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THREE FORMS OF PHYSICAL MEASUREMENT AND THEIR COMPUTABILITY

Published online by Cambridge University Press:  09 September 2014

EDWIN BEGGS ,
JOSÉ FÉLIX COSTA  and
JOHN V TUCKER
EDWIN BEGGS*
Affiliation:
Department of Mathematics, Swansea University
JOSÉ FÉLIX COSTA*
Affiliation:
Instituto Superior Técnico and Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa
JOHN V TUCKER*
Affiliation:
Department of Computer Science, Swansea University
*
*DEPARTMENT OF MATHEMATICS, COLLEGE OF SCIENCE SWANSEA UNIVERSITY, SINGLETON PARK, SWANSEA, SA2 8PP WALES, U.K. E-mail: e.j.beggs@swansea.ac.uk
†INSTITUTO SUPERIOR TÉCNICO UNIVERSIDADE DE LISBOA PORTUGAL and CENTRO DE MATEMÁTICA E APLICAÇÕES FUNDAMENTAIS UNIVERSIDADE DE LISBOA PORTUGAL3 E-mail: fgc@math.tecnico.ulisboa.pt
‡DEPARTMENT OF COMPUTER SCIENCE, COLLEGE OF SCIENCE SWANSEA UNIVERSITY, SINGLETON PARK, SWANSEA, SA2 8PP WALES, U.K. E-mail: j.v.tucker@swansea.ac.uk
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Abstract

We have begun a theory of measurement in which an experimenter and his or her experimental procedure are modeled by algorithms that interact with physical equipment through a simple abstract interface. The theory is based upon using models of physical equipment as oracles to Turing machines. This allows us to investigate the computability and computational complexity of measurement processes. We examine eight different experiments that make measurements and, by introducing the idea of an observable indicator, we identify three distinct forms of measurement process and three types of measurement algorithm. We give axiomatic specifications of three forms of interfaces that enable the three types of experiment to be used as oracles to Turing machines, and lemmas that help certify an experiment satisfies the axiomatic specifications. For experiments that satisfy our axiomatic specifications, we give lower bounds on the computational power of Turing machines in polynomial time using nonuniform complexity classes. These lower bounds break the barrier defined by the Church-Turing Thesis.

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 4 , December 2014 , pp. 618 - 646
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

