Semantics with Applications: An Appetizer

284 Pages·2007·2.3 MB·English

by  Hanne Riis Nielson

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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
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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
(cid:1) (cid:2)
(cid:1) (cid:2)
(cid:1) (cid:2)
(cid:1) (cid:2)
(cid:1) (cid:2)
Chapter 3 Chapter 4
Chapter 5
(cid:1) (cid:2)
(cid:1) (cid:2)
(cid:1) (cid:2)
(cid:1) (cid:2)
(cid:1) (cid:2)
Chapter 6 Chapter 7
Chapter 9
(cid:1)
(cid:1)
(cid:1)
(cid:1)
(cid:1)
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 (cid:1)→ and (cid:3) non-
standard. We use D (cid:1)→ E for the set of partial functions from D to E; this is
because we find that the D (cid:2) E notation is too easily overlooked. Also, we
use R (cid:3) 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
x Contents
5.4 An Equivalence Result ....................................121
6. More on Denotational Semantics ...........................127
6.1 Environments and Stores ..................................127
6.2 Continuations............................................138
7. Program Analysis ..........................................145
7.1 Detection of Signs Analysis: Specification....................149
7.2 Detection of Signs Analysis: Existence.......................161
7.3 Safety of the Analysis.....................................166
7.4 Program Transformation ..................................171
8. More on Program Analysis .................................175
8.1 Data Flow Frameworks....................................177
8.2 Security Analysis.........................................183
8.3 Safety of the Analysis.....................................193
9. Axiomatic Program Verification ............................205
9.1 Direct Proofs of Program Correctness .......................205
9.2 Partial Correctness Assertions..............................212
9.3 Soundness and Completeness...............................220
10. More on Axiomatic Program Verification ...................229
10.1 Total Correctness Assertions ...............................229
10.2 Assertions for Execution Time .............................239
11. Further Reading............................................247
A. Review of Notation.........................................251
B. Implementation of Program Analysis .......................255
B.1 The General and Monotone Frameworks.....................257
B.2 The Completely Additive Framework .......................259
B.3 Iterative Program Schemes ................................262
Bibliography....................................................267
Index...........................................................269