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# D Other ACE Options

### Sections

1. Experimentation Options
2. Options that Modify a Presentation
3. Mode Options
4. Options that Interact with the Operating System
5. Query Options
6. Options that Modify the Coset Table
7. Options for Comments

Here we list all the known ACE options not provided earlier. Most of the options provided here have interactive function alternatives (each such alternative is noted at the end of the section describing the corresponding option and introduced by ``INTERACTIVELY, use ...''). A few options have only limited usefulness from GAP; many options, users will normally only wish to use if generating an input file, by using the option `aceinfile` (see option aceinfile). However all options here are functional, both interactively and non-interactively.

## D.1 Experimentation Options

- `aep:=`val``
  Runs the enumeration for `a`ll `e`quivalent `p`resentations; val is in the integer range 1 to 7.

  The `aep` option runs an enumeration for combinations of relator ordering, relator rotations, and relator inversions.

  The argument val is considered as a binary number. Its three bits are treated as flags, and control relator rotations (the 20 bit), relator inversions (the 21 bit) and relator orderings (the 22 bit), respectively; where 1 means ``active'' and 0 means ``inactive''. (See below for an example).

  The `aep` option first performs a ``priming run'' using the options as they stand. In particular, the `asis` and `messages` options are honoured.

  It then turns `asis` on and `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 a new ``record'' is found, the relators used and the summary result line is printed. See Appendix The Meanings of ACE's output messages for a discussion of the statistics `m` and `t`. To observe these messages either set the `InfoLevel` of `InfoACE` to 3 or non-interactively you can peruse the ACE output file (see option aceoutfile).

  Normally when a non-interactive ACE interface function is called, the option `start` (see option start), is quietly inserted after all the options provided by the user, to initiate a coset enumeration. Since the `aep` option invokes an enumeration, the quiet insertion of the `start` option is neither needed nor done, when a non-interactive ACE interface function is called with the `aep` option.

  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 `aep` option, (not that a non-interactive user cares).

  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 file:

  - the number of runs which yielded a finite index;

  - the total number of runs (excluding the priming run); and

  - the range of values observed for `m` and `t`.

  As an example (drawn from the discussion in HR99a) consider the enumeration of the 448 coset numbers of the subgroup á a2,a-1b ñ of the group

  (8,7 | 2,3) = á a,b | a8 = b7 = (ab)2 = (a-1b)3 = 1 ñ·

  There are 4!=24 relator orderings and 24=16 combinations of relator or inverted relator. Exponents are taken into account when rotating relators, so the relators given give rise to 1, 1, 2 and 2 rotations respectively, for a total of 1·1·2·2=4 combinations. So, for `aep` = 7 (resp. 3), 24·16·4=1536 (resp. 16·4=64) equivalent presentations are tested.

  Notes: There is no way to stop the `aep` option 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 time-wasting 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 search-space, consider firing up many copies of ACE, and starting each with a ``random'' equivalent presentation. Alternatively, you could use the `rep` command.

  INTERACTIVELY, use `ACEAllEquivPresentations` (see ACEAllEquivPresentations).

- `rep:=`val``
- `rep:= [`val`,` Npresentations`]`
  Run the enumeration for `r`andom `e`quivalent `p`resentations; val is in the integer range 1 to 7; Npresentations must be a positive integer.

  The `rep` (random equivalent presentations) option complements the `aep` option. It generates and tests some random equivalent presentations. The argument val acts as for `aep`. It is also possible to set the number Npresentations of random presentations used (by default, eight are used), by using the extended syntax `rep:=[`val`,`Npresentations`]`.

  The routine first turns `asis` on and `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. To observe these messages either set the `InfoLevel` of `InfoACE` to at least 3 or non-interactively you can peruse the ACE output file (see option aceoutfile).

  Normally when a non-interactive ACE interface function is called, the option `start` (see option start), is quietly inserted after all the options provided by the user, to initiate a coset enumeration. Since the `rep` option invokes an enumeration, the quiet insertion of the `start` option is neither needed nor done, when a non-interactive ACE interface function is called with the `rep` option.

  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 `rep` command completes, the presentation active is the last one generated, (not that the non-interactive user cares).

  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 `aep`, we can start up multiple copies of `aep` at random points in the search-space. Manually generating `random` equivalent presentations to serve as starting-points is tedious and error-prone. The `rep` option provides a simple solution; simply run `rep := 7` before `aep := 7`.

  INTERACTIVELY, use `ACERandomEquivPresentations` (see ACERandomEquivPresentations).

