Computability Theory

Turing's Titanic Machine? Embodied and Disembodied Computing at the Turing Centenary

by Barry Cooper

Published in: Communications of the ACM,
Volume 55 Issue 3, March 2012

It is in the nature of things that 1912 is generally more remembered for the sinking of the “unsinkable” Titanic than... more It is in the nature of things that 1912 is generally more remembered for the sinking of the “unsinkable” Titanic than it is for the far more world-changing event: the birth of Alan Mathison Turing in a London nursing home on a Sunday in June of that year. The Turing Centenary presents a timely opportunity for us to remember the startling depth of impact of Turing’s universal machine over the last 75 years.
In this brief review I examine challenges to its continued primacy as a model for computation in daily practice and in the wider universe. In doing this, the Titanic analogy is surprisingly useful, even apt from some points of view. It reminds us of the achievements of science and engineering, and its precarious coexistence with the often surprising unpredictability of events in nature.

More really is different

by Mile Gu

Quantum complexity, Emergence and Computation by Measurement: On what computers reveal about physical laws, and what physical laws reveal about computers

by Mile Gu

Is economic planning hypercomputational? The argument from cantor diagonalisation

by paul cockshott

Boettke Syntax and the Turing Test,

by paul cockshott

Joint paper with Greg Michaelson and Allin Cottrell

a reply by Greg Michaelson, Allin Cottrell and I to a paper on Methodology by Boettke which he published in Cahiers... more a reply by Greg Michaelson, Allin Cottrell and I to a paper on Methodology by Boettke which he published in Cahiers D'Epistemolgie.

Computabilidad, aleatoriedad y la Tesis de Church-Turing Física

by Valentín Muro

Ponencia presentada en las I Jornadas de Estudiantes de Filosofía de la Universidad de Buenos Aires el 19 de octubre de 2011.

«Adam Olszewski (1999) propone que la Tesis de Church-Turing puede ser usada para refutar el platonismo matemático1.... more «Adam Olszewski (1999) propone que la Tesis de Church-Turing puede ser usada para refutar el platonismo matemático1. Para ello postula una máquina que al lanzar una moneda define una función que ―computa el valor (0 o 1) para la moneda n que se haya lanzado y dice que dicha función es efectivamente computable pero no Turing-computable.

Entonces, dado que según el platonismo matemático (PM), toda función de enteros positivos a enteros positivos ya existe, existiría una función efectivamente computable pero no Turing-computable (computable por una máquina de Turing).

Esto contradice a la Tesis de Church-Turing (CT) que dice que toda función efectivamente computable es Turing-computable. Por contraposición, si CT es verdadera, entonces PM es falso. Rafał Urbaniak (2011) critica este argumento desafiando la afirmación de que lo que esta máquina de hecho realiza sea una computación. Revisaré dicha crítica y detallaré algunos puntos de la misma. Además introduciré la noción de Tesis de Church-Turing Física.»

Non-classical computing: feasible versus infeasible.

by paul cockshott

Given at the BCS 2010 conference

Physics sets certain limits on what is and is not computable. These limits are very far from having been reached by... more Physics sets certain limits on what is and is not computable. These limits are very far from having been reached by current technologies. Whilst proposals for hypercomputation are almost certainly infeasible, there are a number of non classical approaches that do hold considerable promise. There are a range of possible architectures that could be implemented on silicon that are distinctly different from the von Neumann model. Beyond this, quantum simulators, which are the quantum equivalent of analogue computers, may be constructable in the near future

A Hardware Relaxation Paradigm for Solving NP-Hard Problems

by paul cockshott

Digital circuits with feedback loops can solve some instances of NP-hard problems
by relaxation: the circuit will... more Digital circuits with feedback loops can solve some instances of NP-hard problems
by relaxation: the circuit will either oscillate or settle down to a stable state that represents a solution to the problem instance. This approach differs from using hardware accelerators to speed up the execution of deterministic algorithms, as it exploits stabilisation properties of circuits with feedback, and it allows a variety of hardware techniques that do not have counterparts in software. A feedback circuit that solves many instances of Boolean satisfiability problems is described, with experimental results from a preliminary simulation using a hardware accelerator.

Constraints on hypercomputation

by paul cockshott

Are there new models of computation? Reply to Wegner and Eberbach

by paul cockshott

TYPE INFERENCE FOR SET THEORY

by Raymond Turner

Theoretical Computer Science archive
Volume 266 Issue 1-2, 09/06/2001
Elsevier Science Publishers Ltd. Essex, UK
table of contents doi>10.1016/S0304-3975(01)00123-2

Weak set theories are employed for set-theoretic specification. We develop and explore type inference systems for such... more Weak set theories are employed for set-theoretic specification. We develop and explore type inference systems for such set theories.

COMPUTABILITY IN SPECIFICATION

by Raymond Turner

J Logic Computation (2006) 16 (6): 841-865.
doi: 10.1093/logcom/exl012
First published online: August 12, 2006

In reference (Foundation of specification. Journal of Logic and Computation, 15, 951–974, 2005), the author introduces... more In reference (Foundation of specification. Journal of Logic and Computation, 15, 951–974, 2005), the author introduces a core specification theory (CST) in order to provide a logical framework for the design and exploration of specification languages. In this article, we formulate two highly expressive extensions of CST. The first (CSTU) is CST + a universe of types and the second (CSTUS) permits specifications themselves to be data items. Finally, we shall explore their metamathematical properties and, in particular, provide an interpretation into first-order arithmetic.

COMPUTABLE MODELS

by Raymond Turner

J Logic Computation (2008) 18 (2): 283-318.
doi: 10.1093/logcom/exm002
First published online: March 3, 2007

We investigate mathematical modelling with theories of data types. We provide a formal setting for the formulation of... more We investigate mathematical modelling with theories of data types. We provide a formal setting for the formulation of such theories (TPL) and use it to introduce the notion of a computational model. We explore the notion and provide several case studies.