Introduction to Domain Theory
, 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.
(denotational semantics, non-termination and bottom,
partially-ordered sets, monotonic functions);
- Recursively defined programs
(chains, directed sets, least upper bound, cpos and
continuous functions, the fixpoint theorem);
- Constructions on cpos
(sums, products, function-spaces, lifting, lazyness
vs strictness, axiomatisations, functors);
- Scott domains
(finite elements, algebraicity, a representation
theorem, consistent completeness, Scott-domains);
- Recursively defined domains
(colimits, cocontinuous functors, the generalised
fixpoint theorem, Scott's inverse limit construction,
modelling the untyped lambda-calculus).
There is also a summary sheet
of the main concepts from lectures 1-4.