The user will probably benefit most from interactive use of ACE by
setting the InfoLevel
of InfoACE
to at least 3
(see SetInfoACELevel), particularly if she uses the messages
option with a nonzero value.
Have you read the various options warnings yet? If not, please take the time to read Section General Warnings regarding the Use of Options which directs you to various important sections of Chapter Options for ACE.
We describe in this chapter the functions that manipulate and initiate interactive ACE processes.
An interactive ACE process is initiated by ACEStart
and
terminated via ACEQuit
; these functions are described in
Section Starting and Stopping Interactive ACE Processes. ACEStart
also has forms that manipulate an already started interactive ACE
process. ACEStart
always returns a positive integer i, which
identifies the interactive ACE process that was initiated or
manipulated.
Most functions (there is one ACEStart
exception), that manipulate an
already started interactive ACE process, have a form where the
first argument is the integer i returned by the initiating
ACEStart
command, and a second form with one argument fewer (where
the integer i is discovered by a default mechanism, namely by
determining the least integer i for which there is a currently
active interactive ACE process). We will thus commonly say that
``for the ith (or default) interactive ACE process'' a certain
function performs a given action. In each case, it is an error, if i
is not the index of an active interactive process, or there are no
current active interactive processes.
Notes:
The global method of passing options (via PushOptions
), should not
be used with any of the interactive functions. In fact, the
OptionsStack
should be empty at the time any of the interactive
functions is called.
On quit
ting GAP, ACEQuitAll();
is executed, which terminates
all active interactive ACE processes. If GAP is killed without
quit
ting, before all interactive ACE processes are terminated,
zombie processes (still living child processes whose parents
have died), will result. Since zombie processes do consume resources,
in such an event, the responsible computer user should seek out and
kill the still living ace
children (e.g. by piping the output of a
ps
with appropriate options, usually aux
or ef
, to grep ace
,
to find the process ids, and then using kill
; try man ps
and man
kill
if these hints are unhelpful).
ACEStart(
fgens,
rels,
sgens [:
options] ) F
ACEStart(
i [:
options] ) F
ACEStart( [:
options] ) F
ACEStart(
i,
fgens,
rels,
sgens [:
options] ) F
ACEStart( 0 [:
options] ) F
ACEStart( 0,
fgens,
rels,
sgens [:
options] ) F
ACEStart( 0,
i [:
options] ) F
ACEStart( 0,
i,
fgens,
rels,
sgens [:
options] ) F
The variables are: i, a positive integer numbering from 1 that represents (indexes) an already running interactive ACE process; fgens, a list of free generators; rels, a list of words in the generators fgens giving relators for a finitely presented group; and sgens, a list of subgroup generators, again expressed as words in the free generators fgens. Each of fgens, rels and sgens are given in the standard GAP format for finitely presented groups (See Chapter Finitely Presented Groups of the GAP Reference Manual).
All forms of ACEStart
accept options described in Chapters Options for ACE and Strategy Options for ACE, and Appendix Other ACE Options, which are listed behind a colon in the usual way (see
Function Calls in the GAP Reference Manual). The reader is
strongly encouraged to read the introductory sections of
Chapter Options for ACE, with regard to options. The global
mechanism (via PushOptions
) of passing options is not recommended
for use with the interactive ACE interface functions; please ensure
the OptionsStack
is empty before calling an interactive ACE
interface function.
The return value (for all forms of ACEStart
) is an integer
(numbering from 1) which represents the running process. It is
possible to have more than one interactive process running at once.
The integer returned may be used to index which of these processes an
interactive ACE interface function should be applied to.
The first four forms of ACEStart
insert a start
(see option start) directive after the user's options to invoke an enumeration.
The last four forms, with 0
as first argument, do not insert a
start
directive. Moreover, the last 3 forms of ACEStart
, with 0
as first argument only differ from the corresponding forms of
ACEStart
without the 0
argument, in that they do not insert a
start
directive. ACEStart(0)
, however, is special; unlike the
noargument form of ACEStart
it invokes a new interactive ACE
process. We will now further describe each form of ACEStart
, in the
order they appear above.
The first form of ACEStart
(on three arguments) is the usual way to
start an interactive ACE process.
When ACEStart
is called with one positive integer argument i it
starts a new enumeration on the ith running process, i.e. it scrubs
a previously generated table and starts from scratch with the same
parameters (i.e. the same arguments and options); except that if new
options are included these will modify those given previously. The
only reason for doing such a thing, without new options, is to perhaps
compare timings of runs (a second run is quicker because memory has
already been allocated). If you are interested in this sort of
information, however, you may be better off dealing directly with the
standalone.
When ACEStart
is called with no arguments it finds the least
positive integer i for which an interactive process is running and
applies ACEStart(
i)
. (Most users will only run one interactive
process at a time. Hence, ACEStart()
will be a useful shortcut for
ACEStart(1)
.)
The fourth form of ACEStart
on four arguments, invokes a new
enumeration on the ith running process, with new generators fgens,
relators rels and subgroup generators sgens. This is provided so a
user can reuse an already running process, rather than start a new
process. This may be useful when pseudottys are a scarce resource.
See the notes for the noninteractive ACECosetTable
(ACECosetTableFromGensAndRels) which demonstrates an application of
a usage of this and the following form of ACEStart
in a loop.
The fifth form of ACEStart
(on the one argument: 0
) initiates an
interactive ACE process, processes any user options, but does not
insert a start
(see option start) directive. This form is mainly
for gurus who are familiar with the ACE standalone and who wish, at
least initially, to communicate with ACE using the primitive
read/write tools of Section Primitive ACE Read/Write Functions. In
this case, after the group generators, relators, and subgroup
generators have been set in the ACE process, invocations of any of
ACEGroupGenerators
(see ACEGroupGenerators), ACERelators
(see ACERelators), ACESubgroupGenerators
(see ACESubgroupGenerators), or ACEParameters
(see ACEParameters) will establish the corresponding GAP values.
Be warned, though, that unless one of the general ACE modes (see
Section General ACE modes): ACEStart
(without a zero argument),
ACERedo
(see ACERedo) or ACEContinue
(see ACEContinue), or one
of the mode options: start
(see option start), redo
(see option redo) or continu
(see option continu), has been invoked since the
last change of any parameter options (see ACEParameterOptions), some
of the values reported by ACEParameters
may well be incorrect.
The sixth form of ACEStart
(on four arguments), is like the first
form of ACEStart
(on three arguments), except that it does not
insert a start
(see option start) directive. It initiates an
interactive ACE process, with a presentation defined by its last 3
arguments.
The seventh form of ACEStart
(on two arguments), is like the second
form of ACEStart
(on one argument), except that it does not insert a
start
(see option start) directive. It processes any new options
for the ith interactive ACE process. ACEStart(0,
i [:
options ] )
, is similar to SetACEOptions(
i [:
options ] )
(see SetACEOptions), but unlike the latter does not invoke a general
mode (see Section General ACE Modes).
The last form of ACEStart
(on five arguments), is like the fourth
form of ACEStart
(on four arguments), except that it does not insert
a start
(see option start) directive. It reuses an existing
interactive ACE process, with a new presentation. There is no form
of ACEStart
with the same functionality as this form, where the i
argument is omitted.
