William M. Hoza Catholic theorem-wielder
About me: I'm a second-year grad student at UT Austin studying theoretical computer science. I have the good fortune to be advised by David Zuckerman. I'm interested in the strengths of different computational resources. To what extent is randomness useful for computation? How about quantum mechanics? Does parallelism always give a huge computational advantage? Is space more useful than time? Is it ever easier to check your work than it is to solve a problem from scratch? All of these questions can be made mathematically precise, and nobody knows the answers. My personal favorite open problem is the "L vs. BPL" problem: Is randomness ever helpful for solving a problem using as little memory as possible?

Simple Optimal Hitting Sets for Small-Success RL
With David Zuckerman. To appear in FOCS 2018.
Not-so-abstract: A "decision algorithm" takes as input a string and either "accepts" or "rejects" the string. A "hitting set" is a set of strings so that if A is an efficient decision algorithm that accepts a decent fraction of inputs, then A accepts some string in the hitting set. For this paper, "efficient" means that A uses only a small amount of memory and reads its input just once from left to right. In this paper, we present a new hitting set. Our hitting set is smaller than any hitting set discovered previously.

Notes: The main result of this paper is covered in these lecture notes for a course taught by Amnon Ta-Shma.
Typically-Correct Derandomization for Small Time and Space
Preprint, 2017.
Not-so-abstract: A "randomized" algorithm tosses coins to make decisions, whereas a "deterministic" algorithm doesn't use any randomness. Most theoretical computer scientists believe that if a problem can be solved by a randomized algorithm, then it can also be solved by a deterministic algorithm that uses about the same amount of time and memory. But nobody knows how to prove it. In this paper, we prove that if a problem can be solved by a fast randomized algorithm, then it can also be solved by a deterministic algorithm that uses about the same amount of memory, with the caveat that the deterministic algorithm might give the wrong answer on a tiny fraction of inputs.

Presentations:
• I presented at the Weizmann Institute's Foundations of Computer Science Seminar, March 2018 [slides]
• I presented at TAU's Theory of Computation Seminar, March 2018 [slides]
• I presented at HUJI's Theory of Computer Science Seminar, March 2018 [slides]
Preprint, 2017.
Not-so-abstract: Quantum algorithms are algorithms that exploit quantum effects, requiring special hardware that, broadly speaking, nobody has managed to build yet. In this paper, we study quantum algorithms for two types of problems. In the first type of problem, Alice has some data x, Bob has some data y, and the two of them would like to figure out whether the pair (x, y) has some property P using as little communication as possible. For example, maybe x and y are numbers between 1 and a million, and P is the property that x > y. The second type of problem is similar to the game 20 Questions. Alice chooses x. If Bob specifies y, then Alice tells him whether (x, y) has property P. Bob's goal is to learn x by making as few queries as possible. We prove that if the same property P is used to formulate each problem, then the two problems can't both have extremely efficient quantum algorithms.
Universal Bell Correlations Do Not Exist
With Cole A. Graham. In PRL, 2017.
Not-so-abstract: Bell's theorem says, roughly, that it follows from the laws of quantum mechanics that two parties can interact instantaneously across arbitrary distances. This phenomenon, called "quantum nonlocality", has been called "the most profound discovery of science". The nature of these "interactions" is notoriously subtle. For example, the no-communication theorem says that quantum nonlocality is not useful for sending genuine signals. In this paper, we rule out one possible approach to characterizing the exact extent of the "nonlocal powers" granted by the laws of quantum mechanics. In particular, aside from quantum nonlocality, another situation in which two parties can interact in a limited way is if there is some discrete, classical device that each party is able to interact with. We prove that quantum nonlocality is not quite equivalent to any such discrete, classical device.

Versions: The PRL version, also available on this site, is much more compact than the earlier arXiv version. The arXiv version has essentially the same results, but it has more detailed definitions and proofs and some suggested open problems. The arXiv version also uses slightly different notation and is missing some references.

Presentations:
• I presented in Scott Aaronson's course "Topics in Quantum and Classical Complexity Theory", December 2016 [slides]
With Adam R. Klivans. To appear in RANDOM 2018.
Not-so-abstract: Suppose you have a randomized algorithm that estimates some numeric quantity, and you plan to use it k times. Suppose the estimation algorithm makes n coin tosses. This situation seems to call for nk coin tosses in total. In this paper, we show that you can actually get away with only making about n + k coin tosses if you use some tricks.

Versions: The arXiv version and the ECCC version are identical.

