IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v586y2022ics0378437121007317.html
   My bibliography  Save this article

Chaitin’s Omega and an algorithmic phase transition

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437121007317
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2021.126458?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. James D. Watson & Emilio Onorati & Toby S. Cubitt, 2022. "Uncomputably complex renormalisation group flows," Nature Communications, Nature, vol. 13(1), pages 1-8, December.
    2. Heinz Langhals, 2017. "Interaction of Components in Molecular Optoelectronics for the Next Generation of IT Devices," Scientific Review, Academic Research Publishing Group, vol. 3(3), pages 17-28, 03-2017.
    3. Bin Yan & Nikolai A. Sinitsyn, 2022. "Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian," Nature Communications, Nature, vol. 13(1), pages 1-12, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:586:y:2022:i:c:s0378437121007317. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.