Related papers: The Algebraic View of Computation
Computation is commonly defined as the execution of abstract algorithms over symbolic representations, with physical systems treated as substrates that realise predefined operations. While effective for engineered machines, this separation…
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering…
Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and…
A coherent mathematical overview of computation and its generalisations is described. This conceptual framework is sufficient to comfortably host a wide range of contemporary thinking on embodied computation and its models.
This paper proposes a specific conceptualization of intelligence as computation. This conceptualization is intended to provide a unified view for all disciplines of intelligence research. Already, it unifies several conceptualizations…
This work continues the development of an intensional approach to computability initiated in previous work, in which programs and computations, rather than functions, constitute the primary objects of study. In this setting, models of…
Computing is a high-level process of a physical system. Recent interest in non-standard computing systems, including quantum and biological computers, has brought this physical basis of computing to the forefront. There has been, however,…
An abstract machine is a theoretical model designed to perform a rigorous study of computation. Such a model usually consists of configurations, instructions, programs, inputs and outputs for the machine. In this paper we formalize these…
Physical processes are computations only when we use them to externalize thought. Computation is the performance of one or more fixed processes within a contingent environment. We reformulate the Church-Turing thesis so that it applies to…
The theory that all processes in the universe are computational is attractive in its promise to provide an understandable theory of everything. I want to suggest here that this pancomputationalism is not sufficiently clear on which problem…
Theory of representations of universal algebra is a natural development of the theory of universal algebra. In the book, I considered representation of universal algebra, diagram of representations and examples of representation. Morphism…
Theory of representations of universal algebra is a natural development of the theory of universal algebra. Morphism of the representation is the map that conserve the structure of the representation. Exploring of morphisms of the…
A model of computation is abstract if, when applied to any algebra, the resulting programs for computable functions and sets on that algebra are invariant under isomorphisms, and hence do not depend on a representation for the algebra.…
A map is an abstract visual representation of a region, taken from a given space, usually designed for final human consumption. Traditional cartography focuses on the mapping of Euclidean spaces by using some distance metric. In this paper…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
Artificial computing machinery transforms representations through an objective process, to be interpreted subjectively by humans, so the machine and the interpreter are different entities, but in the putative natural computing both…
This lecture addresses some general ideas behind numerical computations ranging from representation of numbers in computers to stability and accuracy of standard algorithms for some simple mathematical problems.
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…
This piece plays with the idea of the Computocene: an era defined not merely by the ubiquity of computers, but by their deepening role in how we observe, interpret, and make sense of the world. Rather than emphasizing automation, speed,…