Table Of ContentLecture Notes ni
Computer Science
Edited by G. Goos and .J Hartmanis
42
semaJ .E Donahue
Complementary Definitions of
Programming Language Semantics
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Editorial Board
.P Brinch Hansen • D. Gries • C. Moler • G. Seegm~ller • .J Stoer
N. Wirth
Author
James .E Donahue
Cornell University
Department of Computer Science
Upson Hall
Ithaca, N.Y. 14853/USA
Library of Congress Cataloging iu Publication Data
Donahue, James Edward, 1947-
Complementary definitions ofp rogramming language
semantics.
(Lecture notes in computer science 42) ;
Bibliography: .p
Includes index.
.1 Programming languages (Electronic computers)
.2 Pascal (Computer program language) .3 Semantics.
.I Title. ,II Series.
QA76.7.D66 O01.6'424 Z6-6573
AMS Subject Classifications (1970): 94.50
CR Subject Classifications (1974): 4.22, 5.24
ISBN 3-540-07628-X Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-07628-X Springer-Verlag New York - Heidelberg • Berlin
This work is subject to copyright. All rights are reserved, whether the whole
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© by Springer-Verlag Berlin • Heidelberg 1976
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TABLE OF CONTENTS
Chapter 1 Introduction ................................... 1
Chapter 2 History and Literature Review .................. 5
2.1 Introduction ..................................... 5
2.2 Operational semantics ............................ 6
2.3 Denotational semantics ........................... i0
2.4 Propositional semantics .......................... 15
2.5 Complementary definitions ........................ 18
Chapter 3 Basic Elements of the Lattice-Theoretic Approach 22
3.1 Domains .......................................... 22
3.2 Functions ........................................ 24
3.3 Recursion ........................................ 27
Chapter 4 Complementary Definitions of a PASCAL Subset ... 30
4.1 Introduction ..................................... 30
4.2 An informal description of the PASCAL subset ..... 31
4.3 Abstract syntax of the PASCAL subset ............. 42
4.4 Mathematical semantics of the PASCAL subset ...... 44
4.5 Axiomatic semantics of the PASCAL subset ......... 61
Chapter 5 Consistency of the PASCAL Subset Definitions ... 77
5.1 Introduction ..................................... 77
5.2 The interpretation of assertions ................. 78
5.3 Substitution ..................................... 82
5.4 Interpretation of the axiomatic formulas ......... 84
5.5 Proof of consistency of the definitions .......... 86
5.6 Predicate transformers ........................... ii0
Chapter 6 Extensions to the PASCAL Subset Language ....... ll4
6.1 Introduction ..................................... 114
6.2 "Standard" semantics of the PASCAL subset ........ 115
6.3 The semantics of the extensions .................. 135
6.4 Type- and range-checking in PASCAL ............... 149
Chapter 7 Conclusions and Directions for Future Research .157
7.1 Conclusions ...................................... 157
7.2 Directions for future research ................... 159
Bibliography .............................................. 163
ABSTRACT
This volume describes the use of mathematical and axiomatic
semantics as a means of giving complementary definitions of the
semantics of programming languages. We show that by giving
descriptions of language semantics at different levels of
abstraction, we can both:
i. provide semantic definitions well-suited for the purposes
for which formal descriptions are needed, e.g., proving
properties of programs or showing implementations to be
correct, and
.2 develop a framework in which the issues involved in
language design can be rigorously formulated.
As an example of the use of complementary definitions, we
present mathematical and axiomatic definitions of a large subset of
PASCAL. Using the mathematical definition as a model for the more
abstract axiomatic semantics, we show that the axioms and rules of
inference are valid with respect to the mathematical model, i.e., the
definitions are consistent. We also show that mathematical
semantics provides a useful complement to axiomatic semantics because
of
.i the theory of computation underlying the mathematical
approach, which provides important induction rules for
proving the consistency of the~definitions, and
2. the ability to understand the difficulties involved in
extending the axiomatic definitions in terms of the domain
structure of the mathematical definition.
The remainder of the monograph describes a number of possible extensions
of the PASCAL subset in terms of their effect on the definitions given
earlier. We show that certain constructs are inherently more complex
than those used in the PASCAL subset, in terms of the domains necessary
to give a simple, intuitive semantics of the constructs.
ACKNOWLEDGEMENTS
I would like to thank the following people for their assistance and
encouragement during the production of this manuscript:
Professor J.J. Horning, my thesis supervisor at the
University of Toronto,
Professor R.L. Constable, my external thesis examiner,
Professors S.A. Cook, D.G. Corneil, and D.B. Wortman, the
members of my thesis committee, and
my wife, Marilyn, for all her patience and support.
Also, the financial support of the Department of Computer Science,
University of Toronto, and the National Research Council of Canada
is gratefully acknowledged. And, the expert typing of Cheryl Stark
and Debi Fields is greatly appreciated.
CHAPTER 1
INTRODUCTION
The literature of computer science is replete with ingenious
advances in the use of programming languages to communicate to a
computer more effectively the intent of the programmer. However,
language designers find it far more difficult to find examples in the
literature of useful meta-languages in which they can effectively
communicate the intent of their programming language to the user.
Natural languages have long been the principal medium of com-
munication of programming language descriptions. Although such
descriptions have strong appeal, particularly when they are as well-
written as the Revised Algol Report [Naur 1963], there seem to be
many instances when an informal description of the semantics of a
programming language is inadequate for the needs of users, other
designers, or implementors. The interest in the development and use
of formal meta-languages to describe the semantics of programming
languages is, in large measurer a reaction to the deficiencies of
informal semantic descriptions.
The goal of using formal methods to communicate understanding
about programming languages has certainly not been the only motivation
behind many of the efforts in formal semantics. The classical
problems of program equivalence and implementation correctness also
require formal semantic definitions. It is important to remember,
however, that the primary goal of formal semantics is to provide more
effective communication between the language designer and the various
audiences with an interest in the language.
One of the major problems in effectively communicating this
intent is that we use the semantic description of a language at many
different levels. At one level, a semantic definition should
provide a means of reasoning about how to form programs using the
facilities of the language, including the problems of guaranteeing
the correctness of the programs produced. Semantic descriptions at
a more detailed level are required as tools for language design and
comparison. And, finally, sophisticated users and implementors
may require language descriptions that give detailed descriptions
of the sequence of operations invoked by each construct to judge
the relative efficiency of programs written in the language or to
have a useful implementation model.
The complexities of programming languages also make the choice
of an appropriate level of detail of a semantic description
particularly important. For example, a compiler for a language
provides a formal description of the semantics of the language. Yet
the enormous amount of detail found in most compilers includes many
notions irrelevant to an understanding of the language for many
purposes, e.g., how to establish addressability on a particular
machine. This detail only serves to cloud the essential character-
istics of the language of interest to the casual user or language
designer. Just as we can describe a program at several levels of
abstraction, so it seems necessary to provide semantic descriptions
at various "levels of abstraction" suited for the particular
purpose for which a formal definition is required.
This report develops the proposition, first made by Hoare and
Lauer [1974], that these characteristics of the problem of semantic
description of programming languages make any single definition
technique unsuitable for giving the sole description of the semantics
of the language. Instead, we propose giving semantic descriptions
in terms of complementary definitions, i.e., a set of defintions,
each defining the language at a particular level of abstraction, and
each well-suited for use for particular purposes. We show that by
using the axiomatic approach to semantics developed by Hoare [1969a,
1969b, 1971a] and the mathematical approach of Scott and Strachey
[Scott 1970, Scott 1972c, Scott and Strachey 1972] as complementary
definition techniques, it is possible to give complete descriptions
of useful programming languages that are suitable for a variety of
purposes.
