Table Of ContentProgress in Computer Science and Applied Logic
Volume 15
Editor
JohnC. Cherniavsky,National ScienceFoundation
AssociateEditors
RobertConstable, Cornell University
JeanGallier, UniversityofPennsylvania
Richard Platek, CornellUniversity
Richard Statman, Carnegie-MellonUniversity
Symbolic Rewriting
Techniques
Manuel Bronstein
J ohannes Grabmeier
Volker Weispfenning
Editors
Springer Basel AG
Editors:
Manuel Bronstein Johannes Grabmeier
INRIA IBM Deutschland Informationssysteme GmbH
2004, Route des lucioles, BP 93 Postfach 103068
06902 Sophia Antipolis Cedex 69020 Heidelberg
France Germany
Volker Weispfenning
FMI
Universităt Passau
94030 Passau
Germany
1991 Mathematics Subject Classification 68Q40, 13PlO, 20-04, 65L99, 65P05
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Symbolic rewriting techniques / Manuel Bronstein ... , ed. -
Springer Basel AG, 1998
(Progress in computer science and applied logic ; VoI. 15)
ISBN 978-3-0348-9779-2 ISBN 978-3-0348-8800-4 (eBook)
DOI 10.1007/978-3-0348-8800-4
This work is subject to copyright. AlI rights are reserved, whether the whole or part of
the material is concerned, specificalIy the rights oftranslation, reprinting, re-use of
illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in
data banks. For any kind ofuse whatsoever, permission from the copyright owner must
be obtained.
© 1998 Springer Basel AG
OriginalIy published by Birkhăuser Verlag, Basel, Switzerland in 1998
Softcover reprint ofthe hardcover Ist edition 1998
Printed on acid-free paper produced of chlorine-free pulp. rCF
CX)
ISBN 978-3-0348-9779-2
987654321
Contents
Introduction vii
B. Amrhein, R. Biindgen, W. Kiichlin,
Parallel Completion Techniques 1
J. Apel, The Computation of Grobner Bases Using
an Alternative Algorithm 35
R. Biindgen, Symmetrization Based Completion 47
M. Gobel, On the Reduction ofG-invariant Polynomials for
an Arbitrary Permutation Groups G 71
E. Green, T. Mora, V. Ufnarovski,
The Non-Commutaive Grobner Freaks 93
B. Keller, Alternatives in Implementing Noncommutative
Grobner Basis Systems 105
K. Madlener, B. Reinert, String Rewriting and Grabner Bases -
A General Approach to Monoid and Group Rings 127
D. Mall, Grabner Fans and Projective Schemes........................... 183
C. MarcM, Normalized Rewriting: A Unified View of
Knuth-Bendix Completion and Grobner Bases Computation 195
U. Martin, New Directions for Syntactic Termination Orderings 211
M. Pesch, Two-sided Grobner Bases in Iterated Ore Extensions 227
A. Petho, J. Stein, T. Weis, H. G. Zimmer,
Computing the Torsion Group ofElliptic Curves by
the Method ofGrobner Bases........................................ 247
P. Strogova, Finding a Finite Group presentation Using Rewriting........ 269
A. Widiger, Deciding Degree-Four-Identities for Alternative
Rings by Rewriting 279
Preface
Symbolicrewritingtechniquesare methodsforderivingconsequencesfromsystems
ofequations, andareofgreat usewheninvestigatingthestructureofthesolutions.
Such techniques appear in many important areas of research within computer
algebra:
• the Knuth-Bendix completion for groups, monoids and
general term-rewriting systems,
• the Buchberger algorithm for Grabner bases,
• the Ritt-Wu characteristic set method for ordinary
differential equations,
• the Riquier-Janet method for partial differential equations.
This volume contains invited and contributed papers to the Symbolic Rewriting
Techniques workshop, which was held in the Centro Stefano Franscini in Ascona,
Switzerland, from April 30 to May 4, 1995. That workshop, organized together
by the editors of this volume and Professor Erwin Engeler of the ETH Zurich,
brought together 40 researchers from various areas of rewriting techniques, the
main goal being the investigation ofcommon threads and methods. Following the
workshops, each contribution was formally refereed and 14 papers were selected
for publication.
We are thankful to the Swiss national fund and the canton of Ticino, who
sponsored the workshop through the Centro Stefano Franscini, as well as all the
attendees, speakers, authors and reviewers. It is all your joint efforts that make
such meetings and volumes happen.
