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<h1>C Finer Points with Examples</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP00C.htm#SECT001">Getting Started</a>
<li> <A HREF="CHAP00C.htm#SECT002">Emulating Sims</a>
</ol>

The examples in this chapter are intended to provide the nearest
GAP equivalent of the similarly named sections in Appendix A of
<code>ace3001.ps</code> (the standalone manual in directory <code>standalone-doc</code>).
There is a lot of detail here, which the novice ACE Package user
won't want to know about. Please, despite the name of the first
section of this chapter, read the examples in Appendix <a href="CHAP00B.htm">Examples</a>
first.


<h2><a name="SECT001">C.1 Getting Started</a></h2>

Each of the functions <code>ACECosetTableFromGensAndRels</code>
(see <a href="CHAP001.htm#SSEC002.1">ACECosetTableFromGensAndRels</a>), <code>ACEStats</code> (see <a href="CHAP001.htm#SSEC003.1">ACEStats</a> ---
non-interactive version) and <code>ACEStart</code> (see <a href="CHAP006.htm#SSEC001.1">ACEStart</a>), may be
called with three arguments: <var>fgens</var> (the group generators), <var>rels</var>
(the group relators), and <var>sgens</var> (the subgroup generators). While it
is legal for the arguments <var>rels</var> and <var>sgens</var> to be empty lists, it is
always an error for <var>fgens</var> to be empty, e.g.

<pre>
gap> ACEStats([],[],[]);
Error, fgens (arg[1]) must be a non-empty list of group generators ...
called from
CALL_ACE( "ACEStats", arg[1], arg[2], arg[3] ) called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
type: 'quit;' to quit to outer loop, or
type: 'fgens := <val>; return;' to assign <val> to fgens to continue.
brk> fgens := FreeGeneratorsOfFpGroup(FreeGroup("a"));
[ a ]
brk> return;
rec( index := 0, cputime := 13, cputimeUnits := "10^-2 seconds",
 activecosets := 499998, maxcosets := 499998, totcosets := 499998 )

</pre>

The example shows that the ACE package does allow you to recover
from the <code>break</code>-loop. However, the definition of <code>fgens</code> above is
local to the <code>break</code>-loop, and in any case we shall want two
generators for the examples we wish to consider and raise some other
points; so let us re-define <code>fgens</code> and start again:

<pre>
gap> F := FreeGroup("a", "b");; fgens := FreeGeneratorsOfFpGroup(F);;

</pre>

<a name = "I0"></a>

An <code>ACEStats</code> example

By default, the presentation is not echoed; use the <code>echo</code>
(see <a href="CHAP004.htm#SSEC011.12">option echo</a>) option if you want that. Also, by default, the
ACE binary only prints a results message, but we won't see that
unless we set <code>InfoACE</code> to a level of at least 2
(see <a href="CHAP001.htm#SSEC009.1">InfoACE</a>):

<pre>
gap> SetInfoLevel(InfoACE, 2);

</pre>

Calling <code>ACEStats</code> with arguments <code>fgens</code>, <code>[]</code>, <code>[]</code>, defines a free
froup with 2 generators, since the second argument defines an empty
relator list; and since the third argument is an empty list of
generators, the subgroup defined is trivial. So the enumeration
overflows:

<pre>
gap> ACEStats(fgens, [], []);
#I OVERFLOW (a=249998 r=83333 h=83333 n=249999; l=337 c=0.10; m=249998 t=2499\
98)
rec( index := 0, cputime := 10, cputimeUnits := "10^-2 seconds",
 activecosets := 249998, maxcosets := 249998, totcosets := 249998 )

</pre>

The line starting with ``<code>#I </code>''. is the <code>Info</code>-ed results
message from ACE; see Appendix <a href="CHAP00A.htm">The Meanings of ACE's Output Messages for details on what it means. Observe that since the
enumeration overflowed, ACE's result message has been translated
into a GAP record with <code>index</code> field 0.

To dump out the presentation and parameters associated with an
enumeration, ACE provides the <code>sr</code> (see <a href="CHAP00D.htm#SSEC005.7">option sr</a>) option.
However, you won't see output of this command, unless you set the
<code>InfoACE</code> level to at least 3. Also, to ensure the reliability of the
output of the <code>sr</code> option, an enumeration should precede it; for
<code>ACEStats</code> (and <code>ACECosetTableFromGensAndRels</code>) the directive <code>start</code>
(see <a href="CHAP00D.htm#SSEC003.2">option start</a>) required to initiate an enumeration is inserted
(automatically) after all the user's options, except if the user
herself supplies an option that initiates an enumeration (namely, one
of <code>start</code> or <code>begin</code> (see <a href="CHAP00D.htm#SSEC003.2">option start</a>), <code>aep</code> (see <a href="CHAP00D.htm#SSEC001.1">option aep</a>)
or <code>rep</code> (see <a href="CHAP00D.htm#SSEC001.2">option rep</a>)). Interactively, the equivalent of the
<code>sr</code> command is <code>ACEParameters</code> (see <a href="CHAP006.htm#SSEC005.10">ACEParameters</a>), which gives an
output record that is immediately GAP-usable. With the above in
mind let's rerun the enumeration and get ACE's dump of the
presentation and parameters:

<pre>
gap> SetInfoLevel(InfoACE, 3);
gap> ACEStats(fgens, [], [] : start, sr := 1);
#I ACE 3.001 Wed Oct 31 09:36:39 2001
#I =========================================
#I Host information:
#I name = rigel
#I OVERFLOW (a=249998 r=83333 h=83333 n=249999; l=337 c=0.09; m=249998 t=2499\
98)
#I #--- ACE 3.001: Run Parameters ---
#I Group Name: G;
#I Group Generators: ab;
#I Group Relators: ;
#I Subgroup Name: H;
#I Subgroup Generators: ;
#I Wo:1000000; Max:249998; Mess:0; Ti:-1; Ho:-1; Loop:0;
#I As:0; Path:0; Row:1; Mend:0; No:0; Look:0; Com:10;
#I C:0; R:0; Fi:7; PMod:3; PSiz:256; DMod:4; DSiz:1000;
#I #---------------------------------
rec( index := 0, cputime := 9, cputimeUnits := "10^-2 seconds",
 activecosets := 249998, maxcosets := 249998, totcosets := 249998 )

</pre>

Observe that at <code>InfoACE</code> level 3, one also gets ACE's banner. We
could have printed out the first few lines of the coset table if we
had wished, using the <code>print</code> (see <a href="CHAP00D.htm#SSEC005.8">option print</a>) option, but note as
with the <code>sr</code> option, an enumeration should precede it. Here's what
happens if you disregard this (recall, we still have the <code>InfoACE</code>
level set to 3):

<pre>
gap> ACEStats(fgens, [], [] : print := [-1, 12]);
#I ACE 3.001 Wed Oct 31 09:37:37 2001
#I =========================================
#I Host information:
#I name = rigel
#I ** ERROR (continuing with next line)
#I no information in table
#I ***
#I ***
#I OVERFLOW (a=249998 r=83333 h=83333 n=249999; l=337 c=0.09; m=249998 t=2499\
98)
rec( index := 0, cputime := 9, cputimeUnits := "10^-2 seconds",
 activecosets := 249998, maxcosets := 249998, totcosets := 249998 )

</pre>

Essentially, because ACE had not done an enumeration prior to
getting the <code>print</code> directive, it complained with an ``<code>** ERROR</code>'',
recovered and went on with the <code>start</code> directive automatically
inserted by the <code>ACEStats</code> command: no ill effects at the GAP
level, but also no table.

