Searcharxiv⌕ Search

arXiv subjects

Ryan Williams

Publications and source records attributed to Ryan Williams.

At least 19 recordsLinked to original sources
arXiv2026

Automated Template-free Synthesis of Instruction-Centric Leakage Contracts for Black-Box CPUs

Side-channel attacks pose a significant security threat for modern computing platforms, because they exploit subtle discrepancies in CPU behaviors to leak sensitive information. To model the information leaked by a CPU via microarchitectural side-channels, recent work proposed leakage contracts: an ISA-level security abstraction that provides the foundations for secure CPU programming. Unfortunately, due to the complexity of current microarchitectures, devising a leakage contract for a CPU requires extensive manual effort and thus modern CPUs lack dedicated leakage contracts. We present a methodology to extract instruction-centric leakage contracts for major CPU architectures with minimal manual intervention. We implemented this technique in malcos, the first template-free tool that automates the synthesis of leakage contracts for black-box CPUs. We evaluate malcos on x86 and ARM CPUs, and show that the contracts it synthesizes are precise and sound with respect to all leaks observed during synthesis. Our results demonstrate that learning leakage contracts from black-box CPUs is feasible.

cs.CR↗
arXiv2026

Near-Maximum Circuit Lower Bounds for Exponential Time with Merlin-Arthur Queries

We prove a near-maximum ($2^n / n$) circuit lower bound for the complexity class $\mathsf{E}^{\mathrm{pr}\mathsf{MA}}/_1$, corresponding to exponential time with access to a promise-$\mathsf{MA}$ oracle and one bit of advice. Our proof incorporates the iterative win-win paradigm (Chen--Lu--Oliveira--Ren--Santhanam, FOCS'23), the reduction from the Range Avoidance problem to circuit lower bounds (Je\v{r}\'abek, Ann. Pure Appl. Log. '04; Korten, FOCS'21), and the PCP theorem. Crucial to our proof is the analysis of the complexity class $\mathsf{P}^\mathsf{NP}[{\textsf{#rounds}}=r, {\textsf{length}}=s]$, which is $\mathsf{P}^\mathsf{NP}$ with $r(n)$ adaptive rounds of $\mathsf{NP}$ queries, where each $\mathsf{NP}$ query has witness length $s(n)$.

cs.CC↗
arXiv2026

Beating Trivial Time for Tricky Triangle Tasks

For several well-studied triangle detection problems in the literature, the trivial enumeration algorithms are known to be optimal (up to the exponent) assuming popular fine-grained conjectures. For example, All-Edges Sparse Triangle and Sparse Monochromatic Triangle where each node has degree $n^{\delta}$ for some $\delta < 1$, and the Exact Triangle where edges have arbitrary weights, all have this property under the 3SUM Conjecture. However, as there are slightly nontrivial algorithms for 3SUM, it is natural to wonder if the trivial algorithm for these tricky triangle tasks might also be improved. Applying a variety of techniques from randomized algorithms, circuit complexity, and communication complexity, we present the first improvements over the trivial algorithms for each of these problems in the Word RAM model. Moreover, our algorithms can be implemented with only polysize $AC0$ operations on words. Extending our techniques, we also show how to solve the notorious 4-cycle detection problem on $n$-node graphs in $o(n^2)$ time, in a Word-RAM model with word size $w > \omega(\log^2 n)$. Along the way, we show how to sort $n$ items over a universe of size $2^u$ using only $AC0$ word operations in $O(n u \log n)/w$ time.

