Table Of ContentWORD PROBLEMS I1
The Oxford Book
Edited by
S. I. ADIAN W. W. BOONE
7he Steklou Mathematical Institute University of Illinois
Moscow, USSR Urbana, USA
G. HIGMAN
University of Oxford
Oxford. England
1980
NORTH-HOLLAND PUBLISHING COMPANY
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@ NORTH-HOLLAND PUBLISHING COMPANY - 1980
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Library of Congress Cataloging in Publication Data
Main entry under tit)e:
Word problems 11.
(Studies in logic and the foundations of mathematics ;
,851.
V-
This book grew out of the working conference 'Deci-
sion problems in algebra' held in Oxford the summer
of 196.''
Bibliography: p.
1. Groups, Theory of--Congresses. 2. G6del's
theorem--Congresses. I. Adian, S. I. 11. Boone,
William W. 111. Higman, Graham. IT. Series.
~ 7 1 . ~ 8 6 5I2l.22 79-1P76
ISBN 0-444-85343-X
PRINTED IN THE NETHERLANDS
Dedicated to the memory of
Kurt Godel (1906-1978)
in awe and affection
In angusto vivebamus,
si quicquam esset cogitationibus clausum.
Seneca, Letters 55.11
INTRODUCTION
This book grew out of the working conference “Decision Problems
in Algebra”, held in Oxford the summer of 1976, under the auspices of
the Science Research Council of the United Kingdom. This work is a
sequel to the volume “Word problems: decision problems and the
Burnside problem in group theory”, which itself was the result of a
similar working conference held in Irvine the summer of 1969.
The Oxford conference was organized by the editors of this volume.
The secretary of the conference was Donald J. Collins, without whom
the entire endeavour could not have been successfully carried out.
What is to be said about the present book? Like the Irvine book, a
major intention is that it serves as a means of entry for the reader into
the field of word problem. For this reason we have included various
surveys; but, moreover, it is hoped that various articles which, while on
the very borderline of advancing ideas, have still been presented in
such a way as to be highly accessible to the working mathematician.
The field of “Word problems” would seem to have flourished
between Irvine and Oxford. Unsolvability results have been sharpened
in various ways. Thus, as explained in an expository article herein,
various classical decision problems about groups have been attacked,
but with the class of groups considered now restricted to some familiar
variety; and in certain cases this has led to new unsolvability results -
e.g., in the case of solvable groups. Now, too, the full story is known
about the existence of word problems of the various finer degrees of
Post. But positive results have not been lacking either. Indeed, we
include an article solving the conjugacy problem for matrices with
integer entries. (This was problem 22 of the Irvine book.)
A short article on algebraically closed groups appeared in the Irvine
book, but in subsequent years, this has become a large area of inquiry
both with regard to decision problems and with regard to purely
algebraic questions. An in-depth study and a shorter paper are
included. Much the same thing could be said about a focus of interest
on simple groups within the word problem field, and we include an
article on this matter also.
Small cancellation theory and generalizations (surveyed in the Irvine
book) have truly come of age. We include two papers in the general
area, one of which solves a well-known problem of Kurosch and Bjarni
J6nsson.
vii
viii Introduction
A brief list of open questions is given as well.
We call the reader’s attention to the note by J.L. Britton, “Erratum:
The existence of infinite Burnside groups” in which he says that a
mistake occurs in his article in the Irvine volume. This mistake was
first noted in the book “The Burnside problem and identities in
groups”, by S.I. Adian (Nauka, Moscow, 1975), page 4. The existence
of infinite Burnside groups of large odd exponent was proved in a
joint article by P.S. Novikov and Adian in 1968. The much stronger
proposition of the announcement of Novikov in 1959 must be regarded
as not proved.
We wish to thank all the mathematicians who have helped us by
refereeing individual articles, but to single out here, by name, only
Gerhard Hesse for his especially valuable help in this regard.
Originally, this volume was intended as a Festschrift to mark the
seventieth birthday of Kurt Godel. Now, sadly, we can only dedicate it
to his memory.
