The polynomial hierarchy and alternations

[S]ynthesizing circuits is exceedingly difficulty. It is even more difficult to show that a circuit foundin this way is themosteconomical one to realize a function. The difficulty springs from the large number of essentially different networks available.

– Claude Shannon, 1949 We have already encountered some ways of “capturing” the essence of families of computational problems by showing that they are complete for some natural complexity class. This chapter continues this process by studying another family of natural prob-lems (including one mentioned in Shannon’s quote at the begginning of this chapter) whose essence is not capturedby nondeterminism alone. The correct complexity class that captures these problems is thepolynomial hierarchy, denotedPH, which is a gen-eralization ofP,NPandcoNP. It consists of an infinite number of subclasses (called levels) each of which is important in its own right. These subclasses are conjecturedto be distinct, and this conjecture is a stronger form ofP=NP. This conjecture tends to crop up (sometimes unexpectedly) in many complexity theoretic investigations, including in Chapters6,7, and17of this book.

In this chapter we provide three equivalent definitions of the polynomial hierarchy:

1. In Section5.2we define the polynomial hierarchy as the set of languages defined via polynomial-time predicates combined with a constant number of alternating forall (∀) andexists (∃) quantifiers, generalizing the definitions ofNPandcoNPfrom Chapter2.

2. In Section 5.3 we show an equivalent characterization of the polynomial hierarchy usingalternatingTuring machines, that are a generalization of nondeterministic Turing machines defined in Section2.1.2.

3. In Section5.5we show the polynomial hierarchy can also be defined usingoracleTuring machines (Section3.4).

A fourth characterization using uniform families of circuits will be given in Chapter6. In Section 5.4, we use the different characterizations of the polynomial hierarchy to show an interesting result:SATcannot be solvedusing simultaneously lin-ear time andlogarithmic space. This represents a frontier of current approaches toP versusNP.

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5.1 THE CLASS^P₂

To motivate the study ofPH, we focus on some computational problems that seem to not be capturedbyNP-completeness.

As warmup, let’s recall the followingNPproblemINDSET(see Example2.2), for which wedohave a short certificate of membership:

INDSET=

G,k: graphGhas an independent set of size≥k

Consider a slight modification to this problem, namely, determining the largest independent set in a graph (phrased as a decision problem):

EXACT INDSET=

G,k: the largest independent set inGhas size exactlyk Now there seems to be no short certificate for membership:G,k ∈EXACT INDSET iffthere existsan independent set of sizekinGandevery otherindependent set has size at mostk.

Similarly, consider the problem referred to in Shannon’s quote, namely, to deter-mine the smallest Boolean formulas equivalent to a given formula. For convenience, we state it as a decision problem.

MIN-EQ-DNF=

ϕ,k:∃DNF formulaψof size≤k that is equivalent to the DNF formulaϕ

where a DNF formula is a Boolean formula that is an OR of ANDs andwe say that two formulas are equivalent if they agree on all possible assignments. The complement of this language is referedto in Shannon’s quote, except Shannon is interestedmore generally in small circuits rather than just DNF formulas.

MIN-EQ-DNF=

ϕ,k: ∀DNF formulaψof size≤k

∃assignmentus.t.ϕ(u)=ψ(u)

Again, there is no obvious notion of a certificate of membership forMIN-EQ-DNF.

Thus to capture languages such asEXACT INDSETandMIN-EQ-DNF, we seem to need to allow not only a single “exists” quantifier (as in Definition2.1of NP) or “for all”

quantifier (as in Definition2.20ofcoNP) but a combination of both quantifiers. This motivates the following definition.

Definition 5.1 The class ^p₂ is the set of all languages L for which there exists a polynomial-time TMManda polynomialqsuch that

x∈L⇔ ∃u∈ {0, 1}^q(|x|)∀v∈ {0, 1}^q(|x|)M(x,u,v)=1

for everyx∈ {0, 1}^∗. #

Note that^p₂contains both the classesNPandcoNP.

5.2. The Polynomial Hierarchy 97

EXAMPLE 5.2

The languageEXACT INDSETabove is in^p₂, since, as we notedpreviously, a pairG,k is inEXACT INDSETiffthere existsa size-ksubsetSofG’s vertices such thatfor every S that is a(K+1)-sizedsubset,Sis an independent set inGandSis not an independent set inG. (Exercise5.9shows a finer placement ofEXACT INDSET.)

