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 filling (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).
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
start. This option is inserted anyway, if a user omits it, and
importantly is inserted after the max option is put to the ACE
binary.
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 A5, of order 60. (We saw the group A5 with
subgroup C5 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 Infos 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.
[Up] [Previous] [Next] [Index]
ACE manual