Book contents
- Frontmatter
- Contents
- Preface
- Continuous Functionals of Dependent and Transfinite Types
- Degree-Theoretic Aspects of Computably Enumerable Reals
- Simplicity and Independence for Pseudo-Algebraically Closed Fields
- Clockwork or Turing U/universe? - Remarks on Causal Determinism and Computability
- A Techniques Oriented Survey of Bounded Queries
- Relative Categoricity in Abelian Groups
- Computability and Complexity Revisited
- Effective Model Theory: The Number of Models and Their Complexity
- A Survey on Canonical Bases in Simple Theories
- True Approximations and Models of Arithmetic
- On the Topological Stability Conjecture
- A Mahlo-Universe of Effective Domains with Totality
- Logic and Decision Making
- The Sheaf of Locally Definable Scalars over a Ring
- Human Styles of Quantificational Reasoning
- Recursion Theoretic Memories 1954–1978
- Fields Definable in Simple Groups
- A Combinatory Algebra for Sequential Functionals of Finite Type
- Model Theory of Analytic and Smooth Functions
A Combinatory Algebra for Sequential Functionals of Finite Type
Published online by Cambridge University Press: 17 May 2010
- Frontmatter
- Contents
- Preface
- Continuous Functionals of Dependent and Transfinite Types
- Degree-Theoretic Aspects of Computably Enumerable Reals
- Simplicity and Independence for Pseudo-Algebraically Closed Fields
- Clockwork or Turing U/universe? - Remarks on Causal Determinism and Computability
- A Techniques Oriented Survey of Bounded Queries
- Relative Categoricity in Abelian Groups
- Computability and Complexity Revisited
- Effective Model Theory: The Number of Models and Their Complexity
- A Survey on Canonical Bases in Simple Theories
- True Approximations and Models of Arithmetic
- On the Topological Stability Conjecture
- A Mahlo-Universe of Effective Domains with Totality
- Logic and Decision Making
- The Sheaf of Locally Definable Scalars over a Ring
- Human Styles of Quantificational Reasoning
- Recursion Theoretic Memories 1954–1978
- Fields Definable in Simple Groups
- A Combinatory Algebra for Sequential Functionals of Finite Type
- Model Theory of Analytic and Smooth Functions
Summary
Abstract
It is shown that the type structure of finite-type functionals associated to a combinatory algebra of partial functions from ℕ to ℕ (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from ℕ to ℕ), is isomorphic to the type structure generated by object N (the flat domain on the natural numbers) in Ehrhard's category of “dI-domains with coherence”, or his “hypercoherences”.
Introduction
PCF, “Gödel's T with unlimited recursion”, was defined in Plotkin's paper. It is a simply typed λ-calculus with a type o for integers and constants for basic arithmetical operations, definition by cases and fixed point recursion. More important, there is a special reduction relation attached to it which ensures (by Plotkin's “Activity Lemma”) that all PCF-definable higher-type functionals have a sequential, i.e. non-parallel evaluation strategy. In view of this, the obvious model of Scott domains is not faithful, since it contains parallel functions. A search began for “fully abstract” domain-theoretic models for PCF.
A proliferation of ever more complicated theories of domains saw the light, inducing the father of domain theory, Dana Scott, to lament that “there are too many proposed categories of domains and […] their study has become too arcane”, a judgement with which it is hard to disagree.
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- Information
- Models and Computability , pp. 389 - 406Publisher: Cambridge University PressPrint publication year: 1999
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