The examples in this chapter are intended to provide the nearest
GAP equivalent of the similarly named sections in Appendix A of
`ace3001.ps`

(the standalone manual in directory `standalone-doc`

).
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 Examples
first.

Each of the functions `ACECosetTableFromGensAndRels`

(see ACECosetTableFromGensAndRels), `ACEStats`

(see ACEStats ---
non-interactive version) and `ACEStart`

(see ACEStart), may be
called with three arguments: `fgens` (the group generators), `rels`
(the group relators), and `sgens` (the subgroup generators). While it
is legal for the arguments `rels` and `sgens` to be empty lists, it is
always an error for `fgens` to be empty, e.g.

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 )

The example shows that the ACE package does allow you to recover
from the `break`

-loop. However, the definition of `fgens`

above is
local to the `break`

-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 `fgens`

and start again:

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

By default, the presentation is not echoed; use the `echo`

(see option echo) 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 `InfoACE`

to a level of at least 2
(see SetInfoACELevel):

gap> SetInfoACELevel(2);

Calling `ACEStats`

with arguments `fgens`

, `[]`

, `[]`

, 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:

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 )

The line starting with ```#I `

''. is the `Info`

-ed **results
message** from ACE; see Appendix 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 `index`

field 0.

To dump out the presentation and parameters associated with an
enumeration, ACE provides the `sr`

(see option sr) option.
However, you won't see output of this command, unless you set the
`InfoACELevel`

to at least 3. Also, to ensure the reliability of the
output of the `sr`

option, an enumeration should **precede** it; for
`ACEStats`

(and `ACECosetTableFromGensAndRels`

) the directive `start`

(see option start) 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 `start`

or `begin`

(see option start), `aep`

(see option aep)
or `rep`

(see option rep)). Interactively, the equivalent of the
`sr`

command is `ACEParameters`

(see ACEParameters), 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:

gap> SetInfoACELevel(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 )

Observe that at `InfoACE`

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 `print`

(see option print) option, but note as
with the `sr`

option, an enumeration should **precede** it. Here's what
happens if you disregard this (recall, we still have the `InfoACE`

level set to 3):

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 )

Essentially, because ACE had not done an enumeration prior to
getting the `print`

directive, it complained with an ```** ERROR`

'',
recovered and went on with the `start`

directive automatically
inserted by the `ACEStats`

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 `start`

option before the
`print`

option (the `InfoACE`

level is still 3):

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 )

The values we gave to the `print`

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 `fill`

ing (see option fill) 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 `co:`

tag), since there
are no dead cosets. The coset representatives are simply all possible
freely reduced words, in length plus lexicographic (i.e. `lenlex`

; see
Section Coset Table Standardisation Schemes) order.

**Using ACECosetTableFromGensAndRels**

The essential difference between the functions `ACEStats`

and
`ACECosetTableFromGensAndRels`

is that `ACEStats`

parses the **results
message** from the ACE binary and outputs a GAP record containing
statistics of the enumeration, and `ACECosetTableFromGensAndRels`

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 `ACECosetTableFromGensAndRels`

instead of
`ACEStats`

in our examples above, we would have observed similar
output, except that we would have ended up in a `break`

-loop (because
the enumeration overflows) instead of obtaining a record containing
enumeration statistics. We have already seen an example of that in
Section Using ACE Directly to Generate a Coset Table. So, here we
will consider two options that prevent one entering a `break`

-loop,
namely the `silent`

(see option silent) and `incomplete`

(see option incomplete) options. Firstly, let's take the last
`ACEStats`

example, but use `ACECosetTableFromGensAndRels`

instead and
include the `silent`

option. (We still have the `InfoACE`

level set at
3.)

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

Since, the enumeration overflowed and the `silent`

option was set,
`ACECosetTableFromGensAndRels`

simply returned `fail`

. 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
`incomplete`

option, instead of the `silent`

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
`max`

(see option max) option, in order that we should get a more
modestly sized partial table. Finally, we will use `print := -12`

since it is a shorter equivalent alternative to `print := [-1, 12]`

.
Note that the output here was obtained with GAP 4.3 (and is the
same with GAP 4.4). With GAP 4.2 the output was similar except
that the last `Info`

-ed message (before the final output) states that
the coset table result is incomplete only, since no standardisation is
done. It turns out that the table displayed via the `print`

option is
already in `lenlex`

standard form; so despite the differences in the
GAP versions, each version of GAP since GAP 4.2 has output
the same table.

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 ] ]

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 `max`

to 38, since that was the largest coset
number that appeared in the body of the 12-line table, previously.
Also, note that the `max`

option **preceded** the `start`

option; since
the interface respects the order in which options are put by the user,
the enumeration invoked by `start`

would otherwise have only been
restricted by the size of `workspace`

(see option workspace). The
warning that the coset table is incomplete is emitted at `InfoACE`

or
`InfoWarning`

level 1, i.e. by default, you will see it.

