Exhaustible sets in higher-type computation

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

Validating Brouwer's Continuity Principle for Numbers Using Named Exceptions

This paper extends the Nuprl proof assistant (a system representative of the class of extensional type theories with dependent types) withnamed exceptionsandhandlers, as well as a nominalfreshoperator. Using these new features, we prove a version of Brouwer's continuity principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's metatheory using our formalization of Nuprl in Coq and reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key metatheoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem, we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem, bar induction on monotone bars and the negation of the law of excluded middle.</jats:p

Computing with functionals - computability theory or computer science

We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject