Related papers: Algorithmic Perspective on Toda's Theorem
We establish two new direct product theorems for the randomized query complexity of Boolean functions. The first shows that computing $n$ copies of a function $f$, even with a small success probability of $\gamma^n$, requires $\Theta(n)$…
Grover's algorithm relies on the superposition and interference of quantum mechanics, which is more efficient than classical computing in specific tasks such as searching an unsorted database. Due to the high complexity of quantum…
This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness…
In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical…
Dichotomy theorems, which characterize the conditions under which a problem can be solved efficiently, have helped identify important tractability borders for as probabilistic query evaluation, view maintenance, query containment (among…
Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring…
Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…
The notion of weak truth-table reducibility plays an important role in recursion theory. In this paper, we introduce an elaboration of this notion, where a computable bound on the use function is explicitly specified. This elaboration…
Lloyd's algorithm is an iterative method that solves the quantization problem, i.e. the approximation of a target probability measure by a discrete one, and is particularly used in digital applications. This algorithm can be interpreted as…
We describe a method for inverting Gentzen's cut-elimination in classical first-order logic. Our algorithm is based on first computign a compressed representation of the terms present in the cut-free proof and then cut-formulas that realize…
We focus in this paper on generating models of quantified first-order formulas over built-in theories, which is paramount in software verification and bug finding. While standard methods are either geared toward proving the absence of…
Given a single observation from a Gaussian distribution with unknown mean $\theta$, we design computationally efficient procedures that can approximately generate an observation from a different target distribution $Q_{\theta}$ uniformly…
Many machine learning models involve solving optimization problems. Thus, it is important to deal with a large-scale optimization problem in big data applications. Recently, subsampled Newton methods have emerged to attract much attention…
It has been argued that reduction procedures are closely connected to the question about identity of proofs and that accepting certain reductions would lead to a trivialization of identity of proofs in the sense that every derivation of the…
The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to…
We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both…
We address the decision problem for a fragment of real analysis involving differentiable functions with continuous first derivatives. The proposed theory, besides the operators of Tarski's theory of reals, includes predicates for…
The ``impossibility theorem'' -- which is considered foundational in algorithmic fairness literature -- asserts that there must be trade-offs between common notions of fairness and performance when fitting statistical models, except in two…
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
The possibility of a fundamental consistency between the basic quantum principles and reduction (so-called wave function reduction) is reexamined. The mathematical description of an organized macroscopic device is constructed explicitly as…