The limitation of the functions `ACEStats`

and
`ACECosetTableFromGensAndRels`

(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 `ACEStart`

(see ACEStart) function
allows one to start up an ACE process and maintain a dialogue with
it. Moreover, via the functions `ACEStats`

and `ACECosetTable`

(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) `ACEStats`

exchanges above, using
interactive ACE interface functions. We could do this with:
`ACEStart(0, fgens, [], [] : start, sr := 1);`

where the `0`

first
argument tells `ACEStart`

not to insert `start`

after the options
explicitly listed. Alternatively, we may do the following (note that
the `InfoACE`

level is still 3):

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 )

Observe that the `ACEStart`

call returned an integer (1, here). All 8
forms of the `ACEStart`

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 `1`

to the `ACEParameters`

command which caused ```
sr :=
1
```

(see option sr) to be passed to the interactive ACE process
indexed by 1 (the process we just started), and a record containing
the parameter options (see ACEParameterOptions) is returned. Note
that the ``Run Parameters'': `Group Generators`

, `Group Relators`

and
`Subgroup Generators`

are considered ``args'' (i.e. arguments) and a
record containing these is returned by the `GetACEArgs`

(see GetACEArgs) command; or they may be obtained individually via
the commands: `ACEGroupGenerators`

(see ACEGroupGenerators),
`ACERelators`

(see ACERelators), or `ACESubgroupGenerators`

(see ACESubgroupGenerators).

We can obtain the enumeration statistics record, via the interactive
version of `ACEStats`

(see ACEStats!interactive) :

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 )

To display 12 lines of the coset table with coset representatives
without invoking a further enumeration we could do: ```
ACEStart(0, 1 :
print := [-1, 12]);
```

. Alternatively, we may use the
`ACEDisplayCosetTable`

(see ACEDisplayCosetTable) (the table itself
is emitted at `InfoACE`

level 1, since by default we presumably want
to see it):

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 ------------------------------------------------------------

Still with the same interactive ACE process we can now emulate the
`ACECosetTableFromGensAndRels`

exchange that gave us an incomplete
coset table. Note that it is still necessary to invoke an enumeration
after setting the `max`

(see option max) option. We could just call
`ACECosetTable`

with the argument 1 and the same 4 options we used for
`ACECosetTableFromGensAndRels`

. Alternatively, we can do the
equivalent of the 4 options one (or two) at a time, via their
equivalent interactive commands. Note that the `ACEStart`

command
(without `0`

as first argument) inserts a `start`

directive after the
user option `max`

:

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

Now the following `ACEDisplayCosetTable`

command does the equivalent
of the `print := [-1, 12]`

option.

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 ------------------------------------------------------------

Finally, we call `ACECosetTable`

with 1 argument (which invokes the
interactive version of `ACECosetTableFromGensAndRels`

) with the option
`incomplete`

.

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 ] ]

Observe the line beginning ```#I start = yes,`

'' (the first line in
the output of `ACECosetTable`

). This line appears in response to the
option `mode`

(see option mode) inserted by `ACECosetTable`

after
any user options; it is inserted in order to check that no user
options (possibly made before the `ACECosetTable`

call) have
invalidated ACE's coset table. Since the line also says ```
continue =
yes
```

, the mode `continue`

(the least expensive of the three modes;
see option continu) is directed at ACE which evokes a **results
message**. Then `ACECosetTable`

extracts the incomplete table via a
`print`

(see option print) directive. If you wish to see all the
options that are directed to ACE, set the `InfoACE`

level to 4
(then all such commands are `Info`

-ed behind a ```ToACE> `

'' prompt;
see SetInfoACELevel).

Following the standalone manual, we now set things up to do the
alternating group *A*_{5}, of order 60. (We saw the group *A*_{5} with
subgroup *C*_{5} earlier in Section Example of Using ACE Interactively (Using ACEStart); here we are concerned with observing and remarking
on the output from the ACE binary.) We turn messaging on via the
`messages`

(see option messages) option; setting `messages`

to 1
tells ACE to emit a **progress message** on each pass of its main
loop. In the example following we set `messages := 1000`

, 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 ```
sr
:= 1
```

), no progress messages and the final **results message**. Recall
`F`

is the free group we defined on generators `fgens`

: `"a"`

and
`"b"`

. Here we will be interested in seeing what is transmitted to the
ACE binary; so we will set the `InfoACE`

level to 4 (what is
transmitted to ACE will now appear behind a ```ToACE> `

'' prompt,
and we will still see the messages **from** ACE). Note, that when
GAP prints `F.1`

(= `fgens[1]`

) it displays `a`

, but the
**variable** `a`

is (at the moment) unassigned; so for convenience (in
defining relators, for example) we first assign the variable `a`

to be
`F.1`

(and `b`

to be `F.2`

).

gap> SetInfoACELevel(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

Observe that the `fgens`

and subgroup generators (the empty list)
arguments are transmitted to ACE via the ACE binary's `group`

and `generators`

options, respectively. Observe also, that the relator
`(a*b)^5`

is expanded by GAP to `a*b*a*b*a*b*a*b*a*b`

when
transmitted to ACE and then ACE correctly deduces that it's
`(a*b)^5`

.