Sergei I. Adian
William W. Boone
Graham Higman
S.I. Adian, W.W. Boone, G. Higman, eds., Word Problems I1
@ North-Holland Publishing Company (1980) 1-16
MODULAR MACHINES, THE WORD PROBLEM FOR
FINITELY PRESENTED GROUPS AND COLLINS’ THEOREM
StAl AANDERAA and Daniel E. COHEN
University of Oslo and Queen Mary College, London
We shall define a class of machines which we call modular machines,
related to Minsky machines [15] (these were called r-machines in the
lecture by the first author on which this paper is based). These
machines act on NZ,t he set of pairs of natural numbers, with a very
simple transition function. It will then be almost immediate that any
function computable by a modular machine is partial recursive. On the
other hand, modular machines are defined in such a way that Turing-
computable functions are computable by modular machines (which can
also be regarded as a new way of Godelising Turing machines). This
provides a new proof that Turing-computable functions are partial
recursive. It also provides an easy proof of the normal form theorem
for partial recursive functions, since the data for a modular machine,
being numerical in nature, can easily be encoded by a natural number
with the decodings being primitive recursive. There will be a modular
machine M,, whose halting problem is unsolvable. Readers are invited
to construct their own proof of the undecidability of elementary
number theory using this machine M,, and to see how it compares in
difficulty with standard proofs. (Various results on the degrees of
halting, word and confluence problems for modular machines, similar
to known results for Turing machines, are proved in [9] but are not
needed in this paper.)
However the main aim of this paper is not to apply modular
machines in computability theory and logic, but to give some of their
applications in group theory. In this and the following paper the group-
theoretic results are all known, but the new proofs are very much
easier than any previous proofs. We prove the following theorems.
Theorem A (Novikov-Boone). There is a finitely presented group whose
word problem is unsolvable.
Theorem B (Collins). For any recursively enumerable unbounded truth -
table degree, there is a finitely presented group whose word problem has
that degree.
1
2 S. Aanderaa, D.E. Cohen
Theorem A was first proved by Novikov [16] and Boone [2]
independently. The original proofs were somewhat complicated and
lengthy, but were later simplified. In particular, Britton [5] made very
important simplifications by proving results about HNN extensions.
Lemma 4 of his paper is now referred to as Britton’s Lemma and is a
very important tool in combinatorial group theory, which will be used
in our papers also. Our proof of Theorem A has the same basic idea
as Britton’s, but is very much easier because of the use of modular
machines rather than Turing machines.
Other proofs of Theorem A are known. It can be proved using
Higman’s embedding theorem [0, 131. A very different proof from any
of the others has been given by McKenzie and Thomson [14].
A weaker form of Theorem B, with “unbounded truth-table degree”
replaced by “Turing degree” was proved independently by Boone [3,
41, Clapham [6], and Fridman [ll, 121. Collins [lo] was able to improve
Boone’s proof to obtain Theorem B itself.
In $1 we develop the theory of modular machines. Theorem A is
proved in 02. In 54 we prove Theorem B, having obtained the relevant
degree results for modular machines in 03. The paper following this
used similar techniques to prove the Higman embedding theorem.
01. Modular machines
We shall follow Turing and consider Turing machines as defined by
quintuples q,a,a,q,D (where a,, a, are letters, q,, qs are states, and D
is one of the two symbols L, R) instead of the more common
definition by quadruples due to Post. The two definitions are
equivalent, a non-moving quadruple q,aa ’qs being replaced by a
quintuple q,aa’q,s,R (where q,s. is a new auxiliary state) together with
quintuples qrsaxxqsLf or all letters x. We shall use the word
configuration rather than “instantaneous description”. We write uqau
for the configuration with uau on the tape (u, u being words, a being
a letter), the machine being in state q scanning letter a.
Let T be a Turing machine. We regard its alphabet as consisting of
the natural numbers 0,1,. . . , n (where 0 is the blank) and its states as
consisting of n i- 1,. . . , m - 1 and possibly 0.
