Computable Reducibility of Equivalence Relations and an Effective Jump Operator

Abstract

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

Investigating the computable Friedman-Stanley jump

“…As for many other places of this paper, our starting point is [CCK]. We give a definition of a strong way to reduce a set A Ď ω to an equivalence relation E. This is similar to and inspired by [CCK,Definition 3.3]; whereas they aren't concerned with the image hpxq if x R A (so long as it is E-contained in the image of the reduction for an x P A), we demand only two possible images depending on whether or not x P A.…”

“…For every ceer E, we have E `ď Id `[CCK, Proposition 4.1]. And certainly any light ceer satisfies Id `ď E notations matter in the definition of the jump [CCK, Question 2] and if every jump fixed point must be an upper bound under computable reduction (not just m-reduction) for all HYP equivalence relations [CCK,Question 3]. We answer both in the affirmative in Section 5 and 6: Theorem 1.11.…”

Investigating the Computable Friedman–stanley Jump

“…For a long time, the study of Borel and computable reducibility were conducted independently, despite the clear analogy between the two notions. Yet, there is rapidly emerging a theory of computable reductions which blends ideas from both computability theory and descriptive set theory [13,14,17,21]. In particular, computable reductions are well-suited for assessing the complexity of isomorphism relations on classes of computable structures, as one can encode the atomic diagram of a computable structure by a single natural number.…”

“…Less is known about the complexity of classes that are not on top for ≤ c . To deepen the natural connection between the Borel and the computable setting, the following computable analog of the Friedman-Stanley jump has been introduced (a finitary analog of this jump appeared in [21]): Definition 1.3 (Clemens, Coskey, and Krakoff [13]). For E an equivalence relation on , E is given by x E y if and only if [W x ] E = [W y ] E , where W i is the ith c.e.…”

“…Clemens, Coskey, and Krakoff [13] showed that E > E holds for any hyperarithmetic (HYP) equivalence relation. They also asked several carefully chosen questions (presented in Section 1.2) regarding features of the jump operator .…”