Truth and Light: Physical Algorithmic Randomness

This thesis examines some problems related to Chaitin’s Ω number. In the first section, we describe several new minimalist prefix-free machines suitable for the study of concrete algorithmic information theory; the halting probabilities of these machines are all Ω numbers. In the second part, we show that when such a sequence is the result given by a measurement of a system, the system itself can be shown to satisfy an uncertainty principle equivalent to Heisenberg’s uncertainty principle. This uncertainty principle also implies Chaitin’s strongest form of incompleteness. In the last part, we show that Ω can be written as an infinite product over halting programs; that there exists a “natural, ” or base-free formulation that does not (directly) depend on the alphabet of the universal prefix-free machine; that Tadaki’s generalized halting probability is well-defined even for arbitrary univeral Turing machines and the plain complexity; and finally, that the natural generalized halting probability can be written as an infinite product over primes and has the form of a zeta function whose zeros encode halting information. We conclude with some speculation about physical systems in which partial randomness could arise, and identify many open problems