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Introduction to Domain Theory

Introduction to Domain Theory

Graham Hutton
School of Computer Science and IT
University of Nottingham



Domain theory has its roots in the seminal work of Scott and Strachey in the 1970s on denotational semantics, in particular on solving the technical problems that arise in giving semantics to recursively defined programs and types. As well as being one of the most fascinating topics in the foundations of computing science, domain-theoretic ideas are used in many branches of modern research on programming languages and their semantics. The aim of this course is to teach the basic concepts of domain theory. The course comprises five lectures, each with exercises:

  1. Background and motivation (denotational semantics, non-termination and bottom, partially-ordered sets, monotonic functions);

  2. Recursively defined functions (chains, directed sets, least upper bound, cpos and continuous functions, the fixpoint theorem);

  3. Constructions on types (sums, products, function-spaces, lifting, lazyness vs strictness, axiomatisations, functors);

  4. Computability issues (finite elements, algebraicity, a representation theorem, consistent completeness, Scott-domains);

  5. Recursively defined types (colimits, cocontinuous functors, the generalised fixpoint theorem, Scott's inverse limit construction, modelling the untyped lambda-calculus).

The handwritten slides are suitable for self-study; copies are available on request.