DOCSLIB.ORG

The Complexity and Distribution of Computationally Useful Problems David W

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Complexity and Distribution of Computationally Useful Problems David W Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1994 The complexity and distribution of computationally useful problems David W. Juedes Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Computer Sciences Commons, and the Mathematics Commons Recommended Citation Juedes, David W., "The ompc lexity and distribution of computationally useful problems " (1994). Retrospective Theses and Dissertations. 10613. https://lib.dr.iastate.edu/rtd/10613 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. , UMl MICROFILMED 1994 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMl films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms international A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9424227 The complexity and distribution of computationally useful problems Juedes, David W., Ph.D. Iowa State University, 1994 UMI 300 N. ZeebRcl. Ann Arbor, MI 48106 The complexity and distribution of computationally useful problems by David W. Juedes A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department: Computer Science Major: Computer Science Approved; Signature was redacted for privacy. In Cliarge of Major ^ Signature was redacted for privacy. For the Major Departmipnt Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1994 ii TABLE OF CONTENTS ACKNOWLEDGEMENTS v CHAPTER 1. INTRODUCTION 1 1.1 Complexity in Nature 2 1.2 Accessible Information and Computational Usefulness 5 1.3 Overview 8 CHAPTER 2. FUNDAMENTAL CONCEPTS FROM COMPLEX­ ITY THEORY 13 2.1 Basic Notation and Terminology 13 2.2 Models of Computation and Complexity Classes 15 2.3 Reducibilities and Hardness 18 2.4 Measure and Category 20 2.4.1 Resource-bounded Measure 21 2.4.2 Baire Category 29 2.5 Algorithmic Information and Randomness 33 2.5.1 Resource-bounded Kolmgorov Complexity 34 2.5.2 Algorithmic Information Theory 35 2.5.3 Algorithmic Randomness 46 iii CHAPTER 3. COMPUTATIONAL DEPTH 48 3.1 Strong Computational Depth 49 3.2 Weak Computational Depth 64 CHAPTER 4. THE STRUCTURE OF EXPONENTIAL TIME . 67 4.1 The Distribution of Uniform Complexity 68 4.2 The Complexity of Weakly Hard Problems: Lower Bounds 74 4.3 The Measure of Degrees 76 4.4 The Complexity of Hard Problems: Upper Bounds 82 CHAPTER 5. THE STRUCTURE OF EXPONENTIAL SPACE . 87 5.1 The Distribution of Nonuniform Complexity 88 5.1.1 Kolmogorov Complexity 89 5.1.2 Nonuniform Complexity Cores 95 5.1.3 Incompressibility by Nonuniform Reductions 98 5.2 Nonuniform Complexity of Weakly Hard Problems: Lower Bounds . 103 5.3 Nonuniform Complexity of Hard Problems: Upper Bounds 109 5.4 Distribution of Hardness and Measure of Degrees 115 CHAPTER 6. WEAKLY COMPLETE PROBLEMS 120 6.1 Martingale Diagonalization 121 6.2 Weakly Complete Problems are Not Rare 130 CHAPTER 7. CONCLUSION 137 BIBLIOGRAPHY 140 iv LIST OF FIGURES Figure 1.1: The dependency relation among the chapters 12 Figure 2.1: A self-delimiting Turing machine 36 Figure 2.2: The Turing Machine M used in the proof of Lemma 2.10. 43 Figure 3.1: The Turing machine M' used in the proof of Lemma 3.5. 57 Figure 3.2: A classification of binary sequences. RAND has measure 1, while wkDEEP — strDEEP is comeager 66 Figure 5.1: An algorithm that computes X/i=„ 97 Figure 5.2: The algorithm for Ma in the proof of Theorem 5.13 108 Figure 5.3: The machine M Ill Figure 5.4: The construction of an <^-complete language with high KS a.e 115 V ACKNOWLEDGEMENTS First and foremost I wish to thank my advisor, Jack Lutz, for his constant pa­ tience and support, and his many timely suggestions. I am forever grateful to him for introducing me to structural complexity theory and for always being a good friend. I also wish to thank my many friends and colleagues at Iowa State University who made my years there so interesting and enjoyable, especially Amy Ackerberg, Josef Breutzmann, Ann Grimm, Nicole Lang, James Lathrop, Kesig Lee, David Martin, Susan Mayer (Stapleton), Sunil Nair, Linda Null, Hitendra