Selected Papers

474 Pages·1982·12.251 MB·English

by  Calvin C. Elgot (auth.), Stephen L. Bloom (eds.)

CALVIN C. ELGOT
SELECTED PAPERS
CALVIN C. ELGOT
Calvin C. Elgot
Selected Papers
Edited by Stephen L. Bloom
With a Foreword
by Dana S. Scott
and
'A Glimpse Back'
by Samuel Eilenberg
Springer-Verlag
New York Heidelberg Berlin
Calvin C. Elgot
Mathematical Sciences Department
IBM Watson Research Center
( 1960-1980)
Yorktown Heights, New York 10598
USA
Editor
Stephen L. Bloom
Department of Pure and Applied Mathematics
Stevens Institute of Technology
Hoboken, New Jersey 07030
USA
Library of Congress Cataloging in Publication Data
Elgot, Calvin C.
Calvin C. Elgot selected papers.
Bibliography: p.
1. Mathematics--Collected works. 2. Recursion theory
--Collected works. 3. Machine theory--Collected works.
I. Bloom, Stephen L. II. Title.
QA3.E5825 1982 510 82-19161
With One Halftone and 135 Line Illustrations.
© 1982 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1982
All rights reserved. No part of this book may be translated or
reproduced in any form without written permission from Springer-Verlag,
175 Fifth Avenue, New York, New York 10010, USA.
9 8 765 432 1
ISBN-13: 978-1-4613-8179-2 e-ISBN-13: 978-1-4613-8177-8
DOl: 10.1007/978-1-4613-8177-8
Foreword
Cal Elgot was a very serious and thoughtful researcher, who with great determi
nation attempted to find basic explanations for certain mathematical phenomena
as the selection of papers in this volume well illustrate. His approach was, for the
most part, rather finitist and constructivist, and he was inevitably drawn to studies
of the process of computation. It seems to me that his early work on decision
problems relating automata and logic, starting with his thesis under Roger Lyndon
and continuing with joint work with Biichi, Wright, Copi, Rutledge, Mezei, and
then later with Rabin, set the stage for his attack on the theory of computation
through the abstract treatment of the notion of a machine. This is also apparent in
his joint work with A. Robinson reproduced here and in his joint papers with John
Shepherdson. Of course in the light of subsequent work on decision problems by
Biichi, Rabin, Shelah, and many, many others, the subject has been placed on a
completely different plane from what it was when Elgot left the area. But I feel that
his papers, results-and style-were very definitely influential at the time and may
well have altered the course of the investigation of these problems.
As Sammy Eilenberg explains, the next big influence on Elgot's thinking was
category theory, which gave him a way of expressing his ideas in a sharply algebraic
manner. The joint book with Eilenberg is one illustration of this influence. Note,
however, that Eilenberg himself went on in his own (and, one hopes, soon to be
completed) treatise on automata theory to make that subject a substantial branch of
algebra without very much use of category theory. What Elgot seems to have been
seeking was rather a suitable part of categorical logic, which he found partly in
Lawvere's algebraic theories plus his own addition of iteration. Eilenberg is perhaps
more interested in the analysis of certain structures (as is a common concern in a
large part of abstract ~lgebra), while Elgot wanted to find a way of organizing
certain properties that appear in the study of operations on abstract algorithms.
I first met Cal and his wife Jane in Berkeley in 1954-55 when we took part in the
very stimulating set theory lectures of Alfred Tarski. The year with Tarski seems to
have left a lasting impression on Elgot's work-especially as concerns the use of
rigorous set-theoretical constructions and of model theory. I later visited Elgot
many times at Yorktown Heights over the years. Our last two meetings were at the
Mac Lane Symposium at Aspen, Colorado, and, finally, the next year at Oxford, a
very few days before his most untimely death.
