Throughout this manual for the use of ACE as a GAP package, we shall assume that the reader already knows the basic ideas of coset enumeration, as can be found for example in Neu82. There, a simple proof is given for the fact that a coset enumeration for a subgroup of finite index in a finitely presented group must eventually terminate with the correct result, provided the enumeration process obeys a simple condition (Mendelsohn's condition) formulated in Lemma 1 and Theorem 2 of Neu82. This basic condition leaves room for a great variety of strategies for coset enumeration; two ``classical'' ones have been known for a long time as the Felsch strategy and the HLT strategy (for Haselgrove, Leech and Trotter). Extensive experimental studies on many strategies can be found in CDHW73, Hav91, HR99ace, and HR01, in particular.
A few basic points should be particularly understood:
It will also be necessary to understand some further basic features of the implementation and the corresponding terminology which we will explain in the sequel.
The first main decision for any coset enumeration is in which sequence to make definitions. When a new coset number has to be defined, in ACE there are basically three possible methods to choose from:
The enumeration style is mainly determined by the balance between
C style and R style definitions, which is controlled by the values of
the ct and rt options (see option ct and option rt).
However this still leaves us with plenty of freedom for the design of definition strategies, freedom which can, for example, be used to great advantage in Felsch-type strategies. Though it is not strictly necessary, before embarking on further enumeration, Felsch-type programs generally start off by ensuring that each of the given subgroup generators produces a cycle of coset numbers at coset 1. To explain the idea, an example may help. Suppose a,b are the group generators and w=Abab is a subgroup generator, where A represents the inverse of a; then to say that ``(1,i,j,k) is a cycle of coset numbers produced at coset 1 by w'' means that the successive application of the ``letters'' A,b,a,b of w takes one successively from coset 1, through cosets i, j and k, and back to coset 1, i.e. A applied to coset 1 results in coset i, b applied to coset i results in coset j, a applied to coset j results in coset k, and finally b applied to coset k takes us back to coset 1. In this way, a hypothetical subgroup table is filled first. The use of this and other possibilities leads to the following table of enumeration styles.
Rt value Ct value style name ----------------------------------------- 0 >0 C <0 >0 Cr >0 >0 CR >0 0 R <0 0 R* >0 <0 Rc <0 <0 R/C 0 0 R/C (defaulted) -----------------------------------------
However, in C style, some definitions may be made following a
preferred definition strategy, controlled by the pmode and psize
options (see option pmode and option psize).
ct set to 1000 and rt set to
approximately 2000 divided by the total length of the relators in an
attempt to balance R style and C style definitions when we switch to
CR style.
First, let us broadly discuss strategies and how they influence
``definitions''. By definition we mean the allocation of a coset
number. In a complete coset table each group relator produces a cycle
of cosets numbers at each coset number, in particular, at coset 1;
i.e. for each relator w, and for each coset number i, successive
application of the letters of w trace through a sequence of coset
numbers that begins and ends in i (see Section Enumeration Style
for an example). It has been found to be a good general rule to use
the given group relators as subgroup generators. This ensures the
early definition of some useful coset numbers, and is the basis of the
default strategy (see option default). The number of group
relators included as subgroup generators is determined by the no
option (see option no). Over a wide range of examples the use of
group relators in this way has been shown to produce generally
beneficial results in terms of the maximum number of cosets numbers
defined at any one time and the total number of coset numbers defined.
In CDHW73, it was reported that for some Macdonald group
G(α,β) examples, (pure) Felsch-type strategies (that don't
include the given group relators as subgroup generators) e.g. the
felsch := 0 strategy (see option felsch) defined significantly
more coset numbers than HLT-type (e.g. the hlt strategy, see option hlt) strategies. The comparison of these strategies in terms of total
number of coset numbers defined, in Hav91, for the enumeration
of the cosets of a certain index 40 subgroup of the G(3,21)
Macdonald group were 91 for HLT versus 16067 for a pure Felsch-type
strategy. For the Felsch strategy with the group relators included as
subgroup generators, as for the felsch := 1 strategy (see option felsch) the total number of coset numbers defined reduced markedly to
59.
A deduction occurs when the scanning of a relator results in the
assignment of a coset table body entry. A completed table is only
valid if every table entry has been tested in all essentially
different positions in all relators. This testing can either be done
directly (Felsch strategy) or via relator scanning (HLT strategy). If
it is done directly, then more than one deduction can be waiting to be
processed at any one time. The untested deductions are stored in a
stack. How this stack is managed is determined by the dmode option
(see option dmode), and its size is controlled by the dsize option
(see option dsize).
