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


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.

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C. S. Calude , L. Staiger and S. A. Terwijn (2006) On partial randomness. Ann. Pure Appl. Logic 138 2030.

C. S. Calude and M. A. Stay (2006) Natural halting probabilities, partial randomness, and zeta functions. Inform. and Comput. 204 17181739.

G. J. Chaitin (1975) A theory of program size formally identical to information theory. J. Assoc. Comput. Mach. 22 329340.

A. Nies (2009) Computability and Randomness, Oxford University Press.

J. Reimann and F. Stephan (2006) On hierarchies of randomness tests. In: Proceedings of the 9th Asian Logic Conference, Novosibirsk, Russia, World Scientific215232.

D. Ruelle (1999) Statistical Mechanics, Rigorous Results, 3rd edition, Imperial College Press and World Scientific.

K. Tadaki (2002) A generalization of Chaitin's halting probability Ω and halting self-similar sets. Hokkaido Math. J. 31 219253.

K. Tadaki (2012) Fixed point theorems on partial randomness. Annals of Pure and Applied Logic 163 763774.

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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