## Oracles and Advice as Measurements

Citations: | 5 - 4 self |

### BibTeX

@MISC{Beggs_oraclesand,

author = {Edwin Beggs and José Félix Costa and Bruno Loff and John V Tucker},

title = {Oracles and Advice as Measurements},

year = {}

}

### OpenURL

### Abstract

Abstract. In this paper we will try to understand how oracles and advice functions, which are mathematical abstractions in the theory of computability and complexity, can be seen as physical measurements in Classical Physics. First, we consider how physical measurements are a natural external source of information to an algorithmic computation. We argue that oracles and advice functions can help us to understand how the structure of space and time has information content that can be processed by Turing machines (after Cooper and Odifreddi [10] and Copeland and Proudfoot [11, 12]). We show that non-uniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature. By classifying the information content of such an oracle using Kolmogorov complexity, we obtain a hierarchical structure for advice classes. 1

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Citation Context ...token.Conjecture O (for ‘oracle’). The Universe has non-computable information which may be used as an oracle to build a hypercomputer. The conjecture was popularised by Penrose’s search for (ii) in =-=[19, 20]-=- and much can be written about it. Cooper and Odifreddi [10] have suggested similarities between the structure of the Universe and the structure of the Turing universe. Calude [9] investigates to what... |

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Citation Context ...token.Conjecture O (for ‘oracle’). The Universe has non-computable information which may be used as an oracle to build a hypercomputer. The conjecture was popularised by Penrose’s search for (ii) in =-=[19, 20]-=- and much can be written about it. Cooper and Odifreddi [10] have suggested similarities between the structure of the Universe and the structure of the Turing universe. Calude [9] investigates to what... |

148 |
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(Show Context)
Citation Context ...we apply these notions to the SME and through the interactions between oracle SME in Nature and the Turing machines, develop an inner structure of the advice class P/poly, similar to the one found in =-=[3, 21]-=-. 2 Stonehenge and calculating with an oracle 2.1 Hoyle’s algorithm Stonehenge is an arrangement of massive stones in Wiltshire. Its earliest form dates from 3100 BC and is called Stonehenge I. The As... |

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Citation Context ...ce functions can help us to understand how the structure of space and time has information content that can be processed by Turing machines (after Cooper and Odifreddi [10] and Copeland and Proudfoot =-=[11, 12]-=-). We show that non-uniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature. By classifying the information content of... |

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28 |
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(Show Context)
Citation Context ...ntum randomness can be considered algorithmically random. The search for a physical oracle was proposed by Copeland and Proudfoot [12]. Their article and subsequent work have been severely criticised =-=[13, 14]-=- for historical and technical errors. There is, however, an appealing aesthetical side to what Copeland and Proudfoot proposed. Consider a variation of the Church–Turing thesis: the physical world is ... |

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Citation Context ...theories over abstract algebras ([22]). Oracles are seen as abstract theoretical entities, technical devices whose use is to compare and classify sets by means of degree theories and hierarchies: see =-=[23]-=-. ⋆ Corresponding author.However, here we will argue that it is a useful, interesting, even beautiful, endeavour to develop a computability theory wherein oracles are natural phenomena, and to study ... |

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(Show Context)
Citation Context ...ce functions can help us to understand how the structure of space and time has information content that can be processed by Turing machines (after Cooper and Odifreddi [10] and Copeland and Proudfoot =-=[11, 12]-=-). We show that non-uniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature. By classifying the information content of... |

14 | Experimental computation of real numbers by Newtonian machines
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(Show Context)
Citation Context ... studied in some detail from the computational point of view. The Scatter Machine Experiment (SME) is an experimental procedure that measures the position of a vertex of a wedge to arbitrary accuracy =-=[8]-=-. Since the position may itself be arbitrary, it is possible to analyse the ways in which a simple experiment in Newtonian kinematics can measure or compute an arbitrary real in the interval [0, 1]. I... |

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(Show Context)
Citation Context .... We argue that oracles and advice functions can help us to understand how the structure of space and time has information content that can be processed by Turing machines (after Cooper and Odifreddi =-=[10]-=- and Copeland and Proudfoot [11, 12]). We show that non-uniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature. By cl... |