At both of these last meetings Cal spoke to me at length and with particular
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FOREWORD
intensity about the role he saw for category-theoretic methods. Unfortunately, I do
not think his publications give us clearly enough the picture he was in the process of
forming in the last period of his research. He had come to a synthesis and
unification through category theory, which perhaps the somewhat lengthy detail of
his later papers does not quite let shine through. I wish I had written notes on our
conversations so I could recapture the sweep of his plans. Alas, we planned to meet
and correspond again soon, so I put off thinking about the details; and in Oxford I
was distracted by many duties and personal concerns, so my memory now fails me.
I wrote to both Bloom and Shepherdson, the latter of whom Cal was visiting for
joint work in the weeks before leaving England, but neither felt he could recapture
and state precisely Elgot's final philosophy.
A hint-but I feel only a preliminary hint-is given in the brief paper for the Mac
Lane Symposium. One problem with that paper, as Elgot admits there, is that the
examples given are rather slight. As this paper is not reprinted in this volume, it is
perhaps worth repeating a few of Elgot's paragraphs. First, there is his motivation
(p. 85):
A couple of years ago, despairing that progress in the foundations of the theory of
computation was taking place much too slowly, I came to the conclusion that what was
needed was a "suitable" framework for such a theory. On the one hand sufficient
detail and decisions would have to be worked out in such a way as to promise an
attractive, perspicuous and incisive theory, and on the other hand, the framework
would have to be stlfficiently broad to allow many different kinds of questions to be
asked and answered. Independent studies carried out within the framework would be
insured a measure of cohesiveness. The perspective gained would permit judgements
concerning the importance of and the relationships among the various questions. One
conclusion concerning the nature of the framework which has been slowly evolving in
my mind: the language of categorical algebra would be essential.
There is nothing to quarrel with in Elgot's statement of desires for the framework
of the theory; the conclusion about categorical algebra, however, does not seem to
follow from what he has just said about theory building. In fact, on the next page in
a paragraph called "the anomaly, " Elgot remarks:
The importance that I attached (and still do) to the framework enterprise, together
with the conclusion reached, immediately provided a pressing puzzle: only the most
elementary and basic concepts of category theory were central for the framework.
Indeed the oft-quoted dictum-which it seems to me is quite consistent with present
category-theoretic activity-to the effect "the reason for defining 'category' is to define
'functor,' and the reason for defining 'functor' is to define 'natural transforma
tion' ... " seemed strikingly at odds with both those aspects of category theory which I
found most useful in the past (cf. Monadic computation and iterative algebraic theories,
reprinted in this volume) as well as those aspects which I anticipated would be most
useful in the future. But these latter have been, for the most part, not sufficiently clear
to attempt to elucidate. The conference at Aspen stimulated this first, very small,
attempt to give these matters voice even though the ideas involved have not yet fully
crystallized.
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FOREWORD
Elgot goes on to discuss the "new roles" for categorical algebra, namely: "(1)
category theory as a tool for applied mathematics; (2) category theory as a common
foundation for theoretic computer science and finitely describable mathematics." We
will return in a moment to the discussion of the examples of the use of category
theory that Elgot has in mind. Elgot ends the paper with thoughts on the "common
foundation," first recalling some remarks of F. W. Lawvere, about which he
comments:
Lawvere's approach to the Foundation serves as a first approximation to the goal
(2). It takes the point of view, however, that all categories pre-exist, while I would like
to construct categories as required. Lawvere wants his foundation to be capable of
defining and proving the usual things while I seek an incisive tool for re-examining
much of mathematics that exists and which would serve as a guide in creating new
mathematics (including theoretical computer science). On a more detailed level, I
would like the underlying logic to be quantifier-free and regard the treatment of
composition (in categories) as a ternary relation to be a distortion. The underlying
language I have (only vaguely) in mind would gain power by having available the
ability to describe general algorithmic procedures for introducing new functions and
having available, for proofs, a principle (or principles) of mathematical induction.
