How to build a hypercomputer

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Abstract

We claim that the theoretical hypercomputation problem has already been solved, and that what remains is an engineering problem. We review our construction of the Halting Function (the function that settles the Halting Problem) and then sketch possible blueprints for an actual hypercomputer.

Section snippets

Prologue

The authors have degrees in engineering. Engineers build things. So, the goal of this paper is to sketch a series of steps at whose conclusion we would have an actual, working hypercomputer.

Will it work? We leave that question unanswered. But, as we insist, we are cautiously optimistic that its answer may turn out to be a “yes”.

We take our cue from the following remark by Scarpellini [15]:

In this connection one may ask whether it is possible to construct an analog-computer which is in a

Hypercomputation theory

(We stress that this is just an informal sketch.) If we define a hypercomputer to be a machine that can generate, or decide, all arithmetical truths, then very little is required to build a theory that fits the requirement, namely, Peano Arithmetic (PA) plus Shoenfield’s version of the ω-rule is enough (for a nice presentation of Shoenfield’s rule see [10]).

PA plus Shoenfield’s rule proves all true arithmetic sentences, that is, those that hold of the standard model. The hypercomputation theory

From Richardson’s transforms to the halting function

The present section is based on [1]; for the proofs see [13]. Our presentation splits into several topics:

  • Formalized arithmetic and Turing machines.

  • Richardson’s maps.

  • The Halting Function in formal languages that extend arithmetic.

We refer to [12] for notation and requirements from logic. We use: ¬, “not,” , “or,” , “and,” , “if … then…,” , “if and only if,” x, “there is a x,” x, “for every x”. P(x) is a formula with x free; it roughly means “x has property P”. Finally Tξ means T proves ξ

How to build a hypercomputer

There are two ways we can attack the question. The first one is based on the previous construction of the Halting Function. The second one is based on an alternative, well established technique.

First notice the following:

The Halting Problem is: given an arbitrary Turing machine {e} and an arbitrary input n, can we check whether {e}(n) stops?

Now pick up some e,n and keep it fixed. As it will turn out, there is an algorithmic procedure (sketched below) that allows us to check whether {e}(n) stops

Conclusion

We rest our case. To stress the point:

  • We believe that the theoretical hypercomputation problem is solved. We can — ideally — build an analog machine that settles arbitrary instances of the Halting Problem.

  • The actual construction of such a machine is not a mathematical problem anymore; it is an engineering problem. One has to build a prototype and see which problems will eventually come up.

If one settles arbitrary instances of the Halting Problem, we may now conceive a machine that decides

Acknowledgement

The authors wish to thank Professors C. Calude and J.F. Costa who kindly invited them to present this paper at the Vienna workshop on Unusual Models of Computation, August 2008. They also wish to thank two anonymous referees for their comments and criticisms.

The ongoing research program that led to this text has been sponsored by the Advanced Studies Group, Production Engineering Program, COPPE-UFRJ, Rio, Brazil. The authors also wish to thank the Institute for Advanced Studies at the

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The authors are partially funded by CNPq-Brazil, Philosophy Section. They also acknowledge support from the Brazilian Academy of Philosophy.

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