Structural Complexity II

294 Pages·1990·14.588 MB·English

by  José Luis Balcázar, Josep Díaz, Joaquim Gabarró

EATCS
Monographs on Theoretical Computer Science
Volume 22
Editors: W. Brauer G. Rozenberg A. Salomaa
Advisory Board: G.Ausiello M.Broy S.Even
J.Hartmanis N.Jones T.Leighton M.Nivat
C. Papadimitriou D. Scott
Jose Luis Balcazar
Josep Diaz
Joaquim Gabarr6
Structural Complexity II
With 72 Figures
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo Hong Kong
Authors
Prof. Dr. Jose Luis Balcizar
Prof. Dr. Josep Diaz
Prof. Dr. Joaquim Gabarr6
Facultat d'Infonnatica
Universitat Politecnica de Catalunya
Pau Gargallo, 5, E-08028 Barcelona, Spain
Editors
Prof. Dr. Wilfried Brauer
Institut fUr Infonnatik, Technische Universitiit Miinchen
Arcisstrasse 21, D-8000 Miinchen 2, FRG
Prof. Dr. Grzegorz Rozenberg
Institute of Applied Mathematics and Computer Science
University of Leiden, Niels-Bohr-Weg 1, P. O. Box 9512
NL-2300 RA Leiden, The Netherlands
Prof. Dr.Arto Salomaa
Department of Mathematics, University ofTurku
SF-20 500 Turku 50, Finland
ISBN-13 :978-3-642-75359-6 e-ISBN-13:978-3-642-75357-2
DOl: 10.1007/978-3-642-75357-2
Library of Congress Cataloging-in-Publication Data
(Revised for volume 2)
Balc3zar, Jose Luis.
Structural complexity.
(EATCS monographs on theoretical computer science; v.ll,
Includes bibliographies and indexes.
1. Computational complexity. I. Diaz, J. (Josep), 1950- . II. Gabarr6, Joaquim. III. Title.
IV. Series: EATCS monographs on theoretical computer science; v.ll, etc.
QA267.B34 1988 511.3 87-36933
ISBN-13:978-3-642-75359-6
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© Springer-Verlag Berlin Heidelberg 1990
Softcover reprint of the hardcover 1st edition 1990
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Preface
This is the second volume of a two volume collection on Structural Complexity.
This volume assumes as a prerequisite knowledge about the topics treated in
Volume I, but the present volume itself is nearly self-contained.
As in Volume I, each chapter of this book ends with a section entitled
"Bibliographical Remarks", in which the relevant references for the chapter are
briefly commented upon. These sections might also be of interest to those
wanting an overview of the evolution of the field, as well as relevant related
results which are not included in the text. Each chapter includes a section
of exercises. The reader is encouraged to spend some time on them. Some
results presented as exercises are occasionally used later in the text. A reference
is provided for the most interesting and for the most useful exercises. Some
exercises are marked with a • to indicate that, to the best knowledge of the
authors, the solution has a certain degree of difficulty.
Many topics from the field of Structural Complexity are not treated in depth,
or not treated at all. The authors bear all responsibility for the choice of topics,
which has been made based on the interest of the authors on each topic.
Many friends and colleagues have made suggestions or corrections. In partic
ular we would like to express our gratitude to Richard Beigel, Ron Book, Rafael
Casas, Jozef Gruska, Uwe Schoning, Pekka Orponen, and Osamu Watanabe. Ja
cobo Toran not only checked most of the manuscript but suggested and worked
out some of the proofs in the book. We also would like to thank Springer-Verlag,
especially Dr. H. Wossner and Mrs. I. Mayer, for the assistance and patience
through the elaboration of these two volumes, and Rosa Martin for her assis
tance with the typesetting software. Finally we will like to thank C. Alvarez,
J. Castro, R. Gavalda, A. Lozano, M. J. Serna, A. Torrecillas and B. Valles,
graduate students in our Department, which had to listen to the genesis of most
of the chapters in the book. To them we dedicate this work.