Note:
When an interactive ACE process is initiated by ACEStart
a
process number i is assigned to it (the integer i returned by the
ACEStart
command), an ACE (binary) process (in the UNIX sense) is
started up, a GAP iostream is assigned to communicate with the
ACE (binary) process, and the essential ``defining'' data
associated with the interactive ACE process is saved in
ACEData.io[
i]
(see ACEData for precisely what is saved).
ACEQuit(
i ) F
ACEQuit() F
terminate an interactive ACE process, where i is the integer
returned by ACEStart
when the process was started. If the second
form is used (i.e. without arguments) then the interactive process of
least index that is still running is terminated.
Note:
ACEQuit(
i)
terminates the ACE (binary) process of interactive
ACE process i, and closes its GAP iostream, and unbinds the
record ACEData.io[
i]
(see ACEStart note).
It can happen that the ACE (binary) process, and hence the GAP
iostream assigned to communicate with it, can die, e.g. by the user
typing a CtrlC
while the ACE (binary) process is engaged in a
long calculation. IsACEProcessAlive
(see IsACEProcessAlive) is
provided to check the status of the GAP iostream (and hence the
status of the ACE (binary) process it was communicating with); in
the case that it is indeed dead, ACEResurrectProcess
(see ACEResurrectProcess) may be used to start a new ACE (binary)
process and assign a new GAP iostream to communicate with it, by
using the ``defining'' data of the interactive ACE process saved in
ACEData.io[
i]
.
ACEQuitAll() F
is provided as a convenience, to terminate all active interactive
ACE processes with a single command. It is equivalent to executing
ACEQuit(
i)
for all active interactive ACE processes i
(see ACEQuit).
For our purposes, we define an interactive ACE interface command to
be an ACE mode, if it delivers an enumeration result (see
Section Results messages). Thus, ACEStart
(see ACEStart), except
when called with the argument 0, and the commands ACERedo
and
ACEContinue
which we describe below are ACE modes; we call these
general ACE modes. Additionally, there are two other commands
which deliver enumeration results: ACEAllEquivPresentations
(see ACEAllEquivPresentations) and ACERandomEquivPresentations
(see ACERandomEquivPresentations); we call these ``experimentation''
ACE modes and describe them in Section Experimentation ACE Modes.
Guru Note:
The ACE standalone uses the term mode
in a slightly different
sense. There, the commands: start
, redo
or continue
, put the
ACE enumerator in start mode
, redo mode
or continue mode
. In
this manual, we have used the term to mean the command itself, and
generalised it to include any command that produces enumeration
results.
After changing any of ACE's parameters, one of three general
modes is possible: one may be able to ``continue'' via ACEContinue
(see ACEContinue), or ``redo'' via ACERedo
(see ACERedo), or if
neither of these is possible one may have to re``start'' the
enumeration via ACEStart
(see ACEStart). Generally, the
appropriate mode is invoked automatically when options are changed; so
most users should be able to ignore the following three functions.
ACEModes(
i ) F
ACEModes() F
for the ith (or default) interactive ACE process, return a record
whose fields are the modes ACEStart
, ACEContinue
and ACERedo
,
and whose values are true
if the mode is possible for the process
and false
otherwise.
ACEContinue(
i [:
options] ) F
ACEContinue( [:
options] ) F
for the ith (or default) interactive ACE process, apply any
options and then ``continue'' the current enumeration, building upon
the existing table. If a previous run stopped without producing a
finite index you can, in principle, change any of the options and
continue on. Of course, if you make any changes which invalidate the
current table, you won't be allowed to ACEContinue
and an error will
be raised. However, after quit
ting the break
loop, the interactive
ACE process should normally still be active; after doing so, run
ACEModes
(see ACEModes) to see which of ACERedo
or ACEStart
is
possible.
ACERedo(
i [:
options] ) F
ACERedo( [:
options] ) F
for the ith (or default) interactive ACE process, apply any options and then ``redo'' the current enumeration from coset 1 (i.e., the subgroup), keeping any existing information in the table.
Notes:
The difference between ACEContinue
and ACERedo
is somewhat
technical, and the user should regard it as a mode that is a little
more expensive than ACEContinue
but cheaper than ACEStart
.
ACERedo
is really intended for the case where additional relators
and/or subgroup generators have been introduced; the current table,
which may be incomplete or exhibit a finite index, is still ``valid''.
However, the new data may allow the enumeration to complete, or cause
a collapse to a smaller index. In some cases, ACERedo
may not be
possible and an error will be raised; in this case, quit
the
break
loop, and try ACEStart
, which will discard the current table
and re``start'' the enumeration.
ACEProcessIndex(
i ) F
ACEProcessIndex() F
With argument i, which must be a positive integer, ACEProcessIndex
returns i if it corresponds to an active interactive process, or
raises an error. With no arguments it returns the default active
interactive process or returns fail
and emits a warning message to
Info
at InfoACE
or InfoWarning
level 1.
Note:
Essentially, an interactive ACE process i is ``active'' if
ACEData.io[
i]
is bound (i.e. we still have some data telling us
about it). Also see ACEStart note.
ACEProcessIndices() F
returns the list of integer indices of all active interactive ACE processes (see ACEProcessIndex for the meaning of ``active'').
IsACEProcessAlive(
i ) F
IsACEProcessAlive() F
return true
if the GAP iostream of the ith (or default)
interactive ACE process started by ACEStart
is alive (i.e. can
still be written to), or false
, otherwise. (See the notes
for ACEStart and ACEQuit.)
If the user does not yet have a gap>
prompt then usually ACE is
still away doing something and an ACE interface function is still
waiting for a reply from ACE. Typing a CtrlC
(i.e. holding down
the Ctrl
key and typing c
) will stop the waiting and send GAP
into a break
loop, from which one has no option but to quit;
. The
typing of CtrlC
, in such a circumstance, usually causes the stream
of the interactive ACE process to die; to check this we provide
IsACEProcessAlive
(see IsACEProcessAlive). If the stream of an
interactive ACE process, indexed by i, say, has died, it may
still be possible to recover enough of the state, before the death of
the stream, from the information stored in the ACEData.io[
i]
record (see Section The ACEData Record). For such a purpose, we have
provided ACEResurrectProcess
(see ACEResurrectProcess).
The GAP iostream of an interactive ACE process will also die if
the ACE binary has a segmentation fault. We do hope that this never
happens to you, but if it does and the failure is reproducible, then
it's a bug and we'd like to know about it. Please read the README
that comes with the ACE package to find out what to include in a
bug report and who to email it to.
ACEResurrectProcess(
i [:
options] ) F
ACEResurrectProcess( [:
options] ) F
regenerate the GAP iostream of the ith (or default) interactive
ACE process started by ACEStart
(see ACEStart, the final note,
in particular), and try to recover as much as possible of the previous
state from saved values of the process's arguments and parameter
options. The possible options here are use
and useboth
which are
described in detail below.
The arguments of the ith interactive ACE process are stored in
ACEData.io[
i].args
, a record with fields fgens
, rels
and
sgens
, which are the GAP group generators, relators and subgroup
generators, respectively (see Section The ACEData Record). Option
information is saved in ACEData.io[
i].options
when a user uses an
interactive ACE interface function with options or uses
SetACEOptions
(see SetACEOptions). Parameter option information is
saved in ACEData.io[
i].parameters
if ACEParameters
(see ACEParameters)
is used to extract from ACE the current values of the ACE
parameter options (this is generally less reliable unless one of the
general ACE modes (see Section General ACE Modes), has been run
previously).
