About

Ryan Williams is a Professor of Electrical Engineering and Computer Science at MIT, affiliated with the departments of CS and AI+D. His research areas include the Theory of Computation, where he focuses on the development and analysis of algorithms, computational complexity, and the theoretical foundations of computer science. He is involved in exploring systems that interact with the external world through perception, communication, and action, as well as systems that learn, make decisions, and adapt to changing environments. His work leverages computational, theoretical, and experimental tools to develop groundbreaking sensors, energy transducers, and physical substrates for computation, addressing shared challenges facing humanity.

Research topics

• Algorithm
• Combinatorics
• Discrete mathematics
• Mathematics
• Statistics

Selected publications

• Sovereign-Mohawk: A Formally Verified 10M-Node Federated Learning Architecture with Byzantine Resilience, Differential Privacy, and Recursive Zero-Knowledge Verifiability

  2026-01-01

  articleOpen access1st authorCorresponding
  PublisherDOI

• Structure & Quality: Conceptual and Formal Foundations for the Mind-Body Problem

  ArXiv.org · 2025-04-23

  preprintOpen access1st authorCorresponding

  This paper explores the hard problem of consciousness from a different perspective. Instead of drawing distinctions between the physical and the mental, an exploration of a more foundational relationship is examined: the relationship between structure and quality. Information-theoretic measures are developed to quantify the mutual determinability between structure and quality, including a novel Q-S space for analyzing fidelity between the two domains. This novel space naturally points toward a five-fold categorization of possible relationships between structural and qualitative properties, illustrating each through conceptual and formal models. The ontological implications of each category are examined, shedding light on debates around functionalism, emergentism, idealism, panpsychism, and neutral monism. This new line of inquiry has established a framework for deriving theoretical constraints on qualitative systems undergoing evolution that is explored in my companion paper, Qualia & Natural Selection.

  PublisherOA PDFDOI

• Comment on “SAT requires exhaustive search”

  Frontiers of Computer Science · 2025-09-17

  articleOpen accessSenior author
  PublisherOA PDFDOI

• The Orthogonal Vectors Conjecture and Nonuniform Circuit Lower Bounds

  SIAM Journal on Computing · 2025-10-14

  article1st authorCorresponding
  PublisherDOI

• When Connectivity Is Hard, Random Walks Are Easy with Non-determinism

  2025-06-15

  articleSenior author

  STOC ’25, Prague, Czechia

  PublisherDOI

• Simulating Time With Square-Root Space

  ArXiv.org · 2025-02-25

  preprintOpen access1st authorCorresponding

  We show that for all functions $t(n) \geq n$, every multitape Turing machine running in time $t$ can be simulated in space only $O(\sqrt{t \log t})$. This is a substantial improvement over Hopcroft, Paul, and Valiant's simulation of time $t$ in $O(t/\log t)$ space from 50 years ago [FOCS 1975, JACM 1977]. Among other results, our simulation implies that bounded fan-in circuits of size $s$ can be evaluated on any input in only $\sqrt{s} \cdot poly(\log s)$ space, and that there are explicit problems solvable in $O(n)$ space which require $n^{2-\varepsilon}$ time on a multitape Turing machine for all $\varepsilon > 0$, thereby making a little progress on the $P$ versus $PSPACE$ problem. Our simulation reduces the problem of simulating time-bounded multitape Turing machines to a series of implicitly-defined Tree Evaluation instances with nice parameters, leveraging the remarkable space-efficient algorithm for Tree Evaluation recently found by Cook and Mertz [STOC 2024].

  PublisherOA PDFDOI

• PO-06-146 FIRST LOOK AT CARE DISPARITIES IN PACEMAKER IMPLANTATION RATES AMONG NATIVE AMERICANS

  Heart Rhythm · 2025-04-01

  article
  PublisherDOI

• Simulating Time with Square-Root Space

  2025-06-15 · 1 citations

  articleOpen access1st authorCorresponding

  We show that for all functions t ( n ) ≥ n , every multitape Turing machine running in time t can be simulated in space only \(O(\sqrt {t \log t}) \) . This is a substantial improvement over Hopcroft, Paul, and Valiant’s simulation of time t in O ( t /log t ) space from 50 years ago [FOCS 1975, JACM 1977]. Among other results, our simulation implies that bounded fan-in circuits of size s can be evaluated on any input in only \(\sqrt {s} \cdot \text{poly}(\log s) \) space, and that there are explicit problems solvable in O ( n ) space which require n 2 − ε time on a multitape Turing machine for all ε > 0, thereby making a little progress on the \({\sf P} \) versus \({\sf PSPACE} \) problem. Our simulation reduces the problem of simulating time-bounded multitape Turing machines to a series of implicitly-defined Tree Evaluation instances with nice parameters, leveraging the remarkable space-efficient algorithm for Tree Evaluation recently found by Cook and Mertz [STOC 2024].