Patel, Susan Wahls, Tim­ othy Wahls, and Jihoon Yang. I also wish to thank my Ph.D. committee, Daniel Ashlock, David Fernandez-Baca, Jonathan Smith, and Giora Slutzki, for their input and suggestions on my work. Finally, I wish to thank my parents, brother, and sisters for their support during my long years at school. I will never forget the life and love that they have given me. I gratefully acknowledge Applied Communications, Inc. and the Arthur Collins Foundation for providing support for my graduate education through their generous scholarships. My thesis research was also supported in part by National Science Foun­ dation grants CCR-8809238 and CCR-9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation. Much of the work contained here was performed jointly with Jack Lutz and vi James Lathrop. I am eternally grateful for their contributions to this work and for their permission to include this work here. 1 CHAPTER 1. INTRODUCTION What do a sequence of DNA, an encyclopedia, and the UNIX operating system have in common? All three of these objects contain large amounts of useful informa­ tion, organized in such a fashion as to be readily available to biological, intellectual, or computational processes. Similarly, the solutions of certain natural decision prob­ lems, such as the halting problem or the boolean satisfiability problem, contain large amounts of useful information about computation that is readily available to effi­ cient computational processes. Here we search for the common thread among these objects. This dissertation investigates classes of problems that are computationally useful. In particular, we investigate the complexity and distribution of problems that contain large amounts of useful information about computation. The main results of this dissertation are of the following three general types. (1) Useful problems contain highly organized information. (2) Very useful problems are so highly organized that they are unusually simple and hence rare. (3) Useful problems are, as a whole, not rare and thus are not necessarily simple. Our main result of type (1) has its roots in the search for a measure that accu­ 2 rately reflects the complexity of natural objects. Our results of type (2) and (3) are motivated by the study of intractability in structural complexity theory. We first ex­ amine our main result of type (1). We begin with a brief digression on the complexity of objects in nature. 1.1 Complexity in Nature The spectrum of complexity in nature ranges from the exceedingly simple to the exceedingly complex. On the low end of the spectrum lie very simple objects: perfect crystals, pure liquids, gases, etc. On the high end of the spectrum lie very complex objects: DNA, human beings, societies, etc. In most cases, objects at the high end of the complexity spectrum are the result of the action, over time, of some complex adaptive system. The best example of this phenomenon is the amazing variety of life on earth. Each organism on this planet, including ourselves, is the result of the action of evolution over a span of billions of years. In the light of these phenomena, we are left wonder about the origin, nature, and driving principles of this vague notion of natural complexity. Recently [82, 90], there has been a concerted effort among scientists from various disciplines to uncover the nature and origins of complexity. Part of that investigation has concentrated on the complexity of the binary representations of physical and mathematical objects. This part of the investigation has its origins in algorithmic information theory. Algorithmic information theory provides a rigorous, quantitative measure of the information content of individual binary strings and binary sequences. Briefly, the algorithmic information content of an individual binary string is the length of the 3 shortest complete description of that string. (Algorithmic information theory was developed through the work of SolomonofF [87], Kolmogorov [44, 45, 46], Chaitin [18,19,20, 22], Martin-Lof [70, 71], Levin [48, 49,50, 51,52, 53,92], Schnorr [83], Gacs [28], Shen' [84, 85], and many others.
