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Notions of computability at higher types I
- from TUTORIALS
- Edited by René Cori, Université de Paris VII (Denis Diderot), Alexander Razborov, Institute for Advanced Study, Princeton, New Jersey, Stevo Todorčević, Université de Paris VII (Denis Diderot), Carol Wood, Wesleyan University, Connecticut
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- Book:
- Logic Colloquium 2000
- Published online:
- 27 June 2017
- Print publication:
- 02 March 2005, pp 32-142
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Summary
Abstract. This is the first of a series of three articles devoted to the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers) and establishing the relationships between these notions. In the present paper, we undertake an extended survey of the different strands of research to date on higher type computability, bringing together material from recursion theory, constructive logic and computer science, and emphasizing the historical development of the ideas. The paper thus serves as a reasonably comprehensive survey of the literature on higher type computability.
Introduction. This article is essentially a survey of fifty years of research on higher type computability. It was a great privilege to present much of this material in a series of three lectures at the Paris Logic Colloquium.
In elementary recursion theory, one begins with the question: what does it mean for an ordinary first order function on N to be “computable”? As is well known, many different approaches to defining a notion of computable function — via Turing machines, lambda calculus, recursion equations, Markov algorithms, flowcharts, etc. — lead to essentially the same answer, namely the class of (total or partial) recursive functions. Indeed, Church's thesis proposes that for functions from N to N we identify the informal notion of an “effectively computable” function with the precise mathematical notion of a recursive function.
An important point here is thatmany prima facie independentmathematical constructions lead to the same class of functions. Whilst one can argue over whether this is good evidence that the recursive functions include all effectively computable functions (see Odifreddi [1989] for a discussion), it is certainly good evidence that they represent a mathematically natural and robust class of functions. And since no other serious contenders for a class of effectively computable functions are known, most of us are happy to accept Church's thesis most of the time.
Now suppose we consider second order functions which map first order functions to natural numbers (say), and then third order functions which map second order functions to natural numbers, and so on. We will use the word functional to mean a function that takes functions of some kind as arguments.
A uniform approach to domain theory in realizability models
- JOHN R. LONGLEY, ALEX K. SIMPSON
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- Journal:
- Mathematical Structures in Computer Science / Volume 7 / Issue 5 / October 1997
- Published online by Cambridge University Press:
- 01 October 1997, pp. 469-505
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We propose a uniform way of isolating a subcategory of predomains within the category of modest sets determined by a partial combinatory algebra (PCA). Given a divergence on a PCA (which determines a notion of partiality), we identify a candidate category of predomains, the well-complete objects. We show that, whenever a single strong completeness axiom holds, the category satisfies appropriate closure properties. We consider a range of examples of PCAs with associated divergences and show that in each case the axiom does hold. These examples encompass models allowing a ‘parallel’ style of computation (for example, by interleaving), as well as models that seemingly allow only ‘sequential’ computation, such as those based on term-models for the lambda-calculus. Thus, our approach provides a uniform approach to domain theory across a wide class of realizability models. We compare our treatment with previous approaches to domain theory in realizability models. It appears that no other approach applies across such a wide range of models.