Symmetry and equivalence relations in classical and geometric complexity theory

211 Pages·2012·1.23 MB·English

THE UNIVERSITY OF CHICAGO
SYMMETRY AND EQUIVALENCE RELATIONS IN CLASSICAL AND GEOMETRIC
COMPLEXITY THEORY
A DISSERTATION SUBMITTED TO
THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES
IN CANDIDACY FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
DEPARTMENT OF COMPUTER SCIENCE
BY
JOSHUA ABRAHAM GROCHOW
CHICAGO, ILLINOIS
JUNE 2012
To my parents, Jerrold Marvin Grochow and Louise Barnett Grochow
ABSTRACT
This thesis studies some of the ways in which symmetries and equivalence relations arise in
classical and geometric complexity theory. The Geometric Complexity Theory program is
aimed at resolving central questions in complexity such as P versus NP using techniques
from algebraic geometry and representation theory. The equivalence relations we study
are mostly algebraic in nature and we heavily use algebraic techniques to reason about
the computational properties of these problems. We first provide a tutorial and survey on
Geometric Complexity Theory to provide perspective and motivate the other problems we
study.
One equivalence relation we study is matrix isomorphism of matrix Lie algebras,
which is a problem that arises naturally in Geometric Complexity Theory. For certain cases
of matrix isomorphism of Lie algebras we provide polynomial-time algorithms, and
for other cases we show that the problem is as hard as graph isomorphism. To our
knowledge, this is the first time graph isomorphism has appeared in connection with any
lower bounds program.
Finally, we study algorithms for equivalence relations more generally (joint work with
Lance Fortnow). Two techniques are oftenemployed for algorithmically deciding equivalence
relations: 1)finding acomplete setofeasilycomputableinvariants, or2)findinganalgorithm
which will computeacanonical formforeach equivalence class. Someequivalence relationsin
theliterature have beensolved efficiently by other means aswell. We askwhether these three
conditions—having anefficient solution, having anefficiently computable complete invariant,
and having an efficiently computable canonical form—are equivalent. We show that this
questionrequiresnon-relativizingtechniquestoresolve, andprovidenewconnectionsbetween
this question and factoring integers, probabilistic algorithms, and quantum computation.
iii
ACKNOWLEDGMENTS
First I thank my advisors. I thank Lance Fortnow for his advice, support, guidance, and
collaboration on Chapter 5 and several other projects which have yet to bear fruit. I thank
Ketan Mulmuley for his advice and support, as well as countless hours of discussion through-
out the course of my graduate career. Without these discussions, it would not have been
possible to seriously work on problems related to Geometric Complexity Theory—such as
matrix isomorphism of Lie algebras (Chapter 4)—let alone to write a survey on it
(Chapter 3). I thank Benson Farb for his advice, guidance, and many fruitful discussions,
even when he was not officially my advisor.
I thank my thesis committee for their continued advice, prodding, and editorial support.
I thank Anne Rogers for her support and for the countless decisions regarding my career
trajectory, both large and small, she helped me understand how to make.
IthankSashaRazborovformanyinteresting discussions, andforenforcingamuch-needed
kick-in-the-pants in the middle of my graduate career that ensured I graduated in a timely
fashion. I cannot imagine this duty was much fun for him, but it was a tremendous help to
me.
I thank anonymous reviewers for feedback that improved the quality, clarity, and presen-
tation of the works on which Chapters 4 and 5 are based. In particular, one of the reviewers
pointed out the importance of the complexity of factoring polynomials for Chapter 4. One
of the reviewers suggested that we define some sort of hybrid notion of Cohen and transi-
tive genericity, as well as suggested the notion of UP-transitive genericity that are used in
Chapter 5. I also thank Lane Hemaspaandra—who was our editor for the corresponding
paper—and Paolo Codenotti for useful comments on a draft of Chapter 5. I thank Laci
Babai for useful comments on a draft of Chapter 4, as well as pointing me to several results
[20, 62] related to that chapter, and suggesting that I consider the corresponding questions
for associative algebras.
I thank Stuart Kurtz and Laci Babai for several useful discussions regarding Chapter 5.