M. Bronstein, J. Grabmeier, V. Weispfenning
ProgressinComputerScienceandAppliedLogic,Vol. 15
©1998BirkhiiuserVerlagBasel/Switzerland
Parallel Completion Techniques
Beatrice Amrhein Reinhard Biindgen Wolfgang Kiichlin
Wilhelm-Schickard-Institut, Universitat Tiibingen
Sand 13, D-72076 Tiibingen, Germany
{amrhein,buendgen,kuechlin}@informatik.uni-tuebingen.de
http://wvw-sr.informatik.uni-tuebingen.de/
Abstract. We survey and categorize techniques for the parallelization of
completion procedures. We cover both Knuth-Bendix term completion and
Buchberger's algorithm for Grabner Basis completion. The survey includes
a discussion ofparallel installations ofthese algorithms in our own systems
PaReDuX and GBjPARSAC, running on parallel desktop workstations.
1 Introduction
The purpose of this work is to systematically study sources of parallelism in the
completion computation, and to compare, and categorize, different approaches for
exploitingthis parallelism in practice. We cover the critical pair / completion pro
cedure [Buc85a, Buc87a] in its two most prominent, related forms, Knuth-Bendix
(KB) term completion [KB70] and Grobner Basis (GB) polynomial completion
[Buc65, Buc85b].
Although we attempt a comprehensive overview ofdifferent parallel comple
tion techniques, we do not claim that this is a complete survey of the literature.
Duetothecomplexityofcompletionsoftware, and thewell-knownsubtletiesinthe
behavior ofcompletion algorithms, it is also impossible to give a totally fair com
parison ofthe practical performance ofthe competing approaches. In many cases,
neither the hardware nor the software involved have been described in enough
detail, nor do they exist any more. However, we still feel that this overview and
attempt at a systematic comparison is useful, because it may prevent the replica
tion ofcostly efforts whose descriptions are widely scattered in the literature.
1.1 The Parallel Completion Challenge
The completion procedure can be formulated as a set of logically independent
inference rules. In theory, these can be applied concurrently, but it is well known
that such an installation would be horrendously inefficient due to excess work. In
1
2 B. Amrhein, R. Biindgen and W. Kiichlin
practice, it is a formidable problem to harness rule execution in such a way that
the resulting concrete algorithm is highly efficient on extant parallel machines.
The efficiency benefits of performing only the best next computation (as
selected by a completion strategy) must be balanced against the speedup benefits
of performing multiple such computations in parallel. It is notoriously hard to
obtain significant speedups, because a good sequential implementation including
modern selection heuristics and deletion criteria can avoid much of the work on
which a parallel implementation thrives. Selection heuristics and deletion criteria
favor the sequential algorithm, while the parallel algorithm achieves speedups if
thereare manysimplifications (work parallelism), orifit canimprovetheselection
strategy (search parallelism).
In the framework of a parallel desktop environment, the parallel algorithm
must already perform on a par with a high-quality sequential algorithm when
executed on a uniprocessor workstation. In addition it must be able to utilize
the power of multiple processors if they are available in the same or in other
workstations.
2 Abstract Completion Frameworks
We first present the logical inference system for completion which defines the
abstract framework for all concrete completion algorithms. Any algorithm must
then fix an exact sequence in which the logical inference rules are applied. Such
concrete completion methodologies are treated in Section 3.
The representation of completion as an inference system originated in term
rewriting research [BD87, BD88], where it has become a standard. Therefore we
first introduce the term rewriting view, which is also the more general approach
based on Mathematical Logic, where function symbols are not yet interpreted.
We then present ~ corresponding inference system for polynomial completion in
Subsection 2.2.
In synopsis, the two inference systems also exhibit the important correspon
dences in the two forms ofcompletion, which can then be used to transfer paral
lelization ideas from one form to the other. However, a uniform common model
for completion with mathematically precise mappings to both forms has proved
surprisingly tricky to build and is beyond the scope ofthis paper. For some work
in this direction see [Biin91, Sto95, Biin96a, Biin96b, Biin97].
2.1 Abstract Term-Rewriting Completion
Termsareconstructed from variables, constantsand function symbolsinthe usual
way. The basic operations on terms are instantiation and tests for (structural)
equality,matchingandunification. Atermt' isan instanceoftifitcanbeobtained
by substituting terms for the variables in tj we write t' = where is the
t(J (J
instantiating substitution. A term s matches another term t if all variables in s
Parallel Completion Techniques 3
can be substituted by terms such that the new instance of 8 is equal to t. Two
terms 8 and t unify if8 and t have a common instance. Asubstitution J.l is a most
general unifier ofsand t ifSJ.l = tJ.l and all other common instances of8 and tare
also instances of SJ.l.
