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

Models of Type Theory Based on Moore Paths

This paper introduces a new family of models of intensional Martin-L\"of type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.Comment: This is a revised and expanded version of a paper with the same name that appeared in the proceedings of the 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017

Nominal techniques

This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2893582.2893594 Programming languages abound with features making use of names in various ways. There is a mathematical foundation for the semantics of such features which uses groups of permutations of names and the notion of the support of an object with respect to the action of such a group. The relevance of this kind of mathematics for the semantics of names is perhaps not immediately obvious. That it is relevant and useful has emerged over the last 15 years or so in a body of work that has acquired its own name: nominal techniques. At the same time, the application of these techniques has broadened from semantics to computation theory in general. This article introduces the subject and is based upon a tutorial at LICS-ICALP 2015 [Pitts 2015a]. </jats:p

Relating Two Semantics of Locally Scoped Names

The operational semantics of programming constructs involving locally scoped names typically makes use of stateful "dynamic allocation": a set of currently-used names forms part of the state and upon entering a scope the set is augmented by a new name bound to the scoped identifier. More abstractly, one can see this as a transformation of local scopes by expanding them outward to an implicit top-level. By contrast, in a neglected paper from 1994, Odersky gave a stateless lambda calculus with locally scoped names whose dynamics contracts scopes inward. The properties of "Odersky-style" local names are quite different from dynamically allocated ones and it has not been clear, until now, what is the expressive power of Odersky\u27s notion. We show that in fact it provides a direct semantics of locally scoped names from which the more familiar dynamic allocation semantics can be obtained by continuation-passing style (CPS) translation. More precisely, we show that there is a CPS translation of typed lambda calculus with dynamically allocated names (the Pitts-Stark nu-calculus) into Odersky\u27s lambda-nu-calculus which is computationally adequate with respect to observational equivalence in the two calculi

Typal Heterogeneous Equality Types

The usual homogeneous form of equality type in Martin-L\"of Type Theory contains identifications between elements of the same type. By contrast, the heterogeneous form of equality contains identifications between elements of possibly different types. This paper introduces a simple set of axioms for such types. The axioms are equivalent to the combination of systematic elimination rules for both forms of equality, albeit with typal (also known as "propositional") computation properties, together with Streicher's Axiom K, or equivalently, the principle of uniqueness of identity proofs

Life Cycle Costs for Alaska Bridges

INE/AUTC 15.0

Constructing Initial Algebras Using Inflationary Iteration

An old theorem of Adámek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using “inflationary” iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor’s constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of elementary topos.UK EPSRC PhD studentship 211980