In particular, Stuart suggested the use of the equivalence relation R , which led to Theorem
L
iv
5.3.3, and Laci pointed out the canonical form for subgroup equality of permutation groups
[23]. I thank Scott Aaronson for the observations leading to Section 5.3.1. I thank Andreas
Blass for pointing me to the original two papers he co-authored with Gurevich [56, 57].
I thank my collaborators, on projects both finished and in progress: Lance Fortnow,
La´szlo´ Babai, Paolo Codenotti, Youming “Jimmy” Qiao, Jonah Blasiak, and Thomas
Church. It was and continues to be a pleasure to work with them. In particular, Chap-
ter 5 is based on joint work with Lance, and Jonah helped me clarify my thoughts on
matrix isomorphism of Lie algebras and together realize the equivalence with graph
isomorphism in Chapter 4.
I find it incredibly useful, rewarding, and fun to talk through mathematics with others,
and it is my great pleasure and honor to thank Lance Fortnow, Ketan Mulmuley, Ben-
son Farb, Thomas Church, Ian Shipman, Spencer Dowdall, Anna Marie Bohmann, Daniel
Studenmund, Vipul Naik, Paolo Codenotti, Youming “Jimmy” Qiao, Chris Umans, J. M.
Landsberg, JerzyWeyman, ShrawanKumar,NeerajKayal, ArkadevChatthopadhyay, Pascal
Koiran, Gerald J. Sussman, and Jonah Blasiak for not only useful and interesting discus-
sions, but also fortheir infectious enthusiasm. Many discussions regarding GCTandmatrix
isomorphism of Lie algebras took place at the Brown-ICERM Workshop on Mathe-
matical Aspects of P vs. NP and its Variants in August 2011, for which I would like to
thank ICERM and the organizers of the workshop—J. M. Landsberg, Saugata Basu, and J.
Maurice Rojas—for the invitation and support to attend the workshop.
I would especially like to thank Stuart Kurtz and Gerald J. Sussman for sharing with me
some small portion of their incredible breadth of knowledge and depth of philosophy. They
have both made my research career and my life more interesting.
This thesis was partially supported by K. Mulmuley’s NSF Grant CCF-1017760, L. Fort-
now et al.’s NSF Grant DMS-0652521 and fellowships from the University Chicago Depart-
ment of Computer Science.
I would like to thank the members of the University of Chicago Department of Computer
Science Techstaff. They’ve setup such a great system and were so helpful that I barely
noticed all the technology I was using: I could do what I wanted, how I wanted, when I
wanted. I think this is the mark of a truly great technical staff. I would also like to thank
v
the staff of the University of Chicago Library, especially those in Eckhart Library: I am
likely one of their most frequent patrons.
Finally, I thank my family and extended family. My extended family, who were also my
roommatesatvariouspointsthroughoutmygraduatecareer: SpencerDowdall, IanShipman,
Ann Herbert, Rebecca Lordan, and (honorary roommate) Thomas Church; it’s not so much
that they made graduate school worth the time and effort, but that they made it worthwhile
at least ten times over. I especially thank my grandparents Samuel and Frances Grochow,
and Marvin and Hazel Barnett, my parents Jerrold and Louise Grochow, and my sister,
Rebecca Grochow, for all their love and support in so many ways over the years. Last but
by no means least, I thank my fianc´e Nikki Pfarr. I thank her for her patience, support, and
partnership; for her humor; for her love, romance, and companionship; for her wisdom, wit,
humor, and intelligence; and for her smile.