Now, let's do what we should have done (to get those first few lines
of the coset table), namely, insert the <code>start</code> option before the
<code>print</code> option (the <code>InfoACE</code> level is still 3):

<pre>
gap> ACEStats(fgens, [], [] : start, print := [-1, 12]);
#I ACE 3.001 Wed Oct 31 09:38:28 2001
#I =========================================
#I Host information:
#I name = rigel
#I OVERFLOW (a=249998 r=83333 h=83333 n=249999; l=337 c=0.10; m=249998 t=2499\
98)
#I co: a=249998 r=83333 h=83333 n=249999; c=+0.00
#I coset | a A b B order rep've
#I -------+---------------------------------------------
#I 1 | 2 3 4 5
#I 2 | 6 1 7 8 0 a
#I 3 | 1 9 10 11 0 A
#I 4 | 12 13 14 1 0 b
#I 5 | 15 16 1 17 0 B
#I 6 | 18 2 19 20 0 aa
#I 7 | 21 22 23 2 0 ab
#I 8 | 24 25 2 26 0 aB
#I 9 | 3 27 28 29 0 AA
#I 10 | 30 31 32 3 0 Ab
#I 11 | 33 34 3 35 0 AB
#I 12 | 36 4 37 38 0 ba
#I ***
rec( index := 0, cputime := 10, cputimeUnits := "10^-2 seconds",
 activecosets := 249998, maxcosets := 249998, totcosets := 249998 )

</pre>

The values we gave to the <code>print</code> option, told ACE to print the
first 12 lines and include coset representatives. Note that, since
there are no relators, the table has separate columns for generator
inverses. So the default workspace of 1000000 words allows a table
of 249998 = 1000000/4 − 2 cosets. Since row <code>fill</code>ing (see <a href="CHAP004.htm#SSEC014.1">option fill</a>) is on by default, the table is simply filled with cosets in
order. Note that a compaction phase is done before printing the table,
but that this does nothing here (the lowercase <code>co:</code> tag), since there
are no dead cosets. The coset representatives are simply all possible
freely reduced words, in length plus lexicographic (i.e. <code>lenlex</code>; see
Section <a href="CHAP003.htm#SECT004">Coset Table Standardisation Schemes</a>) order.

<a name = "I1"></a>

Using <code>ACECosetTableFromGensAndRels</code>

The essential difference between the functions <code>ACEStats</code> and
<code>ACECosetTableFromGensAndRels</code> is that <code>ACEStats</code> parses the results
message from the ACE binary and outputs a GAP record containing
statistics of the enumeration, and <code>ACECosetTableFromGensAndRels</code>
after parsing the results message, goes on to parse ACE's coset
table, if it can, and outputs a GAP list of lists version of that
table. So, if we had used <code>ACECosetTableFromGensAndRels</code> instead of
<code>ACEStats</code> in our examples above, we would have observed similar
output, except that we would have ended up in a <code>break</code>-loop (because
the enumeration overflows) instead of obtaining a record containing
enumeration statistics. We have already seen an example of that in
Section <a href="CHAP001.htm#SECT002">Using ACE Directly to Generate a Coset Table</a>. So, here we
will consider two options that prevent one entering a <code>break</code>-loop,
namely the <code>silent</code> (see <a href="CHAP004.htm#SSEC011.3">option silent</a>) and <code>incomplete</code>
(see <a href="CHAP004.htm#SSEC011.6">option incomplete</a>) options. Firstly, let's take the last
<code>ACEStats</code> example, but use <code>ACECosetTableFromGensAndRels</code> instead and
include the <code>silent</code> option. (We still have the <code>InfoACE</code> level set at
3.)

<pre>
gap> ACECosetTableFromGensAndRels(fgens, [], [] : start, print := [-1, 12],
> silent);
#I ACE 3.001 Wed Oct 31 09:40:18 2001
#I =========================================
#I Host information:
#I name = rigel
#I OVERFLOW (a=249998 r=83333 h=83333 n=249999; l=337 c=0.09; m=249998 t=2499\
98)
#I co: a=249998 r=83333 h=83333 n=249999; c=+0.00
#I coset | a A b B order rep've
#I -------+---------------------------------------------
#I 1 | 2 3 4 5
#I 2 | 6 1 7 8 0 a
#I 3 | 1 9 10 11 0 A
#I 4 | 12 13 14 1 0 b
#I 5 | 15 16 1 17 0 B
#I 6 | 18 2 19 20 0 aa
#I 7 | 21 22 23 2 0 ab
#I 8 | 24 25 2 26 0 aB
#I 9 | 3 27 28 29 0 AA
#I 10 | 30 31 32 3 0 Ab
#I 11 | 33 34 3 35 0 AB
#I 12 | 36 4 37 38 0 ba
#I ***
fail

</pre>

Since, the enumeration overflowed and the <code>silent</code> option was set,
<code>ACECosetTableFromGensAndRels</code> simply returned <code>fail</code>. But hang on,
ACE at least has a partial table; we should be able to obtain it in
GAP format, in a situation like this. We can. We simply use the
<code>incomplete</code> option, instead of the <code>silent</code> option. However, if we
did that with the example above, the result would be an enormous table
(the number of active cosets is 249998); so let us also set the
<code>max</code> (see <a href="CHAP004.htm#SSEC017.6">option max</a>) option, in order that we should get a more
modestly sized partial table. Finally, we will use <code>print := -12</code>
since it is a shorter equivalent alternative to <code>print := [-1, 12]</code>.
Note that the output here was obtained with GAP 4.3 (and is the
same with GAP 4.4).

Note: Sinec the order options are passed to ACE behind the colon,
has not been honoured since GAP 4.5 (at about the time ACE 5.1
was released in 2012), the behaviour exhibited below is no longer
observed. To approximately get the behaviour below, omit the option
<code>start</code>. This option is inserted anyway, if a user omits it, and
importantly is inserted after the <code>max</code> option is put to the ACE
binary.

<pre>
gap> ACECosetTableFromGensAndRels(fgens, [], [] : max := 12,
> start, print := -12,
> incomplete);
#I ACE 3.001 Wed Oct 31 09:41:14 2001
#I =========================================
#I Host information:
#I name = rigel
#I OVERFLOW (a=12 r=4 h=4 n=13; l=5 c=0.00; m=12 t=12)
#I co: a=12 r=4 h=4 n=13; c=+0.00
#I coset | a A b B order rep've
#I -------+---------------------------------------------
#I 1 | 2 3 4 5
#I 2 | 6 1 7 8 0 a
#I 3 | 1 9 10 11 0 A
#I 4 | 12 0 0 1 0 b
#I 5 | 0 0 1 0 0 B
#I 6 | 0 2 0 0 0 aa
#I 7 | 0 0 0 2 0 ab
#I 8 | 0 0 2 0 0 aB
#I 9 | 3 0 0 0 0 AA
#I 10 | 0 0 0 3 0 Ab
#I 11 | 0 0 3 0 0 AB
#I 12 | 0 4 0 0 0 ba
#I ***
#I co: a=12 r=4 h=4 n=13; c=+0.00
#I coset | a A b B
#I -------+----------------------------
#I 1 | 2 3 4 5
#I 2 | 6 1 7 8
#I 3 | 1 9 10 11
#I 4 | 12 0 0 1
#I 5 | 0 0 1 0
#I 6 | 0 2 0 0
#I 7 | 0 0 0 2
#I 8 | 0 0 2 0
#I 9 | 3 0 0 0
#I 10 | 0 0 0 3
#I 11 | 0 0 3 0
#I 12 | 0 4 0 0
#I ACECosetTable: Coset table is incomplete, reduced & lenlex standardised.
[ [ 2, 6, 1, 12, 0, 0, 0, 0, 3, 0, 0, 0 ],
 [ 3, 1, 9, 0, 0, 2, 0, 0, 0, 0, 0, 4 ],
 [ 4, 7, 10, 0, 1, 0, 0, 2, 0, 0, 3, 0 ],
 [ 5, 8, 11, 1, 0, 0, 2, 0, 0, 3, 0, 0 ] ]