A rewrite rule is a pair of terms (l,r), usually written as l -+ r. It may be
applied to reduce a term t if t contains an instance l' of l. Then t reduces to t',
wheret' ist withl' replaced byacorrespondinginstanceofr. Aset ofrewrite rules
isa term rewriting system (TRS). ATRSR presentsareductionrelationsuchthat
8 -+R t ifthere is a rule in R that reduces 8 to t. Computing an irreducible term f
where t -+n f is called normalizing t. R is said to be confluent or Church-Rosser
if for any two terms tl and t2 such that t -+n it and t -+n t2 there is a term b
such that tl -+n band t2 -+n b.
We assume that for the TRS considered in this paper normalization is a
terminating procedure. A terminating and confluent TRS is complete; for any
term the result ofthe normalization procedure is then uniquelydetermined. Given
acongruencerelationontermspresentedbyafinitesetofequations£,a completion
procedure computes a complete TRS R for £ that can decide £-equivalences: two
termsareequalmodulo£ iftheir respectiveR-normalforms areequal. Suchaterm
completion procedure was discovered by Knuth and implemented by his student
Bendix, and it found a first application to word problems in group theory [KB70].
(£u{s=s};R)
Delete:
(£;R)
(£u{s=t};R)
Simplify: if t-+RU
(£u{s-u};R)
(£u{s=t};R)
Orient: (E;Ru{s->t}) if s )- t for a terminating ordering )-:-J-++R
(£;Ru{s->t})
Compose: (£;RU{s->u}) if t-+RU
Collapse: ((££;uR{uu{-st-}>;Rt})) if8 -+R u by l -+ r E R where (8,t) [> (l,r)a
£;R)
Deduce: (£u{s=t ;R) if 8 +-R U -+R t
al> is a terminating orderingon term pairs.
Figure 1: Term completion inference rules
A term completion procedure compiles (upon success) a set of equations £
into a canonical TRS R. It does so by repeatedly applying the inference rules
in Figure 1 [BD88] to a pair (£;R) of equations and rules. The core deduction
step Deduce generates a critical pair out oftwo reduction rules, the parents. This
can be interpreted as a local repair to the defect in deduction power caused by
4 B. Amrhein, R. Biindgen and W. Kiichlin
restricting equations to left-to-right application as rules. Completion succeeds if
starting with (£;0) a pair (0;1(.) can be derived such that 1(. is complete; this
means that a finite number of local repairs are sufficient. It may fail due to an
inability to orient existing critical pairs into new rules (a needed local repair is
impossible), or due to non-termination of the loop (an infinite number of local
repairs is needed).
Let l rand l' r' betwo rules where lcontainsa subterm s which unifies
-t -t
with l' such that a most general unifier of sand l' is f..L. Then If..L can be reduced
by each ofthe two rules, and the two terms resulting from the two different one
step reductions are called a critical pair. Knuth and Bendix [KB70] showed that
a terminating TRS is confluent iff all critical pairs are confluent and that only
critical pairs must be considered as new equations in the deduce inference rule.
Extend: (£;R.)·f t
(£u{s-t};R.) I S +-R. U =AC
Figure 2: Extension inference rule for AC-completion
In manyinterestingapplicationssome binaryoperators are known to beboth
associative and commutative (AC). In these cases term rewriting and completion
is performedon AC-equivalenceclasses ofterms inorderto avoid infinite rewrites.
Technically this means that tests for equality, matches, and unifications, must
be performed modulo AC. Extended completion procedures to deal with terms
containing AC-symbols are described in [LB77, PS81, JK86]. Besides the replace
ment ofthe basic operations by their AC-variants, two modifications to standard
completion are necessary for AC-completion. First, AC-equal pairs may be deleted
from £ and second there is a new extension inference rule (cf. Figure 2) that es
sentially computes critical pairs between a rule in1(. and equations specifying the
AC-theory.
For a more detailed discussion and alternative approaches the reader is re
ferred to [DJ90, Klo92, Pla93].
2.2 Abstract Polynomial Completion
SlightlybeforeKnuth discoveredtermcompletion, Buchbergerfound an algorithm
for completing a set ofpolynomial equations into a Grabner Basis°[Buc65]. In the
polynomial case, 1(. is a set of basis polynomials head - rest = which may be
regarded as reduction rules ofthe form head rest. Given two polynomial rules
-t
PI = hI -t rl and P2 = h2-t r2, the least common multiple m = s .hI = t· h2of
hI and h2can be rewritten to both s .rl and t .r2. The core deduction step thus
generatesthe (polynomial) critical pair s·rl = t·r2, in analogy to thecritical pair
ofterm rewriting [Buc85a].
