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%W  basics.tex          ACE documentation - basics       Alexander Hulpke
%W                                                      Joachim Neub"user
%W                                                            Greg Gamble
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%H  $Id: basics.tex,v 1.15 2006/01/26 16:15:05 gap Exp $
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%Y  Copyright (C) 2000  Centre for Discrete Mathematics and Computing
%Y                      Department of Information Tech. & Electrical Eng.
%Y                      University of Queensland, Australia.
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\Chapter{Some Basics}

\index{strategy}\atindex{Felsch strategy}{@Felsch strategy}
\atindex{HLT strategy}{@HLT strategy}
Throughout this manual for the use of {\ACE} as a {\GAP}  package,  we
shall assume that the reader already knows the basic  ideas  of  coset
enumeration, as can be found for  example  in~\cite{Neu82}.  There,  a
simple proof is given for the fact that  a  coset  enumeration  for  a
subgroup of finite index in a finitely presented group must eventually
terminate with the correct result, provided  the  enumeration  process
obeys  a  simple  condition  (Mendelsohn's  condition)  formulated  in
Lemma~1 and Theorem~2 of~\cite{Neu82}.  This  basic  condition  leaves
room for a great variety of *strategies* for  coset  enumeration;  two
``classical'' ones have been known for a  long  time  as  the  *Felsch
strategy* and the *HLT strategy* (for Haselgrove, Leech and  Trotter).
Extensive  experimental  studies  on  many  strategies  can  be  found
in~\cite{CDHW73},  \cite{Hav91},  \cite{HR99a},  and  \cite{HR01},  in
particular.

A few basic points should be particularly understood:

\beginlist%unordered

\item{--} ``Subgroup (generator) and relator tables'' that are used in
the description of coset enumeration in \cite{Neu82}, and to which  we
will also occasionally refer in this manual, do *not* physically exist
in  the  implementation  of  coset  enumeration  in  {\ACE}.   For   a
terminology that is closer to the actual implementation  and  also  to
the  formulations  in  the  manual  for  the  {\ACE}  standalone   see
\cite{CDHW73} and \cite{Hav91}.

\index{cosets!coset numbers}\index{cosets!coset table}\index{holes}
\item{--} Coset enumeration proceeds by defining *coset numbers*  that
really denote possible representatives for cosets written as words  in
the generators of the group. At the time of their generation it is not
guaranteed that any two of these words do indeed  represent  different
cosets. The state of an  enumeration  at  any  time  is  stored  in  a
2-dimensional array known as a *coset table* whose rows are indexed by
coset numbers and whose columns are indexed by  the  group  generators
and their inverses. Entries of  the  coset  table  that  are  not  yet
defined are known as  *holes*  (typically  they  are  filled  with  0,
i.e.~an invalid coset number).

\index{cosets}\index{cosets!coset application}\index{cosets!coset numbers}
\item{--} It is customary in talking about coset enumeration to  speak
of *cosets* when really coset numbers are meant. While we try to avoid
this in this interface manual, in certain word  combinations  such  as
*coset application* we will follow this custom.

\index{deduction}\index{deduction!deduction stack}
\item{--} The definition of a coset number may  lead  to  *deductions*
from the ``closing of rows in subgroup or relator tables''. These  are
kept in a *deduction stack*.

\index{coincidence}\index{coincidence!coincidence queue}
\item{--}  Also  it  may  be  found  that  (different)  words  in  the
generators defining different coset numbers really  lie  in  the  same
coset of the given subgroup. This is called a *coincidence*  and  will
eventually lead to the elimination of the  larger  of  the  two  coset
numbers.  Until  this   elimination   has   been   performed   pending
coincidences are kept in a *queue of coincidences*.

\index{preferred definition}\index{definition!preferred}
\index{preferred definition!preferred definition stack}
\item{--} A definition that will actually close a row in a subgroup or
relator table will immediately yield twice  as  many  entries  in  the
coset  table  as  other  definitions.  Such  definitions  are   called
*preferred definitions*, the places in rows of a subgroup  or  relator
table that they close are also referred to as ``gaps of  length  one''
or minimal gaps. Such gaps can be found  at  little  extra  cost  when
``relators are traced from a given  coset  number''.  {\ACE}  keeps  a
selection of them in a *preferred definition stack* for  use  in  some
definition strategies (see~\cite{Hav91}).

