Related papers: Symbolic Arithmetic and Integer Factorization
Integer sorting is a fundamental problem in computer science. This paper studies parallel integer sort both in theory and in practice. In theory, we show tighter bounds for a class of existing practical integer sort algorithms, which…
Positional games are a mathematical class of two-player games comprising Tic-tac-toe and its generalizations. We propose a novel encoding of these games into Quantified Boolean Formulas (QBFs) such that a game instance admits a winning…
The minimization of propositional formulae is a classical problem in logic, whose first algorithms date back at least to the 1950s with the works of Quine and Karnaugh. Most previous work in the area has focused on obtaining minimal, or…
Quantum circuits which perform integer arithmetic could potentially outperform their classical counterparts. In this paper, a quantum circuit is considered which performs a specific computational pattern on classically represented integers…
The problem of factorising positive integer $N$ into two integer factors $x$ and $y$ is first reformulated as an optimisation problem over the positive integer domain of either of the Diophantine polynomials $Q_N(x,y)=N^2(N-xy)^2 +…
Probabilistic graphical models are a key tool in machine learning applications. Computing the partition function, i.e., normalizing constant, is a fundamental task of statistical inference but it is generally computationally intractable,…
This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer…
Multiresolution Matrix Factorization (MMF) is unusual amongst fast matrix factorization algorithms in that it does not make a low rank assumption. This makes MMF especially well suited to modeling certain types of graphs with complex…
A parallel method for computing Boolean expressions based on the properties of finite free Boolean algebras is presented. We also show how various finite combinatorial objects can be codded in the formalism of Boolean algebras and counted…
The Quantum Fourier Transform (QFT) is a fundamental component of many quantum computing algorithms. In this paper, we present an alternative method for factoring this transformation. Inspired by this approach, we introduce a new quantum…
In this paper, we propose an algebraic formalization of the two important classes of dynamic programming algorithms called forward and forward-backward algorithms. They are generalized extensively in this study so that a wide range of other…
In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…
Boolean matrix factorization (BMF) is a fundamental tool for analyzing binary data and discovering latent information hidden in the data. Formal Concept Analysis (FCA) provides us with an essential insight into BMF and the design of…
Symbolic regression that aims to detect underlying data-driven models has become increasingly important for industrial data analysis. For most existing algorithms such as genetic programming (GP), the convergence speed might be too slow for…
Bayesian optimization (BO) methods are useful for optimizing functions that are expensive to evaluate, lack an analytical expression and whose evaluations can be contaminated by noise. These methods rely on a probabilistic model of the…
We investigate the special class of formulas made up of arbitrary but finite com- binations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such…
We present fast and highly parallelized versions of Shor's algorithm. With a sizable quantum computer it would then be possible to factor numbers with millions of digits. The main algorithm presented here uses FFT-based fast integer…
We propose reductions to quantified Boolean formulas (QBF) as a new approach to showing fixed-parameter linear algorithms for problems parameterized by treewidth. We demonstrate the feasibility of this approach by giving new algorithms for…
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same…
Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an…