Take a configuration . . . blboqac,,cI.. .. Define u, u by u = C. bm’,
u = C c,rn Minsky [ 151 represents this configuration by (u, u, a, q)E N4.
I.
It can also be represented by either of the pairs (urn + a, urn + q) or
(urn + q, um + a). We shall use whichever pair is most convenient at
the time, sometimes using one pair and sometimes the other for the
Modular machines I 3
same configuration. Notice that both mappings from configurations to
pairs are recursive. Also P(T)= {(a,p )E NZ;(a , p) represents a
configuration} is recursive and the map sending (a, p) to the
corresponding configuration is partial recursive with domain P(T).
Let qaa’q‘R be a quintuple of T. It is easy to see that (with u, u as
above) one element of NZc orresponding to the next configuration is
+
(urn’+ a’m q’,u ). Similarly if qaa‘q’l is a quintuple of T then
(u, urnz + a’m + 9’)i s a pair corresponding to the next configuration.
This motivates the definition of a modular machine.
Definition. A modular machine M consists of an integer m > 1, an
integer n with 0 < n < m, and quadruples (a, b, c, R) and (a, b, c, L)
such that 0 S a, b < h, 0 S c < m ’, and, for each a, b, at most one
quadruple begins with the pair a, b.
A configuration of M is an element (a,p ) of N2. Write a = um + a,
p = urn + b, where 0 S a, b < m. If no quadruple begins with a, b we
call (a,p ) terminal. If (a, p) is not terminal we say (a, p) yields
(a’,p’),w ritten (a,p)3 (a‘,p‘),i f either M has a quadruple
(a, b, c, R) and a’= urn2+ c, p’ = u or M has a quadruple (a, b, c, L)
and a’= u, p’ = urn2 + c. If (aI,P I) 3 (a2p,2 )3 *. * 3 (at, pk) we
write (a,,p l)+(ak,pk).
We define a (partial) function gM : N --+ N2b y gM(a,p )= (a’p,’ ) iff
(a, p)+ (a’,p ’) and (a‘,p ‘) is terminal. The class of functions {gM; all
M} is rather strange. For if gM is somewhere defined, there is a pair a,
b such that no quadruple begins with a, b. Then gM(a,p)=( a,P) if
a = a (mod m), p = b (mod m). We shall use the integer n (which has
played no part as yet) to define input and output functions. The name
“modular machine” is given because the action of M on a pair
depends on its class modulo m.
For any r E N we can write r .uniquely as r = Ci b,n’, 1 S b, S n (for
r = 0, take the empty sum). In a Turing machine with alphabet
0,1,. . . , n, this means that b, . . b,, is the tape description
corresponding to r. Let iM : N 4N 2 be given by iMr = (C b,mI, n + 1).
Write a E N as C c,m’, 0 6 c, < m, and take y with c, = 0 but c, # 0
for j < y. Define uM : N2+ N by uM(ap, )= X:a,n’-I. Then uMgMMiM is a
partial function from N to N, which we call the function computed by
M.
We shall associate with a Turing machine T a modular machine M
such that M and T compute the same function. We shall use a slightly
unusud definition of the function computed by T. The class of Turing
computable functions is (without using their characterisation as partial
4 S. Aanderaa, D.E . &hen
recursive functions) easily seen to be the same for both definitions. The
reader who prefers a more usual definition should change the definition
of the input function iM and the output function uM accordingly.
Let T be a Turing machine with alphabet 0, 1,.. . , n, and states
n + 1,. . . ,m - 1 and (perhaps) 0. Each r E N has a tape description as
above. The output of a configuration will be the integer corresponding
to the portion of the tape lying strictly between the scanned square
and the first blank to its left. The function fT : N + N computed by T
+
is defined by fTr = s if T when started in state n 1 on the rightmost
square of the description of r ultimately halts in a configuration with
output s. Note that, if convenient, we may modify T so that whenever
it halts the scanned square is blank.
