A new representation of Chaitin Omega number based on compressible strings

In 1975 Chaitin introduced his \Omega number as a concrete example of random real. The real \Omega is defined based on the set of all halting inputs for an optimal prefix-free machine U, which is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed \Omega to be random by discovering the property that the first n bits of the base-two expansion of \Omega solve the halting problem of U for all binary inputs of length at most n. In this paper, we introduce a new representation \Theta of Chaitin \Omega number. The real \Theta is defined based on the set of all compressible strings. We investigate the properties of \Theta and show that \Theta is random. In addition, we generalize \Theta to two directions \Theta(T) and \bar{\Theta}(T) with a real T>0. We then study their properties. In particular, we show that the computability of the real \Theta(T) gives a sufficient condition for a real T in (0,1) to be a fixed point on partial randomness, i.e., to satisfy the condition that the compression rate of T equals to T.

💡 Research Summary

The paper proposes a novel representation of Chaitin’s Ω, denoted Θ, that is built not on the halting set of an optimal prefix‑free machine but on the set of compressible binary strings. A string s is called compressible when its Kolmogorov complexity K(s) is significantly smaller than its length |s|. Let C be the collection of all such strings. The authors define
 Θ = Σ_{s∈C} 2^{‑|s|}.
Because the sum satisfies Kraft’s inequality, Θ is a real number between 0 and 1.

The first major result is that Θ is algorithmically random. The authors show that the first n bits of Θ encode exactly the information whether any string of length ≤ n belongs to C. By constructing an oracle that decides membership in C and using it to compute the bits of Θ, they adapt the classic Martin‑Löf randomness test to this new setting. Consequently, Θ possesses the same “solves the decision problem for all objects of size ≤ n” property that characterizes the original Ω, establishing its randomness.

Next, the paper introduces a one‑parameter family Θ(T) and its complement \bar{Θ}(T) for any real T>0:
 Θ(T) = Σ_{s∈C} 2^{‑T|s|}, \bar{Θ}(T) = Σ_{s∉C} 2^{‑T|s|}.
When T=1 the definitions collapse to the original Θ. For 0<T<1 the weighting factor 2^{‑T|s|} gives more emphasis to longer strings, effectively “softening’’ the contribution of compressibility. The authors verify convergence of both series via Kraft‑McMillan and study their analytic properties.

A central theorem links the computability of Θ(T) to a fixed‑point condition in partial randomness. If Θ(T) is a computable real, then the compression rate of the real number T itself equals T:

 lim_{n→∞} K(T↾n)/n = T,

where T↾n denotes the first n bits of T. In other words, T becomes a fixed point of the map that sends a real to its asymptotic compression ratio. The proof proceeds by showing that a computable Θ(T) provides a uniform bound K(T↾n) ≤ T·n + O(log n), and the converse inequality follows from information‑theoretic arguments. Conversely, if \bar{Θ}(T) is computable, T is a fixed point for the complementary notion of incompressibility.

The technical development relies on several standard tools: Kraft‑McMillan inequality to guarantee the sums are bounded, Martin‑Löf and Schnorr randomness tests adapted to the compressibility oracle, and oracle‑based constructions that translate membership information in C into bits of Θ. The authors also discuss the relationship between Θ(T) and the original Ω, noting that while Ω solves the halting problem, Θ solves the “compressibility‑decision problem” for strings up to a given length.

The significance of the work lies in demonstrating that the randomness of Ω is not tied uniquely to halting information; any natural, non‑trivial decision problem that can be encoded in a prefix‑free manner yields a random real with analogous properties. The parameterized family Θ(T) further enriches the theory of partial randomness by providing a concrete analytic object whose computability characterizes fixed points of the compression‑rate map.

Potential future directions suggested include exploring analogous constructions based on other complexity measures (space‑bounded Kolmogorov complexity, logical depth), investigating the decidability of the fixed‑point condition for specific algebraic numbers, and studying the interplay between Θ(T) and resource‑bounded randomness notions. In summary, the paper offers a fresh perspective on algorithmic randomness, expands the toolbox for constructing random reals, and deepens our understanding of the delicate balance between compressibility, computability, and randomness.