Oracles and Advice as Measurements
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, using a simple and engaging case study, namely: Hoyle’s algorithm for calculating eclipses at Stonehenge. Next, we argue that oracles and advice functions can help us understand how the structure of space and time has information content that can be processed by Turing machines. Using an advanced case study from Newtonian kinematics, we show that non-uniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature, and that by classifying the information content of such a natural oracle, using Kolmogorov complexity, we obtain a hierarchical structure based on measurements, advice classes and information.
Keywords
Polynomial Time Turing Machine Solar Eclipse Kolmogorov Complexity Summer SolsticePreview
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References
- 1.Balcázar, J.L., Días, J., Gabarró, J.: Structural Complexity I, 2nd edn. Springer, Heidelberg (1995)Google Scholar
- 2.Balcázar, J.L., Gavaldà, R., Hermo, M.: Compressibility of infinite binary sequences. In: Sorbi, A. (ed.) Complexity, logic, and recursion theory. Lecture notes in pure and applied mathematics, vol. 187, pp. 1175–1183. Marcel Dekker, Inc., New York (1997)Google Scholar
- 3.Balcázar, J.L., Gavaldà, R., Siegelmann, H.: Computational power of neural networks: a characterization in terms of Kolmogorov complexity. IEEE Transactions on Information Theory 43(4), 1175–1183 (1997)zbMATHCrossRefGoogle Scholar
- 4.Balcázar, J.L., Gavaldà, R., Siegelmann, H., Sontag, E.D.: Some structural complexity aspects of neural computation. In: Proceedings of the Eighth IEEE Structure in Complexity Theory Conference, pp. 253–265. IEEE Computer Society, Los Alamitos (1993)CrossRefGoogle Scholar
- 5.Beggs, E., Costa, J.F., Loff, B., Tucker, J.: On the complexity of measurement in classical physics. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 20–30. Springer, Heidelberg (2008)CrossRefGoogle Scholar
- 6.Beggs, E., Costa, J.F., Loff, B., Tucker, J.: Computational complexity with experiments as oracles. Proc. Royal Society, Ser. A (in press)Google Scholar
- 7.Beggs, E., Tucker, J.: Experimental computation of real numbers by Newtonian machines. Proc. Royal Society, Ser. A 463(2082), 1541–1561 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
- 8.Calude, C.: Algorithmic randomness, quantum physics, and incompleteness. In: Margenstern, M. (ed.) MCU 2004. LNCS, vol. 3354, pp. 1–17. Springer, Heidelberg (2005)Google Scholar
- 9.Cooper, B., Odifreddi, P.: Incomputability in Nature. In: Cooper, B., Goncharov, S. (eds.) Computability and Models, Perspectives East and West. University series in mathematics, pp. 137–160. Springer, Heidelberg (2003)Google Scholar
- 10.Copeland, J.: The Church–Turing thesis. In: Zalta, E. (ed.) The Stanford Enciclopedia of Phylosophy (published, 2002), http://plato.stanford.edu/archives/fall2002/entries/church-turing/
- 11.Copeland, J., Proudfoot, D.: Alan Turing’s forgotten ideas in Computer Science. Scientific American 280, 99–103 (1999)CrossRefGoogle Scholar
- 12.Davis, M.: The myth of hypercomputation. In: Teuscher, C. (ed.) Alan Turing: the life and legacy of a great thinker, pp. 195–212. Springer, Heidelberg (2006)Google Scholar
- 13.Hodges, A.: The professors and the brainstorms (published, 1999), http://www.turing.org.uk/philosophy/sciam.html
- 14.Hoyle, F.: From Stonehenge to Modern Cosmology. W.H. Freeman, New York (1972)Google Scholar
- 15.Kobayashi, K.: On compressibility of infinite sequences. Technical Report C–34, Research Reports on Information Sciences (1981)Google Scholar
- 16.Loveland, D.W.: A variant of the Kolmogorov concept of complexity. Information and Control 15, 115–133 (1969)CrossRefMathSciNetGoogle Scholar
- 17.Newham, C.A.: The Astronomical Significance of Stonehenge. Coats and Parker Ltd (2000) (First published, 1972) Google Scholar
- 18.Penrose, R.: The Emperor’s New Mind. Oxford University Press, Oxford (1989)Google Scholar
- 19.Penrose, R.: Shadows of the Mind. Oxford University Press, Oxford (1994)Google Scholar
- 20.Siegelmann, H.T.: Neural Networks and Analog Computation: Beyond the Turing Limit. Birkhäuser, Basel (1999)zbMATHGoogle Scholar
- 21.Tucker, J.V., Zucker, J.I.: Computable functions and semicomputable sets on many sorted algebras. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic for Computer Science. University Series in Mathematics, vol. V, pp. 317–523. Oxford University Press, Oxford (2000)Google Scholar
- 22.Tucker, J.V., Zucker, J.I.: Abstract versus concrete computation on metric partial algebras. ACM Transactions on Computational Logic 5, 611–668 (2004)CrossRefMathSciNetGoogle Scholar