Chaitin Ω Numbers and Halting Problems

• Authors: Kohtaro Tadaki
• Venue: CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
• Year: 2009

Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach. , vol. 22, pp. 329---340, 1975] introduced his Ω number as a concrete example of random real. The real Ω 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 Ω to be random by discovering the property that the first n bits of the base-two expansion of Ω 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. It is known that the base-two expansion of Ω and the halting problem are Turing equivalent. We consider elaborations of both the Turing reductions which constitute the Turing equivalence. These elaborations can be seen as a variant of the weak truth-table reduction, where a computable bound on the use function is explicitly specified. We thus consider the relative computational power between the base-two expansion of Ω and the halting problem by imposing the restriction to finite size on both the problems.