Load more

Recommended publications
  • Axiomatizing Resource Bounds for Measure
    Axiomatizing Resource Bounds for Measure ⋆ Xiaoyang Gu1 ⋆⋆, Jack H. Lutz2 ⋆ ⋆ ⋆, Satyadev Nandakumar3, and James S. Royer4 † 1 LinkedIn Corporation, Mountain View, CA 94043, U. S. A. [email protected] 2 Iowa State University Ames, IA 50011, U. S. A. [email protected] 3 I. I. T. Kanpur Kanpur, UP 280 016 India. [email protected] 4 Syracuse University Syracuse, New York 13244, U. S. A. [email protected] Abstract. Resource-bounded measure is a generalization of classical Lebesgue measure that is useful in computational complexity. The central parameter of resource-bounded measure is the resource bound ∆, which is a class of functions. When ∆ is unrestricted, i.e., contains all functions with the specified domains and codomains, resource-bounded measure coincides with classical Lebesgue measure. On the other hand, when ∆ contains functions satisfying some complexity constraint, resource-bounded measure imposes internal measure structure on a corresponding complexity class. Most applications of resource-bounded measure use only the “measure-zero/measure-one fragment” of the theory. For this fragment, ∆ can be taken to be a class of type-one functions (e.g., from strings to rationals). However, in the full theory of resource-bounded measurability and measure, the resource bound ∆ also contains type-two functionals. To date, both the full theory and its zero-one fragment have been developed in terms of a list of example resource bounds chosen for their apparent utility. This paper replaces this list-of-examples approach with a careful investigation of the conditions that suffice for a class ∆ to be a resource bound.
    [Show full text]
  • Contributions to the Study of Resource-Bounded Measure
    Contributions to the Study of ResourceBounded Measure Elvira Mayordomo Camara Barcelona abril de Contributions to the Study of ResourceBounded Measure Tesis do ctoral presentada en el Departament de Llenguatges i Sistemes Informatics de la Universitat Politecnica de Catalunya para optar al grado de Do ctora en Ciencias Matematicas por Elvira Mayordomo Camara Dirigida p or el do ctor Jose Luis Balcazar Navarro Barcelona abril de This dissertation was presented on June st in the Departament de Llenguatges i Sistemes Informatics Universitat Politecnica de Catalunya The committee was formed by the following members Prof Ronald V Bo ok University of California at Santa Barbara Prof Ricard Gavalda Universitat Politecnica de Catalunya Prof Mario Ro drguez Universidad Complutense de Madrid Prof Marta Sanz Universitat Central de Barcelona Prof Jacob o Toran Universitat Politecnica de Catalunya Acknowledgements Iwant to thank sp ecially two p eople Jose Luis Balcazar and Jack Lutz Without their help and encouragement this dissertation would never have b een written Jose Luis Balcazar was my advisor and taughtmemostofwhatIknow ab out Structural Complexity showing me with his example how to do research I visited Jack Lutz in Iowa State University in several o ccasions from His patience and enthusiasm in explaining to me the intuition behind his theory and his numerous ideas were the starting p ointofmostoftheresults in this dissertation Several coauthors have collab orated in the publications included in the dierent chapters Jack Lutz Steven
    [Show full text]
  • Resource Bounded Randomness and Its Applications
    Resource Bounded Randomness and Its Applications D. M. Stull 1 Introduction Martin-L¨ofgave the first meaningful definition of a random infinite binary se- quence [51]. From a classical standpoint, such a notion seems meaningless, as every sequence has probability zero. Martin-L¨of's insight was to use com- putability theory to give an effective version of measure theory. With this effectivization, a sequence A is (Martin-L¨of) random if the singleton fAg is not of effective measure zero. The field of algorithmic randomness has since grown to include many different notions of randomness, all making essential use of computability [18, 58]. With the prominence of complexity theory in computation, a natural step is to impose resource bounds on the computation in algorithmic randomness. Re- source bounded randomness studies the different notions of what it means for a sequence to be \random" relative to a resource bounded observer. Schnorr gave the first definition of resource bounded randomness by imposing time bounds on martingales [65]. However, this work did not have an immediate impact. Instead, resource bounded randomness was largely unexplored until Lutz's, in- dependent, development of resource bounded measure [46], a resource bounded effectivization of Lebesgue measure theory. As noted by Ambos-Spies, et al. [5], resource bounded measure implicitly redefines Schnorr's notion of resource bounded randomness. Lutz showed that resource bounded measure, and there- fore resource bounded randomness, is a valuable tool in complexity theory. This application ignited interest in resource bounded randomness, resulting in signif- icant growth in the area. Nevertheless, many fundamental problems remain to be explored.