By default, ACEResurrectProcess
recovers parameter option
information from ACEData.io[
i].options
if it is bound, or from
ACEData.io[
i].parameters
if it is bound, otherwise. The
ACEData.io[
i].options
record, however, is first filtered for
parameter and strategy options (see Sections ACEParameterOptions
and The ACEStrategyOptions list) and the echo
option (see option echo). To alter this behaviour, the user is provided two options:
use :=
useList
"options"
and "parameters"
. By default, use = ["options",
"parameters"]
.
useboth
useboth = false
.
If useboth = true
, SetACEOptions
(see SetACEOptions) is applied
to the ith interactive ACE process with each
ACEData.io[
i].(
field)
for each field ("options"
or
"parameters"
) that is bound and in useList, in the order implied
by useList. If useboth = false
, SetACEOptions
is applied with
ACEData.io[
i].(
field)
for only the first field that is bound
in useList. The current value of the echo
option is also preserved
(no matter what values the user chooses for the use
and useboth
options).
Notes:
Do not use general ACE options with ACEResurrectProcess
; they
will only be superseded by those options recovered from
ACEData.io[
i].options
and/or ACEData.io[
i].parameters
.
Instead, call SetACEOptions
first (or afterwards). When called prior
to ACEResurrectProcess
, SetACEOptions
will emit a warning that the
stream is dead; despite this, the ACEData.io[
i].options
will be
updated.
ACEResurrectProcess
does not invoke an ACE mode (see
Section General ACE Modes). This leaves the user free to use
SetACEOptions
(which does invoke an ACE mode) to further modify
options afterwards.
ToACEGroupGenerators(
fgens ) F
This function produces, from a GAP list fgens of free group
generators, the ACE directive string required by the group
(see option group) option. (The group
option may be used to define
the equivalent of fgens in an ACE process.)
ToACEWords(
fgens,
words ) F
This function produces, from a GAP list words in free group
generators fgens, a string that represents those words as an
ACE list of words. ToACEWords
may be used to provide suitable
values for the options relators
(see option relators),
generators
(see option generators), sg
(see option sg), and
rl
(see option rl).
Now we describe the two experimentation modes. The term mode was defined in Section General ACE Modes.
ACEAllEquivPresentations(
i,
val ) F
ACEAllEquivPresentations(
val ) F
for the ith (or default) interactive ACE process, generates and tests an enumeration for combinations of relator ordering, relator rotations, and relator inversions; val is in the integer range 1 to 7.
The argument val is considered as a binary number. Its three bits are treated as flags, and control relator rotations (the 2^{0} bit), relator inversions (the 2^{1} bit) and relator orderings (the 2^{2} bit), respectively; where 1 means ``active'' and 0 means ``inactive''. (See below for an example).
Before we describe the GAP output of ACEAllEquivPresentations
let
us spend some time considering what happens before the ACE binary
output is parsed.
The ACEAllEquivPresentations
command first performs a ``priming
run'' using the options as they stand. In particular, the asis
and
messages
options are honoured.
It then turns asis
(see option asis) on and messages
(see option messages) off (i.e. sets messages
to 0), and generates
and tests the requested equivalent presentations. The maximum and
minimum values attained by m
(the maximum number of coset numbers
defined at any stage) and t
(the total number of coset numbers
defined) are tracked, and each time the statistics are better than
what we've already seen, the ACE binary emits a summary result line
for the relators used. See Appendix The Meanings of ACE's output messages for a discussion of the statistics m
and t
. To observe
these messages set the InfoLevel
of InfoACE
to 3; and it's
recommended that you do this so that you get some idea of what the
ACE binary is doing.
The order in which the equivalent presentations are generated and
tested has no particular significance, but note that the presentation
as given after the initial priming run) is the last presentation
to be generated and tested, so that the group's relators are left
``unchanged'' by running the ACEAllEquivPresentations
command.
As discussed by Cannon, Dimino, Havas and Watson CDHW73 and Havas and Ramsay HR01 such equivalent presentations can yield large variations in the number of coset numbers required in an enumeration. For this command, we are interested in this variation.
After the final presentation is run, some additional status information messages are printed to the ACE output:
m
and t
.
As an example (drawn from the discussion in HR99ace) consider the
enumeration of the 448 coset numbers of the subgroup
áa^{2},a^{1}b ñ of the group

Now we describe the output of ACEAllEquivPresentations
; it is a
record with fields:
primingResult
primingStats
index
, cputime
, cputimeUnits
,
activecosets
, maxcosets
and totcosets
, exactly as for the record
returned by ACEStats
(see ACEStats);
equivRuns
rels
enumResult
stats
ACEStats
(see ACEStats);
summary
successes
runs
maxcosetsRange
equivRuns[
i].maxcosets
lies, and
totcosetsRange
equivRuns[
i].totcosets
lies.
Notes:
In general, the length of the equivRuns
field list will be less than
the number of runs executed.
There is no way to stop the ACEAllEquivPresentations
command before
it has completed, other than killing the task. So do a reality check
beforehand on the size of the search space and the time for each
enumeration. If you are interested in finding a ``good'' enumeration,
it can be very helpful, in terms of running time, to put a tight limit
on the number of coset numbers via the max
option. You may also have
to set compaction
= 100 to prevent timewasting attempts to
recover space via compaction. This maximises throughput by causing the
``bad'' enumerations, which are in the majority, to overflow quickly
and abort. If you wish to explore a very large searchspace, consider
firing up many copies of ACE, and starting each with a ``random''
equivalent presentation. Alternatively, you could use the
ACERandomEquivPresentations
command.
ACERandomEquivPresentations(
i,
val ) F
ACERandomEquivPresentations(
val ) F
ACERandomEquivPresentations(
i, [
val] ) F
ACERandomEquivPresentations( [
val] ) F
ACERandomEquivPresentations(
i, [
val,
Npresentations] ) F
ACERandomEquivPresentations( [
val,
Npresentations] ) F
for the ith (or default) interactive ACE process, generates and
tests up to Npresentations (or 8, in the first 4 forms) random
presentations; val, an integer in the range 1 to 7, acts as for
ACEAllEquivPresentations
and Npresentations, when given, should be
a positive integer.
The routine first turns asis
(see option asis) on and messages
(see option messages) off (i.e. sets messages
to 0), and then
generates and tests the requested number of random equivalent
presentations. For each presentation, the relators used and the
summary result line are printed by ACE. To observe these messages
set the InfoLevel
of InfoACE
to at least 3.
ACERandomEquivPresentations
parses the ACE messages, translating
them to GAP, and thus returns a list of records (similar to the
field equivRuns
of the returned record of
ACEAllEquivPresentations
). Each record of the returned list is the
data derived from a presentation run and has fields:
rels
enumResult
stats
ACEStats
(see ACEStats).
Notes:
The relator inversions and rotations are ``genuinely'' random. The
relator permuting is a little bit of a kludge, with the ``quality'' of
the permutations tending to improve with successive presentations.
When the ACERandomEquivPresentations
command completes, the
presentation active is the last one generated.
Guru Notes:
It might appear that neglecting to restore the original presentation
is an error. In fact, it is a useful feature! Suppose that the space
of equivalent presentations is too large to exhaustively test. As
noted in the entry for ACEAllEquivPresentations
, we can start up
multiple copies of ACEAllEquivPresentations
at random points in the
searchspace. Manually generating random
equivalent presentations to
serve as startingpoints is tedious and errorprone. The
ACERandomEquivPresentations
command provides a simple solution;
simply run ACERandomEquivPresentations(
i, 7);
before
ACEAllEquivPresentations(
i, 7);
.