</pre>

Observe, that despite the fact that ACE is able to define coset
representatives for all 12 coset numbers defined, the body of the
coset table now contains a 0 at each place formerly occupied by a
coset number larger than 12 (0 essentially represents ``don't know'').
To get a table that is the same in the first 12 rows as before we
would have had to set <code>max</code> to 38, since that was the largest coset
number that appeared in the body of the 12-line table, previously.
Also, note that the <code>max</code> option preceded the <code>start</code> option; since
the interface respects the order in which options are put by the user,
the enumeration invoked by <code>start</code> would otherwise have only been
restricted by the size of <code>workspace</code> (see <a href="CHAP004.htm#SSEC017.1">option workspace</a>). The
warning that the coset table is incomplete is emitted at <code>InfoACE</code> or
<code>InfoWarning</code> level 1, i.e. by default, you will see it.

<a name = "I2"></a>

Using <code>ACEStart</code>

The limitation of the functions <code>ACEStats</code> and
<code>ACECosetTableFromGensAndRels</code> (on three arguments) is that they do
not interact with ACE; they call ACE with user-defined input,
and collect and parse the output for either statistics or a coset
table. On the other hand, the <code>ACEStart</code> (see <a href="CHAP006.htm#SSEC001.1">ACEStart</a>) function
allows one to start up an ACE process and maintain a dialogue with
it. Moreover, via the functions <code>ACEStats</code> and <code>ACECosetTable</code> (on 1
or no arguments), one is able to extract the same information that we
could with the non-interactive versions of these functions. However,
we can also do a lot more. Each ACE option that provides output
that can be used from within GAP has a corresponding interactive
interface function that parses and translates that output into a form
usable from within GAP.

Now we emulate our (successful) <code>ACEStats</code> exchanges above, using
interactive ACE interface functions. We could do this with:
<code>ACEStart(0, fgens, [], [] : start, sr := 1);</code> where the <code>0</code> first
argument tells <code>ACEStart</code> not to insert <code>start</code> after the options
explicitly listed. Alternatively, we may do the following (note that
the <code>InfoACE</code> level is still 3):

<pre>
gap> ACEStart(fgens, [], []);
#I ACE 3.001 Wed Oct 31 09:42:49 2001
#I =========================================
#I Host information:
#I name = rigel
#I ***
#I OVERFLOW (a=249998 r=83333 h=83333 n=249999; l=337 c=0.10; m=249998 t=2499\
98)
1
gap> ACEParameters(1);
#I #--- ACE 3.001: Run Parameters ---
#I Group Name: G;
#I Group Generators: ab;
#I Group Relators: ;
#I Subgroup Name: H;
#I Subgroup Generators: ;
#I Wo:1000000; Max:249998; Mess:0; Ti:-1; Ho:-1; Loop:0;
#I As:0; Path:0; Row:1; Mend:0; No:0; Look:0; Com:10;
#I C:0; R:0; Fi:7; PMod:3; PSiz:256; DMod:4; DSiz:1000;
#I #---------------------------------
rec( enumeration := "G", subgroup := "H", workspace := 1000000,
 max := 249998, messages := 0, time := -1, hole := -1, loop := 0, asis := 0,
 path := 0, row := 1, mendelsohn := 0, no := 0, lookahead := 0,
 compaction := 10, ct := 0, rt := 0, fill := 7, pmode := 3, psize := 256,
 dmode := 4, dsize := 1000 )

</pre>

Observe that the <code>ACEStart</code> call returned an integer (1, here). All 8
forms of the <code>ACEStart</code> function, return an integer that identifies
the interactive ACE interface function initiated or communicated
with. We may use this integer to tell any interactive ACE interface
function which interactive ACE process we wish to communicate with.
Above we passed <code>1</code> to the <code>ACEParameters</code> command which caused <code>sr :=
1</code> (see <a href="CHAP00D.htm#SSEC005.7">option sr</a>) to be passed to the interactive ACE process
indexed by 1 (the process we just started), and a record containing
the parameter options (see <a href="CHAP004.htm#SSEC012.1">ACEParameterOptions</a>) is returned. Note
that the ``Run Parameters'': <code>Group Generators</code>, <code>Group Relators</code> and
<code>Subgroup Generators</code> are considered ``args'' (i.e. arguments) and a
record containing these is returned by the <code>GetACEArgs</code>
(see <a href="CHAP006.htm#SSEC005.5">GetACEArgs</a>) command; or they may be obtained individually via
the commands: <code>ACEGroupGenerators</code> (see <a href="CHAP006.htm#SSEC005.1">ACEGroupGenerators</a>),
<code>ACERelators</code> (see <a href="CHAP006.htm#SSEC005.2">ACERelators</a>), or <code>ACESubgroupGenerators</code>
(see <a href="CHAP006.htm#SSEC005.3">ACESubgroupGenerators</a>).

We can obtain the enumeration statistics record, via the interactive
version of <code>ACEStats</code> (see <a href="CHAP006.htm#SSEC006.2">ACEStats!interactive</a>) :

<pre>
gap> ACEStats(1); # The interactive version of ACEStats takes 1 or no arg'ts
rec( index := 0, cputime := 10, cputimeUnits := "10^-2 seconds",
 activecosets := 249998, maxcosets := 249998, totcosets := 249998 )

</pre>

To display 12 lines of the coset table with coset representatives
without invoking a further enumeration we could do: <code>ACEStart(0, 1 :
print := [-1, 12]);</code>. Alternatively, we may use the
<code>ACEDisplayCosetTable</code> (see <a href="CHAP006.htm#SSEC005.12">ACEDisplayCosetTable</a>) (the table itself
is emitted at <code>InfoACE</code> level 1, since by default we presumably want
to see it):

<pre>
gap> ACEDisplayCosetTable(1, [-1, 12]);
#I co: a=249998 r=83333 h=83333 n=249999; c=+0.00
#I coset | a A b B order rep've
#I -------+---------------------------------------------
#I 1 | 2 3 4 5
#I 2 | 6 1 7 8 0 a
#I 3 | 1 9 10 11 0 A
#I 4 | 12 13 14 1 0 b
#I 5 | 15 16 1 17 0 B
#I 6 | 18 2 19 20 0 aa
#I 7 | 21 22 23 2 0 ab
#I 8 | 24 25 2 26 0 aB
#I 9 | 3 27 28 29 0 AA
#I 10 | 30 31 32 3 0 Ab
#I 11 | 33 34 3 35 0 AB
#I 12 | 36 4 37 38 0 ba
#I ------------------------------------------------------------