## D.2 Options that Modify a Presentation

- `group:=`grpgens``
  Defines the `group` generators; grpgens should be an integer (that is the number of generators) or a string that is the concatenation of, or a list of, single-lowercase-letter group generator names, i.e. it should be in a form suitable for the ACE binary to interpret. (Shortest abbreviation: `gr`.)

  The group generators should normally be input as one of the arguments of an ACE interface function, though this option may be useful when `ACEStart` (see ACEStart) is called with the single argument 0. This option may also be useful for re-using an interactive process for a new enumeration, rather than using `ACEQuit` to kill the process and `ACEStart` to initiate a new process. If the generators each have names that as strings are single lowercase letters, those same strings are used to represent the same generators by ACE; otherwise, ACE will represent each generator by an integer, numbered sequentially from 1.

  To convert a GAP list fgens of free group generators into a form suitable for the `group` option, use the construction: `ToACEGroupGenerators(`fgens`)` (see ToACEGroupGenerators). It is strongly recommended that users of the `group` option use this construction.

  Notes: Any use of the `group` command which actually defines generators invalidates any previous enumeration, and stays in effect until the next `group` command. Any words for the group or subgroup must be entered using the nominated generator format, and all printout will use this format. A valid set of generators is the minimum information necessary before ACE will attempt an enumeration.

  Guru Notes: The columns of the coset table are allocated in the same order as the generators are listed, insofar as this is possible, given that the first two columns must be a generator/inverse pair or a pair of involutions. The ordering of the columns can, in some cases, affect the definition sequence of cosets and impact the statistics of an enumeration.

- `relators:=`relators``
  Defines the group `relators`; relators must be a string or list of strings that the ACE binary can interpret as words in the group generators. (Shortest abbreviation: `rel`.)

  The group relators should normally be input as one of the arguments of an ACE interface function, but this option may occasionally be useful with interactive processes (see option group). If wordList is an empty list, the group is free.

  To convert a GAP list rels of relators in the free group generators fgens into a form suitable for the `relators` option, use the construction: `ToACEWords(`fgens`,` rels`)` (see ToACEWords).

- `generators:=`subgens``
  Defines the subgroup `generators`; subgens must be a string or list of strings that the ACE binary can interpret as words in the group generators. (Shortest abbreviation: `gen`.)

  The subgroup generators should normally be input as one of the arguments of an ACE interface function, but this option may occasionally be useful with interactive processes (see option group). By default, there are no subgroup generators and the subgroup is trivial. This command allows a list of subgroup generating words to be entered.

  To convert a GAP list sgens of subgroup generators in the free group generators fgens into a form suitable for the `generators` option, use the construction: `ToACEWords(`fgens`,` sgens`)` (see ToACEWords).

- `sg:=`subgens``
  Adds the words in subgens to any `s`ubgroup `g`enerators already present; subgens must be a string or list of strings that the ACE binary can interpret as words in the group generators.

  The enumeration must be (re)`start`ed or `redo`ne, it cannot be `continu`ed.

  To convert a GAP list sgens of subgroup generators in the free group generators fgens into a form suitable for the `generators` option, use the construction: `ToACEWords(`fgens`,` sgens`)` (see ToACEWords).

  INTERACTIVELY, use `ACEAddSubgroupGenerators` (see ACEAddSubgroupGenerators).

- `rl:=`relators``
  Appends the `r`elator `l`ist relators to the existing list of relators present; relators must be a string or list of strings that the ACE binary can interpret as words in the group generators.

  The enumeration must be (re)`start`ed or `redo`ne, it cannot be `continu`ed.

  To convert a GAP list rels of relators in the free group generators fgens into a form suitable for the `rl` option, use the construction: `ToACEWords(`fgens`,` rels`)` (see ToACEWords).

  INTERACTIVELY, use `ACEAddRelators` (see ACEAddRelators).

- `ds:=`list``
  `D`eletes `s`ubgroup generators; list must be a list of positive integers.

  This command allows subgroup generators to be deleted from the presentation. If the generators are numbered from 1 in the output of, say, the `sr` command (see option sr), then the generators listed in ``list`` are deleted; ``list`` must be a strictly increasing sequence.

  INTERACTIVELY, use `ACEDeleteSubgroupGenerators` (see ACEDeleteSubgroupGenerators).

- `dr:=`list``
  `D`eletes relators; list must be a list of positive integers.

  This command allows group relators to be deleted from the presentation. If the relators are numbered from 1 in the output of, say, the `sr` command (see option sr), then the relators listed in ``list`` are deleted; ``list`` must be a strictly increasing sequence.

  INTERACTIVELY, use `ACEDeleteRelators` (see ACEDeleteRelators).

- `cc:=`val``
  Makes `c`oset val `c`oincide with coset 1; val should be a positive integer.

