## How to acknowledge hypercomputation? (2007)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Leitsch07howto,

author = {Alexander Leitsch and Günter Schachner and Karl Svozil},

title = {How to acknowledge hypercomputation?},

year = {2007}

}

### OpenURL

### Abstract

We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models.

### Citations

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Citation Context ...on the input length. In the literature, specific classes of interactive proof systems are investigated as well, for example, the Arthur|Merlin class [31] and the Goldwasser|Micali|Rackoff (GMR) class =-=[32]-=-. The former uses public coin tosses, with the intention of accommodating certain languages in the lowest complexity class possible. The latter uses private coin tosses, with the intention of covering... |

890 | Language identification in the limit - Gold - 1967 |

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(Show Context)
Citation Context ...ac.at We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models. 1. Introduction It is widely acknowledged =-=[1, 2]-=-, that every physical system corresponds to a computational process, and that every computational process, if applicable, has to be physically and operationally feasible in some concrete realization. ... |

320 | Algorithmic Information Theory
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Citation Context ...ar or atomic scales, the physical geometry might turn out to be not as straightforward as it appears from larger scales; for example, the object might turn out to be a fractal. Chaitin’s omega number =-=[4]-=-, which is interpretable as the halting probability of a universal computer, can be “computed in the limit” (without any computable radius of convergence) by a finite-size program in infinite time and... |

311 |
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(Show Context)
Citation Context ...ance in cryptography. (The protocol we presented does not have the zero-knowledge property, unless GNI œ BPP, but can be modified to have it.) For further information on interactive proof systems see =-=[33, 34]-=-. 3.3 Inference of Problems One may confront the hypercomputer with the problem of comparing the solutions of multiple tasks. Such a comparison need not necessarily involve the separate computation of... |

309 | Classical Recursion Theory - Odifreddi - 1989 |

302 | Trading group theory for randomness
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Citation Context ...stance the number of messages depends polynomially on the input length. In the literature, specific classes of interactive proof systems are investigated as well, for example, the Arthur|Merlin class =-=[31]-=- and the Goldwasser|Micali|Rackoff (GMR) class [32]. The former uses public coin tosses, with the intention of accommodating certain languages in the lowest complexity class possible. The latter uses ... |

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182 |
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Citation Context ...ance in cryptography. (The protocol we presented does not have the zero-knowledge property, unless GNI œ BPP, but can be modified to have it.) For further information on interactive proof systems see =-=[33, 34]-=-. 3.3 Inference of Problems One may confront the hypercomputer with the problem of comparing the solutions of multiple tasks. Such a comparison need not necessarily involve the separate computation of... |

114 | Computability and Unsolvability - Davis - 1958 |

107 | A New Kind of Science (Wolfram - Wolfram - 2002 |

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42 |
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Citation Context ...he class of interactive proofs by IP, we have shown that GNI œ IP. Interactive proofs further exist for every language in PSPACE (which is assumed to be much larger than NP). In fact, it can be shown =-=[30]-=- that IP equals PSPACE. This means, in particular, that IP is closed under complement. The protocol in the example given has the property that in each round a constant number of messages is sent. In a... |

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31 |
Information and Randomness:An Algorithmic perspective, 2 nd Ed.; Springer-Verlag
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(Show Context)
Citation Context ... in principle, learn nothing about the individual functional values alone. 3.4 Generation of Random Sequences By implementing Chaitin’s “algorithm” to compute Chaitin’s omega [35] or variants thereof =-=[36]-=-, it would in principle be possible to “compute” the first bits of random sequences. Such random sequences could, in principle, be subject to the usual tests of stochasticity [37, 38]. Note that in qu... |

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25 |
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Citation Context ...he mathematical formalism on the one hand, and the particular physical system that is represented by that formalism on the other hand, demand careful attention. If one insists on operationalizability =-=[3]-=-, one need not go very far in the history of mathematics to encounter suspicious mathematical objects. Surely enough, the number p can be defined and effectively computed as the ratio of the circumfer... |

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22 | Algorithmic Information Theory (Cambridge - Chaitin - 1987 |

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20 | The quantum coin toss—testing microphysical undecidability
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(Show Context)
Citation Context ... or variants thereof [36], it would in principle be possible to “compute” the first bits of random sequences. Such random sequences could, in principle, be subject to the usual tests of stochasticity =-=[37, 38]-=-. Note that in quantum mechanics, the randomness of certain microphysical events, such as the spontaneous decay of excited quantum states [39, 40], or the quantum coin toss experiments in complete con... |

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Citation Context ...ledge Hypercomputation? 133 Motivated by recent proposals to utilize quantum computation for trespassing the Turing barrier [19|22], these accelerating Turing machines have been intensively discussed =-=[23]-=- among other forms of hypercomputation [24|26]. Certainly, the almost ruthless and consequential application of seemingly mind-boggling theories such as quantum mechanics, as far as finitistic methods... |

16 | Why there is no such discipline as hypercomputation - Davis |

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10 | A new measure of the difficulty of problems
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Citation Context ... algorithmically interpreted. For instance, for a particular universal computer, Goldbach’s conjecture and Riemann’s hypothesis could be decided with programs of size 3484 and 7780 bits, respectively =-=[6]-=-. Yet, omega appears to have two features which are normally considered contradictory: it is one of the most informative mathematical numbers imaginable, yet at the same time this information is so co... |

10 | The Myth of Hypercomputation. Alan Turing: Life and Legacy of a Great - Davis - 2004 |

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10 | Forever is a day: Supertasks in Pitowsky and Malament-Hogarth spacetimes - Earman, Norton - 1993 |

10 | The Undecidable (Raven - Davis - 1965 |

9 | Computational universes
- Svozil
(Show Context)
Citation Context ...ac.at We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models. 1. Introduction It is widely acknowledged =-=[1, 2]-=-, that every physical system corresponds to a computational process, and that every computational process, if applicable, has to be physically and operationally feasible in some concrete realization. ... |

9 | Exact approximation of Omega numbers
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(Show Context)
Citation Context ...nce) by a finite-size program in infinite time and with infinite space. Just as for p~the difference being the absence of any computable radius of convergence~the first digits of omega are well known =-=[5]-=-, yet omega has been proved to be algorithmically incompressible and thus random. Nevertheless, presently, for all practical purposes, the statement that “the 10 101010 digit in a decimal expansion of... |

9 | Tasks and supertasks - Thomson - 1954 |

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7 |
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Citation Context ...ciple, be subject to the usual tests of stochasticity [37, 38]. Note that in quantum mechanics, the randomness of certain microphysical events, such as the spontaneous decay of excited quantum states =-=[39, 40]-=-, or the quantum coin toss experiments in complete context mismatches [37], is postulated as an axiom. This postulate is then used as the basis for the production of quantum randomness oracles such as... |

7 | The Church-Turing Thesis as a Guiding Principle for - Svozil |