</pre>

Still with the same interactive ACE process we can now emulate the
<code>ACECosetTableFromGensAndRels</code> exchange that gave us an incomplete
coset table. Note that it is still necessary to invoke an enumeration
after setting the <code>max</code> (see <a href="CHAP004.htm#SSEC017.6">option max</a>) option. We could just call
<code>ACECosetTable</code> with the argument 1 and the same 4 options we used for
<code>ACECosetTableFromGensAndRels</code>. Alternatively, we can do the
equivalent of the 4 options one (or two) at a time, via their
equivalent interactive commands. Note that the <code>ACEStart</code> command
(without <code>0</code> as first argument) inserts a <code>start</code> directive after the
user option <code>max</code>:

<pre>
gap> ACEStart(1 : max := 12);
#I ***
#I OVERFLOW (a=12 r=4 h=4 n=13; l=5 c=0.00; m=12 t=12)
1

</pre>

Now the following <code>ACEDisplayCosetTable</code> command does the equivalent
of the <code>print := [-1, 12]</code> option.

<pre>
gap> ACEDisplayCosetTable(1, [-1, 12]);
#I co: a=12 r=4 h=4 n=13; c=+0.00
#I coset | a A b B order rep've
#I -------+---------------------------------------------
#I 1 | 2 3 4 5
#I 2 | 6 1 7 8 0 a
#I 3 | 1 9 10 11 0 A
#I 4 | 12 0 0 1 0 b
#I 5 | 0 0 1 0 0 B
#I 6 | 0 2 0 0 0 aa
#I 7 | 0 0 0 2 0 ab
#I 8 | 0 0 2 0 0 aB
#I 9 | 3 0 0 0 0 AA
#I 10 | 0 0 0 3 0 Ab
#I 11 | 0 0 3 0 0 AB
#I 12 | 0 4 0 0 0 ba
#I ------------------------------------------------------------

</pre>

Finally, we call <code>ACECosetTable</code> with 1 argument (which invokes the
interactive version of <code>ACECosetTableFromGensAndRels</code>) with the option
<code>incomplete</code>.

<pre>
gap> ACECosetTable(1 : incomplete);
#I start = yes, continue = yes, redo = yes
#I ***
#I OVERFLOW (a=12 r=4 h=4 n=13; l=4 c=0.00; m=12 t=12)
#I co: a=12 r=4 h=4 n=13; c=+0.00
#I coset | a A b B
#I -------+----------------------------
#I 1 | 2 3 4 5
#I 2 | 6 1 7 8
#I 3 | 1 9 10 11
#I 4 | 12 0 0 1
#I 5 | 0 0 1 0
#I 6 | 0 2 0 0
#I 7 | 0 0 0 2
#I 8 | 0 0 2 0
#I 9 | 3 0 0 0
#I 10 | 0 0 0 3
#I 11 | 0 0 3 0
#I 12 | 0 4 0 0
#I ACECosetTable: Coset table is incomplete, reduced & lenlex standardised.
[ [ 2, 6, 1, 12, 0, 0, 0, 0, 3, 0, 0, 0 ],
 [ 3, 1, 9, 0, 0, 2, 0, 0, 0, 0, 0, 4 ],
 [ 4, 7, 10, 0, 1, 0, 0, 2, 0, 0, 3, 0 ],
 [ 5, 8, 11, 1, 0, 0, 2, 0, 0, 3, 0, 0 ] ]

</pre>

Observe the line beginning ``<code>#I start = yes,</code>'' (the first line in
the output of <code>ACECosetTable</code>). This line appears in response to the
option <code>mode</code> (see <a href="CHAP00D.htm#SSEC003.1">option mode</a>) inserted by <code>ACECosetTable</code> after
any user options; it is inserted in order to check that no user
options (possibly made before the <code>ACECosetTable</code> call) have
invalidated ACE's coset table. Since the line also says continue =



yes</code>, the mode <code>continue</code> (the least expensive  of  the  three  modes;

see <a href="CHAP00D.htm#SSEC003.4">option continu</a>) is directed at <font face="Gill Sans,Helvetica,Arial">ACE</font> which  evokes  a  <strong>results

message</strong>. Then <code>ACECosetTable</code> extracts the  incomplete  table  via  a

<code>print</code> (see <a href="CHAP00D.htm#SSEC005.8">option print</a>) directive. If you wish  to  see  all  the

options that are directed to <font face="Gill Sans,Helvetica,Arial">ACE</font>, set  the  <code>InfoACE</code>  level  to  4

(then all such commands are <code>Info</code>-ed behind a ``<code>ToACE>  </code>''  prompt;

see <a href="CHAP001.htm#SSEC009.1">InfoACE</a>).

<p>

Following the standalone manual, we  now  set  things  up  to  do  the

alternating group <i>A</i><sub>5</sub>, of order 60. (We saw the group  <i>A</i><sub>5</sub>  with

subgroup <i>C</i><sub>5</sub> earlier in Section <a href="CHAP00B.htm#SECT003">Example of Using ACE  Interactively (Using ACEStart)</a>; here we are concerned with observing and  remarking

on the output from the <font face="Gill Sans,Helvetica,Arial">ACE</font> binary.) We turn messaging  on  via  the

<code>messages</code> (see <a href="CHAP004.htm#SSEC018.1">option messages</a>) option;  setting  <code>messages</code>  to  1

tells <font face="Gill Sans,Helvetica,Arial">ACE</font> to emit a <strong>progress message</strong> on each  pass  of  its  main

loop. In the example following we set <code>messages := 1000</code>,  which,  for

our example, sets the interval between messages so high that  we  only

get the ``Run Parameters'' block (the same as that obtained  with  <code>sr

:= 1</code>), no progress messages and the final <strong>results  message</strong>.  Recall

<code>F</code> is the free group we defined  on  generators  <code>fgens</code>:  <code>"a"</code>  and

<code>"b"</code>. Here we will be interested in seeing what is transmitted to the

<font face="Gill Sans,Helvetica,Arial">ACE</font> binary; so we will set  the  <code>InfoACE</code>  level  to  4  (what  is

transmitted to <font face="Gill Sans,Helvetica,Arial">ACE</font> will now appear behind a  ``<code>ToACE> </code>''  prompt,

and we will still see the messages <strong>from</strong>  <font face="Gill Sans,Helvetica,Arial">ACE</font>).  Note,  that  when

<font face="Gill Sans,Helvetica,Arial">GAP</font>  prints  <code>F.1</code>  (=  <code>fgens[1]</code>)  it  displays  <code>a</code>,  but  the

<strong>variable</strong> <code>a</code> is (at the moment) unassigned; so for  convenience  (in

defining relators, for example) we first assign the variable <code>a</code> to be

<code>F.1</code> (and <code>b</code> to be <code>F.2</code>).