Since we did not specify a strategy the `default`

(see option default) strategy was followed and hence coset number definitions
were R (i.e. HLT) style, and a total of 76 coset numbers (`t=76`

)
were defined (if we had tried `felsch`

we would have achieved the best
possible: `t=60`

). Note, that ACE already ``knew'' the group
generators and subgroup generators; so, we could have avoided
re-transmitting that information by using the `relators`

(see option relators) option:

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

Note the usage of `ToACEWords`

(see ToACEWords) to provide the
appropriate string value of the `relators`

option. Also, observe the
`Info`

-ed warning of the action triggered by using the `relators`

option, that says that the current values of the ``args'' (i.e. what
would be returned by `GetACEArgs`

; see GetACEArgs) were discarded,
which immediately triggered the action of reinstantiating the value of
`ACEData.io[1].args`

(which is what the `Info`

:

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

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

-ed twice; the first time was due to `ACEStart`

emitting `sr`

with value `1`

to ACE, the response of which is used to
re-instantiate `ACEData.io[1].args`

, and the second is in response to
transmitting `Start`

to ACE.

In particular, GAP no longer thinks `fgens`

are the group
generators:

gap> ACEGroupGenerators(1) = fgens; false

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

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

We now define a non-trivial subgroup, of small enough index, to make
the observation of all progress messages, by setting `messages := 1`

,
a not too onerous proposition. As for defining the relators, we could
use the 1-argument version of `ACEStart`

, in which case we would use
the `subgroup`

(see option subgroup) option with the value
`ToACEWords(fgens, [ a*b ])`

. However, as we saw, in the end we don't
save anything by doing this, since afterwards the variables `fgens`

,
`a`

, `b`

and `F`

would no longer be associated with `ACEStart`

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

back to 3:

gap> SetInfoACELevel(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

Observe that we used `ACERelators(1)`

(see ACERelators) to grab the
value of the relators we had defined earlier. We also used
`ACEGroupGenerators(1)`

(see ACEGroupGenerators) to get the group
generators.

The run ended with 12 active (see Section Coset Statistics Terminology) coset numbers (`a=12`

) after defining a total number of
15 coset numbers (`t=15`

); the definitions occurred at the steps with
progress messages tagged by `AD:`

(coset 1 application definition) and
`SG:`

(subgroup generator phase), and the 13 tagged by `RD:`

(R style
definition). So there must have been 3 coincidences: observe that
there were 3 progress messages with a `CC:`

tag. (See Appendix The Meanings of ACE's Output Messages.)

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

(see ACEDumpStatistics), which `Info`

s the
ACE output of the `statistics`

(see option statistics) at
`InfoACE`

level 1.

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 #---------------------------------

The statistic `qcoinc=3`

states what we had already observed, namely,
that there were three coincidences. Of these, two were primary
coincidences (`rdcoinc=2`

). Since `t=15`

, there were fourteen
non-trivial coset number definitions; one was during the application
of coset 1 to the subgroup generator (`apdefn=1`

), and the remainder
occurred during applications of the coset numbers to the relators
(`rddefn=13`

). For more details on the meanings of the variables you
will need to read the C code comments.

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

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 ------------------------------------------------------------

Note how the pre-printout compaction phase now does some work
(indicated by the upper-case `CO:`

tag), since there were
coincidences, and hence dead coset numbers. Note how `b`

and `B`

head
the first two columns, since ACE requires that the first two
columns be occupied by a generator/inverse pair or a pair of
involutions. The `a`

column is also the `A`

column, as `a`

is an
involution.

We now use `ACEStandardCosetNumbering`

to produce a `lenlex`

standard
table within ACE, but note that this is only `lenlex`

with respect
to the ordering `b, a`

of the generators. Then we call
`ACEDisplayCosetTable`

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 `co:`

tag indicates that
nothing is done (all the recovery of dead coset numbers that could be
done was done earlier).

