Related papers: The Algebraic View of Computation
Present day mathematics is a human construct, where computers are used more and more but do not play a creative role. This situation may change however: computers may become creative, and since they function very differently from the human…
Cellular Automata (CA), as they are presented in the literature, are abstract mathematical models of computation. In this pa- per we present an alternate approach: using the CA as a model or theory of physical systems and devices. While…
We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based…
We present an algebraic characterization of the complexity classes Logspace and Nlogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is rooted in proof theory…
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…
With the help of link diagrams with decorated crossings, I explain computations in emergent algebras, introduced in arXiv:0907.1520, as the kind of computations done in the front end visual system.
The speed and transformative power of human cultural evolution is evident from the change it has wrought on our planet. This chapter proposes a human computation program aimed at (1) distinguishing algorithmic from non-algorithmic…
Computational power can be measured by assigning an algebraic structure to a computational device. Here, we convert a small patch of Conway's Game of Life into a transformation semigroup. The conversion captures not only time evolution but…
In this chapter, concepts related to information and computation are reviewed in the context of human computation. A brief introduction to information theory and different types of computation is given. Two examples of human computation…
By nature, transmissible human knowledge is enumerable: every sentence, movie, audio record can be encoded in a sufficiently long string of 0's and 1's. The works of G\"odel, Turing and others showed that there are inherent limits and…
Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state…
In the first of this pair of papers, it was proven that that no physical computer can correctly carry out all computational tasks that can be posed to it. The generality of this result follows from its use of a novel definition of…
The success of the abstract model of computation, in terms of bits, logical operations, programming language constructs, and the like, makes it easy to forget that computation is a physical process. Our cherished notions of computation and…
Incremental computation aims to compute more efficiently on changed input by reusing previously computed results. We give a high-level overview of works on incremental computation, and highlight the essence underlying all of them, which we…
This paper is a reflexion on the computability of natural language semantics. It does not contain a new model or new results in the formal semantics of natural language: it is rather a computational analysis of the logical models and…
We consider the quantum computational process as viewed by an insider observer: this is equivalent to an isomorphism between the quantum computer and a quantum space, namely the fuzzy sphere. The result is the formulation of a reversible…
Being declarative means that we do computer programming on higher levels of abstraction. This vague definition identifies declarativeness with the act of ignoring details, but it is a special case of abstraction. The unspecified part is…
We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.
Morphisms, structure preserving maps, are everywhere in Mathematics as useful tools for thinking and problem solving, or as objects to study. Here, we argue that the idea of operations being compatible across two domains goes beyond its…
Approaching limitations of digital computing technologies have spurred research in neuromorphic and other unconventional approaches to computing. Here we argue that if we want to systematically engineer computing systems that are based on…