\endlist

It will also be necessary to understand some further basic features of
the  implementation and  the corresponding  terminology which  we will
explain in the sequel.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Enumeration Style}

The first main decision for any coset enumeration is in which sequence
to make definitions. When a new coset number has  to  be  defined,  in
{\ACE} there are basically three possible methods to choose from:

\beginlist%unordered

\atindex{C style}{@C style!definition}
\item{--} One may fill the next empty entry  in  the  coset  table  by
scanning from the left/top of the coset table towards the right/bottom
-- that is, in order row by row. This is called *C  style  definition*
(for *C*oset Table Based definition)  of  coset  numbers.  In  fact  a
procedure needs to follow a method like this to some  extent  for  the
proofs that coset enumeration eventually terminates  in  the  case  of
finite index (see~\cite{Neu82}).

\atindex{R style}{@R style!definition}
\item{--}  For  an  *R  style   definition*   (for   *R*elator   Based
definition),  the  order  in  which  coset  numbers  are  defined   is
explicitly prescribed by the order in  which  rows  of  (the  subgroup
generator  tables  and)  the  relator  tables  are  filled  by  making
definitions.

\index{preferred definition}
\index{preferred definition!preferred definition stack}
\index{strategy!minimal gaps}
\item{--} One may choose  definitions  from  a  *Preferred  Definition
Stack*. In this stack possibilities for definition  of  coset  numbers
are stored that will close a certain row of  a  relator  table.  Using
these *preferred definitions* is  sometimes  also  referred  to  as  a
*minimal gaps strategy*. The idea of using these is that by closing  a
row in a relator table, thus, one will immediately get a  consequence.
We will come back to the obvious question of where  one  obtains  this
``preferred definition stack''.

\endlist

The *enumeration style* is mainly determined by  the  balance  between
C style and R style definitions, which is controlled by the values  of
the `ct' and `rt' options (see~"option ct" and~"option rt").

However this still leaves us with plenty of freedom for the design  of
definition strategies, freedom which can,  for  example,  be  used  to
great advantage in Felsch-type strategies. Though it is  not  strictly
necessary,  before  embarking  on  further  enumeration,   Felsch-type
programs generally start off  by  ensuring  that  each  of  the  given
subgroup generators produces a cycle of coset numbers at coset  1.  To
explain the idea, an example may help. Suppose  $a,b$  are  the  group
generators and $w=Abab$ is a subgroup generator, where $A$  represents
the inverse of $a$; then to say that ``$(1,i,j,k)$ is  a  cycle  of
coset numbers produced at coset 1 by $w$'' means  that  the  successive
application  of  the  ``letters''  $A,b,a,b$  of   $w$   takes   one
successively from coset 1, through cosets $i$, $j$ and $k$,  and  back
to coset 1, i.e.~$A$ applied to coset 1  results  in  coset  $i$,  $b$
applied to coset $i$ results in coset $j$, $a$ applied  to  coset  $j$
results in coset $k$, and finally $b$ applied to coset  $k$  takes  us
back to coset $1$. In this  way,  a  hypothetical  subgroup  table  is
filled first. The use of this and other  possibilities  leads  to  the
following table of *enumeration styles*.

% \begin{table}
% \hrule
% \caption{The styles}
% \label{tab:sty}
% \smallskip
% \renewcommand{\arraystretch}{0.875}
% \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}crrlc} 
% \hline\hline
% & \ttt{Rt} value & \ttt{Ct} value & style name & \\
% \hline
% & $<\!0$ & $<\!0$ & R/C & \\
% & $<\!0$ & $0$    & R*  & \\
% & $<\!0$ & $>\!0$ & Cr  & \\
% & $0$    & $<\!0$ & C   & \\
% & $0$    & $0$    & R/C (defaulted) & \\
% & $0$    & $>\!0$ & C  & \\
% & $>\!0$ & $<\!0$ & Rc & \\
% & $>\!0$ & $0$    & R  & \\
% & $>\!0$ & $>\!0$ & CR & \\
% \hline\hline
% \end{tabular*}
% \end{table}
\begintt
Rt value     Ct value     style name
-----------------------------------------