ACEGroupGenerators(
i ) F
ACEGroupGenerators() F
return the GAP group generators of the ith (or default)
interactive ACE process. If no generators have been saved for the
interactive ACE process, possibly because the process was started
via ACEStart(0);
(see ACEStart), the ACE process is
interrogated, and the equivalent in GAP is saved and returned.
Essentially, ACEGroupGenerators(
i)
interrogates ACE and
establishes ACEData.io[
i].args.fgens
, if necessary, and returns
ACEData.io[
i].args.fgens
. As a sideeffect, if any of the
remaining fields of ACEData.io[
i].args
or
ACEData.io[
i].acegens
are unset, they are also set. Note that
GAP provides GroupWithGenerators
(see GroupWithGenerators
in the GAP Reference Manual) to establish a free group on a given
set of alreadydefined generators.
ACERelators(
i ) F
ACERelators() F
return the GAP relators of the ith (or default) interactive
ACE process. If no relators have been saved for the interactive
ACE process, possibly because the process was started via
ACEStart(0);
(see ACEStart), the ACE process is interrogated,
the equivalent in GAP is saved and returned. Essentially,
ACERelators(
i)
interrogates ACE and establishes
ACEData.io[
i].args.rels
, if necessary, and returns
ACEData.io[
i].args.rels
. As a sideeffect, if any of the remaining
fields of ACEData.io[
i].args
or ACEData.io[
i].acegens
are
unset, they are also set.
ACESubgroupGenerators(
i ) F
ACESubgroupGenerators() F
return the GAP subgroup generators of the ith (or default)
interactive ACE process. If no subgroup generators have been saved
for the interactive ACE process, possibly because the process was
started via ACEStart(0);
(see ACEStart), the ACE process is
interrogated, the equivalent in GAP is saved and returned.
Essentially, ACESubgroupGenerators(
i)
interrogates ACE and
establishes ACEData.io[
i].args.sgens
, if necessary, and returns
ACEData.io[
i].args.sgens
. As a sideeffect, if any of the
remaining fields of ACEData.io[
i].args
or
ACEData.io[
i].acegens
are unset, they are also set.
DisplayACEArgs(
i ) F
DisplayACEArgs() F
display the arguments (i.e. fgens, rels and sgens) of the ith
(or default) process started by ACEStart
. In fact,
DisplayACEArgs(
i)
is just a prettyprinter of the
ACEData.io[
i].args
record. Use GetACEArgs
(see GetACEOptions)
in assignments. Unlike ACEGroupGenerators
(see ACEGroupGenerators), ACERelators
(see ACERelators) and
ACESubgroupGenerators
(see ACESubgroupGenerators),
DisplayACEArgs
does not have the sideeffect of setting any of the
fields of ACEData.io[
i].args
if they are unset.
GetACEArgs(
i ) F
GetACEArgs() F
return a record of the current arguments (i.e. fgens, rels and
sgens) of the ith (or default) process started by ACEStart
. In
fact, GetACEOptions(
i)
simply returns the ACEData.io[
i].args
record, or an empty record if that record is unbound. Unlike
ACEGroupGenerators
(see ACEGroupGenerators), ACERelators
(see ACERelators) and ACESubgroupGenerators
(see ACESubgroupGenerators), GetACEOptions
does not have the
sideeffect of setting any of the fields of ACEData.io[
i].args
if
they are unset.
DisplayACEOptions(
i ) F
DisplayACEOptions() F
display the options, explicitly set by the user, of the ith (or default) process started by
ACEStart
. In fact, DisplayACEOptions(
i)
is just a prettyprinter
of the ACEData.io[
i].options
record. Use GetACEOptions
(see GetACEOptions) in assignments. Please note that novalue ACE
options will appear with the assigned value true
(see
Section Interpretation of ACE Options for how the ACE interface
functions interpret such options).
Notes:
Any options set via ACEWrite
(see ACEWrite) will not be
displayed. Also, recall that if ACE is not given any options it
uses the default
strategy (see Section What happens if no ACE Strategy Option or if no ACE Option is passed). To discover the
various settings of the ACE Parameter Options
(see ACEParameterOptions) in vogue for the ACE process, use
ACEParameters
(see ACEParameters).
GetACEOptions(
i ) F
GetACEOptions() F
return a record of the current options (those that have been
explicitly set by the user) of the ith (or default) process started
by ACEStart
. Please note that novalue ACE options will appear
with the assigned value true
(see Section Interpretation of ACE Options for how the ACE interface functions interpret such
options). The notes applying to DisplayACEOptions
(see DisplayACEOptions) also apply here.
SetACEOptions(
i [:
options] ) F
SetACEOptions( [:
options] ) F
modify the current options of the ith (or default) process started
by ACEStart
. Please ensure that the OptionsStack
is empty before
calling SetACEOptions
, otherwise the options already present on the
OptionsStack
will also be ``seen''. All interactive ACE interface
functions that accept options, actually call an internal version of
SetACEOptions
; so, it is generally important to keep the
OptionsStack
clear while working with ACE interactively.
After setting the options passed, the first mode of the following:
ACEContinue
(see ACEContinue), ACERedo
(see ACERedo) or
ACEStart
(see ACEStart), that may be applied, is automatically
invoked.
Since a user will sometimes have options in the form of a record
(e.g. via GetACEOptions
), we provide an alternative to the
behindthecolon syntax, in a manner .like PushOptions
, for the
passing of options via SetACEOptions
:
SetACEOptions(
i,
optionsRec ) F
SetACEOptions(
optionsRec ) F
In this form, the record optionsRec is used to update the current
options of the ith (or default) process started by ACEStart
. Note
that since optionsRec is a record each field must have an assigned
value; in particular, novalue ACE options should be assigned the
value true
(see Section Interpretation of ACE Options). Please
don't mix these two forms of SetACEOptions
with the previous two
forms; i.e. do not pass both a record argument and options, since
this will lead to options appearing in the wrong order; if you want to
do this, make two separate calls to SetACEOptions
, e.g. let's say
you have a process like that started by:
gap> ACEExample("A5", ACEStart);
then the following demonstrates both usages of SetACEOptions
:
gap> SetACEOptions( rec(echo := 2) ); gap> SetACEOptions( : hlt);
Each of the three commands above generates output; for brevity it has not been included.
Notes:
When ACECosetTableFromGensAndRels
enters a break
loop
(see ACECosetTable), local versions of the second form of each of
DisplayACEOptions
and SetACEOptions
become available. (Even though
the names are similar and their function is analogous they are in fact
different functions.)
ACEParameters(
i ) F
ACEParameters() F
return a record of the current values of the ACE Parameter Options
(see ACEParameterOptions) of the ith (or default) process started
by ACEStart
, according to ACE. Please note that some options may
be reported with incorrect values if they have been changed recently
without following up with one of the modes ACEContinue
, ACERedo
or
ACEStart
. Together the commands ACEGroupGenerators
, ACERelators
,
ACESubgroupGenerators
and ACEParameters
give the equivalent GAP
information that is obtained in ACE with sr := 1
(see option sr), which is the ``Run Parameters'' block obtained in the messaging
output (observable when the InfoLevel
of InfoACE
is set to at
least 3), when messages
(see option messages) is set a nonzero
value.