cs.DS↗
arXiv2026

A Theory for Probabilistic Polynomial-Time Reasoning

In this work, we propose a new bounded arithmetic theory, denoted $APX_1$, designed to formalize a broad class of probabilistic arguments commonly used in theoretical computer science. Under plausible assumptions, $APX_1$ is strictly weaker than previously proposed frameworks, such as the theory $APC_1$ introduced in the seminal work of Jerabek (2007). From a computational standpoint, $APX_1$ is closely tied to approximate counting and to the central question in derandomization, the prBPP versus prP problem, whereas $APC_1$ is linked to the dual weak pigeonhole principle and to the existence of Boolean functions with exponential circuit complexity. A key motivation for introducing $APX_1$ is that its weaker axioms expose finer proof-theoretic structure, making it a natural setting for several lines of research, including unprovability of complexity conjectures and reverse mathematics of randomized lower bounds. In particular, the framework we develop for $APX_1$ enables the formulation of precise questions concerning the provability of prBPP=prP in deterministic feasible mathematics. Since the (un)provability of P versus NP in bounded arithmetic has long served as a central theme in the field, we expect this line of investigation to be of particular interest. Our technical contributions include developing a comprehensive foundation for probabilistic reasoning from weaker axioms, formalizing non-trivial results from theoretical computer science in $APX_1$, and establishing a tailored witnessing theorem for its provably total TFNP problems. As a byproduct of our analysis of the minimal proof-theoretic strength required to formalize statements arising in theoretical computer science, we resolve an open problem regarding the provability of $AC^0$ lower bounds in $PV_1$, which was considered in earlier works by Razborov (1995), Krajicek (1995), and Muller and Pich (2020).

cs.CC↗
arXiv2025

Qualia & Natural Selection: Formal Constraints on the Evolution of Consciousness

This paper explores foundational questions about the relationship of qualia to natural selection. The primary result is a derivation of specific formal conditions under which structural systems subject to natural selection can convey consistent effects in an associated qualitative domain, placing theoretical and empirical constraints on theories of consciousness. In order to achieve this result, information-theoretic measures are developed to quantify the mutual determinability between structure and quality, quantifying fidelity between the two domains. The fidelities represented by that space are then incorporated into the Price Equation to yield key bounds on the transmission of selective effects between domains. Finally, transmission of higher-order structures between domains is explored. Placement within a broader philosophical context can be found in the companion paper Structure & Quality.

q-bio.NC↗
arXiv2025

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

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.

q-bio.NC↗
arXiv2024

Grounded Computation & Consciousness: A Framework for Exploring Consciousness in Machines & Other Organisms

Computational modeling is a critical tool for understanding consciousness, but is it enough on its own? This paper discusses the necessity for an ontological basis of consciousness, and introduces a formal framework for grounding computational descriptions into an ontological substrate. Utilizing this technique, a method is demonstrated for estimating the difference in qualitative experience between two systems. This framework has wide applicability to computational theories of consciousness.

q-bio.NC↗
arXiv2024

Parallel Play Saves Quantifiers

The number of quantifiers needed to express first-order properties is captured by two-player combinatorial games called multi-structural (MS) games. We play these games on linear orders and strings, and introduce a technique we call "parallel play", that dramatically reduces the number of quantifiers needed in many cases. Linear orders and strings are the most basic representatives of ordered structures -- a class of structures that has historically been notoriously difficult to analyze. Yet, in this paper, we provide upper bounds on the number of quantifiers needed to characterize different-sized subsets of these structures, and prove that they are tight up to constant factors, including, in some cases, up to a factor of $1+\varepsilon$, for arbitrarily small $\varepsilon$.

cs.LO↗
arXiv2024

Real-Time Systems Optimization with Black-box Constraints and Hybrid Variables

When optimizing real-time systems, designers often face a challenging problem where the schedulability constraints are non-convex, non-continuous, or lack an analytical form to understand their properties. Although the optimization framework NORTH proposed in previous work is general (it works with arbitrary schedulability analysis) and scalable, it can only handle problems with continuous variables, which limits its application. In this paper, we extend the applications of the framework NORTH to problems with a hybrid of continuous and discrete variables. This is achieved in a coordinate-descent method, where the continuous and discrete variables are optimized separately during iterations. The new framework, NORTH+, improves around 20% solution quality than NORTH in experiments.

