1. The Polynomial Hierarchy

2. Deciding Satifiability
We’ve already seen, that deciding whether a formula is satisfiable…
x1 …xn(x1x2x8)… (x6x3)
x1x2x3… [(x1x2x8)…(x6x3)]
only existential quantifier
existential & universal quantifiers
PSPACE-complete
NP-complete

3. Technical Note x1x2…xk is the same as x=<x1,x2,…,xk>
Thus, allowing several adjacent quantifiers of the same type does not change the problem.

4. i alternating quantifiers
The Hierarchy
Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type:
x1x2 x3… R(x1,x2,x3,…)
i alternating quantifiers

5. i alternating quantifiers
The Hierarchy
Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type:
x1x2x3… R(x1,x2,x3,…)
i alternating quantifiers

6. PH (Polynomial-time Hierarchy)
Definition:
PH = i i

7. Simple Observations “base”: 1=NP
“connection between  and ”: i=coi
“hierarchy”: ii+1 and ii+1
“upper bound”: PHPSPACE

8. Can the Hierarchy Collapse?
Proposition:
If NP=coNP,
then PH=NP.
Proof Idea:
By induction on i,
i=NP.

9. Recall: Arithmetical Hierarchy
co-r.e.
DEC=r.e.co-r.e.
DEC=Decidable Sets

10. Oracle TM
Generalize TM M to “oracle TM” M with states Q,Y,N and special “oracle tape.” When M enters state Q it jumps to Y if word on oracle tape is in B otherwise jumps to N.
Oracle B Y Q N
oracle tape
M will erase the oracle tape after entering Y/N

18. PTIME executions

19. NP executions (configuration trees)

20. Alternating Computation
Configuration Type
Existential
Universal
Negating + - +
Accepting - - + + - + -
Rejecting
Computation Tree

23. Back to Alternation
For an accepting tree there exists a witness subtree for acceptance (and similar for rejection)
Witness subtree contains a single accepting son for every accepting node, and a single rejecting son for every rejecting node
A witness subtree is finite, even when the tree itself is infinite!
Infinite branches are irrelevant!

24. Negating Nodes ?
Create for every node its dual node which yields the “same” transitions
Dual of accepting node is rejecting
Dual of rejecting node is accepting
Dual of universal node is existential
Dual of existential node is universal
Dual of Dual is identity
Replace every negating node by an existential one, dualizing the entire subtree below it (think de Morgan!)
Negating states are unnecessary - by dualizing parts of the computation tree they can be removed.

25. Eliminating Negating Nodes+ + + - - + - - + + + - + + - - + + + - - - + + + - - - + - - + - + - + + + - + - - + + - + + - - - - + + - -
Dualized nodes -