  Prints out the representative of coset ``val``, and adds it to the subgroup generators; i.e., forces coset ``val`` to coincide with coset 1, the subgroup.

  INTERACTIVELY, use `ACECosetCoincidence` (see ACECosetCoincidence).

- `rc:=`val``
- `rc:= [`val`]`
- `rc:= [`val`,` attempts`]`
  Enforce `r`andom `c`oincidences; val and attempts must be positive integers.

  This option attempts upto attempts (or, in the first and second forms, 8) times to find nontrivial subgroups with index a multiple of val by repeatedly making random coset numbers coincident with coset 1 and seeing what happens. The starting coset table must be non-empty, but should not be complete. For each attempt, we repeatedly add random coset representatives to the subgroup and `redo` the enumeration. 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.

  Guru Notes: A coset number can have many different coset representatives. Consider running `standard` before `rc`, to canonicise the table and hence the coset representatives.

  INTERACTIVELY, use `ACERandomCoincidences` (see ACERandomCoincidences).

## D.3 Mode Options

- `mode`
  Prints the possible enumeration `mode`s. (Shortest abbreviation: `mo`.)

  Prints the possible enumeration `mode`s (i.e. which of `continu`, `redo` or `start` are possible (see option continu, option redo and option start).

  INTERACTIVELY, use `ACEModes` (see ACEModes).

- `begin`
- `start`
  Start an enumeration. (Shortest abbreviation of `begin` is `beg`.)

  Any existing information in the table is cleared, and the enumeration starts from coset 1 (i.e., the subgroup).

  Normally when a non-interactive ACE interface function is called, the option `start` (see option start), is quietly inserted after all the options provided by the user, to initiate a coset enumeration; however, this is not done, if the user herself supplies either the `begin` or `start` option.

  INTERACTIVELY, use `ACEStart` (see ACEStart).

- `check`
- `redo`
  `Redo` an extant enumeration, using the current parameters.

  As opposed to `start` (see option start), which clears an existing coset table, any existing information in the table is retained, and the enumeration is restarted from coset 1 (i.e., the subgroup).

  Notes: This option is really intended for the case where additional relators (option `rl`; see option rl) and/or subgroup generators (option `sg`; see option sg) have been introduced. The current table, which may be incomplete or exhibit a finite index, is still valid. However, the additional data may allow the enumeration to complete, or cause a collapse to a smaller index.

  INTERACTIVELY, use `ACERedo` (see ACERedo).

- `continu`
  `Continu`es the current enumeration, building upon the existing table. (Shortest abbreviation: `cont`.)

  If a previous run stopped without producing a finite index you can, in principle, change any of the parameters and `continu`e on. Of course, if you make any changes which invalidate the current table, you won't be allowed to `continue`, although you may be allowed to `redo` (see option redo). If `redo` is not allowed, you must re-`start` (see option start).

  Note: The ACE standalone allows the option `continue`, but this is (as of GAP 4.3) a GAP keyword, and so GAP users must use (mixed-case abbreviations of) `continu`.

  INTERACTIVELY, use `ACEContinue` (see ACEContinue).

## D.4 Options that Interact with the Operating System

- `ai`
- `ai:=`filename``
  `A`lter `i`nput to standard input or filename; filename must be a string.

  By default, commands to ACE are read from standard input (i.e., the keyboard). With no value `ai` causes ACE to revert to reading from standard input; otherwise, the `ai` command closes the current input file, and opens ``filename`` as the source of commands. If ``filename`` can't be opened, input reverts to standard input.

  Notes: If you switch to taking input from (another) file, remember to switch back before the end of that file; otherwise the `EOF` there will cause ACE to terminate.

- `bye`
- `exit`
- `qui`
  Quit ACE. (Shortest abbreviation of `qui` is `q`.)

  This quits ACE nicely, printing the date and the time. An `EOF` (end-of-file; i.e., `^d`) has the same effect, so proper termination occurs if ACE is taking its input from a script file.

  Note that `qui` actually abbreviates the corresponding ACE directive `quit`, but since `quit` is a GAP keyword it is not available via the GAP interface to ACE.

  INTERACTIVELY, use `ACEQuit` (see ACEQuit).

- `system:=`string``
  Does a shell escape, to execute string; string must be a string. (Shortest abbreviation: `sys`.)

  Since GAP already provides `Exec()` for this purpose, this option is unlikely to have a use.

## D.5 Query Options

- `cycles`
  Prints out the table in `cycles`. (Shortest abbreviation: `cy`.)

  This option prints out the permutation representation.

  INTERACTIVELY, use `ACECycles` (see ACECycles).

