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Chaitin’s Omega and an algorithmic phase transition

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  • Schmidhuber, Christof

Abstract

We consider the statistical mechanical ensemble of bit string histories that are computed by a universal Turing machine. The role of the energy is played by the program size. We show that this ensemble has a first-order phase transition at a critical temperature, at which the partition function equals Chaitin’s halting probability Ω. This phase transition has curious properties: the free energy is continuous near the critical temperature, but almost jumps: it converges more slowly to its finite critical value than any computable function. At the critical temperature, the average size of the bit strings diverges. We define a non-universal Turing machine that approximates this behavior of the partition function in a computable way by a super-logarithmic singularity, and discuss its thermodynamic properties. We also discuss analogies and differences between Chaitin’s Omega and the partition function of a quantum mechanical particle, and with quantum Turing machines. For universal Turing machines, we conjecture that the ensemble of bit string histories at the critical temperature has a continuum formulation in terms of a string theory.

Suggested Citation

  • Schmidhuber, Christof, 2022. "Chaitin’s Omega and an algorithmic phase transition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 586(C).
    • Handle: RePEc:eee:phsmap:v:586:y:2022:i:c:s0378437121007317
    DOI: 10.1016/j.physa.2021.126458
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    References listed on IDEAS

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    1. Toby S. Cubitt & David Perez-Garcia & Michael M. Wolf, 2015. "Undecidability of the spectral gap," Nature, Nature, vol. 528(7581), pages 207-211, December.
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