Related papers: Algebraic Proof Complexity: Progress, Frontiers an…

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…

In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third…

Several recent results provide theoretical insights into the phenomena of adversarial examples. Existing results, however, are often limited due to a gap between the simplicity of the models studied and the complexity of those deployed in…

We consider the task of verifying the correctness of quantum computation for a restricted class of circuits which contain at most two basis changes. This contains circuits giving rise to the second level of the Fourier Hierarchy, the lowest…

This study examines clusterability testing for a signed graph in the bounded-degree model. Our contributions are two-fold. First, we provide a quantum algorithm with query complexity $\tilde{O}(N^{1/3})$ for testing clusterability, which…

This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of…

We use the implicitization procedure to generate polynomial equality constraints on the set of distributions induced by local interventions on variables governed by a causal Bayesian network with hidden variables. We show how we may reduce…

This is the full version of a paper submitted to the Computability in Europe (CiE 2023) conference, with all proofs omitted there. In 2012 P. D. Azar and S. Micali introduced a new model of interactive proofs, called "Rational Interactive…

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labelled with powers $x^n$ for binary encoded numbers $n$. It is shown that polynomial identity testing for…

We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over $\mathbb{F}_{2}$. Our main contributions include: 1. In STOC 2020, CHHLZ introduced a new technique to prove…

Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…

Connectivity problems like k-Path and k-Disjoint Paths relate to many important milestones in parameterized complexity, namely the Graph Minors Project, color coding, and the recent development of techniques for obtaining kernelization…

As a natural extension of the SAT problem, an array of proof systems for quantified Boolean formulas (QBF) have been proposed, many of which extend a propositional proof system to handle universal quantification. By formalising the…

A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into…

We study the problem of polynomial identity testing (PIT) for depth 2 arithmetic circuits over matrix algebra. We show that identity testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field F is polynomial time equivalent to…

All current investigations to analyze the derivational complexity of term rewrite systems are based on a single termination method, possibly preceded by transformations. However, the exclusive use of direct criteria is problematic due to…

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…

We determine the complexity of several constraint satisfaction problems using the heuristic algorithm, WalkSAT. At large sizes N, the complexity increases exponentially with N in all cases. Perhaps surprisingly, out of all the models…

An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…

We develop a new technique for proving distribution testing lower bounds for properties defined by inequalities involving the bin probabilities of the distribution in question. Using this technique we obtain new lower bounds for…