    [Show full text]
  • Resource Bounded Randomness and Computational Complexity
    Theoretical Computer Science 237 (2000) 33–55 www.elsevier.com/locate/tcs Resource bounded randomness and computational complexity Yongge Wang Department of Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee, P.O. Box 784, Milwaukee, WI 53201, USA Received February 1996; revised February 1998 Communicated by J. DÃaz Abstract The following is a survey of resource bounded randomness concepts and their relations to each other. Further, we introduce several new resource bounded randomness concepts corresponding to the classical randomness concepts, and show that the notion of polynomial time bounded Ko randomness is independent of the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness. Lutz has conjectured that, for a given time or space bound, the corresponding resource bounded Lutz randomness is a proper reÿnement of resource bounded Schnorr random- ness. This conjecture is answered for the case of polynomial time bound. Moreover, we will show that polynomial time bounded Schnorr randomness is a proper reÿnement of polynomial time bounded Kurtz randomness. In contrast to this result, we show that the notions of poly- nomial time bounded Lutz, Schnorr and Kurtz randomness coincide in the case of recursive sets, thus it suces to study the notion of resource bounded Lutz randomness in the context of complexity theory. The stochastic properties of resource bounded random sequences will be discussed in detail. Schnorr has already shown that the law of large numbers holds for p-random sequences. We will show that another important law in probability theory, the law of the iter- ated logarithm, holds for p-random sequences too.
    [Show full text]
  • 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.
    [Show full text]
  • Nonuniform Reductions and NP-Completeness
    Nonuniform Reductions and NP-Completeness John M. Hitchcock and Hadi Shafei Department of Computer Science University of Wyoming Abstract Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP-completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP. Under various hypotheses, we obtain the following separations: 1. There is a set complete for NP under nonuniform many-one reductions, but not under uniform many-one reductions. This is true even with a single bit of nonuniform advice. 2. There is a set complete for NP under nonuniform many-one reductions with polynomial- size advice, but not under uniform Turing reductions. That is, polynomial nonuniformity is stronger than a polynomial number of queries. 3. For any fixed polynomial p(n), there is a set complete for NP under uniform 2-truth-table reductions, but not under nonuniform many-one reductions that use p(n) advice. That is, giving a uniform reduction a second query makes it more powerful than a nonuniform reduction with fixed polynomial advice. 4. There is a set complete for NP under nonuniform many-one reductions with polynomial advice, but not under nonuniform many-one reductions with logarithmic advice. This hierarchy theorem also holds for other reducibilities, such as truth-table and Turing. We also consider uniform upper bounds on nonuniform completeness. Hirahara (2015) showed that unconditionally every set that is complete for NP under nonuniform truth-table reductions that use logarithmic advice is also uniformly Turing-complete.
    [Show full text]
  • A Generalization of Resource-Bounded Measure, with Application to the Bpp Vs
    2000 Society for Industrial and Applied Mathematin; ~;,;IA~t .J_ ('\ 1:-.tr•1_·T © v.. l. :111. ~" '2. PP- :'i';"d hUt A GENERALIZATION OF RESOURCE-BOUNDED MEASURE, WITH APPLICATION TO THE BPP VS. EXP PROBLEM* HARRY Bl'HHl\!ANt. DIETER VAN l\IELKEBEEKt, KENNETH W. REGANS, D. SIVAKUl\!AR~. AND l\IAHTIN STRAussll Abstract. \Ye introduce resource-hounded betting games and propose a gm:1eralizatio:1 of Lutz:s n·,;oarc,•-bounded measure in which the choice of the next string to bet on is fully adaptive. Lutz s !llart ingales an• equivalent to betting games constrained to bet on string~ in lexicographic ~rder. We slio\\· that if strong pseudorandom number generators exist, then betting games are eqmvalent to martiugalt's for measure on E and EXP. However, we construct betting gan1es that su.ccee.d on certam class<'S whos<' Lutz mea-,ures are important open problems: the class of polynomial-time Tunng­ <'<Hllplel<' languages in EXP and its superclass of polynomial-time Turing-autoreducible l~ngua~es. lf an EXP-martingalt• succeeds on either of these classes, or if betting garnes have the "fimte u1110n prnp••rt.\··· possessed by Lutz's measure, one obtains the nonrelativizable consequence BPP f EXP. \\',.also show that if EXP cfa l\IA, then the polynomial-time truth-table-autoredncible languages have Lutz m<>asure zero, whereas if EXP= BPP. they have measure one. Key words. computational complexity, theory of computation, probabilistic computation, c:om­ pk·xity classPs, resource-bounded measure, betting games, polynomial reductions, pseudo random gt•nerator,.;. "tmpling, autoreducibility Al\·18 subject classification.