Barcelona, September 11, 1989 J. L. Balcazar
J. Dfaz
J. Gabarro
Contents
Introduction 1
1 Vector Machines 4
1.1 Introduction 4
1.2 Vector Machines: Definition and Basic Properties . 5
1.3 Elementary Matrix Algebra on Vector Machines . . 11
1.4 Relation Between Vector Machines and Turing Machines. 21
1.5 Exercises......... 29
1.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . 32
2 The Parallel Computation Thesis 33
2.1 Introduction . . . . . . . . . . . . . . . . 33
2.2 An Array Machine: the APM . . . . . . . 34
2.3 A Multiprocessor Machine: the SIMDAG . 39
2.4 A Tree Machine: the k-PRAM 46
2.5 Further Parallel Models . 52
2.6 Exercises......... 57
2.7 Bibliographical Remarks. 61
3 Alternation 63
3.1 Introduction . . . . . . . . . . . . 63
3.2 Alternating Turing Machines ... 63
3.3 Complexity Classes for Alternation 70
3.4 Computation Graphs of a Deterministic Turing Machine 78
3.5 Determinism Versus Nondeterminism for Linear Time 88
3.6 Exercises......... 91
3.7 Bibliographical Remarks. . . . . . . . . . . . . . . . 94
4 Uniform Circuit Complexity 97
4.1 Introduction . . . . . . . 97
4.2 Uniform Circuits: Basic Definitions . . . . . . . . . . 97
4.3 Relationship with General-Purpose Parallel Computers 103
4.4 Other Uniformity Conditions ............ . 108
VI I I Contents
4.5 Alternating Machines and Uniformity 110
4.6 Robustness of NC and Conclusions 115
4.7 Exercises......... 115
4.8 Bibliographical Remarks. . . . 117
5 Isomorphism and NP-completeness 119
5.1 Introduction . . . . . . . . . . 119
5.2 Polynomial Time Isomorphisms 119
5.3 Polynomial Cylinders 122
5.4 Sparse Complete Sets . . 125
5.5 Exercises......... 131
5.6 Bibliographical Remarks. 132
6 Bi-Immunity and Complexity Cores 134
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 134
6.2 Bi-Immunity, Complexity Cores, and Splitting .. . . 134
6.3 Bi-Immune Sets and Polynomial Time m-Reductions . 136
6.4 Complexity Cores and Polynomial Time m-Reductions . 139
6.5 Levelability, Proper Cores, and Other Properties 143
6.6 Exercises......... 146
6.7 Bibliographical Remarks. 147
7 Relativization 149
7.1 Introduction . . . . . . . . . . . 149
7.2 Basic Results. . . . . . . . . . . 150
7.3 Encoding Sets in NP Relativized 154
7.4 Relativizing Probabilistic Complexity Classes. 156
7.5 Isolating the Crucial Parameters 162
7.6 Refining Nondeterminism . . . . 165
7.7 Strong Separations. . . . . . . . 168
7.8 Further Results in Relativizations 172
7.9 Exercises......... 174
7.10 Bibliographical Remarks. 176
8 Positive Relativizations 178
8.1 Introduction . . . . . . . . . . . . . . . . . . . 178
8.2 A Positive Relativization of the P J PSPACE Problem. 180
8.3 A Positive Relativization of the NP J PSPACE Problem 184
8.4 A Positive Relativization of the P J NP Problem. 186
8.5 A Relativizing Principle . 192
8.6 Exercises......... 196
8.7 Bibliographical Remarks. 197
Contents IX
9 The Low and the High Hierarchies 199
9.1 Introduction . . . . . . . . . . . . . . . . . . . . 199
9.2 Definitions and Characterizations . . . . . . . . . 200
9.3 Relationship with the Polynomial Time Hierarchy 202
904 Some Classes of Low Sets. . . . . . . . 204
9.5 Oracle-Restricted Positive Relativizations 209
9.6 Lowness Outside NP. . . 213
9.7 Exercises......... 215
9.8 Bibliographical Remarks. 217
10 Resource-Bounded Kolmogorov Complexity 219
10.1 Introduction ... . . . . . . . . . . . . 219
10.2 Unbounded Kolmogorov Complexity . . 220
10.3 Resource-Bounded Kolmogorov Complexity 222
lOA Tally Sets, Printability, and Ranking. . . . . 225
10.5 Kolmogorov Complexity of Characteristic Functions 230
10.6 Exercises. . . . . . . . . 232
10.7 Bibliographical Remarks. . . . . . . . . . . . . . . 233
11 Probability Classes and Proof-Systems 235
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 235
11.2 Interactive Proof-Systems: Basic Definitions and Examples 235
11.3 Arthur Against Merlin Games . . . . . . . . . . . . 240
11.4 Probabilistic Complexity Classes and Proof-Systems 241
11.5 Equivalence of AM and IP . 251
11.6 Exercises. . . . . . . . . 253