eess.SY↗
arXiv2024

Joint Optimization of Continuous Variables and Priority Assignments for Real-Time Systems with Black-box Schedulability Constraints

In real-time systems optimization, designers often face a challenging problem posed by the non-convex and non-continuous schedulability conditions, which may even lack an analytical form to understand their properties. To tackle this challenging problem, we treat the schedulability analysis as a black box that only returns true/false results. We propose a general and scalable framework to optimize real-time systems, named Numerical Optimizer with Real-Time Highlight (NORTH). NORTH is built upon the gradient-based active-set methods from the numerical optimization literature but with new methods to manage active constraints for the non-differentiable schedulability constraints. In addition, we also generalize NORTH to NORTH+, to collaboratively optimize certain types of discrete variables (e.g., priority assignments, categorical variables) with continuous variables based on numerical optimization algorithms. We demonstrate the algorithm performance with two example applications: energy minimization based on dynamic voltage and frequency scaling (DVFS), and optimization of control system performance. In these experiments, NORTH achieved $10^2$ to $10^5$ times speed improvements over state-of-the-art methods while maintaining similar or better solution quality. NORTH+ outperforms NORTH by 30% with similar algorithm scalability. Both NORTH and NORTH+ support black-box schedulability analysis, ensuring broad applicability.

eess.SY↗
arXiv2023

Optimizing Logical Execution Time Model for Both Determinism and Low Latency

The Logical Execution Time (LET) programming model has recently received considerable attention, particularly because of its timing and dataflow determinism. In LET, task computation appears always to take the same amount of time (called the task's LET interval), and the task reads (resp. writes) at the beginning (resp. end) of the interval. Compared to other communication mechanisms, such as implicit communication and Dynamic Buffer Protocol (DBP), LET performs worse on many metrics, such as end-to-end latency (including reaction time and data age) and time disparity jitter. Compared with the default LET setting, the flexible LET (fLET) model shrinks the LET interval while still guaranteeing schedulability by introducing the virtual offset to defer the read operation and using the virtual deadline to move up the write operation. Therefore, fLET has the potential to significantly improve the end-to-end timing performance while keeping the benefits of deterministic behavior on timing and dataflow. To fully realize the potential of fLET, we consider the problem of optimizing the assignments of its virtual offsets and deadlines. We propose new abstractions to describe the task communication pattern and new optimization algorithms to explore the solution space efficiently. The algorithms leverage the linearizability of communication patterns and utilize symbolic operations to achieve efficient optimization while providing a theoretical guarantee. The framework supports optimizing multiple performance metrics and guarantees bounded suboptimality when optimizing end-to-end latency. Experimental results show that our optimization algorithms improve upon the default LET and its existing extensions and significantly outperform implicit communication and DBP in terms of various metrics, such as end-to-end latency, time disparity, and its jitter.

eess.SY↗
arXiv2023

Faster Detours in Undirected Graphs

The $k$-Detour problem is a basic path-finding problem: given a graph $G$ on $n$ vertices, with specified nodes $s$ and $t$, and a positive integer $k$, the goal is to determine if $G$ has an $st$-path of length exactly $\text{dist}(s, t) + k$, where $\text{dist}(s, t)$ is the length of a shortest path from $s$ to $t$. The $k$-Detour problem is NP-hard when $k$ is part of the input, so researchers have sought efficient parameterized algorithms for this task, running in $f(k)\text{poly}(n)$ time, for $f$ as slow-growing as possible. We present faster algorithms for $k$-Detour in undirected graphs, running in $1.853^k \text{poly}(n)$ randomized and $4.082^k \text{poly}(n)$ deterministic time. The previous fastest algorithms for this problem took $2.746^k \text{poly}(n)$ randomized and $6.523^k \text{poly}(n)$ deterministic time [Bez\'akov\'a-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact that detecting a path of a given length in an undirected graph is easier if we are promised that the path belongs to what we call a "bipartitioned" subgraph, where the nodes are split into two parts and the path must satisfy constraints on those parts. Previously, this idea was used to obtain the fastest known algorithm for finding paths of length $k$ in undirected graphs [Bj\"orklund-Husfeldt-Kaski-Koivisto, JCSS 2017]. Our work has direct implications for the $k$-Longest Detour problem: in this problem, we are given the same input as in $k$-Detour, but are now tasked with determining if $G$ has an $st$-path of length at least $\text{dist}(s, t) + k.$ Our results for k-Detour imply that we can solve $k$-Longest Detour in $3.432^k \text{poly}(n)$ randomized and $16.661^k \text{poly}(n)$ deterministic time. The previous fastest algorithms for this problem took $7.539^k \text{poly}(n)$ randomized and $42.549^k \text{poly}(n)$ deterministic time [Fomin et al., STACS 2022].

