Randomness as Circuit Complexity (and the Connection to Pseudorandomness)

Abstract

This material was generated for the book Randomness through Computation, edited by Hector Zenil. The format of the book calls for various contributors to give responses to five questions. Why were you initially drawn to the study of computation and randomness? I started out my undergraduate studies majoring in theater. However, I also knew that I enjoyed math and computers, thanks to a terrific high school math teacher (Ed Rolenc) who installed a computer in one of the classrooms in my high school in Mount Pleasant, Iowa in the 1970's. Thus, when I was an undergrad at the University of Iowa and I decided to pick a second major that might make me more employable in case theater didn't work out, I picked Computer Science. My initial computing courses didn't really inspire me very much, and I continued to think of computing as a "reserve" career, until two things happened at more-or-less the same time: (1) I worked in summer theaters for a couple of summers, and I noticed that there were incredibly talented people who were working at the same undistinguished summer theaters where I was working, leading me to wonder how much impact I was likely to have in the field of theater. (2) I took my first courses in theoretical computer science (with Ted Baker and Don Alton), which opened my eyes to some of the fascinating open questions in the field. They gave me some encouragement to consider grad school, and thus (after taking a year off to "see the world" by working as a bellhop in Germany) I entered the doctoral program at Georgia Tech. At that time, in the early-to-mid 1980's, Kolmogorov complexity was finding application in complexity theory. (I'm referring to the material that's now covered in the chapter called "The Incompressibility Method" in the standard text by Li and Vitányi [LV08].) Also, there were influential papers by Mike Sipser [Sip83] and Juris Hartmanis [Har83] that discussed resource-bounded Kolmogorov complexity. Also, the theory of pseudorandom generators was just getting underway, with the work of Yao [Yao82] and of Blum and Micali [BM84]. Thus Kolmogorov complexity and randomness were very much in the air. Please recall that, at the time, most people believed that BPP and RP were likely to contain problems that could not be solved in deterministic polynomial time. (BPP is the class of problems that can be solved efficiently if we have access to randomness, and RP is the natural way to define a probabilistic subclass of NP.) I was struck by the observation that there was some tension between the popular conjectures relating deterministic, probabilistic, and nondeterministic computation. For example, if nondeterministic exponential time has no efficient deterministic simulation, it means