The languageMIN-EQ-DNFis also in^p₂, since a pairϕ,kis inMIN-EQ-DNFiff there existsa DNF formulaψsuch thatfor everyassignmentu,ϕ(u)=ψ(u). It is known to be^p₂-complete [Uma98].

5.2 THE POLYNOMIAL HIERARCHY

The definition of the polynomial hierarchy generalizes those ofNP,coNP, and^p₂. This class consists of every language that can be defined via a combination of a polynomial-time computable predicate and a constant number of∀/∃quantifiers.

Definition 5.3 (Polynomial hierarchy)

For i ≥ 1, a language L is in^p_i if there exists a polynomial-time TM M anda polynomialqsuch that

x∈L⇔ ∃u₁∈ {0, 1}^q(|x|)∀u₂∈ {0, 1}^q(|x|)· · ·Q_iu_i∈ {0, 1}^q(|x|)M(x,u₁,. . .,u_i)=1 whereQ_idenotes∀or∃depending on whetheriis even or odd, respectively.

Thepolynomial hierarchyis the setPH= ∪i^p_i.

Note that^p₁ =NP. For everyi, define^p_i =co^p_i =

L:L∈^p_i

. Thus^p₁ = coNP. Also, for everyi, note that^p_i ⊆^p_i+1⊆^p_i+2, andhencePH= ∪i>0^p_i.

5.2.1 Properties of the polynomial hierarchy

We believe thatP=NPandNP=coNP. An appealing generalization of these con-jectures is that for everyi,^p_i is strictly containedin^p_i+1. This conjecture is usedoften in complexity theory. It is often statedas“the polynomial hierarchy does not collapse,”

where the polynomial hierarchy is saidto collapse if there is someisuch that^p_i =^p_i+1. As we will see below, this wouldimply^p_i = ∪j≥1^p_j = PH. In this case, we say that the polynomial hierarchycollapses to the ithlevel. The smalleriis, the weaker—

andhence more believable—it is to conjecture thatPHdoes not collapse to the ith level.

Theorem 5.4

1. For every i≥1, if^p_i =^p_i thenPH=^p_i; that is, the hierarchy collapses to the ith level.

2. IfP=NPthenPH=P; that is, the hierarchy collapses toP.

Proof: We do the secondpart; the first part is similar andis left as Exercise 5.12.

AssumingP=NP, we prove by induction onithat^p_i,^p_i ⊆P. Clearly this is true for i =1 by assumption since^p₁ =NPand^P₁ = coNP. We assume it is true fori−1 andprove that^p_i ⊆P. Since^p_i consists of complements of languages in^p_i andPis closed under under complementation, it would also then follow that^p_i ⊆P.

Let L ∈ ^p_i. By definition, there is a polynomial-time Turing machineM anda polynomialqsuch that

x∈L⇔ ∃u1∈ {0, 1}^q(|x|)∀u2∈ {0, 1}^q(|x|)· · ·Qiui∈ {0, 1}^q(|x|)M(x,u1,. . .,ui)=1 (5.1) whereQ_iis∃/∀as in Definition5.3. Define the languageLas follows:

x,u1 ∈L⇔ ∀u2∈ {0, 1}^q(|x|)· · ·Qiui ∈ {0, 1}^q(|x|)M(x,u1,u2,. . .,ui)=1 Clearly,L∈^p_i−1andso under our assumptionLis inP. This implies that there is a polynomial-time TMMcomputingL. PluggingMin (5.1), we get

x∈L⇔ ∃u₁∈ {0, 1}^q(|x|)M(x,u₁)=1 .

But this meansL∈NPandhence under our assumption thatP=NP,L∈P.

5.2.2 Complete problems for levels ofPH

Recall that a languageB reducesto a languageCvia a polynomial-time Karp reduction, denoted byB≤p C, if there is a polynomial-time computable functionf : {0, 1}^∗ → {0, 1}^∗ such that x ∈ B ⇔ f(x) ∈ Cfor every x (see Definition2.7). We say that a languageL is^p_i-complete if L ∈ ^p_i andfor everyL ∈ ^p_i, L ≤p L. We define ^p_i-completeness andPH-completeness in the same way. In this section, we show that for everyi∈N, both^p_i and^p_i have complete problems. By contrast, the polynomial hierarchy itself is believednot to have a complete problem, as is shown by the following simple claim.

Claim 5.5 If there exists a language L that isPH-complete, then there exists an i such that