Related papers: The Algebraic View of Computation
Some notions in mathematics can be considered relative. Relative is a term used to denote when the variation in the position of an observer implies variation in properties or measures on the observed object. We know, from Skolem theorem,…
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…
This chapter summarizes quantum computation, including the motivation for introducing quantum resources into computation and how quantum computation is done. Finally, this chapter articulates advantages and limitations of quantum…
We give a definition of partition C*-algebras: To any partition of a finite set, we assign algebraic relations for a matrix of generators of a universal C*-algebra. We then prove how certain relations may be deduced from others and we…
Basic arithmetic is the cornerstone of mathematics and computer sciences. In arithmetic, 'division by zero' is an undefined operation and any attempt at extending logic for algebraic division to incorporate division by zero has resulted in…
It is first pointed out that there is a common mathematical model for the universe and the quantum computer. The former is called the histories approach to quantum mechanics and the latter is called measurement based quantum computation.…
In the present paper, we describe some experiences in using programming, commands and graphical interfaces based on computer algebra systems, as tools for learning Physics and Mathematics.
Anyons obtained from a finite gauge theory have a computational power that depends on the symmetry group. The relationship between group structure and computational power is discussed in this paper. In particular, it is shown that anyons…
We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…
Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics. It concerns a utilization of quantum mechanics to improve the efficiency of computation. Here we present a gentle introduction…
Logic really is just algebra, given one uses the right kind of algebra, and the right kind of logic. The right kind of algebra is abstraction algebra, and the right kind of logic is abstraction logic.
Self-replication is central to all life, and yet how it dynamically emerges in physical, non-equilibrium systems remains poorly understood. Von Neumann's pioneering work in the 1940s and subsequent developments suggest a natural hypothesis:…
The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric…
The dialogue develops arguments for and against adopting a new world system, info-computationalist naturalism, that is poised to replace the traditional mechanistic world system. We try to figure out what the info-computational paradigm…
An abstract mathematical framework is presented in this paper as a unification of several deformed or generalized algebra proposed recently in the context of generalized statistical theories intended to treat certain complex thermodynamic…
Quantum computing is a new model of computation, based on quantum physics. Quantum computers can be exponentially faster than conventional computers for problems such as factoring. Besides full-scale quantum computers, more restricted…
We introduce an axiomatization for the notion of computation. Based on the idea of Brouwer choice sequences, we construct a model, denoted by $E$, which satisfies our axioms and $E \models \mathrm{ P \neq NP}$. In other words, regarding…
The Turing Machine is the paradigmatic case of computing machines, but there are others such as analogical, connectionist, quantum and diverse forms of unconventional computing, each based on a particular intuition of the phenomenon of…
We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the $\Sigma_\beta$ hierarchy. We focus on linear orderings. We show that at the $\Sigma_1$ level all linear…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…