Resource-bounded measure explained
, resource bounded measure gives a method to classify the size of subsets of complexity classes.
For instance, computer scientists generally believe that the complexity class P (the set of all decision problems solvable in polynomial time) is not equal to the complexity class NP (the set of all decision problems checkable, but not necessarily solvable, in polynomial time). Since P is a subset of NP, this would mean that NP contains more problems than P. A stronger hypothesis than "P is not NP" is the statement "NP does not have p-measure 0". Here, p-measure is a generalization of Lebesgue measure to subsets of the complexity class E, in which P is contained. P is known to have p-measure 0, and so the hypothesis "NP does not have p-measure 0" would imply not only that NP and P are unequal, but that NP is, in a measure-theoretic sense, "much bigger than P".
Definition
is the set of all
infinite binary sequences. We can view a
real number in the
unit interval as an infinite binary sequence, by considering its binary expansion. We may also view a
language (a set of binary
strings) as an infinite binary sequence, by setting the
nth
bit of the sequence to 1 if and only if the
nth binary string (in
lexicographical order) is contained in the language. Thus, sets of real numbers in the unit interval and complexity classes (which are sets of languages) may both be viewed as sets of infinite binary sequences, and thus the techniques of
measure theory used to measure the size of sets of real numbers may be applied to measure complexity classes. However, since each computable complexity class contains only a
countable number of elements(because the number of computable languages is countable), each complexity class has
Lebesgue measure 0. Thus, to do measure theory inside of complexity classes, we must define an alternative
measure that works meaningfully on countable sets of infinite sequences. For this measure to be meaningful, it should reflect something about the underlying definition of each complexity class; namely, that they are defined by
computational problems that can be solved within a given resource bound.
The foundation of resource-bounded measure is Ville's formulation of martingales. A martingale is a function
such that, for all finite strings
w,
.(This is Ville's original definition of a martingale, later extended by
Joseph Leo Doob.) A martingale
d is said to
succeed on a sequence
if
\limsupn\toinftyd(S\upharpoonrightn)=infty,
where
is the first
n bits of
S. A martingale
succeeds on a set of sequences
if it succeeds on every sequence in
X.
Intuitively, a martingale is a gambler that starts with some finite amount of money (say, one dollar). It reads a sequence of bits indefinitely. After reading the finite prefix
, it bets some of its current money that the next bit will be a 0, and the remainder of its money that the next bit will be a 1. It doubles whatever money was placed on the bit that appears next, and it loses the money placed on the bit that did not appear. It must bet all of its money, but it may "bet nothing" by placing half of its money on each bit. For a martingale
d,
d(
w) represents the amount of money
d has after reading the string
w. Although the definition of a martingale has the martingale calculating how much money it will have, rather than calculating what bets to place, because of the constrained nature of the game, knowledge the values
d(
w),
d(
w0), and
d(
w1) suffices to calculate the bets that
d placed on 0 and 1 after seeing the string
w. The fact that the martingale is a function that takes as input the string seen so far means that the bets placed are solely a function of the bits already read; no other information may affect the bets (other information being the so-called
filtration in the
generalized theory of martingales).
The key result relating measure to martingales is Ville's observation that a set
has Lebesgue measure 0 if and only if there is a martingale that succeeds on
X. Thus, we can define a measure 0 set to be one for which there exists a martingale that succeeds on all elements of the set.
To extend this type of measure to complexity classes, Lutz considered restricting the computational power of the martingale. For instance, if instead of allowing any martingale, we require the martingale to be polynomial-time computable, then we obtain a definition of p-measure: a set of sequences has p-measure 0 if there is a polynomial-time computable martingale that succeeds on the set. We define a set to have p-measure 1 if its complement has p-measure 0. For example, proving the above-mentioned conjecture, that NP does not have p-measure 0, amounts to proving that no polynomial-time martingale succeeds on all of NP.
Almost complete
A problem is almost complete for a complexity class C if it is in C and "many" other problems in C reduce to it. More specifically, the subset of problems of C which reduce to the problem is a measure one set, in terms of the resource bounded measure. This is a weaker requirement than the problem being complete for the class.
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