We now define a modular machine M associated with T. M will
have the integers m, n previouslyAe6ned for T. M will have two
quadruples (a, q,a ’rn + q’,R (or L))a nd (4,a , a‘rn + q’,R (or L))
corresponding to each quintuple qaa’q’R( or L) of T. The account
preceding the definition of modular machines explaining how to
associate members of N2w ith configurations of T makes it clear that
M simulates T. Precisely, if (a,p ) corresponds to C then (a, p) is
terminal iff C is terminal, while if C 3 C’ then (a, p) 3 (a’,p ‘)
where (a’,p ’) is a pair corresponding to C’. The definition of the
functions computed by T and M now makes it clear that T and M
compute the same function.
The numerical nature of M makes it obvious that the function
computed by M is partial recursive, so we have a proof that Turing
computable functions are partial recursive. We could also obtain the
normal form theorem fairly easily by a Godel numbering of modular
machines.
Define Hfl(M)to be 0 if (0,O) is not terminal for M and Hfl(M=)
{(a,p );( a, p)+ (0,O)) if (0,O) is terminal. It is obvious that Ho(M)i s
r.e. For any r.e. set S there is a Turing machine T such that fT is the
(partial) characteristic function of S. Further we may assume that if fTr
is defined then T halts on a blank tape (see [18], where T is
constructed to simulate the action of a single-register machine
computing the function). If M is the modular machine associated with
T,p lainly Ho(M)i s not recursive if S is not recursive. A stronger
result is proved in 03.
02. Unsolvability
We begin with an account of HNN extensions and Britton’s Lemma
Modular machines I 5
for the reader who is unfamiliar with these topics. A knowledge of free
groups and free products will be assumed.
Let A,, A_,( for i in some index set Z which need not be countable)
be subgroups of a group H. Let cp, : A, + A-, be isomorphisms. Let G
be the free product H**(p,) (i E I). Let N be the normal closure in
G of the set of elements p;la,pi(cp,a,)-fl or all i and all aiE A,. The
quotient group GIN is called the HNN extension of H with stable
letters p, and associated pairs of subgroups A, and A-, ; it is usually
written as (H,p,;p;'A,p,= A-,).A ny element of G can be written as
hop::hI *. . hn-lp::hn,
where n 20, h,, ..., hn E H and ,..., E, = +I.
If for some r we have i, = i,,', E, + = 0 and either E, = - 1 and
h, E A,, or E, = 1 and h, E A_,,w e say this expression has a pinch
(more precisely, that it has a p,,-pinch). In such a case the same
element of GIN is represented by an expression with fewer p-symbols
(since we can replace p;'ap, with a E A, by cp,a without altering the
image) and we call this process pinching out.
We may now state Britton's Lemma (for a proof see [7] or [S]).
Britton's Lemma. Any non-trivial element of G lying in N has a pinch.
The following results are immediate consequences (and are all we
need in this section; the full force of Britton's lemma is used in 54).
-,
(I) H embeds in the HNN extension (by the map H G + GIN;
we regard this as inclusion).
(11) If K is a subgroup of H such that cp, (K n A,)= K r lA _,f or all
i then H n (K,p a( all i))= K. For it is easy to check that if a pinch in G
can be applied to an element of the subgroup (K,p, (all i)) of G the
hypotheses ensure that the new element still lies in this subgroup.
Hence any element of the subgroup (K,p,)o f GIN can be represented
in a form without pinches. By Britton's Lemma it cannot lie in H if
any p, occurs, i.e. if it is in H it is in K. (It also follows that the map
from the HNN extension of K with associated pairs K n A, and
K fl A_, to the group (K,p,)i s an isomorphism. This part is used in
[O], not in this paper.)
(111) In the group (H,p;p-'Ap= A), the isomorphism being the
identity, p-lhp = h (with h E H) only if h E A.
Let A be the group (t,x , y ;x y = yx)= (t)* (x,y ;x y = yx), which is
also an HNN extension of the free group (r,x) with stable letter y. Let
T =( t)^ (the normal subgroup of A generated by t) and let t(r,s)=
y -'x-'tx'y '. We easily obtain the following properties.