  PublisherOA PDFDOI

• Constructive Separations and Their Consequences

  TheoretiCS · 2024-02-15

  articleOpen accessSenior author

  For a complexity class $C$ and language $L$, a constructive separation of $L \notin C$ gives an efficient algorithm (also called a refuter) to find counterexamples (bad inputs) for every $C$-algorithm attempting to decide $L$. We study the questions: Which lower bounds can be made constructive? What are the consequences of constructive separations? We build a case that "constructiveness" serves as a dividing line between many weak lower bounds we know how to prove, and strong lower bounds against $P$, $ZPP$, and $BPP$. Put another way, constructiveness is the opposite of a complexity barrier: it is a property we want lower bounds to have. Our results fall into three broad categories. 1. Our first set of results shows that, for many well-known lower bounds against streaming algorithms, one-tape Turing machines, and query complexity, as well as lower bounds for the Minimum Circuit Size Problem, making these lower bounds constructive would imply breakthrough separations ranging from $EXP \neq BPP$ to even $P \neq NP$. 2. Our second set of results shows that for most major open problems in lower bounds against $P$, $ZPP$, and $BPP$, including $P \neq NP$, $P \neq PSPACE$, $P \neq PP$, $ZPP \neq EXP$, and $BPP \neq NEXP$, any proof of the separation would further imply a constructive separation. Our results generalize earlier results for $P \neq NP$ [Gutfreund, Shaltiel, and Ta-Shma, CCC 2005] and $BPP \neq NEXP$ [Dolev, Fandina and Gutfreund, CIAC 2013]. 3. Our third set of results shows that certain complexity separations cannot be made constructive. We observe that for all super-polynomially growing functions $t$, there are no constructive separations for detecting high $t$-time Kolmogorov complexity (a task which is known to be not in $P$) from any complexity class, unconditionally.

  PublisherOA PDFDOI

• The Orthogonal Vectors Conjecture and Non-Uniform Circuit Lower Bounds

  2024-10-27 · 2 citations

  article1st authorCorresponding

  A line of work has shown how nontrivial uniform algorithms for analyzing circuits can be used to derive non-uniform circuit lower bounds. We show how the non-existence of nontrivial circuit-analysis algorithms can also imply non-uniform circuit lower bounds. Our connections yield new win-win circuit lower bounds, and suggest a potential approach to refuting the Orthogonal Vectors Conjecture in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log n)$</tex> -dimensional case, which would be sufficient for refuting the Strong Exponential Time Hypothesis (SETH). For example, we show that at least one of the following holds: • There is an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\varepsilon&gt;0$</tex> such that for infinitely many <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex>, read-once 2-DNFs on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> variables cannot be simulated by non-uniform <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^{\varepsilon n}$</tex> -size depth-two exact threshold circuits. It is already a notorious open problem to prove that the class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$E^{N P}$</tex> does not have polynomial-size depth-two exact threshold circuits, so such a lower bound would be a significant advance in low-depth circuit complexity. In fact, a stronger lower bound holds in this case: the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^n \times 2^n$</tex> Disjointness Matrix (well-studied in communication complexity) cannot be expressed by a linear combination of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^{o(n)}$</tex> structured matrices that we call “equality matrices”. • For every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$c \geq 1$</tex> and every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\varepsilon&gt;0$</tex>, Orthogonal Vectors on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> vectors in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$c \log n$</tex> dimensions can be solved in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n^{1+\varepsilon}$</tex> uniform deterministic time. This case would provide a strong refutation of the Orthogonal Vectors conjecture, and of SETH: for example, CNF-SAT on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> variables and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(n)$</tex> clauses could be solved in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^{n / 2+o(n)}$</tex> time. Moreover, this case would imply non-uniform circuit lower bounds for the class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$E^{NP}$</tex>, against Valiant series-parallel circuits. Inspired by this connection, we give evidence from SAT/SMT solvers that the first item (in particular, the Disjointness lower bound) may be false in its full generality. In particular, we present a systematic approach to solving Orthogonal Vectors via constant-sized decompositions of the Disjointness Matrix, which already yields interesting new algorithms. For example, using a linear combination of 6 equality matrices that express <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^6 \times 2^6$</tex> Disjointness, we derive an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}\left(n \cdot 6^{d / 6}\right) \leq \tilde{O}\left(n \cdot 1. \dot{35^d}\right)$</tex> time and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n \cdot \operatorname{poly}(\log n, d)$</tex> space algorithm for Orthogonal Vectors on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> vectors in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d$</tex> dimensions. We show similar results for counting pairs of orthogonal vectors.

  PublisherDOI

Recent grants

• AF: Small: Lower Bounds in Complexity Theory Via Algorithms

  NSF · $500k · 2021–2024

• AF: Large: Collaborative Research: Exploiting Duality between Algorithms and Complexity

  NSF · $450k · 2012–2016

• CAREER: Common Links in Algorithms and Complexity

  NSF · $503k · 2017–2021

• CAREER: Common Links in Algorithms and Complexity

  NSF · $210k · 2015–2017

• AF:Small:Limitations on Algebraic Methods via Boolean Complexity Theory

  NSF · $71k · 2017–2017

Frequent coauthors

• Huacheng Yu

  Tsinghua University

  19 shared

• Virginia Vassilevska Williams

  Massachusetts Institute of Technology

  17 shared

• Lijie Chen

  16 shared

• D Litt

  University of North Texas Health Science Center

  16 shared

• Amir Abboud

  14 shared

• Cody D. Murray

  Massachusetts Institute of Technology

  11 shared

• Virginia Vassilevska

  Carnegie Mellon University

  10 shared

• Suguru Tamaki

  10 shared

Education

• PhD, Computer Science

  Carnegie Mellon University

  2007

Awards & honors

• 2025-26 EECS Faculty Award Roundup
• 2023-24 EECS Faculty Award Roundup