Table Of ContentS
Hanne Riis Nielson
(cid:7)(cid:25)(cid:21)(cid:26)(cid:26)(cid:24)(cid:27)(cid:22)(cid:1)(cid:12)(cid:24)(cid:21)(cid:25)(cid:31)(cid:28)(cid:27)
(cid:12)
(cid:24)
(cid:21)
(cid:25)
(cid:31)
(cid:28)
(cid:27)
(cid:1)
(cid:35)
(cid:1)
(cid:12)
(cid:24)
(cid:21)
(cid:25)
(cid:31)
(cid:28)
(cid:27)
(cid:9)(cid:19)(cid:27)(cid:27)(cid:21)(cid:1)(cid:15)(cid:24)(cid:24)(cid:31)(cid:1)(cid:12)(cid:24)(cid:21)(cid:25)(cid:31)(cid:28)(cid:27)(cid:1)(cid:35)(cid:1)(cid:7)(cid:25)(cid:21)(cid:26)(cid:26)(cid:24)(cid:27)(cid:22)(cid:1)(cid:12)(cid:24)(cid:21)(cid:25)(cid:31)(cid:28)(cid:27) UNDERGRADUATE TOPICS
in COMPUTER SCIENCE
Semantics
with
Applications
An Appetizer
(cid:11)(cid:13)(cid:10)(cid:12)(cid:1)(cid:9)(cid:7)(cid:8)(cid:2)(cid:3)(cid:2)(cid:8)(cid:5)(cid:6)(cid:4)(cid:8)(cid:2)(cid:6)(cid:9)(cid:3)(cid:2)(cid:9)
UTiCS
Undergraduate Topics in Computer Science
Undergraduate Topics in Computer Science (UTiCS) delivers high-quality instructional content for
undergraduates studying in all areas of computing and information science. From core foundational and
theoretical material to final-year topics and applications, UTiCS books take a fresh, concise, and modern
approach and are ideal for self-study or for a one- or two-semester course. The texts are all authored by
established experts in their fields, reviewed by an international advisory board, and contain numerous
examples and problems. Many include fully worked solutions.
Also in this series
Iain Craig
Object-Oriented Programming Languages: Interpretation
978-1-84628-773-2
Max Bramer
Principles of Data Mining
978-1-84628-765-7
Hanne Riis Nielson and Flemming Nielson
Semantics with
Applications:
An Appetizer
Hanne Riis Nielson, PhD Flemming Nielson, PhD, DSc
The Technical University of Denmark The Technical University of Denmark
Denmark Denmark
Series editor
Ian Mackie
École Polytechnique, France and King’s College London, UK
Advisory board
Samson Abramsky, University of Oxford, UK
Chris Hankin, Imperial College London, UK
Dexter Kozen, Cornell University, USA
Andrew Pitts, University of Cambridge, UK
Hanne Riis Nielson, Technical University of Denmark, Denmark
Steven Skiena, Stony Brook University, USA
Iain Stewart, University of Durham, UK
David Zhang, The Hong Kong Polytechnic University, Hong Kong
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2006939147
Undergraduate Topics in Computer Science ISSN 1863-7310
ISBN-10: 1-84628-691-3 e-ISBN-10: 1-84628-692-1
ISBN-13: 978-1-84628-691-9 e-ISBN-13: 978-1-84628-692-6
Printed on acid-free paper
© Springer-Verlag London Limited 2007
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,
or in the case of reprographic reproduction in accordance with the terms of licences issued by the
Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to
the publishers.
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of
a specific statement, that such names are exempt from the relevant laws and regulations and therefore
free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the information
contained in this book and cannot accept any legal responsibility or liability for any errors or omissions
that may be made.
9 8 7 6 5 4 3 2 1
Springer Science+Business Media
springer.com
Preface
This book is written out of a tradition that places special emphasis on the
following three approaches to semantics:
– operational semantics,
– denotational semantics, and
– axiomatic semantics.
It is therefore beyond the scope of this introductory book to cover other ap-
proaches such as algebraic semantics, game semantics, and evolving algebras.
We strongly believe that semantics has an important role to play in the fu-
turedevelopmentofsoftwaresystemsanddomain-specificlanguages(andhence
is not confined to the enormous task of specifying “real life” languages such
as C++, Java or C#). We have therefore found the need for an introductory
book that
– presents the fundamental ideas behind these approaches,
– stressestheirrelationship byformulatingandprovingtherelevanttheorems,
and
– illustrates the applications of semantics in computer science.
This is an ambitious goal for an introductory book, and to achieve it, the bulk
of the technical development concentrates on a rather small core language of
while-programs for which the three approaches are developed to roughly the
same level of sophistication; this should enable students to get a better grasp
of similarities and differences among the three approaches.
In our choice of applications, we have selected some of the historically
important application areas as well as some of the more promising candidates
for future applications:
vi Preface
Chapter 1
Chapter 2
✟ ❍
✟ ❍
✟ ❍
✟ ❍
✟ ❍
Chapter 3 Chapter 4
Chapter 5
✟ ❍
✟ ❍
✟ ❍
✟ ❍
✟ ❍
Chapter 6 Chapter 7
Chapter 9
✟
✟
✟
✟
✟
Chapter 10 Chapter 8
Chapter 11
– the use of semantics for validating prototype implementations of program-
ming languages;
– the use of semantics for verifying program analyses that are part of more
advanced implementations of programming languages;
– the use of semantics for verifying security analyses; and
– the use of semantics for verifying useful program properties, including infor-
mation about execution time.
Clearly this only serves as an appetizer to the fascinating area of “Semantics
with Applications”; some pointers for further reading are given in Chapter 11.
Overview. As is illustrated in the dependency diagram, Chapters 1, 2, 5, 9,
and 11 form the core of the book. Chapter 1 introduces the example language
While of while-programs that is used throughout the book. In Chapter 2
we cover two approaches to operational semantics, the natural semantics of
Preface vii
G. Kahn and the structural operational semantics of G. D. Plotkin. Chapter
5 develops the denotational semantics of D. Scott and C. Strachey, including
simple fixed point theory. Chapter 9 introduces program verification based on
operational and denotational semantics and goes on to present the axiomatic
approach due to C. A. R. Hoare. Finally, Chapter 11 contains suggestions for
further reading. Chapters 2, 5, and 9 are devoted to the language While and
cover specification as well as theory; there is quite a bit of attention to the
proof techniques needed for proving the relevant theorems.
Chapters 3, 6, and 10 consider extensions of the approach by incorporat-
ing new descriptive techniques or new language constructs; in the interest of
breadth of coverage, the emphasis is on specification rather than theory. To be
specific, Chapter 3 considers extensions with abortion, non-determinism, par-
allelism, block constructs, dynamic and static procedures, and non-recursive
andrecursiveprocedures.InChapter6weconsiderstaticproceduresthatmay
ormaynotberecursiveandweshowhowtohandleexceptions;thatis,certain
kindsofjumps.Finally,inSection10.1weconsidernon-recursiveandrecursive
procedures and show how to deal with total correctness properties.
Chapters4,7,8,and10covertheapplicationsofoperational,denotational,
andaxiomaticsemanticstothelanguageWhileasdevelopedinChapters2,5,
and9.InChapter4weshowhowtoprovethecorrectnessofasimplecompiler
using the operational semantics. In Chapter 7 we show how to specify and
prove the correctness of a program analysis for “Detection of Signs” using the
denotational semantics. Furthermore, in Chapter 8 we specify and prove the
correctness of a security analysis once more using the denotational semantics.
Finally, in Section 10.2 we extend the axiomatic approach so as to obtain
information about execution time.
AppendixAreviewsthemathematicalnotationonwhichthisbookisbased.
It is mostly standard notation, but some may find our use of ! and non-
→ ⋄
standard. We use D ! E for the set of partial functions from D to E; this is
→
because we find that the D ⇀ E notation is too easily overlooked. Also, we
use R S for the composition of binary relations R and S. When dealing with
⋄
axiomatic semantics we use formulae P S Q for partial correctness
{ } { }
assertions but P S Q for total correctness assertions, hoping that
{ } {⇓ }
the explicit occurrence of (for termination) may prevent the student from
⇓
confusing the two systems.
AppendixBcontainssomefairlydetailedresultsforcalculatingthenumber
of iterations of a functional before it stabilises and produces the least fixed
point.Thisappliestothefunctionalsarisingintheprogramanalysesdeveloped
in Chapters 7 and 8.
viii Preface
Notes for the instructor. Thereadershouldpreferablybeacquaintedwiththe
BNF style of specifying the syntax of programming languages and should be
familiar with most of the mathematical concepts surveyed in Appendix A.
Weprovidetwokindsofexercises.Onekindhelpsthestudentinunderstand-
ingthedefinitions,results,andtechniquesusedinthetext.Inparticular,there
areexercisesthataskthestudenttoproveauxiliaryresultsneededforthemain
results but then the proof techniques will be minor variations of those already
explainedinthetext.Wehavemarkedthoseexerciseswhoseresultsareneeded
later by “Essential”. The other kind of exercises are more challenging in that
they extend the development, for example by relating it to other approaches.
Weuseastartomarkthemoredifficultoftheseexercises.Exercisesmarkedby
two stars are rather lengthy and may require insight not otherwise presented
in the book. It will not be necessary for students to attempt all the exercises,
but we do recommend that they read them and try to understand what the
exercises are about. For a list of misprints and supplementary material, please
consult the webpage http://www.imm.dtu.dk/ riis/SWA/swa.html.
∼
Acknowledgments. This book grew out of our previous book Semantics with
Applications: A Formal Introduction [18] that was published by Wiley in 1992
andanote,Semantics with Applications: Model-Based Program Analysis,writ-
ten in 1996. Over the years,we have obtained many comments from colleagues
and students, and since we are constantly reminded that the material is still
in demand, we have taken this opportunity to rework the book. This includes
using shorter chapters and a different choice of security-related analyses. The
present version has benefitted from the comments of Henning Makholm.
Kongens Lyngby, Denmark, January 2007 Hanne Riis Nielson
Flemming Nielson
Contents
List of Tables ................................................... xi
1. Introduction................................................ 1
1.1 Semantic Description Methods ............................. 1
1.2 The Example Language While............................. 7
1.3 Semantics of Expressions .................................. 9
1.4 Properties of the Semantics ................................ 16
2. Operational Semantics...................................... 19
2.1 Natural Semantics........................................ 20
2.2 Structural Operational Semantics........................... 33
2.3 An Equivalence Result .................................... 41
3. More on Operational Semantics ............................ 47
3.1 Non-sequential Language Constructs........................ 47
3.2 Blocks and Procedures .................................... 54
4. Provably Correct Implementation .......................... 67
4.1 The Abstract Machine .................................... 67
4.2 Specification of the Translation............................. 75
4.3 Correctness.............................................. 78
4.4 An Alternative Proof Technique ............................ 88
5. Denotational Semantics..................................... 91
5.1 Direct Style Semantics: Specification ........................ 92
5.2 Fixed Point Theory....................................... 99
5.3 Direct Style Semantics: Existence...........................115
