Chaitin’s Constant

Chaitin’s constant is a favorite discussion in Algorithmic Information Theory. It’s fascinating how an unquantifiable figure, an uncomputable figure can stand for so much. Chaitin’s constant, in a sense, is regarded as normal and transcendental. It is believed that there is no algorithm that can truly calculate or even guess the number or its digits.

Chaitin’s constant represents the probability that any random program will halt. This random program is assumed to run in a universal, self-delimiting Turing machine. It was first conceived by its namesake, Gregory Chaitin who was a well-known computer scientist and mathematician. Chaitin was an influential figure in Algorithmic Information Theory, having penned his own incompleteness theory, the Kolmogorov complexity and defined the Chaitin’s constant. Chaitin used the omega figure in stating the halting probability. He further stressed that this constant is only definable and not computable.

To better understand Chaitin’s constant, let us look into its different properties.

  • Uncomputable
  • First and foremost, it is uncomputable. Why? Chaitin’s constant computes the halting problem through the algorithm that necessities the n digits. Once the n is specified, the constant will be able to calculate the halting problem up until the nth digit. Its reliance on the unspecified n digit makes it uncomputable.
  • Normal
  • Chaitin’s constant is a normal arithmetical number. This means that the digits are equidistributed.
  • Algorithmically Random
  • Chaitin’s constant cannot work with an algorithm that is characterized with less than n-O(1) bits. Why? n represents the longest bit in the program. The algorithm works this way. It will use the program that has the shortest length in order to compute for the constant. Thus, the algorithm has to have at least n-O(1) bits in size or else, it won’t be able to achieve its purpose. It won’t be able to identify which of the programs would halt.

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