Related papers: Algorithmic Perspective on Toda's Theorem
Factor models are a very efficient way to describe high dimensional vectors of data in terms of a small number of common relevant factors. This problem, which is of fundamental importance in many disciplines, is usually reformulated in…
In this paper we present the formal, computer-supported verification of a functional implementation of Buchberger's critical-pair/completion algorithm for computing Gr\"obner bases in reduction rings. We describe how the algorithm can be…
Decision theories offer principled methods for making choices under various types of uncertainty. Algorithms that implement these theories have been successfully applied to a wide range of real-world problems, including materials and drug…
The Quantum Fourier Transform (QFT) is a key component of many important quantum algorithms, most famously as being the essential ingredient in Shor's algorithm for factoring products of primes. Given its remarkable capability, one would…
Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where non-termination happens, more structure. In this paper we consider…
How can complexity theory and algorithms benefit from practical advances in computing? We give a short overview of some prior work using practical computing to attack problems in computational complexity and algorithms, informally describe…
This article describes an evaluation of Automated Theorem Proving (ATP) systems on problems taken from the QMLTP library of first-order modal logic problems. Principally, the problems are translated to both typed first-order and…
Determining the validity of a quantified Boolean formula (QBF) is a PSPACE-complete problem with rich expressive power. Despite interest in efficient solvers, there is, compared to problems in NP, a lack of positive theoretical results, and…
Algorithms are the engine for reproducible problem-solving. We present a framework automating algorithm discovery by conceptualizing them as sequences of operations, represented as tokens. These computational tokens are chained using a…
Casting machine learning as a type of search, we demonstrate that the proportion of problems that are favorable for a fixed algorithm is strictly bounded, such that no single algorithm can perform well over a large fraction of them. Our…
With the relentless rise of computer power, there is a widespread expectation that computers can solve the most pressing problems of science, and even more besides. We explore the limits of computational modelling and conclude that, in the…
This book can be seen either as a text on theorem proving that uses techniques from general algebra, or else as a text on general algebra illustrated and made concrete by practical exercises in theorem proving. The book considers several…
The problem of deciding the validity (QSAT) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a…
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…
In this thesis, we introduce a new quantum Turing machine (QTM) model that supports general quantum operators, together with its pushdown, counter, and finite automaton variants, and examine the computational power of classical and quantum…
When mining large datasets in order to predict new data, limitations of the principles behind statistical machine learning pose a serious challenge not only to the Big Data deluge, but also to the traditional assumptions that data…
A coarse description of a subset A of omega is a subset D of omega such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse…
In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically…
Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring…
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons…