Table Of ContentAdvances in Computing Science
Advisory Board
R. F. Albrecht, Innsbruck
Z. Bubnicki, Wroclaw
R. E. Burkard, Graz
A. G. Butkovskiy, Moscow
C. Cap, Zurich
M. Dal Cin, Erlangen
W Hackbusch, Kiel
G. Johnson, Fort Collins
W Kreutzer, Christchurch
W. G. Kropatsch, Wien
1. Lovrec, Zagreb
G. Nemeth, Budapest
H. J. Steuer, Wien
Y. Takahara, Chiba
W. Tornig, Darmstadt
1. Troch, Wien
H. P. Zima, Wien
C.Brink
W. Kahl
G. Schmidt (eds.)
Relational Methods
in Computer Science
Springer-Verlag Wien GmbH
Prof. Dr. Chris Brink
Department of Mathematics and Applied Mathematics, University of Cape Town,
Cape Town, South Africa
Dr. Wolfram Kahl
Prof. Dr. Gunther Schmidt
Fakultiit fiir Informatik, Universitiit der Bundeswehr Munchen,
Neubiberg, Federal Republic of Germany
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© 1997 Springer-Verlag Wien
Originally published by Springer-Verlag Wien New York in 1997
Typesetting: Camera ready by authors
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Printed on acid-free and chlorine-free bleached paper
With 30 Figures
ISSN 1433-0113
ISBN 978-3-211-82971-4 ISBN 978-3-7091-6510-2 (eBook)
DOI 10.1007/978-3-7091-6510-2
Advances in Computing Science
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The Publisher
We dedicate this book
to the memory of Ernst Schroder (1841-1902),
in celebration of the first centenary of his
Algebra und Logik der Relative
of 1895
Preface
The calculus of relations has been an important component of the development of
logic and algebra since the middle of the nineteenth century, when Augustus De
Morgan observed that since a horse is an animal we should be able to infer that
the head of a horse is the head of an animal. For this, Aristotelian syllogistic does
not suffice: We require relational reasoning.
George Boole, in his Mathematical Analysis of Logic of 1847, initiated the
treatment of logic as part of mathematics, specifically as part of algebra. Quite
the opposite conviction was put forward early this century by Bertrand Russell
and Alfred North Whitehead in their Principia Mathematica (1910 - 1913): that
mathematics was essentially grounded in logic. Logic thus developed in two
streams.
On the one hand algebraic logic, in which the calculus of relations played a
particularly prominent part, was taken up from Boole by Charles Sanders Peirce,
who wished to do for the "calculus of relatives" what Boole had done for the
calculus of sets. Peirce's work was in turn taken up by Schroder in his Algebra
und Logik der Relative of 1895 (the third part of a massive work on the algebra of
logic). Schroder's work, however, lay dormant for more than 40 years, until revived
by Alfred Tarski in his seminal paper "On the calculus of binary relations" of 1941
(actually his presidential address to the Association for Symbolic Logic). Tarski's
paper is still often referred to today as the best introduction to the calculus of
relations. It gave rise to a whole field of study, that of relation algebras, and
more generally Boolean algebras with operators. These formed part of the study
of algebraic logic, which in its modern form overlaps with universal algebra, model
theory, nonclassical logics, and more recently program semantics and program
development.
In the other stream of development, the theory of relations was accorded a
key role in Principia Mathematica. This important work defined much of the
subsequent development of logic in the 20th century, completely eclipsing for some
time the development of algebraic logic. In this stream of development, relational
calculus and relational methods appear with the development of universal algebra
in the 1930's, and again with model theory from the 1950's onwards. In so far
as these disciplines in turn overlapped with the development of category theory,
relational methods sometimes appear in this context as well.
It is fair to say that the role of the calculus of relations in the interaction
between algebra and logic is by now well understood and appreciated, and that
relational methods are part of the toolbox of the mathematician and the logician.
Over the past twenty years relational methods have however also become of fun
damental importance in computer science. For example, much of the theory of
x Preface
nonclassical logics was used (though sometimes re-invented) in the new so-called
program logics. These arose from the realisation that a program may be thought of
as an input-output relation over some state space: an accessibility relation, in the
sense of modal logic, saying that from a given initial state certain final states are ac
cessible. Not every relation over a state space can be thought of as a program, but
any relation can be thought of as a specification of a program. This point of view,
that the calculus of relations is fundamental to programs, was clearly enunciated
by the Oxford group of Tony Hoare in an influential paper on "Laws of program
ming" in 1987. Also during the 1980's, much of the equational theory of relation
algebras were already being applied to program semantics and program develop
ment. For example, the book Relations and Graphs by Schmidt and Str6hlein (in
German, 1989; in English, 1993) started from the basis of programs as graphs with
Boolean matrices. On the other hand, a topic such as relational databases was
an essentially new development, owing little to previous mathematical work on or
with relations. Thus computer science, as the new application field for relational
methods, has both drawn from and contributed to previous logico/mathematical
work; this is a sign of healthy development.
However, the role of relational methods within computer science is not yet as
well-known or well-understood as it is in logic and mathematics. It is the aim of
this book to help address this situation, by giving an overview of relational methods
in computer science. We have not tried to give an exhaustive survey - indeed, part
of what we claim is that the topic is already too large for an exhaustive survey
to fit into one volume. We hope, however, that the topics included will suffice
to convey an impression of the power and wide-ranging applicability of relational
methods.
Just as we have attempted to bring together under one heading various dis
parate endeavours in using relational methods in computer science, so we have
also attempted to bring together the researchers involved. It became clear to us
in the early 1990's that a core group of researchers using or working on relational
methods was turning up repeatedly at various specialist conferences. It seemed a
good idea, therefore, to organise a meeting specifically on relational methods, so
as to give all those who developed or used them in some area of computer science a
chance to meet and talk across disciplinary boundaries. The outcome of this idea
was the International Seminar on Relational Methods in Computer Science (later
known as RelMiCS 1), held as Seminar 9403 at SchloB Dagstuhl in the Saarland
in January 1994. We gratefully acknowledge the assistance of the Internationales
Begegnungs- und Forschungszentrum fur Informatik in organising and funding this
Seminar. In our preface to the Seminar Report we said:
Since the mid-1970's it has become clear that the calculus of relations is
a fundamental conceptual and methodological tool in computer science
just as much as in mathematics. A number of seemingly distinct areas
ofresearch have in fact this much in common that their concepts and/or
techniques come from the calculus of relations. However, it has also
become clear that many opportunities for cross-pollination are being
lost simply because there was no organised forum of discussion between
Preface xi
researchers who, though they use the same concepts and methods,
nonetheless perceive themselves as working in different fields.
The aim of this Dagstuhl Seminar was, therefore, to bring together
researchers from various subdisciplines of computer science and math
ematics, all of whom use relational methods in their work, and to en
courage the creation of an active network continuing after the Seminar
to exchange ideas and results.
This has become a manifesto of sorts for what is now known as the RelMiCS Group.
The idea of producing this book first originated at Dagstuhl, where a number of
authors were recruited. Some subsequent organisational work was done from Cape
Town and Miinchen, but the book only really took shape in Rio de Janeiro: first
at a planning session in July 1994, then at the RelMiCS 2 meeting in July 1995.
We are pleased to acknowledge our indebtedness to Armando Haeberer for or
ganising these extremely pleasant and productive events. We gratefully acknowl
edge the financial assistance of CNPq (Conselho Nacional de Desenvolvimento
Cientffico e Tecnologico) for the planning session of July 1994, and of FAPERJ
(Funda~ao de Amparo it Pesquisa do Estado do Rio de Janeiro) and IBM Brazil
for RelMiCS 2 in July 1995. It gave us the chance to "workshop" the contents of
the book so as to increase the homogeneity of notation and terminology that we
felt was desirable.
In consequence, this is a multi-author book. However, though each chapter
is written by a different set of authors, it has been our intention that the book
should be more than just a collection of papers: it should have unity of style and
presentation. In this way we hope that the spirit of bringing together topics and
researchers is reflected also in the structure of the book itself.
Chapter 1 sets out the background material needed in the rest of the book, and
the notation to be used. The problem of proliferation of different notations and
terminology has been an impediment to communication about relational methods
between various groups of researchers; it seems justified, therefore, to try, at least
in a book like this, for a measure of coherence. After Chapt. 1, which forms the
introductory Part I of the book, there are four more parts. Part II deals with
current versions of algebras of relations, Part III with applications to logic, Part
IV with applications to programming, and Part V with other application areas.
For lack of space the Bibliography only contains works actually referred to in
the text; a more comprehensive bibliography on the calculus of relations and its
applications is available electronically.
Acknowledgements. In the original submissions for this book almost all au
thors acknowledged an indebtedness to other authors; we cover this by making here
a blanket acknowledgement of thanks for mutual cooperation. We personally owe
a large debt of thanks to Thomas Str6hlein, who undertook and carried out the
arduous task of assembling a bibliography of relational methods based on submis
sions from several coauthors. The bibliography at the end of this book is a small
subset of this comprehensive bibliography. In addition, we gratefully acknowledge
work done on the manuscript by Arne Bayer, Thomas Gritzner, and Peter Kempf.
xii Preface
We are grateful to Rudolf Albrecht for establishing contact with the publishers.
Of course our largest debt is to the authors who actually wrote the book, and who
submitted themselves with good grace to the editorial requirements we imposed
on them.
With over 30 contributors to this book we believe it possible that some errors may
remain; for these we crave the indulgence of the reader, and would be pleased to
receive feedback.
October 1996
Chris Brink Wolfram Kahl, Gunther Schmidt
University of Cape Town Universitiit der Bundeswehr Munchen
Description:The calculus of relations has been an important component of the development of logic and algebra since the middle of the nineteenth century, when Augustus De Morgan observed that since a horse is an animal we should be able to infer that the head of a horse is the head of an animal. For this, Arist