- `dump`
- `dump:=`level``
- `dump:= [`level`]`
- `dump:= [`level`,` detail`]`
  `Dump` the internal variables of ACE; level must be an integer in the range 0 to 2, and detail must be 0 or 1. (Shortest abbreviation: `d`.)

  The value of level determines which of the three levels of ACE to dump. (You will need to read the standalone manual `acce3001.dvi` in the `standalone-doc` directory 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 form (with no arguments) selects ``level`= 0,` detail`= 0`. The second form of this command makes ``detail`= 0`. This option is intended for gurus; the source code should be consulted to see what the output means.

  INTERACTIVELY, use `ACEDumpVariables` (see ACEDumpVariables).

- `help`
  Prints the ACE help screen. (Shortest abbreviation: `h`.)

  This option prints the list of options of the ACE binary. Note that this list is longer than a standard screenful.

- `nc`
- `nc:=`val``
- `normal`
- `normal:=`val``
  Check or attempt to enforce normal closure; val must be 0 or 1.

  This option tests the subgroup for normal closure within the group. If a conjugate of a subgroup generator by a generator, is determined to belong to a coset other than coset 1, it is printed out, and if ``val`= 1`, then any such conjugate is also added to the subgroup generators. With no argument or if ``val`= 0`, ACE does not add any new subgroup generators.

  Notes: The method of determination of whether a conjugate of a subgroup generator is in the subgroup, is by testing whether it can be traced from coset 1 to coset 1 (see `trace`: option trace).

  The resultant subgroup need not be normally closed after executing option `nc` with the value 1. It is still possible that some conjugates of the newly added subgroup generators will not be elements of the subgroup.

  INTERACTIVELY, use `ACEConjugatesForSubgroupNormalClosure` (see ACEConjugatesForSubgroupNormalClosure).

- `options`
  Dumps version information of the ACE binary. (Shortest abbreviation: `opt`.)

  A rather unfortunate name for an option; this command dumps details of the ``options'' included in the version of ACE when the ACE binary was compiled.

  A typical output, is as follows:

      Executable built:
        Sat Feb 27 15:57:59 EST 1999
      Level 0 options:
        statistics package = on
        coinc processing messages = on
        dedn processing messages = on
      Level 1 options:
        workspace multipliers = decimal
      Level 2 options:
        host info = on

  INTERACTIVELY and non-interactively, use the command `ACEBinaryVersion();` (see ACEBinaryVersion) for this information, instead, unless you want it in an ACE standalone input file.

- `oo:=`val``
- `order:=`val``
  Print a coset representative of a coset number with order a multiple of val modulo the subgroup; val must be an integer.

  This option finds a coset with order a multiple of `|`val`|` modulo the subgroup, and prints out its coset representative. If ``val`< 0`, then all coset numbers meeting the requirement are printed. If ``val`> 0`, then just the first coset number meeting the requirement is printed. Also, ``val`= 0` is permitted; this special value effects the printing of the orders (modulo the subgroup) of all coset numbers.

  INTERACTIVELY, use `ACEOrders` (see ACEOrders), for the case ``val`= 0`, or `ACEOrder` (see ACEOrder), otherwise.

- `sr`
- `sr:=`val``
  Print out parameters of the current presentation; val must be an integer in the range 0 to 5.

  No argument, or ``val`= 0`, prints out the `Group Name`, the group's `relators`, `Subgroup Name` and the subgroup's `generators`. If ``val`= 1`, then the `Group Generators` and the current setting of the ``run parameters'' is also printed. The printout is the same as that produced at the start of a run when option `messages` (see option messages) is non-zero. Also, val equal to 2, 3, 4, or 5 print out just the `Group Name`, just the group's `relators`, just the `Subgroup Name`, or just the subgroup's `generators`, respectively.

  Notes: The `sr` option should only be used after an enumeration run; otherwise, the value 0 for some options will be unreliable. To ensure this occurs non-interactively, ensure one of the options that invokes an enumeration: `start` (see option start) or `aep` (see option aep) or `rep` (see option rep), precedes the `sr` option. (When an enumeration-invoking option is included non-interactively the quiet inclusion step of the `start` option is omitted.)

  INTERACTIVELY, use `ACEGroupGenerators` (see ACEGroupGenerators), `ACERelators` (see ACERelators), `ACESubgroupGenerators` (see ACESubgroupGenerators), and `ACEParameters` (see ACEParameters).

- `print`
- `print:=`val``
- `print:= [`val`]`
- `print:= [`val`,` last`]`
- `print:= [`val`,` last`,` by`]`
  Compact and print the coset table; val must be an integer, and last and by must be positive integers. (Shortest abbreviation: `pr`.)

