Feasible Mathematics I: A Mathematical Sciences Institute Workshop, Ithaca, New York, June 1989

352 Pages·1990·10.404 MB·English

by  Samuel R. Buss, Philip J. Scott (eds.)

Progress in Computer Science and Applied Logic
Volume 9
Editor
John C. Cherniavsky, Georgetown University
Associate Editors
Robert Constable, Cornell University
Jean Gallier, University of Pennsylvania
Richard Platek, Cornell University
Richard Statman, Carnegie-Mellon University
Samuel R. Buss Philip J. Scott
Editors
Feasible Mathematics
A Mathematical Sciences
Institute Workshop,
Ithaca, New York, June 1989
1990 Birkhauser
Boston . Basel . Berlin
Samuel R. Buss
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093-0114
USA
Philip J. Scott
Department of Mathematics
University of Ottawa
Ottawa, Ontario
Canada KIN 6N5
Library of Congress Cataloging-in-Publication Data
Feasible mathematics : a Mathematical Sciences Institute workshop, Ithaca,
New York, June 1989 / edited by Samuel R. Buss, Philip 1. Scott.
p. cm.
Papers presented at the Workshop on Feasible Mathematics, held at
Cornell University, sponsored by the Mathematical Sciences
Institute.
1. Computational complexity-Congresses. 2. Mathematics
-Congresses. I. Buss, Samuel R. n. Scott, Philip 1.
III. Workshop on Feasible Mathematics (1989 : Cornell University)
N. Cornell University. Mathematical Sciences Institute.
QA267.7.F43 1990
510-dc20 90-918
Printed on acid-free paper.
© Birkhiiuser Boston, 1990
Softcover reprint of the hardcover 1st edition 1990
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system. or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without prior permission of the copyright owner.
Permission to photocopy for internal or personal use, or the internal or personal use of specific
clients, is granted by Birkhiiuser Boston, Inc., for libraries and other users registered with the
Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per
page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should
be addressed directly to Birkhiiuser Boston, Inc., 675 Massachusetts Avenue, Cambridge, MA
02139, U.S.A.
3483-5/90 $0.00 + .20
ISBN-13: 978-0-8176-3483-4 e -ISBN-13: 978-1-4612-3466-1
DOl: 10.1007/978-1-4612-3466-1
Camera-ready text supplied by the authors.
9 8 7 6 5 4 3 2 I
Preface
A so-called "effective" algorithm may require arbitrarily large finite amounts
of time and space resources, and hence may not be practical in the real world.
A "feasible" algorithm is one which only requires a limited amount of space
and/or time for execution; the general idea is that a feasible algorithm is one
which may be practical on today's or at least tomorrow's computers. There is
no definitive analogue of Church's thesis giving a mathematical definition of
feasibility; however, the most widely studied mathematical model of feasible
computability is polynomial-time computability.
Feasible Mathematics includes both the study of feasible computation from
a mathematical and logical point of view and the reworking of traditional
mathematics from the point of view of feasible computation. The diversity
of Feasible Mathematics is illustrated by the. contents of this volume which
includes papers on weak fragments of arithmetic, on higher type functionals,
on bounded linear logic, on sub recursive definitions of complexity classes, on
finite model theory, on models of feasible computation for real numbers, on
vector spaces and on recursion theory.
The vVorkshop on Feasible Mathematics was sponsored by the Mathematical
Sciences Institute and was held at Cornell University, June 26-28, 1989.
The principal speakers were M. Ajtai, L. Blum, S. Buss, P. Clote, S. Cook,
J. Crossley, J.-Y. Girard, Y. Gurevich, K.-I Ko, D. Leivant, A. Nerode,
J. Remmel, A. Scedrov, G. Takeuti, and A. Urquhart. There were shorter
talks by J.C.E. Dekker, F. Ferriera, J. Foy and J. Krajicek. H. J. Hoover did
not speak at the workshop but contributed a paper to the proceedings.
These proceedings illustrate the diversity of talks at the meeting. vVe would
like to thank the speakers for the lively exchange of ideas during the talks.
