Chaitin’s Omega and an algorithmic phase transition

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. •We study computations of universal Turing machines by means of statistical physics.•The “energy” of a bit string history is defined as its program size complexity.•The partition function reveals a first-order phase transition at a critical temperature.•There, it converges to Chaitin’s Omega more slowly than any computable function.•At the critical point, we conjecture that bit strings are described by a string theory.

Note:

29 pages, 5 figures. Added references, a literature review, and a section on analogies with quantum mechanics and field theory (previous title: "Logical Quantum Field Theory")

Chaitin’s Omega
Complexity
Turing machine
Algorithmic thermodynamics
Phase transition
String theory
critical phenomena
partition function
statistical
history

References(29)

Figures(6)

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