  In the first (no value) form, `print` prints the entire coset table, without orders or coset representatives. In the second and third forms, the absolute value of val is taken to be the last line of the table to be printed (and 1 is taken to be the first); in the fourth and fifth forms, `|`val`|` is taken to be the first line of the table to be printed, and last is taken to be the number of the last line to be printed. In the last form, the table is printed 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 printed also.

  INTERACTIVELY, use `ACEDisplayCosetTable` (see ACEDisplayCosetTable).

- `sc:=`val``
- `stabilising:=`val``
  Print out the coset numbers whose elements normalise the subgroup; val must be an integer. (Shortest abbreviation of `stabilising` is `stabil`.)

  If ``val`> 0`, the first ``val`` non-trivial (i.e. other than coset 1) coset numbers whose elements normalise the subgroup are printed. If ``val`= 0`, all non-trivial coset numbers whose elements normalise the subgroup, plus their representatives, are printed. If ``val`< 0`, the first `|`val`|` non-trivial coset numbers whose elements normalise the subgroup, plus their representatives, are printed.

  Note: The name of this option is an historical hangover. It is named for the property that elements that ``normalise'' a subgroup, may be said to ``stabilise'' that subgroup when they act ``by conjugation''. Also, the option `normal` (see option normal) already performs a different function.

  INTERACTIVELY, use `ACECosetsThatNormaliseSubgroup` (see ACECosetsThatNormaliseSubgroup).

- `statistics`
- `stats`
  Dump enumeration statistics. (Shortest abbreviation of `statistics` is `stat`.)

  If the statistics package is compiled into the ACE code, which it is by default (see the `options` option options option), then this option dumps the statistics accumulated during the most recent enumeration. See the `enum.c` source file for the meaning of the variables.

  INTERACTIVELY, use `ACEDumpStatistics` (see ACEDumpStatistics).

- `style`
  Prints the current enumeration style.

  This option prints the current enumeration style, as deduced from the current `ct` and `rt` parameters (see Enumeration Style).

  INTERACTIVELY, use `ACEStyle` (see ACEStyle).

- `tw:= [`val`,` word`]`
- `trace:= [`val`,` word`]`
  Trace ``word`` through the coset table, starting at coset ``val``; val must be a positive integer, and word must be a word in the group generators.

  This option prints the final coset number of the trace, if the trace completes.

  INTERACTIVELY, use `ACETraceWord` (see ACETraceWord).

## D.6 Options that Modify the Coset Table

- `recover`
- `contiguous`
  `Recover` space used by dead coset numbers. (Shortest abbreviation of `recover` is `reco`, and shortest abbreviation of `contiguous` is `contig`.)

  This option invokes the compaction routine on the table to recover the space used by any dead coset numbers. A `CO` message line is printed if any cosets were recovered, and a `co` line if none were. This routine is called automatically if the `cycles`, `nc`, `print` or `standard` options (see option cycles, option nc, option print and option standard) are invoked.

  INTERACTIVELY, use `ACERecover` (see ACERecover).

- `standard`
  Compacts ACE's coset table and standardises the numbering of cosets, according to the `lenlex` scheme (see Section Coset Table Standardisation Schemes). (Shortest abbreviation: `st`.)

  For a given ordering of the generators in the columns of the table, it produces a canonical numbering of the cosets. This function does not display the new table; use the `print` (see option print) for that. Such a table has the property that 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.

  Notes: In a `lenlex` standard 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, since ACE avoids having an involutory generator in the first column when it can, this lexicographic order does not necessarily correspond with the order in which the generators were first put to ACE. Two tables are equivalent only if their canonic forms are the same. Invoking this option directly does not affect the GAP coset table obtained via `ACECosetTable`; use the `lenlex` (see option lenlex) option, if you want your table `lenlex` standardised. (The `lenlex` option restarts the enumeration, if it is necessary to ensure the generators have not been rearranged.)

  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 interactive example).

  INTERACTIVELY, use `ACEStandardCosetNumbering` (see ACEStandardCosetNumbering).

## D.7 Options for Comments

- `text:=`string``
  Prints string in the output; string must be a string.

  This allows the user to add comments to the output from ACE.

  Note: Please avoid using this option to insert comments starting with three asterisks: `***`, since this string is used as a sentinel internally in flushing output from ACE.

- `aceincomment:=`string``
  Prints comment string in the ACE input; string must be a string. (Shortest abbreviation: `aceinc`.)

  This option prints the comment string behind a sharp sign (`#`) in the input to ACE. Only useful for adding comments (that ACE ignores) to standalone input files.

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