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Measure on Small Complexity Classes, with Applications for BPP

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Measure on Small Complexity Classes, with Applications for BPP 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 k Unfortunately, the formulation of resource- to 2 , where the \intervals" are easy to compute b ounded measure studied by Lutz and others ap- in some sense. plies only to complexity classes at least as large To make this more precise, x an enumeration 1 as E, which greatly limits the range of questions fs g of all words, and let p osx denote the i i=0 that can b e explored in this framework. The goal p osition of x in this ordering. A language L is of the work presented here is to remedy this situa- identi ed with its characteristic sequence: an in- tion by providing a meaningful notion of measure nite sequence ! = ha i with a =1i s 2 L: i i i for P and other sub exp onential time classes, and th Let w [i::j ] denote the string of the i through demonstrate its usefulness. th j bits of w: Let y v w indicate that y is an Section 2 presents Lutz's de nition of resource- initial substring of w . b ounded measure, explains why it do es not ex- The sets we will try to measure are sets of lan- tend in anyobvious waytoprovide a measure guages i.e., sets of sequences, and the analogs of on smaller classes, and then shows how these ob- \intervals" in this context are the sets of the form stacles were overcome to provide a measure on P C = f! w xg, which is called the \cylinder at x and other classes. This section concludes with a x". Thus C is the set of languages with the jxj x discussion of how natural and robust our de ni- initial words according to x and the other words tion of measure is, and a comparison of it to the arbitrary. work of Mayordomo [M,M2], where a measure on The next few de nitions describ e what it PSPACE is de ned. means in this context for a set to b e covered, for Section 3 presents our pro of that sets that are all k ,by a collection of intervals whose sizes sum hard for BPP are abundantinE for every >0, k to 2 . as well as related results for PSPACE. Section 4 presents our results concerning pseu- De nition 1 A density-system is a function dorandom sources for BPP. d f0; 1g ! [0; 1; where{ i ;;i r 1 Section 5 includes a discussion of whether the notion of measure intro duced in this pap er rel- 1. The subscripts are natural numbers. ativizes in a meaningful way. That is, we dis- 2. The argument is consideredapartial charac- cuss the question of whether nonrelativizing re- teristic sequence of a language. sults might b e obtained using this notion of mea- sure. d w 0+d w 1 3. d w = : 2 For Lutz's measure, this de nition is equiva- 2 De ning Resource-Bounded Mea- lent to the de nition in [L92]; see also Subsection sure 2.5. A density-system is called an n-DS accord- ing to the numb er of subscripts. A 0-DS is a 2.1 Background Concerning Measure density function. Note there is no mention here of resource b ounds. In order to present our new de nitions of A density function covers X if X resource-b ounded measure and explain the ob- S C : A nul l cover of X is a 1-DS such stacles to extending Lutz's de nitions to sub ex- w fw j dw 1 g k p onential complexity classes, it is necessary to that for all k , d covers X and d 2 : A k k 2 set of languages has Leb esgue-measure zero i it mo dify Lutz's de nitions, merely using smaller hasanull cover. Of course, all recursive complex- resource b ounds C . For instance, to de ne a ity classes are countable and thus have Leb esgue- measure on P, one would consider density func- O 1 measure zero. Thus it was necessary for Lutz to tions computed in DTIMElog n. This fails restrict the class of density systems in order to to work, b ecause de ne measures on complexity classes. The usual convention see, e.g., [AJ] for 1 In de ning notions of measure for E, ES- having sublinear-time Turing machines com- PACE, and other large complexity classes, Lutz pute some function f is to have them rec- de nes a -measure for various classes of func- ognize the language fx; i; b : bit i of f xis tions . Roughly, the sets with -measure zero bg.However it is easy to see that the class are the sets that havea null cover that can b e of functions computed in this wayby, for approximated very closely by a function in . instance, p olylog-time-b ounded machines, is Ifischosen to b e the p olynomial-time com- not closed under addition and subtraction, putable functions, Lutz obtains a measure on E; which seems to b e necessary for many con- if is chosen to b e the functions computable structions. Similar problems arise when one O 1 log n in space 2 , then he obtains a measure on uses the usual binary notation or scienti c O 1 n DSPACE2 , and in general functions com- notation for the numeric values of the den- puted in space or time ff log n:f 2Cgfor a sity functions. Our solution is to have the suciently nice class C of functions yield a mea- run time b ound the length of the output, and sure on space or time ff n: f 2Cg. to express numb ers as the di erence of two In order to justify his claim that the notion formal sums of p owers of two, which allows he de nes do es, in fact, constitute a reasonable us to p erform the necessary arithmetic op er- notion of measure on a complexity class X , Lutz ations in the restricted time available.
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