As already mentioned a coincidence occurs when it is determined that
two coset numbers in fact represent the same coset. When this occurs
the larger coset number becomes a dead coset number and the
coincidence is placed in a queue. When and how these dead coset
numbers are eventually eliminated is controlled by the options
dmode, path and compaction (see option dmode, option path
and option compaction). The user may also force coincidences to
occur (see Section Finding Subgroups), which, however, may change
the subgroup whose cosets are enumerated.
The key to performance of coset enumeration procedures is good
selection of the next coset number to be defined. Leech
in Lee77 and Lee84 showed how a number of coset
enumerations could be simplified by removing coset numbers needlessly
defined by computer implementations. Human enumerators intelligently
choose which coset number should be defined next, based on the value
of each potential definition. In particular, definitions which close
relator cycles (or at least shorten gaps in cycles) are favoured. A
definition which actually closes a relator cycle immediately yields
twice as many table entries (deductions) as other definitions. The
value of the pmode option (see option pmode) determines which
definitions are preferred; if the value of the pmode option is
non-zero, depending on the pmode value, gaps of length one found
during relator C style (i.e. Felsch-like) scans are either filled
immediately (subject to the value of fill) or noted in the
preferred definition stack. The preferred definition stack is
implemented as a ring of size determined by the psize option
(see option psize). However, making preferred definitions carelessly
can violate the conditions required for guaranteed termination of the
coset enumeration procedure in the case of finite index. To avoid such
a violation ACE ensures a fraction of the coset table is filled
before a preferred definition is made; the reciprocal of this
fraction, the fill factor, is manipulated via the fill option
(see option fill). In Hav91, the felsch := 1 type
enumeration of the cosets of the certain index 40 subgroup of the
G(3,21) Macdonald group was further improved to require a total
number of coset numbers of just 43 by incorporating the use of
preferred definitions.
The ACE package, via its interactive ACE interface functions
(described in Chapter Functions for Using ACE Interactively),
provides the possibility of searching for subgroups. To do this one
starts at a known subgroup (possibly the trivial subgroup). Then one
may augment it by adding new subgroup generators either explicitly via
ACEAddSubgroupGenerators (see ACEAddSubgroupGenerators) or
implicitly by introducing coincidences (see ACECosetCoincidence:
ACECosetCoincidence, or ACERandomCoincidences:
ACERandomCoincidences). Also, one may descend to smaller subgroups
by deleting subgroup generators via ACEDeleteSubgroupGenerators
(see ACEDeleteSubgroupGenerators).
The default standardisation scheme for GAP and the
standardisation scheme provided by ACE is the lenlex scheme, of
Charles Sims Sim94. This scheme numbers cosets first according
to word-length and then according to a lexical ordering of coset
representatives. Each coset representative is a word in an alphabet
consisting of the user-supplied generators and their inverses, and the
lexical ordering of lenlex is that implied by ordering that alphabet
so that each generator is followed by its inverse, and the generators
appear in user-supplied order. See below for an example which gives
the first 20 lines of the lenlex standard coset table of the
(infinite) group with presentation 〈x, y, a, b | x2, y3, a4, b2〉.
In the table each inverse of a generator is represented by the corresponding uppercase letter (X represents the inverse of x etc.), and the lexical ordering of the representatives is that implied by defining an ordering of the alphabet of user-supplied generators and their inverses to be x < X < y < Y < a < A < b < B.
A lenlex standard coset table whose columns correspond, in order, to
the already-described alphabet, of generators and their inverses, has
an important property: a scan of the body of the table row by row from
left to right, encounters new coset numbers in numeric order. Observe
that the table below has this property: the definition of coset 1 is
implicit; the first coset number we encounter in the table body is 2,
then 2 again, 3, 4, 5, 6, 7, then 7 again, etc.
