Relative Constructibility via Generalised Sequential Algorithms

TL;DR

This paper extends Gurevich's sequential algorithms to larger sets, establishing a new framework that aligns relative computability with set-theoretic constructibility.

Contribution

It introduces generalized sequential algorithms and parameters, linking their relative computability to the constructibility relation in set theory.

Findings

Defined GSeqAs and GSeqAPs for larger sets.

Established a computability relation analogous to Turing reducibility.

Proved the equivalence of this relation to set-theoretic constructibility.

Abstract

We modify Gurevich's definition of sequential algorithms, so that it becomes amenable to computation with arbitrarily large sets on a sufficiently intuitive level. As a result, two classes of abstract algorithms are obtained, namely generalised sequential algorithms (GSeqAs) and generalised sequential algorithms with parameters (GSeqAPs). We derive from each class a relative computability relation on sets which is analogous to the Turing reducibility relation on reals. We then prove that the relative computability relation derived from GSeqAPs is equivalent to the relative constructibility relation in set theory.