cs.DS↗
arXiv2022

On the Number of Quantifiers as a Complexity Measure

In 1981, Neil Immerman described a two-player game, which he called the "separability game" \cite{Immerman81}, that captures the number of quantifiers needed to describe a property in first-order logic. Immerman's paper laid the groundwork for studying the number of quantifiers needed to express properties in first-order logic, but the game seemed to be too complicated to study, and the arguments of the paper almost exclusively used quantifier rank as a lower bound on the total number of quantifiers. However, last year Fagin, Lenchner, Regan and Vyas rediscovered the games, provided some tools for analyzing them, and showed how to utilize them to characterize the number of quantifiers needed to express linear orders of different sizes. In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results. First we carefully distinguish minimum number of quantifiers from the more usual descriptive complexity measures, minimum quantifier rank and minimum number of variables. Then, for each positive integer $k$, we give an explicit example of a property of finite structures (in particular, of finite graphs) that can be expressed with a sentence of quantifier rank $k$, but where the same property needs $2^{\Omega (k^2)}$ quantifiers to be expressed.

cs.CC↗
arXiv2022

Constructive Separations and Their Consequences

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.

cs.CC↗
arXiv2021

Smaller ACC0 Circuits for Symmetric Functions

What is the power of constant-depth circuits with $MOD_m$ gates, that can count modulo $m$? Can they efficiently compute MAJORITY and other symmetric functions? When $m$ is a constant prime power, the answer is well understood: Razborov and Smolensky proved in the 1980s that MAJORITY and $MOD_m$ require super-polynomial-size $MOD_q$ circuits, where $q$ is any prime power not dividing $m$. However, relatively little is known about the power of $MOD_m$ circuits for non-prime-power $m$. For example, it is still open whether every problem in $EXP$ can be computed by depth-$3$ circuits of polynomial size and only $MOD_6$ gates. We shed some light on the difficulty of proving lower bounds for $MOD_m$ circuits, by giving new upper bounds. We construct $MOD_m$ circuits computing symmetric functions with non-prime power $m$, with size-depth tradeoffs that beat the longstanding lower bounds for $AC^0[m]$ circuits for prime power $m$. Our size-depth tradeoff circuits have essentially optimal dependence on $m$ and $d$ in the exponent, under a natural circuit complexity hypothesis. For example, we show for every $\varepsilon > 0$ that every symmetric function can be computed with depth-3 $MOD_m$ circuits of $\exp(O(n^{\varepsilon}))$ size, for a constant $m$ depending only on $\varepsilon > 0$. That is, depth-$3$ $CC^0$ circuits can compute any symmetric function in \emph{subexponential} size. This demonstrates a significant difference in the power of depth-$3$ $CC^0$ circuits, compared to other models: for certain symmetric functions, depth-$3$ $AC^0$ circuits require $2^{\Omega(\sqrt{n})}$ size [H{\aa}stad 1986], and depth-$3$ $AC^0[p^k]$ circuits (for fixed prime power $p^k$) require $2^{\Omega(n^{1/6})}$ size [Smolensky 1987]. Even for depth-two $MOD_p \circ MOD_m$ circuits, $2^{\Omega(n)}$ lower bounds were known [Barrington Straubing Th\'erien 1990].

