Related papers: On the resolution of the sensitivity conjecture
Satisfiability solvers are increasingly playing a key role in software verification, with particularly effective use in the analysis of security vulnerabilities. String processing is a key part of many software applications, such as…
One of the major outstanding foundational problems about boolean functions is the sensitivity conjecture, which (in one of its many forms) asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that…
In this tutorial, we will survey known results on the complexity of conjunctive query evaluation in different settings, ranging from Boolean queries over counting to more complex models like enumeration and direct access. A particular focus…
In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the log-rank conjecture for communication complexity, up to log n factor, for any Boolean function composed with AND function as the inner gadget. One of the main tools…
We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open.
Determining the maximal separation between sensitivity and block sensitivity of Boolean functions is of interest for computational complexity theory. We construct a sequence of Boolean functions with bs(f) = 1/2 s(f)^2 + 1/2 s(f). The best…
The sensitivity conjecture is a longstanding conjecture concerning the relationship between the degree and sensitivity of a Boolean function. In 2015, a communication game was formulated by Justin Gilmer, Michal Kouck\'{y}, and Michael Saks…
Inspired by Solomonoffs theory of inductive inference, we propose a prior based on circuit complexity. There are several advantages to this approach. First, it relies on a complexity measure that does not depend on the choice of UTM. There…
Given an $n$-bit Boolean function with a complexity measure (such as block sensitivity, query complexity, etc.) $M(f) = k$, the hardness condensation question asks whether $f$ can be restricted to $O(k)$ variables such that the complexity…
A Boolean function $f({\vec x})$ is sensitive to bit $x_i$ if there is at least one input vector $\vec x$ and one bit $x_i$ in $\vec x$, such that changing $x_i$ changes $f$. A function has sensitivity $s$ if among all input vectors, the…
The study of complexity measures of Boolean functions led Nisan and Szegedy to state the sensitivity conjecture in 1994, claiming a polynomial relation between degree and sensitivity. This problem remained unsolved until 2019, when Huang…
Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, the deterministic query complexity, $D(f)$, is at most quartic in the quantum query complexity, $Q(f)$: $D(f) = O(Q(f)^4)$. This matches the…
Previous work on sensitivity analysis in Bayesian networks has focused on single parameters, where the goal is to understand the sensitivity of queries to single parameter changes, and to identify single parameter changes that would enforce…
One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are…
Unmeasured confounding may undermine the validity of causal inference with observational studies. Sensitivity analysis provides an attractive way to partially circumvent this issue by assessing the potential influence of unmeasured…
Sensitivity analysis informs causal inference by assessing the sensitivity of conclusions to departures from assumptions. The consistency assumption states that there are no hidden versions of treatment and that the outcome arising…
We address the need to generate faithful explanations of "black box" Deep Learning models. Several tests have been proposed to determine aspects of faithfulness of explanation methods, but they lack cross-domain applicability and a rigorous…
The proliferation of agentic systems has thrust the reasoning capabilities of AI into the forefront of contemporary machine learning. While it is known that there \emph{exist} neural networks which can reason through any Boolean task…
In this paper we construct a cyclically invariant Boolean function whose sensitivity is $\Theta(n^{1/3})$. This result answers two previously published questions. Tur\'an (1984) asked if any Boolean function, invariant under some transitive…
Binary neural networks (BNNs) are increasingly deployed in edge computing applications due to their low hardware complexity and high energy efficiency. However, verifying the robustness of BNNs against input perturbations, including…