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The Quantitative Structure of Exponential Time Jack H Computer Science Technical Reports Computer Science 1996 The Quantitative Structure of Exponential Time Jack H. Lutz Follow this and additional works at: http://lib.dr.iastate.edu/cs_techreports Part of the Theory and Algorithms Commons Recommended Citation Lutz, Jack H., "The Quantitative Structure of Exponential Time" (1996). Computer Science Technical Reports. 115. http://lib.dr.iastate.edu/cs_techreports/115 This Article is brought to you for free and open access by the Computer Science at Iowa State University Digital Repository. It has been accepted for inclusion in Computer Science Technical Reports by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. The Quantitative Structure of Exponential Time Abstract Department of Computer Science Iowa State University Ames, Iowa 50010 Recent results on the internal, measure-theoretic structure of the exponential time complexity classes linear polynomial E = DTIME(2 ) and E = DTIME(2 ) 2 are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed. Disciplines Theory and Algorithms This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/cs_techreports/115 This is page 1 Printer: Opaque this The Quantitative Structure of Exp onential Time 1 Jack H. Lutz ABSTRACT Recent results on the internal, measure-theoretic structure of the exp onential time complexity classes E and EXP are surveyed. The mea- sure structure of these classes is seen to interact in informativeways with bi-immunity, complexity cores, p olynomial-time reductions, completeness, circuit-size complexity, Kolmogorov complexity, natural pro ofs, pseudoran- dom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. 1 Intro duction In the past veyears, new developments in resource-b ounded measure have op ened the way for a systematic investigation of the internal, measure- theoretic structure of the exp onential time complexity classes E and EXP. The investigation is very far from complete, but it has already yielded a number of interesting insights and results. This pap er surveys the motiva- tions, ideas, and results of the earliest phase of the investigation, i.e., the part completed by mid-1995. It should b e emphasized that the material surveyed here is the work of several investigators. The ongoing e orts of these investigators, together with the e orts of more recent participants, virtually guarantee that this survey will be incomplete by the time it app ears. At the time of this writing, there are already several pap ers in review and manuscripts in cir- culation that app ear to extend the body of knowledge presented here. Nevertheless, it is to b e hop ed that the \organized snapshot" provided by this survey will provide context and motivation for future research. There are three reasons for our interest in the complexity classes E and EXP. i E and EXP have rich, apparently well-b ehaved, internal structures. These structures have manyinteracting facets, including a varietyof 1 Departmentof Computer Science, Iowa State University, Ames, IA 50011. E-mail: [email protected]. This research was supp orted in part by National Science Foundation Grant CCR-9157382, with matching funds from Ro ckwell International, Microware Systems Corp oration, and Amo co Foundation. 1. The Quantitative Structure of Exp onential Time 2 reducibilities [LLS75], complete languages under these reducibilities [SC79, Wat87b], measure structure [Lut92], and category structure [Lut90,Fen91,Fen95]. ii EXP is the smallest deterministic time complexity class known to contain NP. It also contains PSPACE, and hence the p olynomial-time hierarchy and many other classes of interest in complexity theory.E is a prop er subset of EXP, but it contains P and \the essential part of NP" [Wat87b], i.e., many NP-complete problems. iii E and EXP have b een proven to contain intractable problems [HS65]. From the standp oint of complexity theory, this existence of intractabil- ityisavaluable resource. This is b ecause, in practice, a pro of that a sp eci c language A is intractable pro ceeds by inferring the in- tractabilityof A from the intractability of some language B chosen or constructed for this purp ose. Taken together, i, ii, and iii suggest E and EXP as appropriate spaces in whichtoinvestigate emb ed problems involving NP, PH, PSPACE, and other classes in this range. Until recently, the issues addressed by research on the structure of com- plexity classes have b een largely qualitative rather than quantitative. In- deed, the intro duction to [Sch86b] o ered \qualitative" as a synonym for \structural." This seemed to be an inevitable asp ect of the sub ject. A problem is, or is not, complete for a complexity class. One complexity class is, or is not, contained in another. This was unfortunate, since the ob jec- tive of complexity theory is a quantitative theory of computation. However, since the sets of interest are all countably in nite, there app eared to b e no p ossibility of making quantitativeversions of these judgments. The main ob jective of the work surveyed here is to remedy this situation. Supp ose that a language A f0; 1g is chosen by a random exp eri- ment in which an indep endent toss of a fair coin is used to decide whether each string is in A. Then classical Leb esgue measure theory describ ed in [Hal50, Oxt80], for example identi es certain measurable sets of lan- guages also called events and assigns to each measurable set X a mea- sure X , which is the probability that A 2 X in this exp eriment. A set X of languages is then small in the sense of measure if it has measure 0. E ective measure theory, whichsays what it means for a set of decidable languages to have measure 0 as a subset of the set of all such languages, has b een investigated byFreidzon [Fre72], Mehlhorn [Meh74], and others. The resource-bounded measure theory intro duced by Lutz [Lut92, Lutb] has the classical and e ective theories as sp ecial cases, but also de nes measurabil- ity and measure for subsets of many complexity classes. The smal l subsets of such a complexity class are then the measure 0 sets; the large subsets are the measure 1 sets complements of measure 0 sets. We say that almost 1. The Quantitative Structure of Exp onential Time 3 every language in a complexity class C has a given prop ertyif the set of languages in C exhibiting the prop erty is a measure 1 subset of C . Thus, resource-b ounded measure provides a means of investigating the sizes of various subsets of E and EXP. This is a priori a hop eful devel- opment, b oth b ecause quantitative results are more informative and be- cause Leb esgue measure has b een so useful in analysis, probability, and mathematical physics. However, much of the ongoing motivation for this work arises not from a priori considerations, but rather from the fact that resource-b ounded measure turns out to interact informatively with many prop erties of interest in computational complexity. Suchinteractions sur- veyed in this pap er involve bi-immunity section 4, complexity cores sec- tions 5, 7, and 8, the structure of E and EXP under p olynomial-time reductions sections 6,7, and 8, circuit-size complexity and time-b ounded Kolmogorov complexity section 9, natural pro ofs and pseudorandom gen- erators section 9, the densityof hard languages section 11, and other prop erties that had b een extensively studied prior to the advent of resource- b ounded measure. It is to b e hop ed that sustained, systematic investigation along these lines will lead to a detailed, quantitative understanding of E and EXP. From the standp oint of classical mathematics and recursion theory, classes likeP,NP, PH, and PSPACE are all negligibly small, hence dicult to dis- tinguish by quantitative structural means. From the standp oint of E and EXP, matters maybevery di erent. If EXP is, indeed, the smallest deter- ministic time class containing NP, then there maywell b e a natural \notion of smallness" for subsets of EXP such that P is a small subset of EXP, but NP is not. Similarly, it may be that P is a small subset of E, but that NP \ E is not. It is p ossible that resource-b ounded measure already provides such a notion of smallness. It is certainly the case that P has measure 0 in E and EXP [Lut92]. In section 12 we discuss the reasonableness and known consequences of the hyp othesis that NP is not small in this sense. This is a very strong hyp othesis that app ears to havemuch more explanatory p ower than traditional, qualitativehyp otheses, suchasP6= NP or the separation of the p olynomial-time hierarchy. Only further investigation will determine whether this hyp othesis is reasonable. 2 Preliminaries In this pap er, [[ ]] denotes the Boolean value of the condition , i.e., [[ ]]= if then 1 else 0. All languages here are sets of binary strings, i.e., sets A f0; 1g . We 1 identify each language A with its characteristic sequence 2 f0; 1g A 1. The Quantitative Structure of Exp onential Time 4 de ned by =[[s 2 A]][[s 2 A]][[s 2 A]]:::; A 0 1 2 where s = , s = 0, s = 1, s = 00; ::: is the standard enumeration 0 1 2 3 1 of f0; 1g .
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