I am fully in agreement with Elgot's desire for a workable mathematical language
for algorithms which would allow proofs as well as definitions (since programming
languages are too focused on the latter), and I can see reasons why the develop
ments in category theory may be pointing the way, but-unless I have completely
misunderstood his point-his strictures on logic seem far too harsh. Elgot ends the
last section of the Mac Lane paper with a quote from Hermann Weyl on
Brouwerian intuitionism and from Aleksandrov on abstract concepts in mathemat
ics, and there is great justice in his suggestion that a balance is needed between
being constructive and being abstract (and in his criticism of Lawvere's early hopes
for category theory as a foundation for mathematics). Where I am unable to follow
him is in the suggestion that the logic has to be quantifier free. It is true that if you
have a powerful definition language, then a great portion of a (constructive) proof
can be reduced to algebraic calculation plus free-variable inductive arguments. This
has been demonstrated by a number of contributions by logicians to proof theory.
But ordinary mathematical argument does not seem to proceed this way, and recent
renewed interest in intuitionistic mathematics shows that such a restricted logic is
not necessary for remaining constructive. But what is needed is a more extensive
development of "programming logic." There are a number of starts but not as yet
anything very comprehensive, I feel. It is a subject that interests me a great deal,
and Elgot's work will certainly be significant for the future developments. I only
wish I could discuss it with him.
Returning to Elgot's remarks about the desire for constructing categories, the
example he gives in the paper is that of the stack. Without going into any
mathematical details, it seems to me that the point he is making is that any
many-sorted algebra can be viewed as a category with "objects" or "types," just
those needed to explain what structures of the sorts are-and there are usually only
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FOREWORD
finitely many needed. If he had recalled at this point something he knew about
functors, the existence of many of these algebras could have been conveniently
proved by invoking general theorems about initial objects in categories of algebras
associated with these functors. And there are ways of expressing the constructive
character of this existence proof. This remark may not at once justify the use of
natural transformations, but it does go a long way to pointing out-in answer to
Elgot's "anomaly"-why it is good to think in terms of functors. But surely Elgot
knew this.
One of the biggest questions that confronts me in the programme of "re-doing"
parts of theoretical computer science is finding the right choice of category. As the
reader will see in the articles in this volume, Elgot made the choice of a type of
category called "an iterative theory," which has quite nice algebraic properties;
however, these properties have not yet come to a fully satisfactory proof theory, in
my view. For instance, in the joint paper with Bloom and Wright reprinted here, it
takes quite a bit of trouble to make the iteration operation everywhere defined in
such a way that certain desirable laws are preserved. Such a theorem is just a start
at seeing what sort of a logic we can have for these notions. My feeling is that the
last word has not been said on the subject of "fixed-point algebra."
Another indicator of the problems concerning the question about the choice of
category can be appreciated by reference to the recent paper by Jerzy Tiuryn [4].
Here it is shown that iterative theories (or, as the author prefers, iterative algebras)
can be extended to partially ordered "regular" algebras in such a way that the
iteration becomes a suitable least fixed point in the partial ordering. Tiuryn shows,
however, that the extension (which can be made "minimal" and unique) is not
always faithful in that a homomorphism of the given iterative algebra may be
necessary. The consequences of this result for the proof theory must be significant.
Are there equations true in the ordered algebras that are unreasonable for Elgot's
algebra? Perhaps the examples are at hand, but I do not see them. In any case, more
investigation of laws is surely needed.
As a further example of the scope for choice of category, we may refer to the
recent papers of Arbib and Manes [1,2,3], which take good account of Elgot's
writings. What I think that these authors have found is much more smooth-going
categorical axiomatization of the diagramatic operations used by Elgot. They
introduce a calculus of formal power series in their categories that seems very easy
to use in proofs. Moreover, they show with the aid of these power-series expansions
that there is a unique "canonical" fixed-point (or iteration) operator-where
"canonical" has a simple categorical definition. These definitions and investigations
seem to me, therefore, to fit exactly into Elgot's programme. Note, however, that
Arbib and Manes use infinite expansions expressed in the usual mathematical way.
