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##

Denotational Semantics

*Lecturer: Dr M.P. Fiore*

*No. of lectures:* 10 *(including revision)*

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

**Aims**

The aims of this course are to introduce domain theory and denotational semantics, and 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.**Products of domains. Function domains. Flat 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.**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 with 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**

Books:

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

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

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

Papers:

Fiore, M., Jung, A., Moggi, E., O'Hearn, P., Riecke, J., Rosolini, G.,
and Stark, I. (1996).
Domains and denotational semantics: History, accomplishments and
open problems.
*Bulletin of EATCS, 59:227-256*.

Milner, R. (1977).
Fully abstract models of typed lambda-calculi.
*Theoretical Computer Science, 4:1-22*.

Ong, C.-H. (1995).
Correspondence between operational and denotational semantics.
*Handbook of Logic in Computer Science, Vol. 4, pp. 269-356*.

Plotkin, G. (1977).
LCF considered as a programming language.
*Theoretical Computer Science, 5:223-256*.

Scott, D. (1969).
A type-theoretical alternative to CUCH, ISWIM, OWHY.
(In *Theoretical Computer Science, 121:411-440, 1993.*)

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