11.7 Bibliographical Remarks. . 254
Appendix: Complementation via Inductive Counting 257
1 Nondeterministic Space is Closed Under Complement 257
2 Bibliographical Remarks. . . . . . . . . . . . . . . . 262
References 263
Author Index 274
Symbol Index 277
Subject Index 280
Introduction
The notion of algorithm is very rich and can be studied under many different
approaches. One of them is given by structural complexity. In the sixties the
notion of feasible algorithm was developed and gradually was identified with P
problems. In the early seventies the class NP was raised and the polynomial time
reductions were defined. From this moment on these ideas were enlarged and
studied more deeply. In structural complexity we study and classify the inherent
properties of the problems by means of resource-bounded reducibilities; special
attention is paid to complete problems. It was necessary to bring together some
of the most important or fundamental material in a "uniform" way.
We present the material of our choice in two volumes; the first one contains
the basic topics, and the second one treats more advanced ones. The topics,
coinciding with chapters, covered in the first volume are:
1. Basic Notions About Models of Computation. We present some concepts
of formal languages, set theoretic operations, boolean formulas and models
of computations like finite automata and deterministic, nondeterministic
and oracle Turing machines.
2. Time and Space Bounded Computations. We present the basic theorems
about complexity classes. These classes are defined by imposing bounds
on the time and space used by the different models of Turing machines.
3. Central Complexity Classes. We present the basic complexity classes and
their known relationships. We define m-reducibility and related concepts
like closure, completeness, and hardness. Some NP and PSPACE complete
problems are presented. We finish with the technique of "padding" and
the basic facts about logarithmic space nt-reducibility.
4. Time Bounded Turing Reducibilities. We introduce T-reducibility and the
relativized complexity classes. We give a glimpse of the "sparseness"
question and its influence on the relationship between several complexity
classes. Finally we consider a more general form of reducibility known
as SN-reducibility.
5. Nonuniform Complexity. We introduce the nonuniform approach, which
is a tool for dealing with finite sets. Instead of measuring resources used
by algorithms, we measure sizes of algorithms accepting finite sets. To
establish a connection between the uniform and the nonuniform approach
we introduce the "advice functions".
2 Introduction
6. Probabilistic Algorithms. A probabilistic algorithm is a procedure that be
haves in a deterministic way, except that it eventually takes decisions with
a fixed probability distribution. The probabilistic Turing machine gives
us a formal definition for probabilistic algorithms. We study complexity
classes defined by imposing bounds on the time of computation. Also the
concept of the probability of error is considered.
7. Uniform Diagonalization. We present a technique for proving the ex
istence of certain "diagonal" recursive sets. It allows one to prove the
existence of non-complete problems in NP - P and the existence of infi
i
nite hierarchies of incomparable sets in NP - P, provided that NP P.
8. The Polynomial Time Hierarchy. This hierarchy is a polynomial version
of the Kleene hierarchy studied in Recursive Function Theory. The Poly
nomial Time Hierarchy lies in between P and PSPACE. We consider also
the relation between some probabilistic classes and this hierarchy.
This second volume contains more advanced material and provides a more
detailed picture of Structural Complexity. In this volume we consider the fol
lowing major topics: parallel complexity classes, recursion-theoretic aspects of
complexity, relativizations, Kolmogorov complexity, interactive proof-systems,
and inductive counting. We give now a brief description of the contents.