   0           >0         C
  <0           >0         Cr
  >0           >0         CR

  >0            0         R
  <0            0         R*
  >0           <0         Rc
  <0           <0         R/C
   0            0         R/C (defaulted)

-----------------------------------------
\endtt

\beginitems

\atindex{C style}{@C style}
*C style* &
In this style, most definitions are made in the next empty coset table
slot and are  (in  principle)  tested  in  all  essentially  different
positions in the relators; i.e.~this is a Felsch-like style.

However, in  C  style,  some  definitions  may  be  made  following  a
preferred definition strategy, controlled by the `pmode'  and  `psize'
options (see~"option pmode" and~"option psize").

\atindex{Cr style}{@Cr style}
*Cr style* &
is like C style except that a single R style pass is  done  after  the
initial C style pass.

\atindex{CR style}{@CR style}
*CR style* &
In this style, alternate passes of C style and R style are performed.

\atindex{R style}{@R style}
*R  style* &
In this style,  all  the  definitions  are  made  via  relator  scans;
i.e.~this is an HLT-like style.

\atindex{R\* style}{@R\* style}
*R\* style* &
makes definitions the same as R style, but tests  all  definitions  as
for C style.

\atindex{Rc style}{@Rc style}
*Rc style* &
is like R style, except that a single C style pass is done  after  the
initial R style pass.

\atindex{R/C style}{@R/C style}
*R/C style* &
In this style, we  run  in  R  style  until  an  overflow,  perform  a
lookahead on the entire table, and then switch to CR style.

\atindex{Defaulted  R/C  style}{@Defaulted  R/C  style}
\atindex{R/C (defaulted) style}{@R/C (defaulted) style}
*Defaulted R/C* ($={}$*R/C (defaulted)* $\,$) *style* &
is the default style  used  if  you  call  {\ACE}  without  specifying
options. In it, we use R/C style with `ct' set to 1000 and `rt' set to
approximately $2000$ divided by the total length of the relators in an
attempt to balance R style and C style definitions when we  switch  to
CR style.

\enditems

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Finding Deductions, Coincidences, and Preferred Definitions} 

\index{definition}
First, let us  broadly  discuss  strategies  and  how  they  influence
``definitions''. By *definition* we mean the  allocation  of  a  coset
number. In a complete coset table each group relator produces a  cycle
of cosets numbers at each coset number, in  particular,  at  coset  1;
i.e.~for each relator $w$, and for each coset number  $i$,  successive
application of the letters of $w$ trace through a  sequence  of  coset
numbers that begins and ends in $i$ (see  Section~"Enumeration  Style"
for an example). It has been found to be a good general  rule  to  use
the given group relators as  subgroup  generators.  This  ensures  the
early definition of some useful coset numbers, and is the basis of the
`default'  strategy  (see~"option  default").  The  number  of   group
relators included as subgroup generators is  determined  by  the  `no'
option (see~"option no"). Over a wide range of  examples  the  use  of
group relators in  this  way  has  been  shown  to  produce  generally
beneficial results in terms of the maximum number  of  cosets  numbers
defined at any one time and the total number of coset numbers defined.
In~\cite{CDHW73}, it  was  reported  that  for  some  Macdonald  group
$G(\alpha,\beta)$ examples, (pure) Felsch-type strategies (that  don't
include the given group  relators  as  subgroup  generators)  e.g.~the
`felsch := 0' strategy  (see~"option  felsch")  defined  significantly
more coset numbers than HLT-type (e.g.~the `hlt' strategy, see~"option
hlt") strategies. The comparison of these strategies in terms of total
number of coset numbers defined, in~\cite{Hav91}, for the  enumeration
of the cosets  of  a  certain  index  40  subgroup  of  the  $G(3,21)$
Macdonald group were 91 for HLT versus 16067 for  a  pure  Felsch-type
strategy. For the Felsch strategy with the group relators included  as
subgroup generators, as for the `felsch :=  1'  strategy  (see~"option
felsch") the total number of coset numbers defined reduced markedly to
59.

