# Denotational Semantics

**Principal lecturer:** Prof Marcelo Fiore**Taken by:** Part II**Past exam questions**

No. of lectures: 10

Suggested hours of supervisions: 3

## 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.