Equivalent characterizations of partial randomness for a recursively enumerable real

A real number \alpha is called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to \alpha. The randomness of a recursively enumerable real \alpha can be characterized in various ways using each of the notions; program-size complexity, Martin-L"{o}f test, Chaitin’s \Omega number, the domination and \Omega-likeness of \alpha, the universality of a computable, increasing sequence of rational numbers which converges to \alpha, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real number T\in(0,1]. We thus present several equivalent characterizations of partial randomness for a recursively enumerable real number.

💡 Research Summary

The paper extends the classical equivalence results for randomness of recursively enumerable (r.e.) real numbers to a parametrized notion called partial randomness, indexed by a real parameter T∈(0,1]. After reviewing basic notions of algorithmic information theory—optimal computers, program‑size complexity H, Chaitin’s halting probability Ω_V, and universal probability m—the authors introduce weak Chaitin T‑randomness (H(α_n) ≥ T·n − c) and Martin‑Löf T‑randomness (tests weighted by 2^{−T|s|}). They prove these two definitions are equivalent for computable T (Theorem 2.8).

The core contribution is a set of new concepts: a sequence {a_n} is T‑convergent if Σ_n (a_{n+1}−a_n)^T converges; an r.e. real is T‑convergent if it has such a computable increasing rational approximation. Lemma 4.2 shows that T‑convergence can be transferred between real and rational sequences. Using this, Theorem 4.3 establishes that for computable T, the generalized halting probability Ω_V(T)=∑_{p∈dom V}2^{−|p| T} is itself an r.e. T‑convergent real.

Ω(T)‑likeness is defined as domination of all T‑convergent r.e. reals, while T‑universality describes increasing sequences that dominate every T‑convergent sequence up to a constant factor. The main result, Theorem 4.6, states that for any computable T and any r.e. real α∈(0,1), the following nine conditions are equivalent: (i) α is weakly Chaitin T‑random; (ii) α is Martin‑Löf T‑random; (iii) α is Ω(T)‑like; (iv) for every T‑convergent r.e. β, H(β_n) ≤ H(α_n)+O(1); (v) α can be expressed as β + q·γ with β ≥ 0, γ > 0 T‑convergent; (vi) α = β + Ω_V(T) for some β ≥ 0; (vii) α = ∑_{s} m(s)^T for a universal probability m; (viii) every computable increasing rational sequence converging to α is T‑universal; (ix) there exists a T‑universal computable increasing sequence converging to α.

The proof builds on the classical equivalences (Theorem 3.4) by adapting each implication to the T‑parameterized setting, using Lemma 4.2, the convergence of Ω_V(T), and domination arguments. Sections 5 and 6 flesh out the technical details and explore further properties of T‑convergence, while Section 7 outlines future directions. By providing a full suite of equivalent characterizations for partial randomness, the work deepens our understanding of the fine‑grained structure of algorithmic randomness and offers new tools for studying the spectrum between computable and fully random reals.