For p ra c tic a l p u rp o ses it is d esirab le to keep th e n u m b e r of new g e n e ra to rs sm all. T h is is w h a t th e ta ils ro u tin e achieves by p re c o m p u tin g a w ord in Tpc for each g e n e ra to r t £ Tconj using c e rta in checks of th e con sisten cy th e o re m .
T h e ta ils ro u tin e as d escrib e d h ere is a g e n e ra lisa tio n of th e ta ils ro u tin e used in th e ANU P Q p ro g ra m (fo rm erly C a n b e rra N ilp o te n t Q u o tie n t P ro g ra m ). It
differs fro m th e tails ro u tin e as used in th a t p ro g ra m in tw o resp ects: T h e first difference is th a t th e tails ro u tin e in a n ilp o te n t q u o tie n t a lg o rith m h as to b e able to h a n d le c o n ju g ate re la tio n s th a t involve inverses of g e n e ra to rs, a co n d itio n th a t does n o t a p p e a r in th e ANU P Q . T h e second difference is th a t th e w eight fu n c tio n in a n ilp o te n t q u o tie n t a lg o rith m is slig htly different fro m th e w eight fu n c tio n u sed in th e ANU P Q . All g e n e ra to rs in th e n ilp o te n t q u o tie n t a lg o rith m we are d escrib in g are defined as c o m m u ta to rs of e a rlie r g e n e ra to rs. T h is a s s u m p tio n c a n n o t be m a d e in th e ANU P Q since here g e n e ra to rs can also be defined as p -th pow ers. If all g e n e ra to rs are defined as c o m m u ta to rs , th e ta ils ro u tin e is easier to describ e b eca u se th e re are fewer sub cases to deal w ith . T h e d etails of th e ta ils ro u tin e im p lem en tew d in th e ANU P Q have n o t b een p u b lish e d b u t will b eco m e available in th e n e a r fu tu re as a jo in t research re p o rt (C eller, N ew m an , N ickel, N iem eyer, 1993).
W e fix tw o g e n e ra to rs a/ an d ak w ith iu ( a z) > w ( d k ) > 1. If 6 is th e su m
of th e w eights of ak a n d a/, th e n 6 > 4. We w an t to co m p u te th e ta ils of th e re la tio n s
a - ' a i a k = +
if k 0 / ,
a ^ 1a ^ 1ak = a i w ^ f , if / £ / ,
d k a f 1^ 1 = a i w ü ~ t ^ ~ , if k , l # I .
F or th is we assu m e th a t th e tails of th e follow ing c o n ju g a te re la tio n s are w ords in
T • -i p c • a i 1(13a t = aJWtj+ t t / r a i a j a ~ l = a ~ l a ~ l CLi — diCL~l a ~ l = a j W ~ ~ t ~ ~ for i < j a n d w(a{) + w( aj ) > b a n d > for i < ji, w ( a i ) + w ( c L j ) — b a n d w ( d j ) > w ( d i ) .
N ote th a t by th e d efinition of Tpc th e ta ils of all p o sitiv e c o n ju g a te re la tio n s w here th e c o n ju g a tin g g e n e ra to r h as w eight 1 a n d th e ta ils of all pow er re la tio n s are w ords (of le n g th 1) in Tpc. If w is a w ord of w eight a t le ast [6 /2 J -f 1, th e n collecting w to n o rm al fo rm only involves re la tio n s w hose ta ils are w ords in Tpc. Let d i 1 d j d { = d j d k be th e definition of d k . T h is m e an s th a t a z h as w eight 1 a n d
aj h as w eight w ( d k ) — 1. T h e w eight of ai is b ig g er th a n or eq u al to th e w eight of a*.; hence w ( a i) > 6/2.
W e will collect th e w ord a/a^a^ in tw o different w ays. In each case we have to go th r o u g h th e collection carefully a n d see w hich re la tio n s are u sed . T h e q u o tie n t for th e tw o re su lts will be a red u ce d w ord for th e id e n tity . T h is re d u c e d w o rd will b e a w ord in Tpc U { t ^ + } in w hich will have ex p o n e n t —1 a n d can th e re fo re be u sed to exp ress as a w ord in Tpc.
F ir s t, we collect di ( dj di ) a n d o b ta in , by a p p ly in g tw o e le m e n ta ry collection step s,
ö'i(ö'jöj) d\ CLi d j d k d[d [ZV ^ ^ H Q'jQ'k'
H ere, th e g e n e ra to r ay has to be m oved to th e left. T h e w ord + h a s w eight
w( d i ) + l . In m oving dj across, only c o n ju g ate re la tio n s w ith left h a n d side d ~ 1x d j
are u sed w ith w ( x ) + w ( d j ) > w( di ) + 1 + w ( d j ) = 6. T h e ta il of each of th ese c o n ju g a te re la tio n s is a w ord in Tpc by a ss u m p tio n b eca u se e ith e r w ( x ) + w ( d j ) > b
or, if w ( x ) + w ( d j) = 6, w ( x ) > w( di ) . A fter a series of e le m e n ta ry collectio n steps we get th e w ord
didlWidjU)2dk'Wz
w here uq is an in itia l segm ent of w u an d th e w ords w2 a n d are w o rd s of w eight a t le ast w( di ) + 1 > [6/2J + 1. T h ere fo re , a p p ly in g e le m e n ta ry collection step s to th e w ord w 2df. w3 involves only re la tio n s w hose ta ils are w ords in Tpc. C o n tin u in g collection we get th e follow ing s itu a tio n ju s t b efore dj is going to be m oved p a s t di
d i di dj W4d/cW5
w here w 4 a n d w5 are w ords of w eight a t le ast -ie(a/) + l . In m oving d3 p a s t a/ th e re la tio n d J Jdi dj = d i w^ i s u sed. T h e su m of th e w eights w ( d j) a n d u>(a/) is 6 — 1. T h ere fo re , using th is re la tio n in tro d u c e s a ta il w hich is n o t a w ord in Tpc. F ro m h ere collection of th e w ord