Hypercomputation

Visiting professor at IVA Jack Copeland gives the first of two lectures on hypercomputation and Alan Turing.

Abstract

Hypercomputation is the computation of functions or numbers that cannot be computed in the sense of Turing's 1936 paper 'On Computable Numbers', i.e. cannot be computed with paper and pencil in a finite number of steps by a human clerk working effectively. It is often forgotten that, in 1936, a computer was not a machine at all, but a human being, a mathematical assistant who calculated by rote. This is the sense in which Turing used the term computer and its cognates, such as computable, in his 1936 paper; and the Turing machine (or as Turing called it, 'computing machine') is an idealization of the human computer.

A hypercomputer, on the other hand, is any information-processing machine, notional or real, that is able to achieve more than the traditional human clerk working by rote. Hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that lie beyond the reach of the standard universal Turing machine of 1936. The additional computational power of a hypercomputer may arise because certain of the restrictions customarily imposed on the human computer are absent—for example, the restriction that data take the form of symbols on paper, or that all data be supplied in advance of the computation, or that the rules followed by the computer remain fixed for the duration of the computation. Or the additional power may arise because the machine has, among its repertoire of primitive capacities, one or more operations that no human rote-worker can carry out. All computation takes place relative to some set or other of primitive capacities, and the richer the capacities that are available, the greater the extent of the computable.

In this lecture I explore some simple logical models of hypercomputation. I also examine some apriori objections to the possibility of hypercomputation. Will hypercomputation ever be observed? I argue that this is an empirical question, whose answer is not yet known.

Biography

Jack Copeland FRS NZ is Distinguished Professor in Humanities at the University of Canterbury, New Zealand, where he is Director of the Turing Archive for the History of Computing. He is also Honorary Research Professor of Philosophy at the University of Queensland, Australia, and in 2012 was Royden B. Davis Visiting Chair of Interdisciplinary Studies in the Department of Psychology at Georgetown University, Washington DC. His books include The Essential Turing (Oxford University Press), Colossus: The Secrets of Bletchley Park’s Codebreaking Computers (Oxford University Press), Alan Turing’s Electronic Brain (Oxford University Press), Computability: Turing, Gödel, Church, and Beyond (MIT Press), Logic and Reality (Oxford University Press), and Artificial Intelligence (Blackwell); and he has published more than 100 articles on the philosophy and history of computing, and mathematical and philosophical logic.

He is recognised as a leading authority on Turing's work, and in June of 2004, the 50th anniversary of Turing’s death, he delivered the first annual Turing Memorial Lecture at Bletchley Park National Museum and also lectured on Turing’s life and work at the Royal Institution of London. He received the Scientific American Sci/Tech Web Award for his on-line archive www.AlanTuring.net. He has been script advisor, co-writer, and scientific consultant for a number of documentaries on Turing. One of these, the BBC's Code-Breakers: Bletchley Park's Lost Heroes, won two BAFTAs (British Academy of Film and Television Arts awards) in 2012, and was listed as one of the year's three best historical documentaries at the 2013 Media Impact Awards in New York City.

A Londoner by birth, Jack earned a B.Phil. with Distinction from the University of Oxford—where he was taught by Turing's great friend Robin Gandy—and followed by a D.Phil. in mathematical logic. Jack has been a visiting scholar at the University of California at Los Angeles, a visiting professor at the universities of Sydney, Aarhus, Melbourne, and Portsmouth, a senior fellow of the Dibner Institute for the History of Science and Technology at the Massachusetts Institute of Technology, and most recently Gastprofessor in the Departments of Philosophy and Computer Science at the Swiss Federal Institute of Technology (ETH Zurich) and, of course, Visiting Professor at IVA. He is a past president of the U.S.-based Society for Machines and Mentality and is the founding editor of the Rutherford Journal for the History and Philosophy of Science and Technology.

Oxford University Press published his latest paperback last month, a highly accessible biography of Turing titled Turing, Pioneer of the Information Age.