Algorithmic Probability and the Halting Constant in Universal Prediction — E8 Intelligence Research

Abstract

FINDING: Solomonoff induction formalizes universal prediction via algorithmic probability, where the prior over hypotheses decays as 2^(-K(x)) — a geometric/power-law distribution over program lengths, with the halting probability Ω as a normal, non-computable constant. | MATH: - Algorithmic probability: \( P(x) = \sum_{p: U(p)=x} 2^{-\ell(p)} \) (prefix-free code, Kraft inequality ensures sum ≤ 1). - Prior over hypotheses: \( P(h) \propto 2^{-K(h)} \), where \( K(h) \) is Kolmogorov complexity — a geometric decay in code length. - Halting probability (Chaitin's Ω): \( \Omega = \sum_{p: U(p) \text{ halts}} 2^{-\ell(p)} \) — a normal number in base 2 (and in any base b, Ω_b is normal), but non-computable. - Power-law tail: For random programs, the distribution of halting times and output complexities follows a heavy tail (Pareto-like), not exponential — hence "heavy-tailed distributions" in the Sandberg lecture. - Geometric PMF: \( P(k) = (1-p)^{k-1}p \) — the discrete analogu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com