\index{deduction}
A *deduction* occurs when the scanning of a  relator  results  in  the
assignment of a coset table body entry.  A  completed  table  is  only
valid if  every  table  entry  has  been  tested  in  all  essentially
different positions in all relators. This testing can either  be  done
directly (Felsch strategy) or via relator scanning (HLT strategy).  If
it is done directly, then more than one deduction can be waiting to be
processed at any one time. The untested deductions  are  stored  in  a
stack. How this stack is managed is determined by the  `dmode'  option
(see~"option dmode"), and its size is controlled by the `dsize' option
(see~"option dsize").

\index{coincidence}\index{dead coset (number)}
As already mentioned a *coincidence* occurs when it is determined that
two coset numbers in fact represent the same coset. When  this  occurs
the larger  coset  number  becomes  a  *dead  coset  number*  and  the
coincidence is placed in a  queue.  When  and  how  these  dead  coset
numbers  are  eventually  eliminated  is  controlled  by  the  options
`dmode', `path' and `compaction' (see~"option  dmode",  "option  path"
and~"option compaction"). The user  may  also  force  coincidences  to
occur (see Section~"Finding Subgroups"), which,  however,  may  change
the subgroup whose cosets are enumerated.

\index{preferred definition!preferred definition stack}
The key  to  performance  of  coset  enumeration  procedures  is  good
selection  of  the  next   coset   number   to   be   defined.   Leech
in~\cite{Lee77}  and~\cite{Lee84}  showed  how  a  number   of   coset
enumerations could be simplified by removing coset numbers  needlessly
defined by computer implementations. Human  enumerators  intelligently
choose which coset number should be defined next, based on  the  value
of each potential definition. In particular, definitions  which  close
relator cycles (or at least shorten gaps in cycles)  are  favoured.  A
definition which actually closes a relator  cycle  immediately  yields
twice as many table entries (deductions)  as  other  definitions.  The
value of the `pmode'  option  (see~"option  pmode")  determines  which
definitions are *preferred*; if the value of  the  `pmode'  option  is
non-zero, depending on the `pmode' value, gaps  of  length  one  found
during relator C style  (i.e.~Felsch-like)  scans  are  either  filled
immediately  (subject  to  the  value  of  `fill')  or  noted  in  the
*preferred  definition  stack*.  The  preferred  definition  stack  is
implemented as a  ring  of  size  determined  by  the  `psize'  option
(see~"option psize"). However, making preferred definitions carelessly
can violate the conditions required for guaranteed termination of  the
coset enumeration procedure in the case of finite index. To avoid such
a violation {\ACE} ensures a fraction of the  coset  table  is  filled
before  a  preferred  definition  is  made;  the  reciprocal  of  this
fraction, the `fill factor', is  manipulated  via  the  `fill'  option
(see~"option  fill").  In~\cite{Hav91},  the  `felsch   :=   1'   type
enumeration of the cosets of the certain  index  40  subgroup  of  the
$G(3,21)$ Macdonald group was further  improved  to  require  a  total
number of coset numbers  of  just  43  by  incorporating  the  use  of
preferred definitions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Finding Subgroups}

\index{coincidence}
The {\ACE} package, via its  interactive  {\ACE}  interface  functions
(described  in  Chapter~"Functions  for  Using  ACE   Interactively"),
provides the possibility of searching for subgroups. To  do  this  one
starts at a known subgroup (possibly the trivial subgroup).  Then  one
may augment it by adding new subgroup generators either explicitly via
`ACEAddSubgroupGenerators'     (see~"ACEAddSubgroupGenerators")     or
implicitly by introducing *coincidences*  (see  `ACECosetCoincidence':
"ACECosetCoincidence",           or           `ACERandomCoincidences':
"ACERandomCoincidences"). Also, one may descend to  smaller  subgroups
by  deleting  subgroup  generators  via  `ACEDeleteSubgroupGenerators'
(see~"ACEDeleteSubgroupGenerators").