cs.CC↗
arXiv2021

MAJORITY-3SAT (and Related Problems) in Polynomial Time

Majority-SAT is the problem of determining whether an input $n$-variable formula in conjunctive normal form (CNF) has at least $2^{n-1}$ satisfying assignments. Majority-SAT and related problems have been studied extensively in various AI communities interested in the complexity of probabilistic planning and inference. Although Majority-SAT has been known to be PP-complete for over 40 years, the complexity of a natural variant has remained open: Majority-$k$SAT, where the input CNF formula is restricted to have clause width at most $k$. We prove that for every $k$, Majority-$k$SAT is in P. In fact, for any positive integer $k$ and rational $\rho \in (0,1)$ with bounded denominator, we give an algorithm that can determine whether a given $k$-CNF has at least $\rho \cdot 2^n$ satisfying assignments, in deterministic linear time (whereas the previous best-known algorithm ran in exponential time). Our algorithms have interesting positive implications for counting complexity and the complexity of inference, significantly reducing the known complexities of related problems such as E-MAJ-$k$SAT and MAJ-MAJ-$k$SAT. At the heart of our approach is an efficient method for solving threshold counting problems by extracting sunflowers found in the corresponding set system of a $k$-CNF. We also show that the tractability of Majority-$k$SAT is somewhat fragile. For the closely related GtMajority-SAT problem (where we ask whether a given formula has greater than $2^{n-1}$ satisfying assignments) which is known to be PP-complete, we show that GtMajority-$k$SAT is in P for $k\le 3$, but becomes NP-complete for $k\geq 4$. These results are counterintuitive, because the ``natural'' classifications of these problems would have been PP-completeness, and because there is a stark difference in the complexity of GtMajority-$k$SAT and Majority-$k$SAT for all $k\ge 4$.

cs.CC↗
arXiv2020

Fast Low-Space Algorithms for Subset Sum

We consider the canonical Subset Sum problem: given a list of positive integers $a_1,\ldots,a_n$ and a target integer $t$ with $t > a_i$ for all $i$, determine if there is an $S \subseteq [n]$ such that $\sum_{i \in S} a_i = t$. The well-known pseudopolynomial-time dynamic programming algorithm [Bellman, 1957] solves Subset Sum in $O(nt)$ time, while requiring $\Omega(t)$ space. In this paper we present algorithms for Subset Sum with $\tilde O(nt)$ running time and much lower space requirements than Bellman's algorithm, as well as that of prior work. We show that Subset Sum can be solved in $\tilde O(nt)$ time and $O(\log(nt))$ space with access to $O(\log n \log \log n+\log t)$ random bits. This significantly improves upon the $\tilde O(n t^{1+\varepsilon})$-time, $\tilde O(n\log t)$-space algorithm of Bringmann (SODA 2017). We also give an $\tilde O(n^{1+\varepsilon}t)$-time, $O(\log(nt))$-space randomized algorithm, improving upon previous $(nt)^{O(1)}$-time $O(\log(nt))$-space algorithms by Elberfeld, Jakoby, and Tantau (FOCS 2010), and Kane (2010). In addition, we also give a $\mathrm{poly} \log(nt)$-space, $\tilde O(n^2 t)$-time deterministic algorithm. We also study time-space trade-offs for Subset Sum. For parameter $1\le k\le \min\{n,t\}$, we present a randomized algorithm running in $\tilde O((n+t)\cdot k)$ time and $O((t/k) \mathrm{polylog} (nt))$ space. As an application of our results, we give an $\tilde{O}(\min\{n^2/\varepsilon, n/\varepsilon^2\})$-time and $\mathrm{polylog}(nt)$-space algorithm for "weak" $\varepsilon$-approximations of Subset Sum.

cs.DS↗