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 ------------------------------------------------------------

Of course, the table above is not `lenlex`

with respect to the order
of the generators we had originally given to ACE; to get that, we
would have needed to specify `lenlex`

(see option lenlex) at the
enumeration stage. The effect of the `lenlex`

option at the
enumeration stage is the following: behind the scenes it ensures that
the relator `a^2`

is passed to ACE as `aa`

and then it sets the
option `asis`

to 1; this bit of skulduggery stops ACE treating `a`

as an involution, allowing `a`

and `A`

(the inverse of `a`

) to take up
the first two columns of the coset table, effectively stopping ACE
from reordering the generators. To see what is passed to ACE, at
the enumeration stage, we set the `InfoACELevel`

to 4, but since we
don't really want to see messages this time we set `messages := 0`

.

gap> SetInfoACELevel(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 ------------------------------------------------------------

You may have noticed the use of ACE's `text`

option several times
above; this just tells ACE to print the argument given to `text`

(as a comment). This is used by the GAP interface as a sentinel;
when the string appears in the ACE output, the GAP interface
knows not to expect anything else.

Here we consider the various `sims`

strategies (see option sims),
with respect to duplicating Sims' example statistics of his strategies
given in Section 5.5 of Sim94, and giving approximations of his
even-numbered strategies.

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 Sim94, since
definitions are made from the front in Sims' routines and from the
rear in ACE. Also, in instances where
`IsACEGeneratorsInPreferredOrder(`

`gens``, `

`rels``)`

returns `false`

, for
group generators `fgens` and relators `rels`, one will need to apply
the `lenlex`

option to stop ACE from re-ordering the generators and
relators (see IsACEGeneratorsInPreferredOrder and option lenlex).
In general, we can match Sims' values for the `sims := 1`

and ```
sims :=
3
```

strategies (the R style and R* style Sims strategies with
`mendelsohn`

off) and for the `sims := 9`

(C style) strategy, but
sometimes we may not exactly match Sims' statistics for the ```
sims :=
5
```

and `sims := 7`

strategies (the R style and R* style Sims
strategies with `mendelsohn`

on); Sims does not specify an order for
the (Mendelsohn) processing of cycled relators and evidently ACE's
processing order is different to the one Sims used in his CHLT
algorithm to get his statistics (see option mendelsohn).

**Note:**
HLT as it appears in Table 5.5.1 of Sim94 is achieved in ACE
with the sequence ```hlt, lookahead := 0`

'' and CHLT is (nearly)
equivalent to ```hlt, lookahead := 0, mendelsohn`

''; also Sims'
`save`` = false`

equates to R style (`rt`

positive, `ct := 0`

) in
ACE, and `save`` = true`

, for Sims' HLT and CHLT, equates to R*
style (`rt`

negative, `ct := 0`

) in ACE. Sims' Felsch strategy
coincides with ACE's `felsch := 0`

strategy, i.e. `sims := 9`

is
identical to `felsch := 0`

. (See the options option hlt, option lookahead, option mendelsohn, option ct, option rt and option felsch.)

The following duplicates the ``Total'' (`totcosets`

in ACE) and
``Max. Active'' (`maxcosets`

in ACE) statistics for Example 5.2 of
Sim94, found in Sims' Table 5.5.3, for the `sims := 3`

strategy.

gap> SetInfoACELevel(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 )

By replacing `sims := 3`

with `sims := `

`i`` for `

`i` equal to 1, 5, 7
or 9, one may verify that for `i` equal to 1 or 9, Sims' statistics
are again duplicated, and observe a slight variance with Sims'
statistics for `i` equal to 5 or 7.

Now, we show how one can approximate any one of Sims' even-numbered
strategies. Essentially, the idea is to start an interactive ACE
process using `ACEStart`

(see ACEStart) with `sims := `

`i``, for `

`i`
equal to 1, 3, 5, 7 or 9, and `max`

set to some low value `maxstart`
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 `ACEStandardCosetNumbering`

and `ACEContinue`

,
progressively increasing the value of `max`

by some value `maxstep`.
The general algorithm is provided by the `ACEEvenSims`

function
following.

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;;

It turns out that one can duplicate the Sims' strategy 4 statistics in
Table 5.5.3 of Sim94, with `i`` = 3`

(so that `i`` + 1 = 4`

),
`maxstart`` = 14`

and `maxstep`` = 50`

:

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 )

Setting `maxstep`` = 60`

(and leaving the other parameters the same)
also gives Sims' statistics, but `maxstart`` = 64`

with `maxstep````
=
80
```

does better:

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 )

Even though the (`lenlex`

) standardisation steps in the above examples
produce a significant improvement over the `sims := 3`

statistics,
this does not happen universally. Sims Sim94 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 `sims := 2`

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 **no** general rules. It's a bit of a
game. Let's hope you win most of the time.

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ACE manual

January 2012