference (checking satisfiability) based on model checking (which are quite effective in practice): • Complete backtracking search algorithms • DPLL algorithm (Davis, Putnam, Logemann, Loveland) • Incomplete local search algorithms • WalkSAT algorithm

20. The DPLL algorithm • Determine if an input propositional logic sentence (in CNF) is satisfiable. • Improvements over truth table enumeration: • Early termination A clause is true if any of its literals is true. A sentence is false if any clause is false. • Pure symbol heuristic Pure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), (B C), (C  A), A and B are pure, C is impure. Make a pure symbol literal true. • Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true.

21. Branch, recursive call Early termination Pure Symbol Unit Propagation The DPLL algorithm

22. DPLL • Basic algorithm for state-of-the-art SAT solvers;; • Several enhancements: • - data structures; • - clause learning; • - randomization and restarts; Check: http://www.satlive.org/

23. Learning in Sat

24. The WalkSAT algorithm • Incomplete, local search algorithm • Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses • Balance between greediness and randomness

25. The WalkSAT algorithm

26. Lots of solvers and information about SAT, theory and practice: http://www.satlive.org/

27. Standard Algorithmic Approach: Too Pessimistic. Ideal Approach, but… What distribution? Computational Complexity of SAT How does an algorithm scale? Analyzable Realistic Spectrum of hardness

28. SAT Complexity • NP-Complete - worst-case complexity • (2n possible assignments) • “Average” Case Complexity (I) • Constant Probability Model – Goldberg 79; Goldberg et al 82 • N variables; L clauses • p - fixed probability of a variable in a clause (literals: 0.5 +/-) (i.e., average clause length is pN) • Eliminate empty and unit clauses • Empirically, on average, SAT can be easily solved - O(n2) Key problem: easy distribution; random guesses find a solution in a constant number of tries Franco 86; Franco and Ho 88

29. Hard satisfiability problems • Consider random 3-CNF sentences. e.g., • (D B  C)  (B A C)  (C B  E)  (E D  B)  (B  E C) m = number of clauses n = number of symbols

30. Typical Case Analysis SAT Complexity • “Average” Case Complexity (II) • Fixed-clause Length Model – Random K-SATFranco 86; • N variables; L clauses; K number of literals per clause • Randomly choose a set of K variables per clause (literals: 0.5 +/-) • Expected time – O(2n) Can we provide a finer characterization beyond worst-case results?

31. Typical-Case Complexity • Typical-case complexity: a more detailed picture • Characterization of the spectrum of hardness of instances as we vary certain interesting instance parameters • e.g. for SAT: clause-to-variable ratio. • Are some regimes easier than others? • What about a majority of the instances?

32. Typical Case Analysis:3 SATAll clauses have 3 literals Median Runtime Selman et al. 92,96

33. Median • Hard problems seem to cluster near m/n = 4.3 (critical point)