<p>

<pre>

gap> SetInfoLevel(InfoACE, 4);

gap> a := F.1;; b := F.2;;

gap> # Enumerating A_5 = < a, b | a^2, b^3, (a*b)^5 >

gap> # over Id (trivial subgp)

gap> ACEStart(1, fgens, [a^2, b^3, (a*b)^5], []

>                                     # 4th arg empty (to define Id)

>             : enumeration := "A_5", # Define the Group Name

>               subgroup := "Id",     # Define the Subgroup Name

>               max := 0,             # Set `max' back to default (no limit)

>               messages := 1000);    # Progress messages every 1000 iter'ns

#I  ToACE> group:ab;

#I  ToACE> relators:a^2,b^3,a*b*a*b*a*b*a*b*a*b;

#I  ToACE> generators;

#I  ToACE> enumeration:A_5;

#I  ToACE> subgroup:Id;

#I  ToACE> max:0;

#I  ToACE> messages:1000;

#I  ToACE> text:***;

#I  ***

#I  ToACE> text:***;

#I  ***

#I  ToACE> Start;

#I    #--- ACE 3.001: Run Parameters ---

#I  Group Name: A_5;

#I  Group Generators: ab;

#I  Group Relators: (a)^2, (b)^3, (ab)^5;

#I  Subgroup Name: Id;

#I  Subgroup Generators: ;

#I  Wo:1000000; Max:333331; Mess:1000; Ti:-1; Ho:-1; Loop:0;

#I  As:0; Path:0; Row:1; Mend:0; No:3; Look:0; Com:10;

#I  C:0; R:0; Fi:6; PMod:3; PSiz:256; DMod:4; DSiz:1000;

#I    #---------------------------------

#I  INDEX = 60 (a=60 r=77 h=1 n=77; l=3 c=0.00; m=66 t=76)

1



</pre>

<p>

Observe that the <code>fgens</code> and  subgroup  generators  (the  empty  list)

arguments are transmitted to <font face="Gill Sans,Helvetica,Arial">ACE</font> via the  <font face="Gill Sans,Helvetica,Arial">ACE</font>  binary's  group

and <code>generators</code> options, respectively. Observe also, that the relator

<code>(a*b)^5</code>  is  expanded  by  <font face="Gill Sans,Helvetica,Arial">GAP</font>  to   <code>a*b*a*b*a*b*a*b*a*b</code>   when

transmitted to <font face="Gill Sans,Helvetica,Arial">ACE</font> and then  <font face="Gill Sans,Helvetica,Arial">ACE</font>  correctly  deduces  that  it's

<code>(a*b)^5</code>.

<p>

Since we  did  not  specify  a  strategy  the  <code>default</code>  (see <a href="CHAP005.htm#SSEC001.1">option default</a>) strategy was followed and  hence  coset  number  definitions

were R (i.e. HLT) style, and a total of 76  coset  numbers  (<code>t=76</code>)

were defined (if we had tried <code>felsch</code> we would have achieved the best

possible: <code>t=60</code>).  Note,  that  <font face="Gill Sans,Helvetica,Arial">ACE</font>  already  ``knew''  the  group

generators  and  subgroup  generators;  so,  we  could  have   avoided

re-transmitting that information by using the <code>relators</code>  (see <a href="CHAP00D.htm#SSEC002.2">option relators</a>) option:

<p>

<pre>

gap> ACEStart(1 : relators := ToACEWords(fgens, [a^2, b^3, (a*b)^5]),

>                 enumeration := "A_5",

>                 subgroup := "Id",

>                 max := 0,

>                 messages := 1000);

#I  Detected usage of a synonym of one (or more) of the options:

#I      `group', `relators', `generators'.

#I  Discarding current values of args.

#I  (The new args will be extracted from ACE, later).

#I  ToACE> relators:a^2,b^3,a*b*a*b*a*b*a*b*a*b;

#I  ToACE> enumeration:A_5;

#I  ToACE> subgroup:Id;

#I  ToACE> max:0;

#I  ToACE> messages:1000;

#I  No group generators saved. Setting value(s) from ACE ...

#I  ToACE> sr:1;

#I    #--- ACE 3.001: Run Parameters ---

#I  Group Name: A_5;

#I  Group Generators: ab;

#I  Group Relators: (a)^2, bbb, ababababab;

#I  Subgroup Name: Id;

#I  Subgroup Generators: ;

#I  Wo:1000000; Max:333331; Mess:1000; Ti:-1; Ho:-1; Loop:0;

#I  As:0; Path:0; Row:1; Mend:0; No:3; Look:0; Com:10;

#I  C:0; R:0; Fi:6; PMod:3; PSiz:256; DMod:4; DSiz:1000;

#I    #---------------------------------

#I  ToACE> text:***;

#I  ***

#I  ToACE> Start;

#I    #--- ACE 3.001: Run Parameters ---

#I  Group Name: A_5;

#I  Group Generators: ab;

#I  Group Relators: (a)^2, (b)^3, (ab)^5;

#I  Subgroup Name: Id;

#I  Subgroup Generators: ;

#I  Wo:1000000; Max:333331; Mess:1000; Ti:-1; Ho:-1; Loop:0;

#I  As:0; Path:0; Row:1; Mend:0; No:3; Look:0; Com:10;

#I  C:0; R:0; Fi:6; PMod:3; PSiz:256; DMod:4; DSiz:1000;

#I    #---------------------------------

#I  INDEX = 60 (a=60 r=77 h=1 n=77; l=3 c=0.00; m=66 t=76)

1



</pre>

<p>

Note the usage  of  <code>ToACEWords</code>  (see <a href="CHAP006.htm#SSEC003.6">ToACEWords</a>)  to  provide  the

appropriate string value of the <code>relators</code> option. Also,  observe  the

<code>Info</code>-ed warning of the action  triggered  by  using  the  <code>relators</code>

option, that says that the current values of the  ``args''  (i.e. what

would be returned by <code>GetACEArgs</code>; see <a href="CHAP006.htm#SSEC005.5">GetACEArgs</a>)  were  discarded,

which immediately triggered the action of reinstantiating the value of

<code>ACEData.io[1].args</code> (which is what the <code>Info</code>:

<p>

<pre>

#I  No group generators saved. Setting value(s) from ACE ...

</pre>

<p>

was all about). Also observe that the  ``Run  Parameters''  block  was

<code>Info</code>-ed twice; the first time was due to  <code>ACEStart</code>  emitting  <code>sr</code>

with  value  <code>1</code>  to  <font face="Gill Sans,Helvetica,Arial">ACE</font>,  the  response  of  which  is  used   to

re-instantiate <code>ACEData.io[1].args</code>, and the second is in response  to

transmitting <code>Start</code> to <font face="Gill Sans,Helvetica,Arial">ACE</font>.