    [Show full text]
  • Inseparability and Strong Hypotheses for Disjoint NP Pairs
    Inseparability and Strong Hypotheses for Disjoint NP Pairs Lance Fortnow∗ Jack H. Lutzy Elvira Mayordomoz Abstract This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational com- plexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2nk )-inseparable. We also relate these conditions to strong hypotheses concerning randomness and generic- ity of disjoint pairs. 1 Introduction The main objective of complexity theory is to assess the intrinsic difficulties of naturally arising computational problems. It is often the case that a problem of interest can be formulated as a decision problem, or else associated with a decision problem of the same complexity, so much of complexity theory is focused on decision problems. Nevertheless, other types of problems also require investigation. ∗Northwestern University, EECS Department, Evanston, Illinois, USA. fort- [email protected]. Research supported in part by NSF grants CCF-0829754 and DMS-0652521. yDepartment of Computer Science, Iowa State University, Ames, IA 50011 USA. [email protected]. Research supported in part by National Science Foundation Grants 0344187, 0652569, and 0728806. zDepartamento de Inform´aticae Ingenier´ıade Sistemas, Instituto de Investigaci´onen Ingenier´ıade Arag´on,Mar´ıade Luna 1, Universidad de Zaragoza, 50018 Zaragoza, SPAIN. elvira at unizar.es. Research supported in part by Spanish Government MICINN Project TIN2008-06582-C03-02. 1 This paper concerns promise problems, a natural generalization of de- cision problems introduced by Even, Selman, and Yacobi [7].
    [Show full text]
  • Results on Resource-Bounded Measure
    Results on ResourceBounded Measure Harry Buhrman and Stephen Fenner and Lance Fortnow 1 Centrum vo or Wiskunde en Informatica 2 University of Southern Maine 3 CWI The University of Chicago Abstract We construct an oracle relative to which NP has pmeasure p but D has measure in EXP This gives a strong relativized negative answer to a question p osed by Lutz Lut Secondly we give strong evidence that BPP is small We show that BPP has pmeasure unless EXP MA and thus the p olynomialtime hierarchy collapses This con trasts with the work of Regan et al RSC where it is shown that Ppol y do es not have pmeasure if exp onentially strong pseudorandom generators exist Intro duction Since the intro duction of resourceb ounded measure by Lutz Lut many re searchers investigated the size measure of complexity classes in exp onential time EXP A particular p oint of interest is the hypothesis that NP do es not have pmeasure Recent results have shown that many reasonable conjectures in computational complexity theory follow from the hyp othesis that NP is not small ie NP and hence it seems to b e a plausible scientic hyp oth p esis LM Lut P He shows In Lut Lutz shows that if NP then BPP is low for p P that this even follows from the seemingly weaker hyp othesis that p He asks whether the latter assumption is weaker or equivalent to NP p In this pap er we show that relative to some oracle the two assumptions are not equivalent p We show a relativized world where D EXP whereas NP has no Pbi immune sets This immediately
    [Show full text]
  • Compressibility and Resource Bounded Measure
    Compressibility and Resource Bounded Measure Harry Buhrman*1 and Luc Longpre**2 1 CWI. PO Box 94079, 1090 GB Amsterdam, The Netherlands. E-mail: [email protected] 2 Computer Science Department, University of Texas at El Paso, El Paso TX 79968, USA. E-mail: [email protected]. Abstract. We give a new definition of resource bounded measure based on compressibility of infinite binary strings. We prove that the new defi­ nition is equivalent to the one commonly used. This new characterization offers us a different way to look at resource bounded measure, shedding more light on the meaning of measure zero results and providing one more tool to prove such results. The main contribution of the paper is the new definition and the proofs leading to the equivalence result. We then show how this new character­ ization can be used to prove that the class of linear auto-reducible sets has p-measure 0. We also prove that the class of sets that are truth-table reducible to a p-selective set has p-measure 0 and that the class of sets that Turing reduce to a. sub-polynomial dense set hasp-measure 0. This strengthens various results. 1 Introduction While Lebesgue measure has been used in mathematics since the previous cen­ tury, and while it has been used earlier this century to study randomness in infinite strings [ML66], its notable appearance in complexity theory is in the formalization of "random oracles". Bennett and Gill [BG81] showed for example that relative to a random oracle A, pA -:/; NPA. The statement about the ran­ dom oracle is formalized as follows: the class of sets A such that pA = NPA has Lebesgue measure zero.
