On Chaitin's Heuristic Principle and Halting Probability
This addresses foundational issues in algorithmic information theory and computability, with incremental contributions to understanding halting probabilities.
The paper tackles the problem of reviving Chaitin's Heuristic Principle for weighing theories and sentences, and shows that Chaitin's constant Omega is not a halting probability under any infinite discrete measure, proposing alternative definitions.
It would be a heavenly reward if there were a method of weighing theories and sentences in such a way that a theory could never prove a heavier sentence (Chaitin's Heuristic Principle). Alas, no satisfactory measure has been found so far, and this dream seemed too good ever to come true. In the first part of this paper, we attempt to revive Chaitin's lost paradise of heuristic principle as much as logic allows. In the second part, which is a joint work with M. Jalilvand and B. Nikzad, we study Chaitin's well-known constant Omega and show that this number is not a probability of halting the randomly chosen input-free programs under any infinite discrete measure. We suggest several methods for defining halting probabilities using various measures.