vi
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Computational problems and complexity measures . . . . . . . . . . . 2
1.1.2 Degrees of complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Continuous symmetries and Lie algebras . . . . . . . . . . . . . . . . 9
1.3.2 Symmetry-based equivalence relations . . . . . . . . . . . . . . . . . . 9
1.4 Symmetry and equivalence relations in complexity . . . . . . . . . . . . . . . 12
1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1 Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 Computational problems . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.2 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 Complexity classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.4 Circuit complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.5 Algebraic complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.6 Barriers: relativization, algebrization, and natural proofs . . . . . . . 31
2.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3 Rings, fields, and modules . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.4 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
vii
3 A TUTORIAL AND SURVEY OF GEOMETRIC COMPLEXITY THEORY . . 49
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 The 1,000-foot view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1 The plan of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 On the necessity of algebraic geometry, representation theory, and al-
gorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.3 The plan of attack II: a few details . . . . . . . . . . . . . . . . . . . 56
3.3 The 100-foot view: from computational reductions to orbit closures . . . . . 57
3.3.1 Background: group actions and orbits . . . . . . . . . . . . . . . . . . 57
3.3.2 Equivalence of lower bounds and orbit closure containment . . . . . . 61
3.3.3 Algebraic versus Boolean complexity . . . . . . . . . . . . . . . . . . 66
3.4 The 10-foot view: characterization by symmetries . . . . . . . . . . . . . . . 70
3.4.1 Background: stabilizers in group actions . . . . . . . . . . . . . . . . 71
3.4.2 Symmetry-characterization and self-reduction: the Flip Theorem . . . 73
3.4.3 Symmetry-characterization avoids the Razborov–Rudich barrier . . . 83
3.4.4 An algorithmic consequence of symmetry-characterization . . . . . . 84
3.5 The view from the ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.1 Using the zeroes of a function to understand its orbit closure . . . . . 86
3.5.2 The relationship between the Mulmuley–Sohoni Conjecture and per-
manent versus determinant . . . . . . . . . . . . . . . . . . . . . . . . 90
4 MATRIX ISOMORPHISM OF MATRIX LIE ALGEBRAS . . . . . . . . . . . . . 94
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.2 A note on finding roots of single-variable polynomials . . . . . . . . . 98
4.1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 Warm-up: diagonalizable Lie algebras and linear code equivalence . . . 101
4.3 Basic algorithms for Lie algebras and their representations . . . . . . . . . . 105
4.3.1 Describing Lie algebras and representations as input to algorithms . . 105
4.3.2 Abstract isomorphism of semisimple Lie algebras . . . . . . . . . . . 106
4.3.3 Equivalence and decomposition of representations . . . . . . . . . . . 107
4.4 Semisimple Lie algebras and graph isomorphism . . . . . . . . . . . . . . 108
4.5 Completely reducible Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6 Application to equivalence of polynomials . . . . . . . . . . . . . . . . . . . 122
4.7 Application to abstract isomorphism of Lie algebras . . . . . . . . . . . . . . 127
4.8 Twisted code equivalence reduces to graph isomorphism . . . . . . . 132
4.9 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.9.1 Other fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.9.2 Connections with finite group isomorphism . . . . . . . . . . . . 139
4.9.3 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
viii
5 THE COMPLEXITY OF EQUIVALENCE RELATIONS . . . . . . . . . . . . . . 148
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.2 Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3 Evidence for Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3.1 New Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3.2 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4 Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.4.1 Preliminaries on Generic Oracles . . . . . . . . . . . . . . . . . . . . 165
5.4.2 Oracles for PEq, Ker, and CF . . . . . . . . . . . . . . . . . . . . . 168
5.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.5.1 Logarithmic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.5.2 Additional Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
ix
LIST OF FIGURES
1.1 Some relationships between symmetry, equivalence relations, and
computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Are these two graphs “the same?” . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 A labeling of the vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Two graphs with the same number of vertices and edges that are not the
same. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Some shapes with varying degrees of symmetry: a circle, an equilateral
triangle, an isosceles triangle, a general triangle. . . . . . . . . . . . . . . . 8
1.6 A geometric figure with an infinite but discrete group of symmetries. . . . 9
1.7 Under the full symmetry group of an equilateral triangle, the points marked
by circles are all equivalent to one another. The midpoints of the sides,
marked by squares, are all equivalent to one another, but are not equivalent
to the points marked by circles. . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 The action of S on n-vertex graphs is by isomorphisms . . . . . . . . . . . 58
n
3.2 Orbits of points on an equilateral triangle under the action of the dihedral
group. Each shape (square or circle) corresponds to a single orbit. . . . . . 59
3.3 The padded permanent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Two matrix isomorphic faithful representations of a Lie algebra yield an
1 L
automorphism of by going around the triangle clockwise: ρ c ρ . . 110
L −2 ◦ A ◦ 1
4.2 Color gadget encoding the action of the groups acting on the columns. In
twisted code equivalence these are the twisting groups; from the Lie
algebra point of view these are the outer automorphism groups of the simple
direct summands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
x