Presentations:
• Adam presented at Simons Institute "Proving and Using Pseudorandomness" workshop, March 2017 [video]
• I presented at Caltech's CS Theory Seminar, May 2017 [slides]
Targeted Pseudorandom Generators, Simulation Advice Generators, and Derandomizing Logspace
With Chris Umans. In STOC 2017.
Erratum (May 2017): In Lemma 2, the space complexity should be O(s + log w + log(1/δ) + log(1/γ)). That is, there is a missing O(log(1/δ)) term in the lemma statement. This does not affect anything else in the paper.

Not-so-abstract: There's a well-known conjecture in complexity theory ("L = BPL") that says that randomness isn't useful for solving problems using a small amount of computer memory. The most promising approach to proving this conjecture is to design a suitable kind of "pseudorandom generator". But in principle, one could hope to prove L = BPL using a totally different approach. In this paper, we prove a theorem that can be roughly stated as follows. If every true statement along the lines of L = BPL can be proven using the pseudorandom generator approach, then in fact L = BPL (actually the conclusion is slightly weaker).

Versions: The STOC proceedings version and the arXiv version are the same except for formatting.

Presentations:
Notes: This paper was invited to the SIAM Journal of Computing special issue for STOC 2017. Oded Goldreich has commented on this paper. In June 2018, I discovered these slides for a talk that Omer Reingold gave in 2010; one of the slides already mentions a statement that is essentially the main theorem proven in this paper.
The Adversarial Noise Threshold for Distributed Protocols
With Leonard J. Schulman. In SODA 2016.
Not-so-abstract: Imagine that many people (or computers) are connected by a network of communication channels. They would like to have a conversation, but unfortunately, an adversary is altering their messages to confuse them. The parties may deal with this noise by encoding their conversation in whatever way they like. In this paper, we determine the maximum amount of noise that can be tolerated in a couple different models as a function of the particular communication network.

Versions: I recommend that you read the SODA proceedings version. The SODA reviewers' feedback never got incorporated into the arXiv version.

Presentations:
• I presented at SODA, January 2016 [slides]
• I presented at Caltech's CMS "Celebration of Undergraduate Research", May 2015 [poster]
Notes: This paper is summarized in Section 7.3 of this survey article by Ran Gelles.

What's My Pro-Life Line?
With Alicia Torres. Ongoing, started 2018.
Description: This is a social media page. We're on Instagram and Facebook. We suggest messages and responses you can use to promote and defend the pro-life position with love and kindness.
Top 10 Blatant Contradictions in Math
Unpublished, 2016.
Description: This is a satirical listicle.
101 Illustrated Real Analysis Bedtime Stories
With Laura Shou and others. Unfinished, last updated 2016.
Description: This is a recreational book. We present real analysis in a way that is supposed to be enjoyable to read.
A "Can" of Worms
Unpublished, 2016.
Description: This is an essay I wrote as an undergrad for a class on free will taught by Christopher Hitchcock. In the essay, I respond to a specific objection to compatibilism, the thesis that free will is possible even in a deterministic universe.

Notes: This essay won the Gordon McClure Memorial Communication Prize in Philosophy from Caltech's HSS division.

Philosophical opinions:
Political opinions: Just in case you read my philosophical opinions and you're still looking for something to disagree with me about :)
• I support a ban on abortion. I think we should use nonviolent methods to support pregnant women. I support government-funded pregnancy centers and legally mandated paid maternity leave.
• I support repealing the second amendment. In the meantime, I support strict gun control laws that do not violate the second amendment.
• I support a ban on physician-assisted suicide and all forms of euthanasia.
• I support extensive government regulations and programs designed to protect the environment.
• I support a ban on any stem cell research that involves intentionally destroying human embryos.
• I support a ban on capital punishment.
• I oppose civil same-sex "marriage" constructs. I recognize that having homosexual inclinations does not mean that you've done anything wrong. I support legislation that protects the LGBT community from unjust discrimination and harrassment. I try to be a good ally.
• I oppose torture. More generally, I oppose almost all violent actions performed by the U.S. military. I am grateful for the service of those members of the military who honorably and ethically defend the country.
• I support originalism.
• I support making it sufficiently easy to legally immigrate to the U.S. that there is essentially no temptation to immigrate here illegally. I want the border between the U.S. and Mexico to be similar to the border between Texas and New Mexico.
• I support a ban on human cloning.
• I support expanded welfare programs.
• I support giving every citizen the right to vote, regardless of age or criminal history.
• I oppose efforts by the NSA to weaken cryptographic software.
Video games, etc.: I occasionally masquerade as a programmer, in order to provide you with ways to waste your time:
Contact: whoza@utexas.edu
As for wisdom, where does she come from?
Where is the place of understanding?
She is hidden from the eyes of every living thing;
even from the birds of the air she is concealed.
(Job 28:20-21)