Arithmetic complexity via effective names for random sequences

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals.

💡 Research Summary

The paper investigates the enumerability properties of classes of sets that admit recursive, lexicographically increasing approximations—so‑called left‑r.e. sets. By focusing on left‑r.e. numberings (effective names) the authors bridge three areas: algorithmic randomness, genericity, and the arithmetic hierarchy.

First, the authors examine whether various randomness notions admit left‑r.e. presentations. They show that the class of Martin‑Löf random sets and the class of Kurtz non‑random sets each have a left‑r.e. numbering, meaning there exists a uniformly recursive enumeration of all members via increasing approximations. In contrast, the classes of Schnorr randoms, Kurtz randoms, and weak‑1‑generic sets do not admit such numberings. This dichotomy highlights how subtle differences in randomness definitions affect the possibility of effective naming.

The core technical contribution is a pair of characterizations of the third and fourth levels of the arithmetical hierarchy purely in terms of left‑r.e. objects with a “shift‑persistent” property. A set is shift‑persistent if, after any finite binary prefix is removed, the remaining tail still belongs to the same class. The authors prove:

• Σ₃⁰ characterization – A set lies in Σ₃⁰ exactly when there exists a shift‑persistent left‑r.e. set whose existence is equivalent to the Σ₃⁰ property. In other words, Σ₃⁰ predicates can be captured by the existence of a suitably persistent left‑r.e. approximation.

• Π₄⁰ characterization – A set lies in Π₄⁰ precisely when there is a shift‑persistent left‑r.e. set together with a complete left‑r.e. numbering for the class. Completeness means every left‑r.e. set can be mapped to an index in this numbering. Thus Π₄⁰ predicates correspond to universal left‑r.e. presentations that survive arbitrary finite shifts.

These results provide a novel, “effective‑name‑only” viewpoint on the arithmetic hierarchy, eliminating the need for traditional Gödel‑coding tricks.

Beyond the hierarchy, the paper explores the relationship between arithmetic complexity and the existence of numberings for classes of left‑r.e. sets that contain shift‑persistent elements. It establishes an equivalence: a class has a certain arithmetical complexity if and only if there exists a left‑r.e. numbering with the shift‑persistence property for that class.

A striking negative result is that no “canonical” or “acceptable” left‑r.e. numbering exists for any class of left‑r.e. random sequences. Even though Martin‑Löf randoms can be enumerated left‑r.e., any attempt to produce a universal, acceptable numbering (in the sense of Rogers) fails. This reflects an inherent non‑uniformity in the structure of random sets.

Finally, the authors compare left‑r.e. numberings for sets versus for real numbers. While sets can often be approximated by increasing finite strings, reals—especially random reals—require additional continuity constraints. Consequently, left‑r.e. numberings for reals behave differently, and the paper points out fundamental distinctions that affect how randomness is captured in the real‑valued setting.

In summary, the work advances our understanding of how effective naming schemes interact with algorithmic randomness and the arithmetic hierarchy. It provides both positive constructions (left‑r.e. numberings for certain randomness classes) and strong limitations (absence of canonical numberings for random classes, and the necessity of shift‑persistence for higher‑level arithmetical characterizations). These insights open new avenues for studying complexity via enumerative methods and deepen the conceptual link between computability, randomness, and logical hierarchies.