Notes:
One use for this function might be to determine the options required
to replicate a previous run, but be sure that, if this is your
purpose, any recent change in the parameter option values has been
followed by an invocation of one of ACEContinue
(see ACEContinue),
ACERedo
(see ACERedo) or ACEStart
(see ACEStart).
As a sideeffect, for ACE process i, any of the fields of
ACEData.io[
i].args
or ACEData.io[
i].acegens
that are unset,
are set.
IsCompleteACECosetTable(
i ) F
IsCompleteACECosetTable() F
return, for the ith (or default) process started by ACEStart
,
true
if ACE's current coset table is complete, or false
otherwise.
Note:
The completeness of the coset table of the ith interactive ACE
process is determined by checking whether
ACEData.io[
i].stats.index
is positive; a value of zero indicates
the last enumeration failed to complete. The record
ACEData.io[
i].stats
is what is returned by ACEStats(
i)
(see ACEStats).
ACEDisplayCosetTable(
i ) F
ACEDisplayCosetTable() F
ACEDisplayCosetTable(
i, [
val] ) F
ACEDisplayCosetTable( [
val] ) F
ACEDisplayCosetTable(
i, [
val,
last] ) F
ACEDisplayCosetTable( [
val,
last] ) F
ACEDisplayCosetTable(
i, [
val,
last,
by] ) F
ACEDisplayCosetTable( [
val,
last,
by] ) F
compact and display the (possibly incomplete) coset table of the ith
(or default) process started by ACEStart
; val must be an integer,
and last and by must be positive integers. In the first two forms
of the command, the entire coset table is displayed, without orders or
coset representatives. In the third and fourth forms, the absolute
value of val is taken to be the last line of the table to be
displayed (and 1 is taken to be the first); in the fifth and sixth
forms, 
val
is taken to be the first line of the table to be
displayed, and last is taken to be the number of the last line to be
displayed. In the last two forms, the table is displayed from line

val
to line last in steps of by. If val is negative, then
the orders modulo the subgroup (if available) and coset
representatives are displayed also.
Note:
The coset table displayed will normally only be lenlex
standardised
if the call to ACEDisplayCosetTable
is preceded by
ACEStandardCosetNumbering
(see ACEStandardCosetNumbering). The
options lenlex
(see option lenlex) and semilenlex
(see option semilenlex) are only executed by ACECosetTable
(see ACECosetTable). The ACE binary does not provide semilenlex
standardisation, and hence ACEDisplayCosetTable
will never display a
semilenlex
standard coset table.
ACECosetRepresentative(
i,
n ) F
ACECosetRepresentative(
n ) F
return, for the ith (or default) process started by ACEStart
, the
coset representative of coset n of the current coset table held by
ACE, where n must be a positive integer.
ACECosetRepresentatives(
i ) F
ACECosetRepresentatives() F
return, for the ith (or default) process started by ACEStart
, the
list of coset representatives of the current coset table held by
ACE.
ACETransversal(
i ) F
ACETransversal() F
return, for the ith (or default) process started by ACEStart
, the
list of coset representatives of the current coset table held by
ACE, if the current table is complete, and fail
otherwise.
Essentially, ACETransversal(
i) = ACECosetRepresentatives(
i)
for
a complete table.
ACECycles(
i ) F
ACECycles() F
ACEPermutationRepresentation(
i ) F
ACEPermutationRepresentation() F
return, for the ith (or default) process started by ACEStart
, a
list of permutations corresponding to the group generators, (i.e., the
permutation representation), if the current coset table held by ACE
is complete or fail
, otherwise. In the event of failure a message is
emitted to Info
at InfoACE
or InfoWarning
level 1.
ACETraceWord(
i,
n,
word ) F
ACETraceWord(
n,
word ) F
for the ith (or default) interactive ACE process started by
ACEStart
, trace word through ACE's coset table, starting at
coset n, and return the final coset number if the trace completes,
and fail
otherwise. In Group Theory terms, if the cosets of a
subgroup H in a group G are the subject of interactive ACE
process i and the coset identified by that process by the integer
n corresponds to some coset Hx, for some x in G, and word
represents the element g of G, then, providing the current coset
table is complete enough, ACETraceWord(
i,
n,
word )
returns
the integer identifying the coset Hxg.
Notes:
You may wish to compact ACE's coset table first, either explicitly
via ACERecover
(see ACERecover), or, implicitly, via any function
call that invokes ACE's compaction routine (see ACERecover note).
If you actually wanted ACE's coset representative, then, for a
compact table, feed the output of ACETraceWord
to
ACECosetRepresentative
(see ACECosetRepresentative).
ACEOrders(
i ) F
ACEOrders() F
ACEOrders(
i : suborder :=
suborder ) F
ACEOrders(: suborder :=
suborder ) F
for the ith (or default) interactive ACE process started by
ACEStart
, search for all coset numbers whose representatives' orders
(modulo the subgroup) are either finite, or, if invoked with the
suborder
option, are multiples of suborder, where suborder
should be a positive integer. ACEOrders
returns a (possibly empty)
list of records, each with fields coset
, order
and rep
, which
are respectively, the coset number, its order modulo the subgroup, and
a representative for each coset number satisfying the criteria of the
search.
Note:
You may wish to compact ACE's coset table first, either explicitly
via ACERecover
(see ACERecover), or, implicitly, via any function
call that invokes ACE's compaction routine (see ACERecover note).
ACEOrder(
i,
suborder ) F
ACEOrder(
suborder ) F
for the ith (or default) interactive ACE process started by
ACEStart
, search for coset number(s) whose coset representatives
have order modulo the subgroup a multiple of suborder. When
suborder is a positive integer, ACEOrder
returns just one record
with fields coset
, order
and rep
, which are respectively, the
coset number, its order modulo the subgroup, and a representative for
the first coset number satisfying the criteria of the search, or
fail
if there is no such coset number. The value of suborder may
also be a negative integer, in which case, ACEOrder(
i,
suborder
)
is equivalent to ACEOrders(
i : suborder := 
suborder)
; or
suborder may be zero, in which case, ACEOrder(
i, 0 )
is
equivalent to ACEOrders(
i )
.
Note:
You may wish to compact ACE's coset table first, either explicitly
via ACERecover
(see ACERecover), or, implicitly, via any function
call that invokes ACE's compaction routine (see ACERecover note).
ACECosetOrderFromRepresentative(
i,
cosetrep ) F
ACECosetOrderFromRepresentative(
cosetrep ) F
for the ith (or default) interactive ACE process return the order (modulo the subgroup) of the coset with representative cosetrep, a word in the free group generators.
Note:
ACECosetOrderFromRepresentative
calls ACETraceWord
to determine
the coset (number) to which cosetrep belongs, and then scans the
output of ACEOrders
to determine the order of the coset (number).
ACECosetsThatNormaliseSubgroup(
i,
n ) F
ACECosetsThatNormaliseSubgroup(
n ) F
for the ith (or default) interactive ACE process started by
ACEStart
, determine nontrivial (i.e. other than coset 1) coset
numbers whose representatives normalise the subgroup.
coset
and
rep
which represent the coset number and a representative,
respectively, of the first n nontrivial coset numbers whose
representatives normalise the subgroup is returned.
coset
and
rep
which represent the coset number and a representative,
respectively, of all nontrivial coset numbers whose representatives
normalise the subgroup is returned.