13 | Computational complexity with experiments as oracles
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(Show Context)
Citation Context .... Conversely, using bisection, we can determine the wedge position to within a given accuracy, and if the wedge position is a good encoding, we can find the original sequence to any given length (see =-=[6, 5]-=-). Theorem 6.1. An error–free analog–digital scatter machine can determine the first n binary places of the wedge position x in polynomial time in n. Theorem 6.2. The class of sets decided by error–fr... |

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8 |
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(Show Context)
Citation Context ...Since the position may itself be arbitrary, it is possible to analyse the ways in which a simple experiment in Newtonian kinematics can measure or compute an arbitrary real in the interval [0, 1]. In =-=[7]-=-, we examined three ways in which the SME can be used as an oracle for Turing machines and established the complexity classes of sets they defined. With this technical knowledge, we can go on to consi... |

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Citation Context ...uracy [8]. Since the position may itself be arbitrary, it is possible to analyse the ways in which a simple experiment in Newtonian kinematics can measure or compute an arbitrary real in the interval =-=[0, 1]-=-. In [7], we examined three ways in which the SME can be used as an oracle for Turing machines and established the complexity classes of sets they defined. With this technical knowledge, we can go on ... |

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Citation Context ...al numbers, in the sense of Turing (i.e. all the binary digits are computable). All the characteristic functions of recursively enumerable sets are in K[log]. This proof was done by Kobayashi in 1981 =-=[16]-=- and by Loveland in 1969 [17] for a variant of the definition of Kolmogorov complexity. The Kolmogorov complexity of a real is provided by the following definition. Definition 5.6. A real is in a give... |

4 |
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(Show Context)
Citation Context ...we apply these notions to the SME and through the interactions between oracle SME in Nature and the Turing machines, develop an inner structure of the advice class P/poly, similar to the one found in =-=[3, 21]-=-. 2 Stonehenge and calculating with an oracle 2.1 Hoyle’s algorithm Stonehenge is an arrangement of massive stones in Wiltshire. Its earliest form dates from 3100 BC and is called Stonehenge I. The As... |

2 |
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(Show Context)
Citation Context ...onehenge I to make good predictions of celestial events, such as the azimuth of the rising Sun and of the rising Moon, or that we can use this Astronomical Observatory as a predictor of eclipses (see =-=[18]-=- for a short introduction). Consider the prediction of eclipses, especially the solar eclipse. This is done by a process of counting time but also requires celestial checks and making corrections. The... |

1 |
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(Show Context)
Citation Context ...ice classes log (k) we add the limit advice class log (ω) = ∩k≥1log (k) . Then proposition 4.1 allows us to take the infinite descending chain of advice function sizes log (ω) ≺ . . . ≺ log (3) ≺ log =-=(2)-=- ≺ log ≺ poly and turn it into a strictly descending chain of sets P/log (ω) ⊂ . . . ⊂ P/log (3) ⊂ P/log (2) ⊂ P/log ⊂ P/poly To show that log (ω) is not trivial, we note that the function log ∗ , def... |

1 |
The computational complexity of the analog-digital scatter machine
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(Show Context)
Citation Context ...iven the limit of traverse of the cannon. The wedge is sufficiently rigid so that the particle cannot movethe wedge from its position. We make the further assumption, without loss of generality (see =-=[5]-=-) that the vertex of the wedge is not a dyadic rational. Suppose that x is the arbitrarily chosen, but non–dyadic and fixed, position of the point of the wedge. For a given dyadic rational cannon posi... |

1 |
The professors and the brainstorms. Published online at http:// www.turing.org.uk /philosophy /sciam.html
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(Show Context)
Citation Context ...ntum randomness can be considered algorithmically random. The search for a physical oracle was proposed by Copeland and Proudfoot [12]. Their article and subsequent work have been severely criticised =-=[13, 14]-=- for historical and technical errors. There is, however, an appealing aesthetical side to what Copeland and Proudfoot proposed. Consider a variation of the Church–Turing thesis: the physical world is ... |