DETERMINISTIC SIMULATIONS AND HIERARCHY THEOREMS FOR RANDOMIZED ALGORITHMS by Jeffrey J. Kinne A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Computer Sciences) at the UNIVERSITY OF WISCONSIN–MADISON 2010 i ABSTRACT In this dissertation, we present three research directions related to the question whether all randomized algorithms can be derandomized, i.e., simulated by deterministic algorithms with a small loss in efficiency. Typically-Correct Derandomization A recent line of research has considered “typically- correct” deterministic simulations of randomized algorithms, which are allowed to err on few inputs. Such derandomizations may be easier to obtain and/or be more efficient than full derandomizations that do not make mistakes. We further the study of typically-correct derandomization in two ways. First, we develop a generic approach for constructing typically-correct derandomizations based on seed-extending pseudorandom generators, which are pseudorandom generators that reveal their seed. We use our approach to obtain both conditional and unconditional typically-correct derandomization results in various algorithmic settings. For example, we present a typically-correct polynomial-time simulation for every language in BPP based on a hardness assumption that is weaker than the ones used in earlier work. Second, we investigate whether typically-correct derandomization of BPP implies circuit lower bounds. We establish a positive answer for small error rates and in doing so provide a proof for the zero-error setting that is simpler and scales better than earlier arguments. Monotone Computations Short of derandomizing all efficient randomized algorithms, we can ask to derandomize more restricted classes of randomized algorithms. Because a strong connection has been proved between circuit lower bounds and derandomization, and ii there has been success proving worst-case circuit lower bounds for monotone circuits, ran- domized monotone computations are a natural candidate to consider. We show that, in fact, any derandomization of randomized monotone computations would derandomize all ran- domized algorithms, whether monotone or not. We prove similar results in the settings of pseudorandom generators and average-case hard functions – that a pseudorandom generator secure against monotone circuits is also secure with somewhat weaker parameters against general circuits, and that an average-case hard function for monotone circuits is also hard with somewhat weaker parameters for general circuits. Hierarchy Theorems For any computational model, a fundamental question is whether machines with more resources are strictly more powerful than machines with fewer resources. Such results are known as hierarchy theorems. The standard techniques for proving hierarchy theorems fail when applied to bounded-error randomized machines and for other so-called “semantic” models of computation for which a machine must satisfy some promise to be valid. If all randomized algorithms can be efficiently derandomized in a uniform way, hierar- chies for bounded-error randomized algorithms would follow from the deterministic hierarchy theorems. But can hierarchies be proved short of proving derandomization? A recent line of work has made progress by proving time hierarchies for randomized and other semantic models that use one bit of advice. We adapt the techniques to prove results in the setting of space, proving space hierarchy results that are as tight as possible for typical space bounds between logarithmic and linear for randomized and other semantic models that use one bit of advice. iii ACKNOWLEDGMENTS I feel that a key part of being human is to seek the good of our community – family, friends, workplace, city, etc. My life has been full of others who share this view. In particular, I owe the completion of this dissertation to the many who provided support, encouragement and more. In acknowledging some of these people I give them thanks, but also I encourage all of us to echo the best of their attributes. Let me begin with my advisor Dieter. When I think of the qualities I would most want in an ideal advisor, I think of: honesty, integrity, selflessness, openness, dedication, patience, and having the highest standards for research and teaching. Those who know Dieter will realize I have just described him! Whether in research or teaching, Dieter always places the best interests of the students first – even if that means extending class by “a few” minutes to finish the day’s topic or going through “a few” revisions of a paper to achieve the desired quality ;). Dieter, thank you so much for everything. It has been an absolute pleasure working together. Next, my wife Devon is the person most responsible for me seeking