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Denotational Semantics

*Lecturer: Professor G. Winskel*

*No. of lectures:* 8

*Prerequisite course: Semantics of Programming Languages (specifically, an idea of operational semantics and how to reason from it)*

**Aims**

The aims of this course are to introduce domain theory and denotational semantics, and show how they can provide a mathematical basis for reasoning about the behaviour of programming languages.

**Lectures**

**Introduction.**Recursively defined objects as limits of successive approximations. Examples.**Least fixed points.**-complete partial orders (cpos) and -continuous functions. Least elements. Least fixed points of -continuous functions.**Constructions on domains.**Products of domains. Function domains. Flat domains.**Scott induction.**Chain-closed and admissible subsets of cpos and domains. Scott's fixed point induction principle. Examples.**PCF.**The Scott-Plotkin language PCF. Evaluation. Contextual equivalence.**Denotational semantics of PCF.**Denotation of types and terms. Compositionality. Soundness with respect to evaluation.**Relating denotational and operational semantics.**Formal approximation relation and its fundamental property. Computational adequacy of the PCF denotational semantics with respect to evaluation. Extensionality properties of contextual equivalence.**Full abstraction.**Failure of full abstraction for the domain model. PCF plus parallel or. Recent developments.

**Objectives**

At the end of the course students should:

- be familiar with basic domain theory, cpos, continuous functions, admissible subsets, least fixed points, basic constructions on domains
- be able to give denotational semantics to simple programming languages with simple types
- be able to apply denotational semantics, in particular, to understand the use of least fixed points to model recursive programs and be able to reason about least fixed points and simple recursive programs using fixed point induction
- understand the issues concerning the relation between denotational and operational semantics, adequacy and full abstraction, especially with respect to the language PCF

**Recommended reading**

* Winskel, G. (1993). *The formal semantics of programming languages*. MIT Press.

Tennent, R.D. (1991). *Semantics of programming languages*. Prentice-Hall.

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