🤖 AI Summary
This paper investigates Chaitin’s heuristic principle—that no formal system can prove the truth of statements “heavier” (i.e., of higher Kolmogorov complexity) than itself—and its relationship with Chaitin’s halting probability constant Ω. Specifically, it addresses whether the principle admits a rigorous formalization within axiomatic systems, and whether Ω can be interpreted as the halting probability of a universal Turing machine with no input under *any* infinite discrete measure.
Method: Integrating algorithmic information theory, computability theory, measure theory, and formal logic, the authors systematically reconstruct and rigorously evaluate the principle within a logical framework—its first such treatment.
Contributions: (1) They precisely characterize the boundary conditions under which the principle is realizable in formal systems; (2) they formally refute the universality claim: Ω cannot represent the halting probability under *any* infinite discrete measure; (3) they introduce a novel multi-measure paradigm for halting probabilities and construct nontrivial instances, establishing a foundational framework for modeling stochastic program behavior.
📝 Abstract
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 to ever 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 some methods for defining the halting probabilities by various measures.
