Chaitin's Constant - Definition

Definition

Let PF be the domain of a prefix-free universal computable function F. The constant ΩF is then defined as

,

where denotes the length of a string p. This is an infinite sum which has one summand for every p in the domain of F. The requirement that the domain be prefix-free, together with Kraft's inequality, ensures that this sum converges to a real number between 0 and 1. If F is clear from context then ΩF may be denoted simply Ω, although different prefix-free universal computable functions lead to different values of Ω.

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