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Chaitin Omega numbers and halting problems | Kohtaro Tadaki
; | Date: |
7 Apr 2009 | Abstract: | Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach., vol.22, pp.329-340, 1975]
introduced Omega number as a concrete example of random real. The real Omega
is defined as the probability that an optimal computer halts, where the optimal
computer is a universal decoding algorithm used to define the notion of
program-size complexity. Chaitin showed Omega to be random by discovering the
property that the first n bits of the base-two expansion of Omega solve the
halting problem of the optimal computer for all binary inputs of length at most
n. In the present paper we investigate this property from various aspects. We
consider the relative computational power between the base-two expansion of
Omega and the halting problem by imposing the restriction to finite size on
both the problems. It is known that the base-two expansion of Omega and the
halting problem are Turing equivalent. We thus consider an elaboration of the
Turing equivalence in a certain manner. | Source: | arXiv, 0904.1149 | Services: | Forum | Review | PDF | Favorites |
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