Chapters 1 and 2 give a detailed approach to elementary parallel models of
computations. The first chapter deals with vector machines, parallel algorithms
and their relation with Turing machines. The second deals with other parallel
machines like array machines, SIMDAGS, and tree machines. All the models
defiped verify the parallel computation thesis, which states that classes defined
by space bounds on sequential models correspond to classes defined by time
bounds on parallel models.
Chapter 3 and 4 analyse more deeply parallel models of computation. Chap
ter 3 combines the power of nondeterminism with parallelism to define the
alternating models of computations and the associated classes. A particularly
important result in this chapter, is the strict inclusion DUN C NUN. Chapter 4
is devoted to the study of the parallelism with a feasible number of processors;
special attention is given to the class NC.
Chapters 5 and 6 deal with complexity-theoretic concepts arising from Re
cursive Function Theory. Chapter 5 discusses the so called "isomorphism con
jecture", which is an analogue to Myhill's theorem for the recursively enumer
able sets. A related, very interesting result is also discussed: unless P = NP,
no sparse NP-complete sets exist. Chapter 6 studies various notions related to
bi-immune sets. Useful tools for studying reducibilities are obtained.
Chapter 7 develops several results regarding the equalities or inequalities
of complexity classes in the presence of oracle sets. It is shown that many
unknown relationships can be solved for appropriately constructed oracles, but
that frequently this can be done in contradictory ways for different oracles.
Chapter 8 tries to give explanations of the phenomena studied in Chapter 7
Introduction 3
by comparing the power of the different computational models with respect to
oracle access.
Chapter 9 discusses the "low" and the "high" sets in NP, by comparing the
power of the sets taken as oracles for the polynomial time hierarchy. These
concepts are inspired also in notions from Recursive Function Theory, although
their properties are substantially different. Several properties that imply lowness
or highness are identified. Any lowness property, such as a set being sparse,
and any highness property, such as a set being T -complete, are seen to be
incompatible provided that the polynomial time hierarchy is proper up to some
level (most frequently up to the second or third level).
Chapter 10 discusses Kolmogorov complexity of strings and their relationship
to the complexity of the sets they belong to. Resource-bounded variants of
Kolmogorov complexity are used to characterize the sets isomorphic to tally
sets and certain nonuniform complexity classes.
Chapter 11 presents a brief introduction to the probabilistic classes from the
structural complexity point of view, placing emphasis on the Interactive Proof
Systems and Arthur-Merlin games. These concepts provide complexity classes
corresponding to alternations between probabilistic computation and nondeter
ministic computation, and can be characterized in several ways. As a conse
quence, the relationship with the polynomial time hierarchy as well as other
properties are shown.
The Appendix contains an important result: the closure under complemen
tation of nondeterministic space-bounded complexity classes. It also serves as
an introduction to the technique of inductive counting, which has applications in
the study of other complexity classes. This technique should have been included
in Volume I, but at the time Volume I was written, the technique had not yet
been published.
Let us give some pedagogical and practical hints. The chapters of this
volume are less interrelated than in Volume 1. In fact the "inherent complexity"
of the book is rather "constant". If you are interested in a particular topic try
to study it directly. You have a good chance of understanding it. In fact we
think this is almost a "RAM-book". The first two chapters are quite elementary
and self-contained. Chapter 3 is independent from Chapters 1 and 2. Almost
all sections are very readable. Section 3.4 contains a proof of the difficult
result DLIN =I NLIN. This section is more technical and difficult to read. The
material contained in Chapter 4 is easily readable if preceding chapters are well
understood. The remaining chapters are quite independent of the first four ones.
Chapters 5 and 6 can be read almost on their own; only the proof of Mahaney's
theorem in Chapter 5 is somewhat involved. Chapters 7 to 9 are assumed to
be read in sequence; some of the constructions are a bit complicated. Chapters
10 and 11 are again nearly independent of the preceding ones. As a whole,
this volume is "almost" self contained. This means that it can be read with an
elementary but solid background on complexity, as provided by Volume 1.