Therefore, they are not giving induction axioms of the kind Elgot was asking for,
since they use ordinary mathematical induction for their proofs in the standard
style. Hence, there is still something to be done in making the proof theory more
abstract. (These papers mentioned, by the way, are only a very small sample of the
large literature that has grown up in the last 15 years. I did not mean to suggest that
these are representative, but both sets of authors address specific concerns of Elgot
as explained.)
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FOREWORD
One of the main objectives in my own research has been to have a category that
is cartesian closed, that is, with function spaces. There are many reasons, especially
for programming language semantics, why I also wish to solve "domain equations"
involving the function-space functor. In papers too numerous to mention, it is
shown that there are many categories for which this is possible. So again we have a
problem of choice of category and of knowing the relation of the fixed-point theory
in these categories to those Elgot was concerned with. I hope, therefore, that it will
not be regarded as casting any shadow on Elgot's memory if I say that I view
Elgot's work as but a first chapter of a theory of iteration and fixed points of which
we will see many more future chapters from many hands, alas, without Cal's
criticism and guidance.
DANA S. SCOTT
REFERENCES
l. M. A. Arbib and E. G. Manes, Partially additive categories and flow diagram
semantics. Journal of Algebra 62 (1980), 203-227.
2. M. A. Arbib and E. G. Manes, Partially additive monoids, graph growing
and the algebraic semantics of recursive calls. Springer-Verlag Lecture Notes in
Computer Science, 73 (1979), 127-138.
3. M. A. Arbib and E. G. Manes, The pattern-of-calls expansion is the canoni
cal fixpoint for recursive definitions. Journal of the Association for Computing
Machinery 29 (1982), 577-602. '
4. J. Tiuryn, Unique fixed points vs least fixed points. Theoretical Computer
Science 12 (1980), 229-254.
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Preface
Calvin C. Elgot (1922-1980) became a mathematician relatively late in life. He
spent 1943-1945 in the U.S. Army, lectured at the Pratt Institute from 1945 to 1948
and received a B.S. in mathematics from the City College of New York in 1948. He
worked for the U.S. Ordnance Lab for several years before entering graduate school
at the University of Michigan in 1955. He wrote his dissertation, "Decision
problems of finite automata design and related arithmetics", under the direction of
Roger Lyndon, and was awarded the Ph.D. in 1960. From that time until his death
he was a member of the Mathematical Sciences Department of the IBM Watson
Research Center at Yorktown Heights.
In conversation Elgot was quiet, warm and generous. He was a pleasure to be
with, especially at scientific meetings, where his delight in people and good food
would cheer everyone. His easygoing manner was in striking contrast to the
forcefulness with which he would defend his strong opinions on technical matters.
The combination of his warm personality and his sharp opinions had a profound
influence on those fortunate enough to know him. Elgot came to Stevens Institute in
1969 as an Adjunct Professor. He had recently finished "Recursiveness" with
Samuel Eilenberg, and I attended his lectures, which were based on this book. It
was not just his enthusiasm for this topic but the spare beauty of his mathematical
formulations that impressed me. Looking at the mathematical world through
category theoretic glasses was a new and strange experience for me, and Cal was
extremely patient in showing me the benefits of this viewpoint. After a year or so I
was "hooked" and became his collaborator on several papers. I learned only after
reading Eilenberg's piece in this volume that Cal was responsible also for Eilen
berg's becoming interested in the theory of computation. Of course, the influence
was mutual, since Eilenberg seems to be the one who introduced Cal to categories.
Elgot's interests in the theory of computation clearly differed from those of most
U.S. scientists. Sometimes he would describe his research as mathematics that arose
from an analysis of certain aspects of computation. This work was not "doing"
computer science but "redoing" it. He was aiming at finding the right framework, at
good formulations, elegant proofs, concise explanations. On occasion he would
criticize a piece of work because it did not "fit together gracefully". He admired the
European computer science community where there are many mathematicians
whose work seems motivated by the same concerns. (In turn, he was much
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