The editors would also like to thank the Mathematical Sciences Institute of
v
vi Preface
Cornell University, and especially Anil Nerode, for their financial and logistic
help in making this meeting possible. We would also like to thank the staff of
MSI for their help in organizing this meeting.
Most of the papers in this volume have been refereed; however, four of the
speakers, L. Blum, Y. Gurevich, D. Leivant, and A. Urquhart submitted
abstracts of their talks which were not refereed. We would like to thank the
referees for their consciencious efforts in reviewing the rest of the articles.
Samuel R. Buss and Philip J. Scott
Table of Contents
Preface ... v
MIKLOS AJTAI
Parity and the Pigeonhole Principle 1
LENORE BLUM
Computing over the Reals (or an Arbitrary Ring) Abstract . . . . . . 25
SAMUEL R. Buss
On Model Theory for Intuitionistic Bounded Arithmetic
with Applications to Independence Results . . . . . . . . . . . . . 27
PETER CLOTE
Sequential, Machine Independent Characterizations of the
Parallel Complexity Classes AlogTIME, ACk, NCk and NC . . . . . . 49
STEPHEN A. COOK AND BRUCE M. KAPRON
Characterizations of the Basic Feasible Functionals of
Finite Type ........ 71
STEPHEN A. COOK AND ALASDAIR URQUHART
Functional Interpretations of Feasibly Constructive
Arithmetic - Abstract . . . . . . . . . . . . . . . . . . 97
JOHN N. CROSSLEY AND JEFF B. REMMEL
Polynomial-time Combinatorial Operators are Polynomials . . . . . . 99
J.C.E. DEKKER
Isols and Kneser Graphs 131
FERNANDO FERREIRA
Stockmeyer Induction . 161
JOHN Foy AND ALAN R. WOODS
Probabilities of Sentences about Two Linear Orderings 181
vii
viii Table of Contents
JEAN-YVES GIRARD, ANDRE SCEDROV AND PHILIP J. SCOTT
Bounded Linear Logic: a Modular Approach to
Polynomial Time Computability, Extended Abstract . . . . . 195
YURI GUREVICH
On Finite Model Theory (Extended Abstract) . 211
H. JAMES HOOVER
Computational Models for Feasible Real Analysis 221
KER-I Ko
Inverting a One-to-One Real Function is Inherently
Sequential . . . . . . . . . . . . . . . . . .
JAN KRAJiCEK AND GAISI TAKEUTI
On Bounded Z=} Polynomial Induction
DANIEL LEIVANT
Subrecursion and Lambda Representation over Free
Algebras (Preliminary Summary) . . . . . . . . 201
ANIL NERODE AND JEFF B. REMMEL
Complexity-Theoretic Algebra: Vector Space Bases
JEFF B. REMMEL
When is every Recursive Linear Ordering of Type fL
Recursively Isomorphic to a Polynomial Time Linear
Ordering over the Natural Numbers in Binary Form? . :)21
PARITY AND THE PIGEONHOLE PRINCIPLE
M. Ajtai
INTRODUCTION. ,]'he Pigeonhole Principle is t.he stat.ement. that
there is no one-to-one map of a set of size n into a set of size n - 1.
This is a theorem of Peano Arithmetic that is it can be proved using the
axioms of complete induction. A weaker version of Penno Arithmetic is
I ~o where we allow only bounded formuins in the induction axioms, that
is for each bounded formula ljJ(x, y) the corresponding induction axiom
Vx((IjJ(x, 0) 1\ V(IjJ(x, y) ---t ljJ(x, Y + 1))) ---t VzljJ(x, z)), where a formula is
called bounded if it contains only quantifiers of the type Vx < y or 3x < y.
A. Woods (see [Wo] or [PWW]) proved that the existence of infinitely
many prime numbers can be proved in I ~o if the pigeonhole principle
is a theorem of this system. A. Wilkie (see [Wi] or [PWW]) has found a
weaker version of the Pigeonhole Principle which indeed can be proved, and
still implies the existence of infinite number of primes, in a system which
is somewhat. stronger than I ~(), but the question about PHP remained
unsolved. Paris and Wilkie [PW] asked whether PHP can be proved in
I ~o(J). (If it can be proved in this extended system then it can be proved
in I ~o too. We get I ~o(J) from T~ o by adding a unary function symbol
f and allowing to use it in the induction axioms. Now the Pigeonhole
Principle is t.he statement that for any x the restriction of f onto x is not a
one-to-one map of x into x -1). The answer is negative (c.r. Ajtai [Aj2] ),
the Pigeonhole Principle cannot be proved in I ~o(f), actually it is possible