The Polynomial Hierarchy

We next look at some problems that seem not to be captured by `NP`-completeness, but seem to be weaker than all of `PSPACE`.
Recall the problem `INDSET = {(G,k) | mbox(graph G has an independent set of size ) ge k }`. We already showed this problem is `NP`-complete.
Consider now the following natural variant on this problem: `EXACT` `INDSET`, the set of ordered pairs `(G, k)` such that the largest independent set in `G` has `k` elements.
`INDSET` had a short certificate for membership, but the latter does not.
Instead `(G,k)` is in `EXACT` `INDSET` iff there exists an independent set of size `k` in `G` such that for every other independent `S` the size of `S` is less than or equal to `k`.

The Polynomial Hierarchy - Definition 1

Definition. For `i ge 1`, a language `L` is in `Sigma_i^p` if there exists a `p`-time TM M and a polynomial `q` such that
`x in L \iff exists u_1 \in {0,1}^(q(|x|))forall u_2 \in {0,1}^(q(|x|)) ... Q_iu_i\in {0,1}^(q(|x|)) M(x, u_1 ..., u_i) = 1`.
where `Q_i` is either `exists` or `forall` depending on whether `i` is even or odd. We write `\Pi^p_i` for the languages which have the rolls of `exists` and `forall` interchanged in the above. Alternatively, it consists of the class of complements of `Sigma_i^p` languages. Lastly, we define`PH = cup_i Sigma_i^p`.

It should be reasonably clear that `Sigma_i^p subseteq Pi_(i+1)^p subseteq Sigma_(i+2)^p`. Also, `NP = Sigma_1^p` `coNP = Pi_1^p`

Theorem. `PH` is contained in `PSPACE`.

(Sketch). Use auxiliary tapes to cycle over possibilities for `u_i`'s.

Quiz

Which of the following are not ruled out by our current state of knowledge?

Nondeterministic `PSPACE` might equal `EXP`
`NL` might equal `EXP`.
`L` is strictly contained in `NL`.

On Collapse

Theorem.
(1) For every `i ge 1`, if `Sigma_i^p = Pi_i^p` then `PH = Sigma_i^p`.
(2) If `P = NP` then `PH =P`; that is, the hierarchy collapses to `P`.

Proof. (1) The idea is a basically a proof by induction on `j ge i`. We show `Sigma_j^p = Pi_i^p`. The base case `j=i` is by assumption. So consider some `j > i`. Let `L` be a language in `Pi_(j+1)^p`, it can be defined as a formula consisting of a `forall` followed by a `Sigma_j^p` formula. Rewrite the `Sigma_j^p` formula as a `Pi_i^p` formula using the induction hypothesis, then collapse the adjacent `forall` quantifiers to make the whole thing `Pi_i^p`. This shows `Pi_(j+1)^p` is in `Pi_i^p`. As we are assuming `Pi_i^p =Sigma_i^p`, we are closed under complement so we also get `Sigma_(j+1)^p = Pi_i^p`. (2) Follows by (1) noting that `P` is closed under complement.

On Collapse II

Theorem. If there exists a language `L` that is `PH`-complete, then there exists an `i` such that `PH = Sigma_i^p`.

Proof. Since `L in PH`, `L` is in `Sigma_i^p` for some `i`. Since `L` is PH-complete we can reduce any problem in `PH` to `L`. But every language `p`-reducible to `Sigma_i^p` is itself `Sigma_i^p`.

Alternating Time

It is possible to generalize NDTMs to give an alternative definition of PH.
An alternating TM is (ATM) is similar to an NDTM except every internal state is labelled as either `exists` or `forall`. We say `x` is accepted by an ATM `M` if when it is in an exists state then there is a one of the two transition function which when applied leads to the machine ending up in an accepting configuration; and if when it is in a universal state, both transition function when applied lead to an accepting configuration.

Definition. (Alternating Time) For every `T: NN -> NN`, we say that an alternating TM runs in time `T(n)` if for every input `x in {0,1}^star` and for every possible sequence of transition function choices, `M` halts after at most `T(|x|)` steps. We write `ATIME(T(n))` for the languages accepted by ATM's in `c cdot T(n)` steps.