Although we could view the axiomatic definition as simply a
useful abstraction of the mathematical definition, we will consider
the definitions as truly complementary for two main reasons. First,
as we will discuss further in the next chapter, the definitions
given using each approach are definitely well-suited for different
pruposes. Moreover, in Chapter ,4 we will use the implicit
assumptions of the axiomatic approach to structure the domains used
in the mathematical definition. Thus we not only use the mathematical
definition to interpret the axiomatic definition, but we also use the
axiomatic definition to impose limitations on the mathematical model.
The monograph presents a mathematical and axiomatic treatment of a
large subset of the programming language PASCAL [Wirth 1973a]. The
major results include:
i. Complete mathematical and axiomatic definitions of the
PASCAL subset are given. The mathematical definition
uses a domain structure which reflects the implicit
assumptions of the axiomatic approach, and which is less
powerful and less "machine-oriented" than the more common
mathematical model used for similar languages, i.e., the
so-called "standard" semantics [Ligler 1975a]. Additionally,
the axiomatic definition includes a new treatment of the
semantics of input and output.
.2 The axioms and rules of inference given by the axiomatic
definition are proven to be valid with respect to the
model provided by the mathematical definition. We also
suggest how the proofs given could be mechanically checked,
or even generated, using an LCF-like [Milner 1972] formal
logic. The use of mathematical semantics as a model of
axiomatic definitions is similar to work recently done by
George Ligler at Oxford [Ligler 1975a, 1975b].
.3 An interpretation of Dijkstra's "predicate transformers"
[1973] is given and the properties of "healthy" predicate
transformers are proven as theorems about the interpretat-
ion.
4. The domain structure of mathematical definitions is used to
analyze the limitations of the axiomatic approach and the
"complexity" of certain language constructs.
The organization of the volume is as follows. Chapter 2 provides
a review of the literature and a history of the development of various
approaches to semantics. Chapter 3 provides the basic elements of
the lattice-theoretic approach to the theory of computation used in
the remainder of the thesis to give the mathematical semantics of the
PASCAL subset. Chapter 4 presents the definition of the PASCAL
subset using the axiomatic and mathematical approaches. In Chapter 5
we give an interpretation of axiomatic formulas in terms of the model
provided by the mathematical definition and present the proofs of
consistency of the axiomatic definition and the mathematical
definition of the PASCAL subset. Additionally, a Similar interpretat-
ion of predicate transformers is presented. Chapter 6 discusses
extensions to the language in terms of their effect on the semantic
model used in the definitions~ Finally, we give summary, conclusions,
and directions for future research in Chapter .7
CHAPTER 2
HISTORY AND LITERATURE REVIEW
2.1 Introduction
The most obvious characteristic of the literature on the
semantics of programming languages is its diversity of viewpoint.
The purpose of this chapter is to organize some of the basic ideas
present in semantics research and to give a developmental context
for the work described in the thesis.
The many different approaches to semantics can be roughly
characterized as being of two different sorts:
°i "Compiler-oriented." This form of semantics is exemplified
by [Knuth 1968, Lee 1972, van Wijngaarden 1966]. In
compiler-oriented semantic definitions, meaning is given
in terms of a set of translations performed on some abstract
representation of the syntactic structure (e.g., a parse
tree) of programs in the language.
2. "Interpreter-oriented." In interpreter-oriented definitions,
the meaning is given in terms of the "transformations"
that may be specified by the syntactically valid programs
in the language. The program text appears in the
definitions more or less directly and transformations of
the text are not included in the definition. All of the
semantic models discussed in the remainder of this thesis
will be interpreter-oriented models.
Specifically, by an interpreter-oriented formal semantics of a
programming language, we mean:
.i a formal description of the "universe of discourse" for the
language, i.e., the class of objects that may be manipulated
by programs in the language. These objects could include
"machine states," functions, normal form lambda expressions,
assertions, or any other class of objects that may be
formally specified.
.2 a set of rules describing the ways in which the basic
expressions in the language can be combined and giving the
associated outcome ("value") of each of the combinations in