<p>

In  particular,  <font face="Gill Sans,Helvetica,Arial">GAP</font>  no  longer  thinks  <code>fgens</code>  are  the   group

generators:

<p>

<pre>

gap> ACEGroupGenerators(1) = fgens;

false



</pre>

<p>

Groan! We will just have to re-instantiate everything:

<p>

<pre>

gap> fgens := ACEGroupGenerators(1);;

gap> F := GroupWithGenerators(fgens);; a := F.1;; b := F.2;;



</pre>

<p>

We now define a non-trivial subgroup, of small enough index,  to  make

the observation of all progress messages, by setting <code>messages :=  1</code>,

a not too onerous proposition. As for defining the relators, we  could

use the 1-argument version of <code>ACEStart</code>, in which case we  would  use

the  <code>subgroup</code>  (see <a href="CHAP004.htm#SSEC019.2">option  subgroup</a>)  option   with   the   value

<code>ToACEWords(fgens, [ a*b ])</code>. However, as we saw, in the end we  don't

save anything by doing this, since afterwards the  variables  <code>fgens</code>,

<code>a</code>, <code>b</code> and <code>F</code> would no longer be associated with <code>ACEStart</code> process

1. Instead, we will use the more convenient 4-argument form, and  also

switch the <code>InfoACE</code> level back to 3:

<p>

<pre>

gap> SetInfoLevel(InfoACE, 3);

gap> ACEStart(1, ACEGroupGenerators(1), ACERelators(1), [ a*b ]

>             : messages := 1);

#I  ***

#I  ***

#I    #--- ACE 3.001: Run Parameters ---

#I  Group Name: A_5;

#I  Group Generators: ab;

#I  Group Relators: (a)^2, (b)^3, (ab)^5;

#I  Subgroup Name: Id;

#I  Subgroup Generators: ab;

#I  Wo:1000000; Max:333331; Mess:1; Ti:-1; Ho:-1; Loop:0;

#I  As:0; Path:0; Row:1; Mend:0; No:3; Look:0; Com:10;

#I  C:0; R:0; Fi:6; PMod:3; PSiz:256; DMod:4; DSiz:1000;

#I    #---------------------------------

#I  AD: a=2 r=1 h=1 n=3; l=1 c=+0.00; m=2 t=2

#I  SG: a=2 r=1 h=1 n=3; l=1 c=+0.00; m=2 t=2

#I  RD: a=3 r=1 h=1 n=4; l=2 c=+0.00; m=3 t=3

#I  RD: a=4 r=2 h=1 n=5; l=2 c=+0.00; m=4 t=4

#I  RD: a=5 r=2 h=1 n=6; l=2 c=+0.00; m=5 t=5

#I  RD: a=6 r=2 h=1 n=7; l=2 c=+0.00; m=6 t=6

#I  RD: a=7 r=2 h=1 n=8; l=2 c=+0.00; m=7 t=7

#I  RD: a=8 r=2 h=1 n=9; l=2 c=+0.00; m=8 t=8

#I  RD: a=9 r=2 h=1 n=10; l=2 c=+0.00; m=9 t=9

#I  CC: a=8 r=2 h=1 n=10; l=2 c=+0.00; d=0

#I  RD: a=9 r=5 h=1 n=11; l=2 c=+0.00; m=9 t=10

#I  RD: a=10 r=5 h=1 n=12; l=2 c=+0.00; m=10 t=11

#I  RD: a=11 r=5 h=1 n=13; l=2 c=+0.00; m=11 t=12

#I  RD: a=12 r=5 h=1 n=14; l=2 c=+0.00; m=12 t=13

#I  RD: a=13 r=5 h=1 n=15; l=2 c=+0.00; m=13 t=14

#I  RD: a=14 r=5 h=1 n=16; l=2 c=+0.00; m=14 t=15

#I  CC: a=13 r=6 h=1 n=16; l=2 c=+0.00; d=0

#I  CC: a=12 r=6 h=1 n=16; l=2 c=+0.00; d=0

#I  INDEX = 12 (a=12 r=16 h=1 n=16; l=3 c=0.00; m=14 t=15)

1



</pre>

<p>

Observe that we used <code>ACERelators(1)</code> (see <a href="CHAP006.htm#SSEC005.2">ACERelators</a>) to grab  the

value  of  the  relators  we  had  defined  earlier.  We   also   used

<code>ACEGroupGenerators(1)</code> (see <a href="CHAP006.htm#SSEC005.1">ACEGroupGenerators</a>) to  get  the  group

generators.

<p>

The  run  ended  with  12  active   (see   Section <a href="CHAP003.htm#SECT005">Coset   Statistics Terminology</a>) coset numbers (<code>a=12</code>) after defining a total number  of

15 coset numbers (<code>t=15</code>); the definitions occurred at the steps  with

progress messages tagged by <code>AD:</code> (coset 1 application definition) and

<code>SG:</code> (subgroup generator phase), and the 13 tagged by <code>RD:</code> (R  style

definition). So there must have  been  3  coincidences:  observe  that

there were 3 progress messages with a <code>CC:</code>  tag.  (See  Appendix <a href="CHAP00A.htm">The Meanings of ACE's Output Messages.)

<p>

We can dump out the  statistics  accumulated  during  the  run,  using

<code>ACEDumpStatistics</code>  (see <a href="CHAP006.htm#SSEC005.24">ACEDumpStatistics</a>),  which   <code>Info</code>s   the

<font face="Gill Sans,Helvetica,Arial">ACE</font>  output  of  the  <code>statistics</code>  (see <a href="CHAP00D.htm#SSEC005.10">option  statistics</a>)   at

<code>InfoACE</code> level 1.

<p>

<pre>

gap> ACEDumpStatistics();

#I    #- ACE 3.001: Level 0 Statistics -

#I  cdcoinc=0 rdcoinc=2 apcoinc=0 rlcoinc=0 clcoinc=0

#I    xcoinc=2 xcols12=4 qcoinc=3

#I    xsave12=0 s12dup=0 s12new=0

#I    xcrep=6 crepred=0 crepwrk=0 xcomp=0 compwrk=0

#I  xsaved=0 sdmax=0 sdoflow=0

#I  xapply=1 apdedn=1 apdefn=1

#I  rldedn=0 cldedn=0

#I  xrdefn=1 rddedn=5 rddefn=13 rdfill=0

#I  xcdefn=0 cddproc=0 cdddedn=0 cddedn=0

#I    cdgap=0 cdidefn=0 cdidedn=0 cdpdl=0 cdpof=0

#I    cdpdead=0 cdpdefn=0 cddefn=0

#I    #---------------------------------



</pre>

<p>

The statistic <code>qcoinc=3</code> states what we had already observed,  namely,

that there  were  three  coincidences.  Of  these,  two  were  primary

coincidences  (<code>rdcoinc=2</code>).  Since  <code>t=15</code>,   there   were   fourteen

non-trivial coset number definitions; one was during  the  application

of coset 1 to the subgroup generator (<code>apdefn=1</code>), and  the  remainder

occurred during applications of the  coset  numbers  to  the  relators

(<code>rddefn=13</code>). For more details on the meanings of the  variables  you

will need to read the C code comments.

<p>

Now let us display  all  12  lines  of  the  coset  table  with  coset

representatives.

<p>

<pre>

gap> ACEDisplayCosetTable([-12]);

#I  CO: a=12 r=13 h=1 n=13; c=+0.00

#I   coset |      b      B      a   order   rep've

#I  -------+--------------------------------------

#I       1 |      3      2      2

#I       2 |      1      3      1       3   B

#I       3 |      2      1      4       3   b

#I       4 |      8      5      3       5   ba

#I       5 |      4      8      6       2   baB

#I       6 |      9      7      5       5   baBa

#I       7 |      6      9      8       3   baBaB

#I       8 |      5      4      7       5   bab

#I       9 |      7      6     10       5   baBab

#I      10 |     12     11      9       3   baBaba

#I      11 |     10     12     12       2   baBabaB

#I      12 |     11     10     11       3   baBabab

#I  ------------------------------------------------------------



</pre>

<p>

Note  how  the  pre-printout  compaction  phase  now  does  some  work

(indicated  by  the  upper-case   <code>CO:</code>   tag),   since   there   were

coincidences, and hence dead coset numbers. Note how <code>b</code> and <code>B</code>  head

the first two columns,  since  <font face="Gill Sans,Helvetica,Arial">ACE</font>  requires  that  the  first  two

columns  be  occupied  by  a  generator/inverse  pair  or  a  pair  of

involutions. The <code>a</code> column is also the  <code>A</code>  column,  as  <code>a</code>  is  an

involution.

