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PSPACE
PH
P
P
1
NP P
1
co-NP
P
2P
2
Figure 1: The polynomial hierarchy.
3
We have already hinted that the levels of the polynomial
hierarchy correspond to k-
alternating Turing machines. The next theorem makes this
correspondence explicit, and
also gives us a third equivalent characterization.
Theorem 2.1 For any language A, the following are
equivalent:
1. A
Pk.
2. A is decided in polynomial time by a k-alternating Turing
machine that starts in an
existential state.
3. There exists a language B
P and a polynomial p such that for all x, x
A if and
only if
y1 : y1
p
x
y2 : y2
p
x
Qyk : yk
p
x
x
y1
yk
B
where the quantifier Q is
if k is odd,
if k is even.
In Section 8 of Chapter 28, we discussed some of the startling
consequences that would
follow ifNP were included in P/poly, but observed that this
inclusion was not known
to imply P NP. It is known, however, that ifNP
P/poly, then PH collapses to its
second level, P2 [Karp and Lipton, 1982]. It is generally
considered likely that PH does
not collapse to any level, and hence that all of its levels are
distinct. Hence this result is
considered strong evidence that NP is not a subset ofP/poly.Also
inside the polynomial hierarchy is the important class BPP of
problems that can
be solved efficiently and reliably by probabilistic algorithms,
to which we now turn.
3 Probabilistic Complexity Classes
Since the 1970s, with the development of randomized algorithms
for computational prob-
lems (see Chapter 15), complexity theorists have placed
randomized algorithms on a firm
intellectual foundation. In this section, we outline some basic
concepts in this area.A probabilistic Turing machine M can be
formalized as a nondeterministic Turing
machine with exactly two choices at each step. During a
computation, M chooses each
possible next step with independent probability 1
2. Intuitively, at each step, M flips a
fair coin to decide what to do next. The probability of a
computation path of t steps is
1
2t. The probability that M accepts an input string x, denoted by
pM
x
, is the sum of the
probabilities of the accepting computation paths.
Throughout this section, we consider only machines whose time
complexity t
n
is
time-constructible. Without loss of generality, we may assume
that every computation path
of such a machine halts in exactly t steps.Let A be a language.
A probabilistic Turing machine M decides A with
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