    [Show full text]
  • Results on Resource-Bounded Measure
    Results on Resource-Bounded Measure Harry Buhrman*1, and Stephen Fenner**2 , and Lance Fortnow***3 1 Cent.rum voor Wiskunde en Informatica 2 University of Southern Maine 3 CWI & The University of Chicago Abstract. We construct an oracle relative to which NP hasp-measure o but DP has measure 1 in EXP. This gives a strong relativized negative answer to a question posed by Lutz [Lut96]. Secondly, we give strong evidence that BPP is small. We show that BPP has p-measure 0 unless EXP = !v!A and thus the polynomial-time hierarchy collapses. This con­ trasts with the work of Regan et. al. [RSC95], where it is shown that p /poly does not have p-measure 0 if exponentially strong pseudorandom generators exist. 1 Introduction Since the introduction of resource-bounded measure by Lutz [Lut92], many re­ searchers investigated the size (measure) of complexity classes in exponential time (EXP). A particular point of interest is the hypothesis that NP does not have µ-measure 0. Recent results have shown that many reasonable conjectures in computational complexity theory follow from the hypothesis that NP is not small (i.e., /Lp(NP) i- 0), and hence it seems to be a plausible scientific hypoth­ esis [D196, Lut96]. In [Lut96J, Lutz shows that if µp(NP) :j:. 0 then BPP is low for d{. He shows that this even follows from the seemingly weaker hypothesis that /Lp ( i11i,J) :j:. 0. He asks whether the latter assumption is weaker or equivalent to /l·p(NP) :/:- 0. In this paper we show that, relative to some oracle, the two assumptions are not equivalent.
    [Show full text]
  • Resource Bounded Measure
    Resource-Bounded Measure ∗ Jack H. Lutz Department of Computer Science Iowa State University Ames, Iowa 50011 U.S.A. Abstract A general theory of resource-bounded measurability and measure is developed. Starting from any feasible probability measure ν on the Cantor space C (the set of all decision problems) and any suitable complexity class C ⊆ C, the theory identifies the subsets of C that are ν-measurable in C and assigns measures to these sets, thereby endowing C with internal measure-theoretic structure. Classes C to which the theory applies include various exponential time and space complexity classes, the class of all decidable languages, and the Cantor space C itself, on which the resource-bounded theory is shown to agree with the classical theory. The sets that are ν-measurable in C are shown to form an algebra relative to which ν-measure is well-behaved (monotone, additive, etc.). This algebra is also shown to be complete (subsets of measure 0 sets are measurable) and closed under sufficiently uniform infinitary unions and intersections, and ν-measure in C is shown to have the appropriate additivity and monotone convergence properties with respect to such in- finitary operations. A generalization of the classical Kolmogorov zero-one law is proven, showing that when ν is any feasible coin-toss (i.e., product) probability measure on C, every set that is ν-measurable in C and (like most complexity classes) invariant under finite alterations must have ν-measure 0 or ν-measure 1 in C. arXiv:1101.5455v4 [cs.CC] 31 Jan 2012 The theory is presented here is based on resource-bounded martingale splitting operators, which are type-2 functionals, each of which maps N × Dν into Dν × Dν, where Dν is the set of all ν-martingales.
    [Show full text]