The Failure of Selman’s Theorem for Hyperenumeration Reducibility
Year: 2026
Type: article
Abstract: Hyperenumeration reducibility was first introduced by Sanchis [11]. The relationship between hyperenumeration and hyperarithmetic reducibility shares many parallels with the relationship between enumeration and Turing reducibility. We ask if this relationship can be pushed to prove and analog of Selman’s Theorem for hyperenumeration reducibility. By studying e-pointed trees in Baire space we are able to get a counter example. An e-pointed tree T is a tree with no dead ends and the property that every path in T enumerates T . We prove that if T is an e-pointed tree then for all X if T is $\Pi ^1_1$ in X then $\overline {T}$ is $\Pi ^1_1$ in X . We build an e-pointed tree T such that $\overline {T}$ is not hyperenumeration reducible to T .
Source: Journal of Symbolic Logic
Author: JOSIAH JACOBSEN-GROCOTT
Institution: Nanyang Technological University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7154335269
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