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Coset Table Standardisation Schemes}

\atindex{lenlex standardisation scheme}{@\noexpand`lenlex' %
standardisation scheme} 
The default standardisation scheme for {\GAP} from {\GAP}~4.3 and  the
standardisation scheme provided by {\ACE} is the `lenlex'  scheme,  of
Charles Sims~\cite{Sim94}. This scheme numbers cosets first  according
to word-length and then according  to  a  lexical  ordering  of  coset
representatives. Each coset representative is a word  in  an  alphabet
consisting of the user-supplied generators and their inverses, and the
lexical ordering of `lenlex' is that implied by ordering that alphabet
so that each generator is followed by its inverse, and the  generators
appear in user-supplied order. See below for an  example  which  gives
the first 20 lines  of  the  `lenlex'  standard  coset  table  of  the
(infinite) group with presentation $\langle x, y, a, b \mid x^2,  y^3,
a^4, b^2\rangle$.

In the table each  inverse  of  a  generator  is  represented  by  the
corresponding uppercase letter ($X$  represents  the  inverse  of  $x$
etc.), and the lexical ordering of the representatives is that implied
by defining an ordering of the alphabet  of  user-supplied  generators
and  their  inverses  to  be  $x\<X\<y\<Y\<a\<A\<b\<B$.  

A `lenlex' standard coset table whose columns correspond, in order, to
the already-described alphabet, of generators and their inverses,  has
an important property: a scan of the body of the table row by row from
left to right, encounters new coset numbers in numeric order.  Observe
that the table below has this property: the definition of coset  1  is
implicit; the first coset number we encounter in the table body is  2,
then 2 again, 3, 4, 5, 6, 7, then 7 again, etc.

With the `lenlex' option (see~"option lenlex"), the coset table output
by `ACECosetTable' or `ACECosetTableFromGensAndRels'  is  standardised
according to the `lenlex' scheme.

\begintt
 coset no. ||      x      X      y      Y      a      A      b      B   rep've
-----------+------------------------------------------------------------------
         1 ||      2      2      3      4      5      6      7      7
         2 ||      1      1      8      9     10     11     12     12    x
         3 ||     13     13      4      1     14     15     16     16    y
         4 ||     17     17      1      3     18     19     20     20    Y
         5 ||     21     21     22     23     24      1     25     25    a
         6 ||     26     26     27     28      1     24     29     29    A
         7 ||     30     30     31     32     33     34      1      1    b
         8 ||     35     35      9      2     36     37     38     38    xy
         9 ||     39     39      2      8     40     41     42     42    xY
        10 ||     43     43     44     45     46      2     47     47    xa
        11 ||     48     48     49     50      2     46     51     51    xA
        12 ||     52     52     53     54     55     56      2      2    xb
        13 ||      3      3     57     58     59     60     61     61    yx
        14 ||     62     62     63     64     65      3     66     66    ya
        15 ||     67     67     68     69      3     65     70     70    yA
        16 ||     71     71     72     73     74     75      3      3    yb
        17 ||      4      4     76     77     78     79     80     80    Yx
        18 ||     81     81     82     83     84      4     85     85    Ya
        19 ||     86     86     87     88      4     84     89     89    YA
        20 ||     90     90     91     92     93     94      4      4    Yb
\endtt

\atindex{semilenlex standardisation scheme}{@\noexpand`semilenlex' %
standardisation scheme}
Another standardisation scheme for coset tables (the default scheme of
versions of {\GAP} up to  {\GAP}~4.2),  numbers  cosets  according  to
coset representative word-length in the group generators  and  lexical
ordering  imposed  by  the  user-supplied  ordering   of   the   group
generators; it is known as `semilenlex' since  though  like  `lenlex',
generator inverses do not feature. Here again is 20 lines of the coset
table of the group with presentation $\langle x, y,  a,  b  \mid  x^2,
y^3, a^4, b^2\rangle$, this time `semilenlex' standardised.

\begintt
 coset no. ||      x      y      a      b    rep've
-----------+--------------------------------------
         1 ||      2      3      4      5
         2 ||      1      6      7      8    x
         3 ||      9     10     11     12    y
         4 ||     13     14     15     16    a
         5 ||     17     18     19      1    b
         6 ||     20     21     22     23    xy
         7 ||     24     25      2     26    xa
         8 ||     27     28     29      2    xb
         9 ||      3     30     31     32    yx
        10 ||     33      1     34     35    yy
        11 ||     36     37     38     39    ya
        12 ||     40     41     42      3    yb
        13 ||      4     43     44     45    ax
        14 ||     46     47     48     49    ay
        15 ||     50     51     52     53    aa
        16 ||     54     55     56      4    ab
        17 ||      5     57     58     59    bx
        18 ||     60     61     62     63    by
        19 ||     64     65     66     67    ba
        20 ||      6     68     69     70    xyx
\endtt

The term `semilenlex' was  coined  by  Edmund  Robertson  and  Joachim
Neub{\accent127u}ser, for the  scheme's  applicability  to  semigroups
where generator inverses need not exist. This scheme ensures  that  as
one scans the  columns  corresponding  to  the  group  generators  (in
user-supplied order) row by row, one encounters new coset  numbers  in
numeric order. 

Observe that the representatives are ordered according to  length  and
then the lexical ordering implied by defining $x\<y\<a\<b$ (with  some
words omitted due to their equivalence to words that precede  them  in
the ordering). Also observe that as one scans the body  of  the  table
row by row from left to right new  coset  numbers  appear  in  numeric
order without gaps (2, 3, 4,  5,  then  1  which  we  have  implicitly
already seen, 6, 7, etc.).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Coset Statistics Terminology}

\indextt{activecosets}\indextt{maxcosets}\indextt{totcosets}
\index{coincidence}\index{dead coset (number)}\index{alive coset number}
There are three statistics involved in the counting  of  coset  number
definitions: `activecosets', `maxcosets' and  `totcosets';  these  are
three of  the  fields  of  the  record  returned  by  `ACEStats'  (see
Section~"Using ACE  Directly  to  Test  whether  a  Coset  Enumeration
Terminates"), and they correspond to the `a', `m' and  `t'  statistics
of an {\ACE} ``results message'' (see Appendix~"The Meanings of  ACE's
Output Messages"). As already described, coset enumeration proceeds by
defining coset numbers; `totcosets' counts *all* such definitions made
during  an  enumeration.  During  the   coset   enumeration   process,
*coincidences*  usually  occur,  resulting  in  the  larger  of   each
coincident  pair  becoming  a  *dead  coset  number*.  The   statistic
`activecosets' is the count of coset numbers  left  *alive*  (i.e.~not
dead) at the end of an enumeration; and  `maxcosets'  is  the  maximum
number of *alive* cosets at any point of an enumeration.

In practice, the statistics `maxcosets' and `totcosets' tend to be  of
a similar order, though, of course, `maxcosets' can never be more than
`totcosets'.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Other Terminology}

\index{loop}\index{pass}\index{state (machine)}
In various places in this manual, we will refer to a (main) *loop*  or
a *pass* through such a loop.  We  don't  intend  to  give  a  precise
meaning to these terms. The reader will need to forgive us for  giving
somewhat circular definitions in our attempt to make these terms  less
nebulous. It is sufficient to appreciate that the {\ACE} enumerator is
organised as a state machine, where each *state* is  a  value  of  the
coset table held internally by  {\ACE}  at  the  end  of  each  ``main
loop''. Each step from  one  state  to  the  next  (i.e.~each  passage
through the main loop)  performs  an  ``action''  (i.e.,  `lookahead',
`compaction'; see~"option lookahead"  and~"option  compaction")  or  a
block of actions (i.e., `|ct|' coset number definitions, `|rt|'  coset
number applications). {\ACE} counts the number of passes  through  the
main loop; if the option  `loop'  (see~"option  loop")  is  set  to  a
positive integer, {\ACE} makes an early return  when  the  loop  count
hits the value of `loop'.

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