1. LECTURE 2-7 Complete Problems in PH

2. THEOREM
PROOF

3. BOUNDED HALTING PROBLEM (BHP)
THEOREM
PROOF

4. A
BHP
LEMMA
PROOF

5. LEMMA
PROOF

6. DEFINITION
THEOREM
PROOF by induction on k

7. Machine Related Problem
Natural Problem
(Machine unrelated)
Machine Related Problem

8. SAT k
THEOREM

9. PROOF (odd k)

13. 3-CNF SAT k
THEOREM

14. Generalized Ramsey Number

15. THEOREM
PROOF
Construction
Such that

16. Variables

17. Variables
Clauses

18. Variables
Clauses

19. Variables
Clauses

20. Variables
Clauses

21. Variables
Clauses

22. Variables
Clauses

23. Variables
Clauses

25. Case 1
Answer:

26. Case 2
Answer:

27. Case 2
Answer:

29. Consider
To prove
Define a coloring c’:
No red clique of size K exists.

31. Consider
To prove
Define a coloring c’:
No red clique of size K exists.
There must exist a blue clique Q of size K.

33. PPROPERTY OF BLUE CLIQUE Q
Assignment for y
Then F(X,Y) is satisfied

35. Thanks, end.