Work

Low $_2$ Computably Enumerable Sets Have Hyperhypersimple Supersets

Year: 2025

Type: article

Abstract: A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal {L}^*(A)$ , of a low $_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal {L}^*(A)\cong {\mathcal E}^*$ . In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e. A is low $_2$ then A has an atomless hyperhypersimple superset. In fact, if A is c.e. and low $_2$ , then for any $\Sigma _3$ -Boolean algebra B there is some c.e. $H\supseteq A$ such that $\mathcal {L}^*(H)\cong B$ .

Source: Journal of Symbolic Logic

DOI: 10.1017/jsl.2025.10146

Authors: PETER CHOLAK, RODNEY DOWNEY, NOAM GREENBERG

Institutions: University of Notre Dame, University of Victoria, Victoria University

Language: English

Topic: Computability, Logic, AI Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

OpenAlex ID: W4414448915

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