Note:
You may wish to compact ACE's coset table first, either explicitly
via ACERecover
(see ACERecover), or, implicitly, via any function
call that invokes ACE's compaction routine (see ACERecover note).
ACEStyle(
i ) F
ACEStyle() F
returns the current enumeration style as one of the strings: "C"
,
"Cr"
, "CR"
, "R"
, "R*"
, "Rc"
, "R/C"
, or "R/C (defaulted)"
(see Section Enumeration Style).
The next two functions of this section are really intended for ACE standalone gurus. To fully understand their output you will need to consult the standalone manual and the C source code.
ACEDumpVariables(
i ) F
ACEDumpVariables() F
ACEDumpVariables(
i, [
level] ) F
ACEDumpVariables( [
level] ) F
ACEDumpVariables(
i, [
level,
detail] ) F
ACEDumpVariables( [
level,
detail] ) F
dump the internal variables of ACE of the ith (or default)
process started by ACEStart
; level should be one of 0, 1, or 2,
and detail should be 0 or 1.
The value of level determines which of the three levels of ACE to
dump. (You will need to read the standalone manual to understand what
Levels 0, 1 and 2 are all about.) The value of detail determines the
amount of detail (detail
= 0
means less detail). The first two
forms of ACEDumpVariables
(with no list argument) selects level
=
0,
detail = 0
. The third and fourth forms (with a list argument
containing the integer level) makes detail
= 0
. This command is
intended for gurus; the source code should be consulted to see what
the output means.
ACEDumpStatistics(
i ) F
ACEDumpStatistics() F
dump ACE's internal statistics accumulated during the most recent
enumeration of the ith (or default) process started by ACEStart
,
provided the ACE binary was built with the statistics package
(which it is by default). Use ACEBinaryVersion();
(see ACEBinaryVersion) to check for the inclusion of the statistics
package. See the enum.c
source file for the meaning of the
variables.
ACEBinaryVersion(
i ) F
ACEBinaryVersion() F
for the ith (or default) process started by ACEStart
, print, via
Info
(at InfoACE
level 1), version details of the ACE binary
you are currently running, including what compiler flags were set when
the executable was built, and also returns the version number of the
binary as a string. Essentially the information obtained is what is
obtained via ACE's options
option (see option options), and the
returned value is what is stored in ACEData.version
(see ACEData).
A typical output, illustrating the default build, is:
gap> ACEBinaryVersion(); #I No interactive ACE sessions are currently active #I ACE Binary Version: 3.001 #I ACE 3.001 Sat Feb 27 11:27:15 2016 #I ========================================= #I Host information: #I name = banksia #I ACE 3.001 executable built: #I Wed Feb 24 15:25:26 AWST 2016 #I Level 0 options: #I statistics package = on #I coinc processing messages = on #I dedn processing messages = on #I Level 1 options: #I workspace multipliers = decimal #I Level 2 options: #I host info = on "3.001"
Notes:
The ACE binary's banner may also appear in the output (if it has
not already appeared). Unlike other ACE interface functions, the
information obtained via ACEBinaryVersion();
is absolutely
independent of any enumeration. For this reason, we make it
permissible to run ACEBinaryVersion();
when there are no currently
active interactive ACE processes; and, in such a case,
ACEBinaryVersion();
emits a warning that there are no interactive
ACE sessions currently active and initiates (and closes again) its
own stream to obtain the information from the ACE binary. For the
current version of the ACE package (the GAP code component) use
ACEPackageVersion();
(see ACEPackageVersion).
ACECosetTable(
i [:
options] ) F
ACECosetTable( [:
options] ) F
return a coset table as a GAP object, in standard form (for
GAP). These functions perform the same function as
ACECosetTableFromGensAndRels
and ACECosetTable
on three arguments
(see ACECosetTable), albeit interactively, on the ith (or default)
process started by ACEStart
. If options are passed then an internal
version of ACEModes
is run to determine which of the general ACE
modes (see Section General ACE Modes) ACEContinue
, ACERedo
or
ACEStart
is possible; and (an internal version of) the first mode of
these that is allowed is executed, to ensure the resultant table is
correct for the current options.
ACEStats(
i [:
options] ) F
ACEStats( [:
options] ) F
perform the same function as ACEStats
on three arguments
(see ACEStats  noninteractive version), albeit interactively, on
the ith (or default) process started by ACEStart
. If options are
passed then an internal version of ACEModes
is run to determine
which of the general ACE modes (see Section General ACE Modes)
ACEContinue
, ACERedo
or ACEStart
is possible; and (an internal
version of) the first mode of these that is allowed is executed, to
ensure the resultant statistics are correct for the current options.
See Section Example of Using ACE Interactively (Using ACEStart) for an example demonstrating both these functions within an interactive process.
IsACEGeneratorsInPreferredOrder(
i ) F
IsACEGeneratorsInPreferredOrder() F
for the ith (or default) interactive ACE process started by
ACEStart
, return true
if the group generators of that process, are
in an order that will not be changed by ACE, and false
otherwise.
This function has greatest relevance to users who call ACECosetTable
(see ACECosetTable!interactive), with the lenlex
option
(see option lenlex). For more details, see the discussion for the
noninteractive version of IsACEGeneratorsInPreferredOrder
(IsACEGeneratorsInPreferredOrder), which is called with two
arguments.
ACERecover(
i ) F
ACERecover() F
invoke the compaction routine on the coset table of the ith (or
default) interactive ACE process started by ACEStart
, in order to
recover the space used by the dead coset numbers. A CO
message line
is printed if any rows of the coset table were recovered, and a co
line if none were. (See Appendix The Meanings of ACE's output messages for the meanings of these messages.)
Note:
The compaction routine is called automatically when any of
ACEDisplayCosetTable
(see ACEDisplayCosetTable),
ACECosetRepresentative
(see ACECosetRepresentative),
ACECosetRepresentatives
(see ACECosetRepresentatives),
ACETransversal
(see ACETransversal), ACECycles
(see ACECycles), ACEStandardCosetNumbering
(see ACEStandardCosetNumbering), ACECosetTable
(see ACECosetTable) or ACEConjugatesForSubgroupNormalClosure
(see ACEConjugatesForSubgroupNormalClosure), is invoked.
ACEStandardCosetNumbering(
i ) F
ACEStandardCosetNumbering() F
compact and then do a lenlex
standardisation (see Section Coset Table Standardisation Schemes) of the numbering of cosets in the
coset table of the ith (or default) interactive ACE process
started by ACEStart
. That is, for a given ordering of the generators
in the columns of the table, they produce a canonic table. A table
that includes a column for each generator inverse immediately
following the column for the corresponding generator, standardised
according to the lenlex
scheme, has the property that a rowmajor
scan (i.e. a scan of the successive rows of the body of the table
row by row, from left to right) encounters previously unseen cosets in
numeric order. This function does not display the new table; use
ACEDisplayCosetTable
(see ACEDisplayCosetTable) for that.
Notes:
In a lenlex
canonic table, the coset representatives are ordered
first according to length and then the lexicographic order defined by
the order the generators and their inverses head the columns. Note
that, unless special action is taken, ACE avoids having an
involutory generator in the first column (by swapping the first two
generators), except when there is only one generator, or when the
second generator is also an involution; so the lexicographic order
used by ACE need not necessarily correspond with the order in which
the generators were first put to ACE. (We have used the term
``involution'' above; what we really mean is a generator x
for
which there is a relator x*x
or x^2
. Such a generator may, of
course, turn out to actually be the identity.) The function
IsACEGeneratorsInPreferredOrder
(see
IsACEGeneratorsInPreferredOrder) detects cases when ACE would
swap the first two generators.
