Course pages 2011–12

**Subsections**

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Denotational Semantics

*Lecturer: Professor A.M. Pitts*

*No. of lectures + exercise classes:* 10 + 2

### Aims

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

### Lectures

**Introduction.**The denotational approach to the semantics of programming languages. Recursively defined objects as limits of successive approximations.**Least fixed points.**Complete partial orders (cpos) and least elements. Continuous functions and least fixed points.**Constructions on domains.**Flat domains. Product domains. Function domains.**Scott induction.**Chain-closed and admissible subsets of cpos and domains. Scott’s fixed-point induction principle.**PCF.**The Scott-Plotkin language PCF. Evaluation. Contextual equivalence.**Denotational semantics of PCF.**Denotation of types and terms. Compositionality. Soundness with respect to evaluation. [2 lectures].**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. [2 lectures].**Full abstraction.**Failure of full abstraction for the domain model. PCF with parallel or.

### 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: an introduction*. MIT Press.

Gunter, C. (1992). *Semantics of programming languages: structures and techniques*. MIT Press.

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