
Fixpoints and relative precompleteness
We study relative precompleteness in the context of the theory of number...
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Finitarybased Domain Theory in Coq: An Early Report
In domain theory every finite computable object can be represented by a ...
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Effective Kan fibrations in simplicial sets
We introduce the notion of an effective Kan fibration, a new mathematica...
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Data
In this article, the data notion is mathematically conceptualized as typ...
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Computability in partial combinatory algebras
We prove a number of elementary facts about computability in partial com...
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Differencerestriction algebras of partial functions: axiomatisations and representations
We investigate the representation and complete representation classes fo...
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A Constructive, TypeTheoretic Approach to Regression via Global Optimisation
We examine the connections between deterministic, complete, and general ...
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Partial Functions and Recursion in Univalent Type Theory
We investigate partial functions and computability theory from within a constructive, univalent type theory. The focus is on placing computability into a larger mathematical context, rather than on a complete development of computability theory. We begin with a treatment of partial functions, using the notion of dominance, which is used in synthetic domain theory to discuss classes of partial maps. We relate this and other ideas from synthetic domain theory to other approaches to partiality in type theory. We show that the notion of dominance is difficult to apply in our setting: the set of Σ_0^1 propositions investigated by Rosolini form a dominance precisely if a weak, but nevertheless unprovable, choice principle holds. To get around this problem, we suggest an alternative notion of partial function we call disciplined maps. In the presence of countable choice, this notion coincides with Rosolini's. Using a general notion of partial function, we take the first steps in constructive computability theory. We do this both with computability as structure, where we have direct access to programs; and with computability as property, where we must work in a programinvariant way. We demonstrate the difference between these two approaches by showing how these approaches relate to facts about computability theory arising from topostheoretic and typetheoretic concerns. Finally, we tie the two threads together: assuming countable choice and that all total functions ℕ→ℕ are computable (both of which hold in the effective topos), the Rosolini partial functions, the disciplined maps, and the computable partial functions all coincide. We observe, however, that the class of all partial functions includes noncomputable partial functions.
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