Standardising the coset numbering within ACE does not affect the
GAP coset table obtained via ACECosetTable
(see ACECosetTable).
If ACECosetTable
is called without the lenlex
option GAP's
default standardisation is applied after conversion of ACE's
output, which undoes an ACE standardisation. On the other hand, if
ACECosetTable
is called with the lenlex
option then after a check
and special action, if required, the equivalent of a call to
ACEStandardCosetNumbering
is invoked, irrespective of whether it has
been done by the user beforehand. The check that is done is a call to
IsACEGeneratorsInPreferredOrder
(see
IsACEGeneratorsInPreferredOrder) to ensure that ACE has not
swapped the first two generators. The special action taken when the
call to IsACEGeneratorsInPreferredOrder
returns false
, is the
setting of the asis
option (see option asis) to 1 and the
resubmission of the relators to ACE taking care not to submit the
relator that determines the first generator as an involution as that
generator squared (these two actions together avert ACE's swapping
of the first two generators), followed by the restart
ing of the
enumeration.
Guru Notes: In five of the ten standard enumeration strategies of Sims Sim94 (i.e. the five Sims strategies not provided by ACE), the table is standardised repeatedly. This is expensive computationally, but can result in fewer cosets being necessary. The effect of doing this can be investigated in ACE by (repeatedly) halting the enumeration (via restrictive options), standardising the coset numbering, and continuing (see Section Emulating Sims for an example).
ACEAddRelators(
i,
wordlist ) F
ACEAddRelators(
wordlist ) F
add, for the ith (or default) interactive ACE process started by
ACEStart
, the words in the list wordlist to any relators already
present, and automatically invoke ACERedo
, if it can be applied, or
otherwise ACEStart
. Note that ACE sorts the resultant relator
list, unless the asis
option (see option asis) has been set to 1;
don't assume, unless asis = 1
, that the new relators have been
appended in userprovided order to the previously existing relator
list. ACEAddRelators
also returns the new relator list. Use
ACERelators
(see ACERelators) to determine the current relator
list.
ACEAddSubgroupGenerators(
i,
wordlist ) F
ACEAddSubgroupGenerators(
wordlist ) F
add, for the ith (or default) interactive ACE process started by
ACEStart
, the words in the list wordlist to any subgroup
generators already present, and automatically invoke ACERedo
, if it
can be applied, or otherwise ACEStart
. Note that ACE sorts the
resultant subgroup generator list, unless the asis
option
(see option asis) has been set to 1; don't assume, unless asis =
1
, that the new subgroup generators have been appended in
userprovided order to the previously existing subgroup generator
list. ACEAddSubgroupGenerators
also returns the new subgroup
generator list. Use ACESubgroupGenerators
(see ACESubgroupGenerators) to determine the current subgroup
generator list.
ACEDeleteRelators(
i,
list ) F
ACEDeleteRelators(
list ) F
for the ith (or default) interactive ACE process started by
ACEStart
, delete list from the current relators, if list is a list
of words in the group generators, or those current relators indexed by
the integers in list, if list is a list of positive integers, and
automatically invoke ACEStart
. ACEDeleteRelators
also returns the
new relator list. Use ACERelators
(see ACERelators) to determine
the current relator list.
ACEDeleteSubgroupGenerators(
i,
list ) F
ACEDeleteSubgroupGenerators(
list ) F
for the ith (or default) interactive ACE process started by
ACEStart
, delete list from the current subgroup generators, if
list is a list of words in the group generators, or those current
subgroup generators indexed by the integers in list, if list is a
list of positive integers, and automatically invoke ACEStart
.
ACEDeleteSubgroupGenerators
also returns the new subgroup generator
list. Use ACESubgroupGenerators
(see ACESubgroupGenerators) to
determine the current subgroup generator list.
ACECosetCoincidence(
i,
n ) F
ACECosetCoincidence(
n ) F
for the ith (or default) interactive ACE process started by
ACEStart
, return the representative of coset n, where n must be
a positive integer, and add it to the subgroup generators; i.e.,
equates this coset with coset 1, the subgroup. ACERedo
is
automatically invoked.
ACERandomCoincidences(
i,
subindex ) F
ACERandomCoincidences(
subindex ) F
ACERandomCoincidences(
i, [
subindex] ) F
ACERandomCoincidences( [
subindex] ) F
ACERandomCoincidences(
i, [
subindex,
attempts] ) F
ACERandomCoincidences( [
subindex,
attempts] ) F
for the ith (or default) interactive ACE process started by
ACEStart
, attempt up to attempts (or, in the first four forms, 8)
times to find nontrivial subgroups with index a multiple of subindex
by repeatedly making random coset numbers coincident with coset 1 and
seeing what happens. The starting coset table must be nonempty, but
must not be complete (use ACERandomlyApplyCosetCoincidence
(see ACERandomlyApplyCosetCoincidence) if your table is already
complete). For each attempt, by applying ACE's rc
option
(see option rc) random coset representatives are repeatedly added to
the subgroup and the enumeration redo
ne. If the table becomes too
small, the attempt is aborted, the original subgroup generators
restored, and another attempt made. If an attempt succeeds, then the
new set of subgroup generators is retained. ACERandomCoincidences
returns the list of new subgroup generators added. Use
ACESubgroupGenerators
(see ACESubgroupGenerators) to determine the
current subgroup generator list.
Notes:
ACERandomCoincidences
may add subgroup generators even if it failed
to determine a nontrivial subgroup with index a multiple of
subindex; in such a case, the original status may be restored by
applying ACEDeleteSubgroupGenerators
(see ACEDeleteSubgroupGenerators) with the list returned by
ACERandomCoincidences
.
ACERandomCoincidences
applies the rc
option (see option rc) of
ACE which takes the line that if an enumeration has already
obtained a finite index then either, subindex is already a divisor
of that finite index, or the request is impossible. Thus an invocation
of ACERandomCoincidences
, in the case where the coset table is
already complete, is an error.
Guru Notes: A coset can have many different representatives.
Consider running ACEStandardCosetNumbering
(see ACEStandardCosetNumbering) before ACERandomCoincidences
, to
canonicise the table and the representatives.
ACERandomlyApplyCosetCoincidence(
i [:
controlOptions]) F
ACERandomlyApplyCosetCoincidence( [:
controlOptions]) F
for the ith (or default) interactive ACE process started by
ACEStart
, try to find a larger proper subgroup (i.e. a subgroup of
smaller but nontrivial index), by repeatedly applying
ACECosetCoincidence
(see ACECosetCoincidence) and seeing what
happens; ACERandomlyApplyCosetCoincidence
returns the (possibly
empty) list of new subgroup generators added. The starting coset table
must already be complete (use ACERandomCoincidences
(see ACERandomCoincidences) if your table is not already complete).
ACERandomlyApplyCosetCoincidence
provides the following four options
(controlOptions).
subindex :=
subindex
hibound :=
hibound
lobound :=
lobound
attempts :=
attempts
ACECosetCoincidence
should be at most attempts; attempts must be
a positive integer.
