Malament-Hogarth spacetime
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A Malament-Hogarth (M-H) spacetime, named after David B. Malament and Mark Hogarth, is a relativistic spacetime that possesses the following property: there exists a worldline λ of infinite proper length that lies to the past of some event p. The significance of M-H spacetimes is that they allow for the implementation of certain non-Turing computable tasks (hypercomputation). The idea is for an observer at some event in p's past to set a computer (Turing machine) to work on some task and then have the Turing machine travel on λ, computing for all eternity. Since λ lies in p's past, the Turing machine can signal (a solution) to p at any stage of this never-ending task. Meanwhile, the observer takes a quick trip (finite proper time) through spacetime to p, to pick up the solution. The set-up can be used to decide the halting problem, which is known to be undecidable by an ordinary Turing machine. All the observer needs to do is to prime the Turing machine to signal to p if and only if the Turing machine halts.
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Related articles
Etesi, G., and Nemeti, I., 2002 'Non-Turing computations via Malament-Hogarth space-times', Int.J.Theor.Phys. 41 (2002) 341-370, online
Earman, J., 1995, Bangs Crunches Whimpers and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. Oxford: Oxford University Press.
Earman, J. and Norton, J., 1993, ‘Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes’, Philosophy of Science, 5, 22-42.
Earman, J. and Norton, J., 1994, ‘Infinite Pains: The Trouble with Supertasks’, to appear in S. Stich (ed), Paul Benacerraf: The Philosopher and His Critics. New York: Blackwell.
Hogarth, M., 1992, ‘Does General Relativity Allow an Observer to View an Eternity in a Finite Time?’, Foundations of Physics Letters, 5, 173-181.
Hogarth, M., 1994, ‘Non-Turing Computers and Non-Turing Computability’, in D. Hull, M. Forbes, and R. M. Burian (eds), PSA 1994, Vol. 1. East Lansing: Philosophy of Science Association, 126-138. [1]
Hogarth, M., 1996, 'Computability, Predicability and Spacetime', Ph.D. Thesis, University of Cambridge [2].
Hogarth, M. 2004, ‘Deciding Arithmetic Using SAD Computers’, The British Journal for the Philosophy of Science 55: 681-691.[3]
Welch, P.D., 2006, 'The Extentent of Computation in Malament-Hogarth Spacetimes', preprint.[4]
