Exhaustible sets in higher-type computation

TL;DR

This paper explores the properties of exhaustible and searchable sets in higher-type computation, establishing their topological and computational characteristics, and providing a comprehensive framework for understanding their structure and relationships.

Contribution

It characterizes exhaustible and searchable sets in higher-type computation, linking them to topological compactness and providing uniform constructions of selection functionals.

Findings

Searchable sets are exhaustible and vice versa for hereditarily total elements.

Exhaustible sets are topologically compact.

Searchable sets are closed under intersections, images, and products.

Abstract

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces. We also show that, in the non-empty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications.