The effective theory of Borel equivalence relations

Abstract

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and Harrington-Kechris-Louveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on P(ω). In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene's O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington (see [6]) establishing for any recursive ordinal α the existence of Π 0 1 singletons whose α-jumps are Turing incomparable. * The authors acknowledge the generous support of the FWF through project P 19375-N18 ( * ) There are effectively Borel sets A and B such that for no effectively Borel function f does one have f [A] ⊆ B or f [B] ⊆ A.

Wadge hardness in Scott spaces and its effectivization

“…Theorem 3.5 does not have an effective version because determinacy is not guaranteed in the effective world. As pointed to us by an anonymous referee, the arguments given by Fokina et al (2010) yield the following result. Proof.…”

“…Proof. We follow the arguments in the proof of Theorem 9 in Fokina et al (2010). Their Theorem 6 establishes that there is a uniform sequence (A n ) n∈N of nonempty Π 0 1 subsets of N ω such that, for each n, there is no hyperarithmetical function…”

“…Since Borel determinacy gives highly non-computable game strategies, the main tool used in the classical Wadge theory is not available. As pointed out to us by an anonymous referee, it follows from Fokina et al (2010) that the structure of effective Wadge degrees is very different, even for the Baire space: the ordered structure of inclusion on computably enumerable subsets of N can be embedded into the effective Wadge hierarchy of Σ 0 2 subsets of the Baire space; cf. Proposition 3.8 infra.…”

OnΔ20-categoricity of equivalence relations

“…The theory of effective equivalence relations has grown to a rather broad area; we cite [1,18,9] for recent results on this subject. Many results of this paper can be stated in terms of Δ 0 2 -embeddings between effectively presented equivalence structures.…”

Isomorphism relations on computable structures

“…The complexity of the isomorphism relation on various classes of countable structures was measured using the idea of effective transformations. In the recent work [11] the general theory of effectively Borel (i.e., ∆ 1 1 ) equivalence relations on effectively presented Polish spaces was developed via the notion of effective Borel reducibility. The resulting structure turned out to be much more complex than in the classical case.…”