<p>

We now use <code>ACEStandardCosetNumbering</code> to produce a <code>lenlex</code>  standard

table within <font face="Gill Sans,Helvetica,Arial">ACE</font>, but note that this is only <code>lenlex</code> with  respect

to  the  ordering   <code>b, a</code>   of   the   generators.   Then   we   call

<code>ACEDisplayCosetTable</code> again to see  it.  Observe  that  at  both  the

standardisation and coset table display steps a  compaction  phase  is

invoked but on both occasions the lowercase <code>co:</code> tag  indicates  that

nothing is done (all the recovery of dead coset numbers that could  be

done was done earlier).

<p>

<pre>

gap> ACEStandardCosetNumbering();

#I  co/ST: a=12 r=13 h=1 n=13; c=+0.00

gap> ACEDisplayCosetTable([-12]);

#I  co: a=12 r=13 h=1 n=13; c=+0.00

#I   coset |      b      B      a   order   rep've

#I  -------+--------------------------------------

#I       1 |      2      3      3

#I       2 |      3      1      4       3   b

#I       3 |      1      2      1       3   B

#I       4 |      5      6      2       5   ba

#I       5 |      6      4      7       5   bab

#I       6 |      4      5      8       2   baB

#I       7 |      8      9      5       5   baba

#I       8 |      9      7      6       5   baBa

#I       9 |      7      8     10       3   babaB

#I      10 |     11     12      9       3   babaBa

#I      11 |     12     10     12       3   babaBab

#I      12 |     10     11     11       2   babaBaB

#I  ------------------------------------------------------------



</pre>

<p>

Of course, the table above is not <code>lenlex</code> with respect to  the  order

of the generators we had originally given to <font face="Gill Sans,Helvetica,Arial">ACE</font>; to get  that,  we

would have needed to specify <code>lenlex</code>  (see <a href="CHAP004.htm#SSEC011.4">option  lenlex</a>)  at  the

enumeration  stage.  The  effect  of  the  <code>lenlex</code>  option   at   the

enumeration stage is the following: behind the scenes it ensures  that

the relator <code>a^2</code> is passed to <font face="Gill Sans,Helvetica,Arial">ACE</font> as <code>aa</code> and  then  it  sets  the

option <code>asis</code> to 1; this bit of skulduggery stops <font face="Gill Sans,Helvetica,Arial">ACE</font> treating  <code>a</code>

as an involution, allowing <code>a</code> and <code>A</code> (the inverse of <code>a</code>) to take up

the first two columns of the coset table, effectively stopping  <font face="Gill Sans,Helvetica,Arial">ACE</font>

from reordering the generators. To see what is passed  to  <font face="Gill Sans,Helvetica,Arial">ACE</font>,  at

the enumeration stage, we set the <code>InfoACE</code> level to 4,  but  since  we

don't really want to see messages this time we set messages := 0.

<p>

<pre>

gap> SetInfoLevel(InfoACE, 4);

gap> ACEStart(1, ACEGroupGenerators(1), ACERelators(1), [ a*b ]

>             : messages := 0, lenlex);

#I  ToACE> group:ab;

#I  ToACE> relators:aa, b^3,a*b*a*b*a*b*a*b*a*b;

#I  ToACE> generators:a*b;

#I  ToACE> asis:1;

#I  ToACE> messages:0;

#I  ToACE> text:***;

#I  ***

#I  ToACE> text:***;

#I  ***

#I  ToACE> Start;

#I  INDEX = 12 (a=12 r=17 h=1 n=17; l=3 c=0.00; m=15 t=16)

1

gap> ACEStandardCosetNumbering();

#I  ToACE> standard;

#I  CO/ST: a=12 r=13 h=1 n=13; c=+0.00

gap> # The capitalised `CO' indicates space was recovered during compaction

gap> ACEDisplayCosetTable([-12]);

#I  ToACE> print:-12;

#I  ToACE> text:------------------------------------------------------------;

#I  co: a=12 r=13 h=1 n=13; c=+0.00

#I   coset |      a      A      b      B   order   rep've

#I  -------+---------------------------------------------

#I       1 |      2      2      3      2

#I       2 |      1      1      1      3       2   a

#I       3 |      4      4      2      1       3   b

#I       4 |      3      3      5      6       5   ba

#I       5 |      7      7      6      4       5   bab

#I       6 |      8      8      4      5       2   baB

#I       7 |      5      5      8      9       5   baba

#I       8 |      6      6      9      7       5   baBa

#I       9 |     10     10      7      8       3   babaB

#I      10 |      9      9     11     12       3   babaBa

#I      11 |     12     12     12     10       3   babaBab

#I      12 |     11     11     10     11       2   babaBaB

#I  ------------------------------------------------------------



</pre>

<p>

You may have noticed the use of <font face="Gill Sans,Helvetica,Arial">ACE</font>'s text option  several  times

above; this just tells <font face="Gill Sans,Helvetica,Arial">ACE</font> to print the argument  given  to  <code>text</code>

(as a comment). This is used by the <font face="Gill Sans,Helvetica,Arial">GAP</font> interface  as  a  sentinel;

when the string appears in the <font face="Gill Sans,Helvetica,Arial">ACE</font>  output,  the  <font face="Gill Sans,Helvetica,Arial">GAP</font>  interface

knows not to expect anything else.

<p>

<p>

<h2><a name="SECT002">C.2 Emulating Sims</a></h2>

<p><p>

Here we consider the various <code>sims</code>  strategies  (see <a href="CHAP005.htm#SSEC001.8">option  sims</a>),

with respect to duplicating Sims' example statistics of his strategies

given in Section 5.5 of <a href="biblio.htm#Sim94"><cite>Sim94</cite></a>, and giving approximations of his

even-numbered strategies.

<p>

In order to duplicate Sims' maximum active  coset  numbers  and  total

coset numbers statistics, one needs to work with the  formal  inverses

of the relators  and  subgroup  generators  from  <a href="biblio.htm#Sim94"><cite>Sim94</cite></a>,  since

definitions are made from the front in Sims'  routines  and  from  the

rear      in      <font face="Gill Sans,Helvetica,Arial">ACE</font>.      Also,      in      instances      where

<code>IsACEGeneratorsInPreferredOrder(</code><var>gens</var><code>, </code><var>rels</var><code>)</code> returns <code>false</code>, for

group generators <var>fgens</var> and relators <var>rels</var>, one will need  to  apply

the <code>lenlex</code> option to stop <font face="Gill Sans,Helvetica,Arial">ACE</font> from re-ordering the generators and

relators (see <a href="CHAP001.htm#SSEC002.3">IsACEGeneratorsInPreferredOrder</a> and <a href="CHAP004.htm#SSEC011.4">option  lenlex</a>).