a PhD in the first place and for finishing the task. We have truly lived a dream together. From the time we began dating in high school, we mapped out what our life would look like. Our predictions were amazingly accurate, in large part to Devon’s love and perseverance! She has always placed others’ interests first (the children, myself, her students, the children’s school, etc.) before her own. Our family has thrived due to her love, dedication, kindness, strong values, sincerity, graciousness, and humility. For example, on the day of my defense Devon dropped off Andrew and William at school, lugged Matthew around picking up and setting out refreshments, stayed for most of the defense while holding a sleeping Matthew, then appropriately discretely iv left the room after Matthew woke up reaching for me with a smile and “da-da, da-da, da-da”. I thank you eternally Devon, you truly are the better half. As for the kids, I am grateful for them for many reasons, not the least of which is their sense of wonder and inquisitiveness that is so contagious. Thank you to the members of my committee – Eric Bach, Jin-Yi Cai, Shuchi Chawla, Steffen Lempp, and Lance Fortnow. Besides graciously helping me now at the end, you have been teachers to me in the classroom, have shared insights into the research process and academia, have written letters of support, and served on my preliminary defense committee. I would not be where I am today without you. I also very much appreciate the mentoring and encouragement from those who I have been a teaching assistant for – Dieter, John Strikwerda, and Rebecca Hasti. I thank my co-authors Dieter and Ronen Shaltiel, who have exemplified the way research ought to be done – openly and with the true goal of bettering the community. These co- authors and a few others, notably Scott Diehl and Matt Anderson, have been invaluable as people to bounce ideas off of and discuss research. Life as a graduate student would be far less enjoyable without the camaraderie of others to share successes and failures. Thanks go to Randy Smith and Margaret Richey for being good friends that helped us adjust to life in Madison and as graduate students. I also thank all of the theory students, which at various times included Denis Charles, Venkat Chakravarthy, Chris Kaiserlian, Rakesh Kumar, Yunpeng Li, Aparna Das, Giordano Fusco, Bess Berg, Eric Skaug, Mike Kowalczyk, Scott, Matt, Jurgen Van Gael, Anand Sinha, Vinay Choudhary, Tom Watson, Siddarth Barman, David Malec, William Umboh, Dalibor Zelen´y, Adeel Pervez, Baris Aydinlioglu, Balu Sivan, Tyson Williams, and Thea Hinkle. I will miss the lunches, reading groups, taking breaks to watch the construction out the window, trips to conferences together, practice presentations, etc. I will miss the conversations about all manner of topics – trying to solve homework problems, studying for the qual, sports, Indian versus American culture, religion, politics, child-raising strategies, “inventions”, “points”, “memorials” to past theory students, etc. v My life as a whole has been influenced by many many kind and generous people. Foremost are my parents Tim and Cathy Kinne, but this also includes the rest of my family, coworkers, and friends from Madison, Xavier, and St. X. A special thanks goes to the CS professors at Xavier – Michael Goldweber, Gary Lewandowski, and Liz Johnson – for encouraging my spark of interest in computer science and ultimately sending me off to graduate school! Thank you to all of you! Finally, this research would not have been possible without the financial support of the University of Wisconsin-Madison, National Science Foundation grants CCR-0133693 and CCR-0728809, and a Cisco Systems Distinguished Graduate Fellowship. vi TABLE OF CONTENTS Page ABSTRACT ....................................... i 1 Introduction ..................................... 1 1.1 The Power of Randomized Algorithms ..................... 5 1.2 Typically-Correct Derandomization ....................... 8 1.2.1 Applications of Our Approach ...................... 10 1.2.2 Typically-Correct Derandomization and Circuit Lower Bounds .......................... 11 1.3 Derandomization of Monotone Computations .................. 13 1.3.1 Our Results ................................ 15 1.4 Space Hierarchy Theorems ............................ 17 1.4.1 Randomized Models with Advice .................... 19 1.4.2 Generic Semantic Models with Advice ................. 21 1.4.3 Promise Problems for Generic Semantic Models ............ 22 1.5 Organization ................................... 23 2 Preliminaries ..................................... 25 2.1 Deterministic Algorithms and Turing Machines ................ 25 2.2 Randomized Algorithms and Turing Machines ................