Hostname: page-component-6b989bf9dc-md2j5 Total loading time: 0 Render date: 2024-04-14T20:23:01.541Z Has data issue: false hasContentIssue false
  • English
  • Français

A statistical mechanical interpretation of algorithmic information theory III: composite systems and fixed points†

Published online by Cambridge University Press:  06 September 2012

KOHTARO TADAKI
KOHTARO TADAKI*
Affiliation:
Research and Development Initiative, Chuo University, JST CREST, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan Email: tadaki@kc.chuo-u.ac.jp Website: http://www2.odn.ne.jp/tadaki/
Get access Rights & Permissions [Opens in a new window]

Abstract

The statistical mechanical interpretation of algorithmic information theory (AIT for short) was introduced and developed in our previous papers Tadaki (2008; 2012), where we introduced into AIT the notion of thermodynamic quantities, such as the partition function Z(T), free energy F(T), energy E(T) and statistical mechanical entropy S(T). We then discovered that in the interpretation, the temperature T is equal to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity, namely, for each of the thermodynamic quantities above, the computability of its value at temperature T gives a sufficient condition for T ∈ (0, 1) to be a fixed point on partial randomness. In this paper, we develop the statistical mechanical interpretation of AIT further and pursue its formal correspondence to normal statistical mechanics. The thermodynamic quantities in AIT are defined on the basis of the halting set of an optimal prefix-free machine, which is a universal decoding algorithm used to define the notion of program-size complexity. We show that there are infinitely many optimal prefix-free machines that give completely different sufficient conditions for each of the thermodynamic quantities in AIT. We do this by introducing the notion of composition of prefix-free machines into AIT, which corresponds to the notion of the composition of systems in normal statistical mechanics.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Special Issue 5: Computability of the Physical , October 2012 , pp. 752 - 770
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Callen, H. B. (1985) Thermodynamics and an Introduction to Thermostatistics, 2nd edition, John Wiley and Sons.Google Scholar
Calude, C. S., Staiger, L. and Terwijn, S. A. (2006) On partial randomness. Ann. Pure Appl. Logic 138 20–30.CrossRefGoogle Scholar
Calude, C. S. and Stay, M. A. (2006) Natural halting probabilities, partial randomness, and zeta functions. Inform. and Comput. 204 1718–1739.CrossRefGoogle Scholar
Chaitin, G. J. (1975) A theory of program size formally identical to information theory. J. Assoc. Comput. Mach. 22 329–340.CrossRefGoogle Scholar
Chaitin, G. J. (1987) Algorithmic Information Theory, Cambridge University Press.CrossRefGoogle Scholar
Dirac, P. A. M. (1958) The Principles of Quantum Mechanics, 4th edition, Oxford University Press.CrossRefGoogle Scholar
Downey, R. G. and Griffiths, E. J. (2004) Schnorr randomness. J. Symbolic Logic 69 533–554.CrossRefGoogle Scholar
Downey, R. G. and Hirschfeldt, D. R. (2010) Algorithmic Randomness and Complexity, Springer-Verlag.CrossRefGoogle Scholar
Gács, P. (1974) On the symmetry of algorithmic information. Soviet Math. Dokl. 15 1477–1480. (Correction, ibid. (1974) 15 1480.)Google Scholar
Levin, L. A. (1974) Laws of information conservation (non-growth) and aspects of the foundations of probability theory. Problems of Inform. Transmission 10 206–210.Google Scholar
Nies, A. (2009) Computability and Randomness, Oxford University Press.CrossRefGoogle Scholar
Reimann, J. and Stephan, F. (2006) On hierarchies of randomness tests. In: Proceedings of the 9th Asian Logic Conference, Novosibirsk, Russia, World Scientific 215–232.Google Scholar
Ruelle, D. (1999) Statistical Mechanics, Rigorous Results, 3rd edition, Imperial College Press and World Scientific.CrossRefGoogle Scholar
Tadaki, K. (1999) Algorithmic information theory and fractal sets. In: Proceedings of 1999 Workshop on Information-Based Induction Sciences, IBIS'99, Syuzenji, Shizuoka, Japan 105–110. (In Japanese.)Google Scholar
Tadaki, K. (2002) A generalization of Chaitin's halting probability Ω and halting self-similar sets. Hokkaido Math. J. 31 219–253.CrossRefGoogle Scholar
Tadaki, K. (2007) A statistical mechanical interpretation of instantaneous codes. In: Proceedings of 2007 IEEE International Symposium on Information Theory, ISIT2007, Nice, France, 1906–1910.CrossRefGoogle Scholar
Tadaki, K. (2008) A statistical mechanical interpretation of algorithmic information theory. In: Local Proceedings of Computability in Europe 2008, CiE 2008, University of Athens, Greece, 425–434. (Electronic version available at http://www.cs.swan.ac.uk/cie08/cie2008-local.pdf.)Google Scholar
Tadaki, K. (2009) Fixed points on partial randomness. In: Proceedings of the 6th Workshop on Fixed Points in Computer Science, FICS 2009, Coimbra, Portugal 100–107. (Electronic version available at http://cs.ioc.ee/fics09/fics09proc.pdf.)Google Scholar
Tadaki, K. (2010) A statistical mechanical interpretation of algorithmic information theory: Total statistical mechanical interpretation based on physical argument. In: Proceedings of Kyoto RIMS workshop: ‘Mathematical Aspects of Generalized Entropies and their Applications’. Journal of Physics: Conference Series (JPCS) 201 012006.Google Scholar
Tadaki, K. (2012) Fixed point theorems on partial randomness. Annals of Pure and Applied Logic 163 763–774.CrossRefGoogle Scholar
Toda, M., Kubo, R. and Saitô, N. (1992) Statistical Physics I: Equilibrium Statistical Mechanics, 2nd edition, Springer-Verlag.CrossRefGoogle Scholar