In general, we can match Sims' values for the sims := 1 and sims :=

3</code> strategies  (the  R  style  and  R*  style  Sims  strategies  with

<code>mendelsohn</code> off) and for the <code>sims :=  9</code>  (C  style)  strategy,  but

sometimes we may not exactly match Sims' statistics for the  sims  :=

5</code> and <code>sims :=  7</code>  strategies  (the  R  style  and  R*  style  Sims

strategies with <code>mendelsohn</code> on); Sims does not specify an  order  for

the (Mendelsohn) processing of cycled relators and evidently  <font face="Gill Sans,Helvetica,Arial">ACE</font>'s

processing order is different  to  the  one  Sims  used  in  his  CHLT

algorithm to get his statistics (see <a href="CHAP004.htm#SSEC013.5">option mendelsohn</a>).

<p>

<strong>Note:</strong> 

HLT as it appears in Table 5.5.1 of <a href="biblio.htm#Sim94"><cite>Sim94</cite></a> is achieved in <font face="Gill Sans,Helvetica,Arial">ACE</font>

with the sequence ``<code>hlt, lookahead  :=  0</code>''  and  CHLT  is  (nearly)

equivalent to  ``<code>hlt,  lookahead  :=  0,  mendelsohn</code>'';  also  Sims'

<code></code><var>save</var><code> = false</code> equates to R style (<code>rt</code>  positive,  <code>ct  :=  0</code>)  in

<font face="Gill Sans,Helvetica,Arial">ACE</font>, and <code></code><var>save</var><code> = true</code>, for Sims' HLT and CHLT,  equates  to  R*

style (<code>rt</code> negative, <code>ct := 0</code>)  in  <font face="Gill Sans,Helvetica,Arial">ACE</font>.  Sims'  Felsch  strategy

coincides with <font face="Gill Sans,Helvetica,Arial">ACE</font>'s felsch := 0 strategy, i.e. sims  :=  9  is

identical to <code>felsch := 0</code>. (See  the  options <a href="CHAP005.htm#SSEC001.5">option  hlt</a>,  <a href="CHAP004.htm#SSEC015.2">option lookahead</a>, <a href="CHAP004.htm#SSEC013.5">option mendelsohn</a>, <a href="CHAP004.htm#SSEC013.2">option ct</a>, <a href="CHAP004.htm#SSEC013.3">option rt</a>  and <a href="CHAP005.htm#SSEC001.3">option felsch</a>.)

<p>

The following duplicates the ``Total''  (<code>totcosets</code>  in  <font face="Gill Sans,Helvetica,Arial">ACE</font>)  and

``Max. Active'' (<code>maxcosets</code> in <font face="Gill Sans,Helvetica,Arial">ACE</font>) statistics for Example 5.2  of

<a href="biblio.htm#Sim94"><cite>Sim94</cite></a>,  found  in  Sims'  Table 5.5.3,  for  the  sims  :=  3

strategy.

<p>

<pre>

gap> SetInfoLevel(InfoACE, 1); # No behind-the-scenes info. please

gap> F := FreeGroup("r", "s", "t");; r := F.1;; s := F.2;; t := F.3;;

gap> ACEStats([r, s, t], [(r^t*r^-2)^-1, (s^r*s^-2)^-1, (t^s*t^-2)^-1], []

>             : sims := 3);

rec( index := 1, cputime := 0, cputimeUnits := "10^-2 seconds", 

  activecosets := 1, maxcosets := 673, totcosets := 673 )



</pre>

<p>

By replacing <code>sims := 3</code> with <code>sims := </code><var>i</var><code></code> for <var>i</var> equal to 1,  5,  7

or 9, one may verify that for <var>i</var> equal to 1  or 9,  Sims'  statistics

are again  duplicated,  and  observe  a  slight  variance  with  Sims'

statistics for <var>i</var> equal to 5 or 7.

<p>

Now, we show how one can approximate any one of  Sims'  even-numbered

strategies. Essentially, the idea is to start  an  interactive  <font face="Gill Sans,Helvetica,Arial">ACE</font>

process using <code>ACEStart</code> (see <a href="CHAP006.htm#SSEC001.1">ACEStart</a>) with <code>sims := </code><var>i</var><code></code>, for  <var>i</var>

equal to 1, 3, 5, 7 or 9, and <code>max</code> set to some low  value  <var>maxstart</var>

so that the enumeration stops after only completing a few rows of  the

coset table. Then, to  approximate  Sims'  strategy  i  +  1,  one

alternately  applies  <code>ACEStandardCosetNumbering</code>  and  <code>ACEContinue</code>,

progressively increasing the value of <code>max</code> by some  value  <var>maxstep</var>.

The general  algorithm  is  provided  by  the  <code>ACEEvenSims</code>  function

following.

<p>

<pre>

gap> ACEEvenSims := function(fgens, rels, sgens, i, maxstart, maxstep)

>      local j;

>      j := ACEStart(fgens, rels, sgens : sims := i, max := maxstart);

>      while ACEStats(j).index = 0 do

>        ACEStandardCosetNumbering(j);

>        ACEContinue(j : max := ACEParameters(j).max + maxstep);

>      od;

>      return ACEStats(j);

>    end;;



</pre>

<p>

It turns out that one can duplicate the Sims' strategy 4 statistics in

Table 5.5.3 of <a href="biblio.htm#Sim94"><cite>Sim94</cite></a>, with <code></code><var>i</var><code> = 3</code> (so that <code></code><var>i</var><code> + 1  =  4</code>),

<code></code><var>maxstart</var><code> = 14</code> and <code></code><var>maxstep</var><code> = 50</code>:

<p>

<pre>

gap> ACEEvenSims([r, s, t], [(r^t*r^-2)^-1, (s^r*s^-2)^-1, (t^s*t^-2)^-1],

>                [], 3, 14, 50);

rec( index := 1, cputime := 0, cputimeUnits := "10^-2 seconds", 

  activecosets := 1, maxcosets := 393, totcosets := 393 )



</pre>

<p>

Setting <code></code><var>maxstep</var><code> = 60</code> (and leaving the other parameters  the  same)

also gives Sims' statistics, but maxstart = 64 with  maxstep  =

80</code> does better:

<p>

<pre>

gap> ACEEvenSims([r, s, t], [(r^t*r^-2)^-1, (s^r*s^-2)^-1, (t^s*t^-2)^-1],

>                [], 3, 64, 80);

rec( index := 1, cputime := 0, cputimeUnits := "10^-2 seconds", 

  activecosets := 1, maxcosets := 352, totcosets := 352 )



</pre>

<p>

Even though the (<code>lenlex</code>) standardisation steps in the above examples

produce a significant improvement over the  <code>sims  :=  3</code>  statistics,

this  does  not  happen  universally.  Sims  <a href="biblio.htm#Sim94"><cite>Sim94</cite></a>  gives  many

examples  where  the  even-numbered  strategies  fail  to   show   any

significant improvement over  the  odd-numbered  strategies,  and  one

example (see Table 5.5.7) where <code>sims := 2</code> gives a  performance  that

is very much worse than any of the other Sims strategies. As with  any

of the strategies, what works well for some groups may not work at all

well with other groups. There are <strong>no</strong> general rules. It's a bit of  a

game. Let's hope you win most of the time.

<p>

<p>

[<a href = "chapters.htm">Up</a>] [<a href ="CHAP00B.htm">Previous</a>] [<a href ="CHAP00D.htm">Next</a>] [<a href = "theindex.htm">Index</a>]

<P>

<address>ACE manual<br>April 2025

</address></body></html>

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