Measure on Small Complexity Classes, with Applications for BPP
Martin Strauss Eric Allender
Department of Mathematics Department of Computer Science
Rutgers University Rutgers University
New Brunswick, NJ 08903 New Brunswick, NJ 08903
[email protected] [email protected]
Abstract not have prop erty Q has measure zero in E.
This sometimes presents itself as an easier
We present a notion of resource-boundedmea-
task than explicitly building a set in E having
sure for P and other subexponential-time classes.
prop erty Q. For example, Lutz and Mayor-
This generalization is based on Lutz's notion of
domo [LM] were able to use this version of
measure, but overcomes the limitations that cause
the probabilistic metho d to prove stronger
Lutz's de nitions to apply only to classes at least
results concerning the density of hard lan-
as large as E.Wepresent many of the basic prop-
guages than had b een proved previously.
erties of this measure, and use it to explore the
Certain interesting notions of intractability
class of sets that are hard for BPP.
can b e formulated only in the context of
Bennett and Gil l showed that almost al l sets are
resource-b ounded measure. Lutz and Juedes
hard for BPP; Lutz improved this from Lebesgue
[JL] studied \weakly complete" sets in E,
measuretomeasureon ESPACE. We use our
which are sets in E that are hard for more
measure to improve this stil l further, showing that
than a measure-zero subset of E.
for al l > 0, almost every set in E is hard
S
n
for BPP, where E = DTIME2 ; which
<
Many separation results and hierarchy the-
is the best that can be achieved without showing
orems in complexity theory have stronger
that BPP is properly containedinE. A number
versions provable in the context of resource-
of relatedresults are also obtained in this way.
p
b ounded measure. For example, the -
m
complete sets are a measure-zero subset of
cn
E [JL], and DTIME2 is a measure-zero
1 Intro duction
subset of E [L92]. However, for many other
separations, no corresp onding measure result
Resource-b ounded measure theory was intro-
is known. Often, as in [JL], when a measure-
duced in [L90a] and has proved to b e a useful
theoretic strengthening of a separation result
to ol in complexity theory. Among the reasons for
is p ossible, the pro of gives enlightening addi-
studying resource-b ounded measure, we mention
tional information concerning the complexity
the following:
classes involved.
The probabilistic metho d can b e applied only
Unproven but plausible hyp otheses suchas
if there is a meaningful notion of probability
\P 6= NP" or the stronger hyp othesis \the
on the underlying space. To show that a set
p olynomial hierarchy is in nite" provide
inEhaving some prop erty Q exists, it is suf-
useful information concerning complexity-
cient to show that the class of sets that do
theoretic prop ositions. That is, \X the
Research supp orted by NSF grant CCR-9204874. p olynomial hierarchy collapses" is taken as
evidence for :X . Lutz [L93a] argues that sketch Lutz's formulation. For more details, con-
the hyp othesis \NP is not a measure zero sult [L92].
O 1
n
subset of DTIME2 " a stronger hy- The intuition b ehind Lutz's formulation of
p othesis than P 6= NP is plausible and o ers
resource-b ounded measure is that a measure-zero
a great deal of explanatory p ower.
set is a set such that, for all k , the set can b e cov-
ered by a collection of intervals whose sizes sum