With the lenlex option (see option lenlex), the coset table output
by ACECosetTable or ACECosetTableFromGensAndRels is standardised
according to the lenlex scheme.
coset no. | x X y Y a A b B rep've
-----------+------------------------------------------------------------------
1 | 2 2 3 4 5 6 7 7
2 | 1 1 8 9 10 11 12 12 x
3 | 13 13 4 1 14 15 16 16 y
4 | 17 17 1 3 18 19 20 20 Y
5 | 21 21 22 23 24 1 25 25 a
6 | 26 26 27 28 1 24 29 29 A
7 | 30 30 31 32 33 34 1 1 b
8 | 35 35 9 2 36 37 38 38 xy
9 | 39 39 2 8 40 41 42 42 xY
10 | 43 43 44 45 46 2 47 47 xa
11 | 48 48 49 50 2 46 51 51 xA
12 | 52 52 53 54 55 56 2 2 xb
13 | 3 3 57 58 59 60 61 61 yx
14 | 62 62 63 64 65 3 66 66 ya
15 | 67 67 68 69 3 65 70 70 yA
16 | 71 71 72 73 74 75 3 3 yb
17 | 4 4 76 77 78 79 80 80 Yx
18 | 81 81 82 83 84 4 85 85 Ya
19 | 86 86 87 88 4 84 89 89 YA
20 | 90 90 91 92 93 94 4 4 Yb
Another standardisation scheme for coset tables numbers cosets according to
coset representative word-length in the group generators and lexical
ordering imposed by the user-supplied ordering of the group
generators; it is known as semilenlex since though like lenlex,
generator inverses do not feature. Here again is 20 lines of the coset
table of the group with presentation 〈x, y, a, b | x2, y3, a4, b2〉, this time semilenlex standardised.
coset no. | x y a b rep've
-----------+--------------------------------------
1 | 2 3 4 5
2 | 1 6 7 8 x
3 | 9 10 11 12 y
4 | 13 14 15 16 a
5 | 17 18 19 1 b
6 | 20 21 22 23 xy
7 | 24 25 2 26 xa
8 | 27 28 29 2 xb
9 | 3 30 31 32 yx
10 | 33 1 34 35 yy
11 | 36 37 38 39 ya
12 | 40 41 42 3 yb
13 | 4 43 44 45 ax
14 | 46 47 48 49 ay
15 | 50 51 52 53 aa
16 | 54 55 56 4 ab
17 | 5 57 58 59 bx
18 | 60 61 62 63 by
19 | 64 65 66 67 ba
20 | 6 68 69 70 xyx
The term semilenlex was coined by Edmund Robertson and Joachim
Neubüser, for the scheme's applicability to semigroups
where generator inverses need not exist. This scheme ensures that as
one scans the columns corresponding to the group generators (in
user-supplied order) row by row, one encounters new coset numbers in
numeric order.
Observe that the representatives are ordered according to length and then the lexical ordering implied by defining x < y < a < b (with some words omitted due to their equivalence to words that precede them in the ordering). Also observe that as one scans the body of the table row by row from left to right new coset numbers appear in numeric order without gaps (2, 3, 4, 5, then 1 which we have implicitly already seen, 6, 7, etc.).
There are three statistics involved in the counting of coset number
definitions: activecosets, maxcosets and totcosets; these are
three of the fields of the record returned by ACEStats (see
Section Using ACE Directly to Test whether a Coset Enumeration Terminates), and they correspond to the a, m and t statistics
of an ACE ``results message'' (see Appendix The Meanings of ACE's Output Messages). As already described, coset enumeration proceeds by
defining coset numbers; totcosets counts all such definitions made
during an enumeration. During the coset enumeration process,
coincidences usually occur, resulting in the larger of each
coincident pair becoming a dead coset number. The statistic
activecosets is the count of coset numbers left alive (i.e. not
dead) at the end of an enumeration; and maxcosets is the maximum
number of alive cosets at any point of an enumeration.
In practice, the statistics maxcosets and totcosets tend to be of
a similar order, though, of course, maxcosets can never be more than
totcosets.
In various places in this manual, we will refer to a (main) loop or
a pass through such a loop. We don't intend to give a precise
meaning to these terms. The reader will need to forgive us for giving
somewhat circular definitions in our attempt to make these terms less
nebulous. It is sufficient to appreciate that the ACE enumerator is
organised as a state machine, where each state is a value of the
coset table held internally by ACE at the end of each ``main
loop''. Each step from one state to the next (i.e. each passage
through the main loop) performs an ``action'' (i.e., lookahead,
compaction; see option lookahead and option compaction) or a
block of actions (i.e., |ct| coset number definitions, |rt| coset
number applications). ACE counts the number of passes through the
main loop; if the option loop (see option loop) is set to a
positive integer, ACE makes an early return when the loop count
hits the value of loop.
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