BIBLIOGRAPHY

Balcázar, J. L., Días, J., & Gabarró, J. (1995). Structural complexity I (second edition). Berlin: Springer-Verlag.Google Scholar
Balcázar, J. L., Días, J., & Gabarró, J. (1990). Structural complexity II. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Balcázar, J. L., & Hermo, M. (1998). The structure of logarithmic advice complexity classes. Theoretical Computer Science, 207(1), 217–244.Google Scholar
Basilevsky, A., Anderson, A. B., & Hum, D. P. J. (1983). Measurement: Theory and techniques. In Rossi, P. H., Wright, J.D., and Anderson, A. B., editors. Handbook of survey research. New York: Academic Press, pp. 231–287.Google Scholar
Beggs, E., Costa, J. F., Loff, B., & Tucker, J. V. (2008a). Computational complexity with experiments as oracles. Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences), 464(2098), 2777–2801.Google Scholar
Beggs, E., Costa, J. F., Loff, B., & Tucker, J. V. (2008b). On the complexity of measurement in classical physics. In Agrawal, M., Du, D., Duan, Z., & Li, A., editors. Theory and applications of models of computation (TAMC 2008), Lecture Notes in Computer Science, Vol. 4978 , Berlin: Springer, pp. 20–30.Google Scholar
Beggs, E., Costa, J. F., Loff, B., & Tucker, J. V. (2008c). Oracles and advice as measurements. In Calude, C. S., Costa, J. F., Freund, R., Oswald, M., & Rozenberg, G., editors. Unconventional computation (UC 2008), Lecture Notes in Computer Science, Vol. 5204 , Berlin: Springer-Verlag, pp. 33–50.Google Scholar
Beggs, E., Costa, J. F., Loff, B., & Tucker, J. V. (2009). Computational complexity with experiments as oracles II. Upper bounds. Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences), 465(2105), 1453–1465.Google Scholar
Beggs, E., Costa, J. F., Poças, D., & Tucker, J. V. (2013a). On the power of threshold measurements as oracles. In Mauri, G., Dennunzio, A., Manzoni, L., & Porreca, A. E., editors. Unconventional computation and natural computation (UCNC 2013), Lecture Notes in Computer Science, Vol. 7956 , Berlin: Springer, pp. 6–18.Google Scholar
Beggs, E., Costa, J. F., Poças, D., & Tucker, J. V. (2013b). Oracles that measure thresholds: The Turing machine and the broken balance. Journal of Logic and Computation, 23(6), 1155–1181.CrossRefGoogle Scholar
Beggs, E., Costa, J. F., Poças, D., & Tucker, J. V. (2014). Computations with oracles that measure vanishing quantities, Submitted.Google Scholar
Beggs, E., Costa, J. F., & Tucker, J. V. (2010a). Computational models of measurement and Hempel’s axiomatization. In Carsetti, A., editor. Causality, meaningful complexity and knowledge construction, Theory and Decision Library A, Vol. 46 , Berlin: Springer, pp. 155–184.Google Scholar
Beggs, E., Costa, J. F., & Tucker, J. V. (2010b). Limits to measurement in experiments governed by algorithms. Mathematical Structures in Computer Science (Special issue on Quantum Algorithms, ed. Salvador Elías Venegas-Andraca), 20(06), 1019–1050.Google Scholar
Beggs, E., Costa, J. F., & Tucker, J. V. (2010c). Physical oracles: The Turing machine and the Wheatstone bridge Studia Logica (Special issue on Contributions of Logic to the Foundations of Physics, eds. Aerts, D., Smets, S., & Van Bendegem, J. P.), 95( 1–2), 279–300.Google Scholar
Beggs, E., Costa, J. F., & Tucker, J. V. (2010d). The Turing machine and the uncertainty in the measurement process. In Guerra, H., editor. Physics and computation, P&C 2010, CMATI – Centre for Applied Mathematics and Information Technology, University of Azores, Ponta Delgada, pp. 62–72.Google Scholar
Beggs, E., Costa, J. F., & Tucker, J. V. (2012a). Axiomatising physical experiments as oracles to algorithms. Philosophical Transactions of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences), 370(12), 3359–3384.Google Scholar
Beggs, E., Costa, J. F., & Tucker, J. V. (2012b). The impact of models of a physical oracle on computational power. Mathematical Structures in Computer Science (Special issue on Computability of the Physical, eds. Calude, C. S. & Cooper, S. B.), 22(5), 853–879.Google Scholar
Beggs, E., Costa, J. F., & Tucker, J. V. (2012c). Unifying science through computation: Reflections on computability and physics. In Pombo, O., Manuel Torres, J., Symons, J., & Rahman, S., editors. New approaches to the unity of science, Vol. II: Special sciences and the unity of science, Logic, Epistemology, and the Unity of Science, vol. 24, Berlin: Springer, pp. 53–80.Google Scholar
Beggs, E. & Tucker, J. V. (2007). Experimental computation of real numbers by Newtonian machines. Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences), 463(2082), 1541–1561.Google Scholar
Bohm, D. (1951, 1979, 1989). Quantum theory, New York: Dover.Google Scholar
Born, M. & Wolf, E. (1964). Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (second (revised) edition), Oxford: Pergamon Press.Google Scholar
Bournez, O. & Cosnard, M. (1996). On the computational power of dynamical systems and hybrid systems. Theoretical Computer Science, 168(2), 417–459.CrossRefGoogle Scholar
Campbell, N. R. (1957). Foundations of science, the philosophy of theory and experiment, New York: Dover.Google Scholar
Carnap, R. (1966). Philosophical foundations of physics, New York: Basic Books.Google Scholar
Geroch, R. & Hartle, J. B. (1986). Computability and physical theories. Foundations of Physics, 16(6), 533–550.Google Scholar
Hempel, C. G. (1952). Fundamentals of concept formation in empirical science. International Encyclopedia of Unified Science, 2(7).Google Scholar
Jain, S., Osherson, D. N., Royer, J. S., & Sharma, A. (1999). Systems That Learn. An Introduction to Learning Theory (second edition), Cambridge, MA: The MIT Press.CrossRefGoogle Scholar
Krantz, D. H., Luce, D. R., Suppes, P., & Tversky, A. (2007a). Foundations of Measurement. Vol. I: Additive and polynomial representations, New York: Dover.Google Scholar
Krantz, D. H., Suppes, P., Luce, D. R., & Tversky, A. (2007b). Foundations of Measurement. Vol. III: Representation, axiomatization and invariance, New York: Dover.Google Scholar
Siegelmann, H. T. (1999). Neural networks and analog computation: Beyond the Turing limit, Switzerland: Birkhäuser.CrossRefGoogle Scholar
Suppes, P. (1951). A set of independent axioms for extensive quantities. Portugaliæ Mathematica, 10(2), 163–172.Google Scholar
Suppes, P., Krantz, D. H., Luce, D. R., & Tversky, A. (2007). Foundations of Measurement. Vol. II: Geometric, threshold and probabilistic representations, New York: Dover.Google Scholar
Turing, A. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42, 230–265.Google Scholar