By default, subindex
= 1
, hibound is the existing subgroup
index, lobound
= 1
and attempts
= 8
. If after an attempt the
new index is a multiple of subindex, less than hibound and greater
than lobound then the process terminates (and the list of new
subgroup generators is returned). Otherwise, if an attempt reaches a
stage where the criteria cannot be satisfied, the attempt is aborted,
the original subgroup generators restored, and another attempt made.
If no attempt is successful an empty list is returned. Use
ACESubgroupGenerators
(see ACESubgroupGenerators) to determine the
current subgroup generator list.
ACEConjugatesForSubgroupNormalClosure(
i ) F
ACEConjugatesForSubgroupNormalClosure() F
ACEConjugatesForSubgroupNormalClosure(
i : add ) F
ACEConjugatesForSubgroupNormalClosure(: add ) F
for the ith (or default) interactive ACE process started by
ACEStart
, test each conjugate of a subgroup generator by a group
generator for membership in the subgroup, and return the (possibly
empty) list of conjugates that were determined to not belong to the
subgroup (coset 1); and, if called with the add
option, these
conjugates are also added to the existing list of subgroup generators.
Notes: A conjugate of a subgroup generator is tested for membership
of the subgroup, by checking whether it can be traced from coset 1 to
coset 1 (see ACETraceWord
: ACETraceWord). For an incomplete
coset table, such a trace may not complete, in which case
ACEConjugatesForSubgroupNormalClosure
may return an empty list even
though the subgroup is not normally closed within the group.
The add
option does not guarantee that the resultant subgroup is
normally closed. It is still possible that some conjugates of the
newly added subgroup generators will not be elements of the subgroup.
Example:
To demonstrate the usage and features of
ACEConjugatesForSubgroupNormalClosure
, let us consider an example
where we know pretty well what to expect.
Let G be the group, isomorphic to the symmetric group S_{6}, with
the presentation

First we start an enumeration with max
set to 80.
gap> F := FreeGroup( "a", "b" );; gap> a := F.1;; b := F.2;; gap> fgens := GeneratorsOfGroup( F );; gap> rels := [ a^2, b^6, (a*b^1*a*b)^3, (a*b^1*a*b^2)^4, (a*b)^5, > (a*b^2*a*b^2)^2 ];; gap> sgens := [ a*b^3 ];; gap> i := ACEStart( fgens, rels, sgens : max := 80 );; gap> IsCompleteACECosetTable( i ); false gap> ACEConjugatesForSubgroupNormalClosure( i : add ); #I ACEConjugatesForSubgroupNormalClosure: All (traceable) conjugates in subgp [ ]
Though we know that H is not equal to its normal closure, we did not
get any new elements (we had warned above of such a possibility).
Apparently our incomplete table is too small. So let us increase max
to 100 and continue.
gap> ACEContinue( i : max := 100 );; gap> IsCompleteACECosetTable( i ); false gap> ACEConjugatesForSubgroupNormalClosure( i : add ); [ b^1*a*b^4 ] gap> IsCompleteACECosetTable( i ); true gap> ACEStats( i ).index; 20
This time we got a new element, and after adding it to the subgroup
generators we obtained a complete table. However the resulting
subgroup need not yet be the normal closure of H (and in fact we
know that it is not). So we continue with another call to the function
ACEConjugatesForSubgroupNormalClosure
.
gap> ACEConjugatesForSubgroupNormalClosure( i : add ); [ b^2*a*b^5, b*a*b^2 ] gap> ACEStats( i ).index; 2
Now we have the index that we expected. Another call to the function
ACEConjugatesForSubgroupNormalClosure
should not yield any more
conjugates. We ensure that this is indeed the case and then display
the resulting list of subgroup generators.
gap> ACEConjugatesForSubgroupNormalClosure( i : add ); #I ACEConjugatesForSubgroupNormalClosure: All (traceable) conjugates in subgp [ ] gap> ACESubgroupGenerators( i ); [ a*b^3, b*a*b^2, b^1*a*b^4, b^2*a*b^5 ]
For those familiar with the workings of the ACE standalone we
provide primitive read/write tools to communicate directly with an
interactive ACE process, started via ACEStart
(possibly with
argument 0, but this is not essential). For the most part, it is up to
the user to translate the output strings from ACE into a form
useful in GAP. However, after the group generators, relators, and
subgroup generators have been set in the ACE process, via
ACEWrite
, invocations of any of ACEGroupGenerators
(see ACEGroupGenerators), ACERelators
(see ACERelators),
ACESubgroupGenerators
(see ACESubgroupGenerators), or
ACEParameters
(see ACEParameters) will establish the corresponding
GAP values. Be warned though, that unless one of the modes
ACEStart
(without a zero argument; see ACEStart), ACERedo
(see ACERedo) or ACEContinue
(see ACEContinue), or their
equivalent for the standalone ACE (start;
, redo;
, or
continue;
), has been invoked since the last change of any parameter
options (see ACEParameterOptions), some of the values reported by
ACEParameters
may well be incorrect.
ACEWrite(
i,
string ) F
ACEWrite(
string ) F
write string to the ith or default interactive ACE process;
string must be in exactly the form the ACE standalone expects.
The command is echoed via Info
at InfoACE
level 4 (with a
``ToACE>
'' prompt); i.e. SetInfoACELevel(4);
will allow you to
see what is transmitted to the ACE binary. ACEWrite
returns true
if successful in writing to the stream of the interactive ACE process,
and fail
otherwise.
Note:
If ACEWrite
returns fail
(which means that the ACE process has
died), you may like to try resurrecting the interactive ACE process
via ACEResurrectProcess
(see ACEResurrectProcess).
ACERead(
i ) F
ACERead() F
read a complete line of ACE output, from the ith or default
interactive ACE process, if there is output to be read and returns
fail
otherwise. When successful, the line is returned as a string
complete with trailing newline character. Please note that it is
possible to be ``too quick'' (i.e. the return can be fail
purely
because the output from ACE is not there yet), but if ACERead
finds any output at all, it waits for a complete line.
ACEReadAll(
i ) F
ACEReadAll() F
read and return as many complete lines of ACE output, from the
ith or default interactive ACE process, as there are to be read,
at the time of the call, as a list of strings with the trailing
newlines removed and returns the empty list otherwise. ACEReadAll
also writes each line read via Info
at InfoACE
level 3. Whenever
ACEReadAll
finds only a partial line, it waits for the complete
line, thus increasing the probability that it has captured all the
output to be had from ACE.
ACEReadUntil(
i,
IsMyLine ) F
ACEReadUntil(
IsMyLine ) F
ACEReadUntil(
i,
IsMyLine,
Modify ) F
ACEReadUntil(
IsMyLine,
Modify ) F
read complete lines of ACE output, from the ith or default
interactive ACE process, ``chomps'' them (i.e. removes any trailing
newline character), emits them to Info
at InfoACE
level 3, and
applies the function Modify (where Modify is just the identity
map/function for the first two forms) until a ``chomped'' line line
for which IsMyLine
(
Modify(
line) )
is true. ACEReadUntil
returns the list of Modifyed ``chomped'' lines read.
Notes: When provided by the user, Modify should be a function that accepts a single string argument.
IsMyLine should be a function that is able to accept the output of Modify (or take a single string argument when Modify is not provided) and should return a boolean.
If IsMyLine
(
Modify(
line) )
is never true, ACEReadUntil
will
wait indefinitely.
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