KBMAG (pronounced ``Kaybeemag'') stands for KnuthBendix on
Monoids, and Automatic Groups. It is a standalone package written
in C
, for use under UNIX, with an interface to GAP. This chapter
describes its use as an external library from within GAP.
There are interfaces for the use of KBMAG with finitely
presented groups, monoids and semigroups defined within GAP.
The package also contains a collection of routines for manipulating
finite state automata, which can be accessed via the GAP
interface.
To use this package effectively, some knowledge of the underlying theory and algorithms is advisable. The KnuthBendix algorithm is described in various places in the literature. Good general references that deal with the applications to groups and monoids are LeC86 and the first few chapters of Sims94. For the theory of automatic groups see the multiauthor book ECHLPT92. The algorithms employed by KBMAG are described more specifically in EHR91 and Holt94.
The manual for the standalone KBMAG package (which can be found in
the standalone/doc
directory of the package) provides more detailed
information on the external C
programs that are called from GAP.
Suppose that G is a finitely presented semigroup, monoid or group defined as a quotient of the free structure F. The overall objective of KBMAG is to construct a normal form for the elements of G in terms of the generators of F, together with a word reduction algorithm for calculating the normal form representative of an element in G, given by a word in the generators of F. If this can be achieved, then it is also possible to enumerate the words in normal form up to a given length, and to determine the order of G, by counting the number of words in normal form. In most serious applications, this will be infinite, since (for example) finite groups are (with some exceptions) usually handled better by ToddCoxeter related methods. In fact a finite state automaton W is calculated that accepts precisely the language of words in the monoid generators of F that are in normal form, and W is used for the enumeration and counting functions.
The normal form of an element g Î G is defined to be the least word in the generators of F (and their inverses) that represents g, with respect to a specified ordering on the set of all words in the generators of F. The available orderings are described in Setting the ordering below.
KBMAG offers two possible means of achieving these objectives. The first is to apply the KnuthBendix algorithm to the presentation, with one of the available orderings on words, and hope that the algorithm will complete with a finite confluent presentation. (If G is finite, then it is guaranteed to complete eventually but, like the ToddCoxeter procedure, it may take a long time, or require more space than is available.) The second is to use the automatic group program, which is only applicable to groups (not to monoids or semigroups). This also uses the KnuthBendix procedure as one component of the algorithm, but it aims to compute certain finite state automata rather than to obtain a finite confluent rewriting system, and it completes successfully on many examples for which such a finite system does not exist. In the current standalone implementation, its use is restricted to the ``shortlex" ordering on words. That is, words are ordered first by increasing length, and then words of equal length are ordered lexicographically, using the specified ordering of the generators. However, there are now some GAP procedures available in the package written by Sarah Rees that enable it be used also for the ``wtlex" ordering, and the ``wreathprod" ordering. See below for further details of these orderings.
For both of the above procedures, the first step is to create a GAP object known as a KnuthBendix rewriting systemR from the finitely presented structure G. There are functions available that can be used to specify the input parameters for the external programs, such as the ordering on words to be used by the KnuthBendix procedure. One of the two external programs is then run on R. If successful, it updates R, which can then be used to reduce words in the generators of F to normal form, and to count and enumerate the words in normal form.
There are also now some routines available for performing corresponding operations with the cosets of a specified subgroup H of the group G. (These are not currently available for semigroups or monoids.) The words in normal form then correspond to minimal representatives under the ordering of the system of the right cosets of H in G. If successful, the index of H in G can be determined. The KnuthBendix routines also allow a confluent rewriting system for H to be computed, whereas the automatic groups routines allow a presentation of H to be computed (although not yet on a userspecified generating set).
In the descriptions of the functions that follow, it is important to distinguish between irreducible words, and words in normal form. As already stated, a word is in normal form if it is the least word under the ordering of the rewriting system that defines a particular group element or coset. So there is always a unique word in normal form for each group element or coset, and it is determined by the group generators and the ordering on words in the group generators. A word in a rewriting system is said to be irreducible if it does not contain the left hand side of any of the reduction rules in the system as a subword. Words in normal form are always irreducible, but the converse is true if and only if the rewriting system is confluent. The automatic groups programs provide a method of reducing words to normal form without obtaining a finite confluent rewriting system (which may not even exist).
Various levels of diagnostic output from the GAP procedures can
be turned on by
setting the Info variable InfoRWS
to 1, 2 or 3.
First the user should be aware of a technicality. The words in a
rewriting system created in GAP for use by KBMAG are defined over an
alphabet that consists of the generators of a free monoid, called the
wordmonoid
of the system. Suppose, as before, that the rewriting system
is defined from the semigroup, monoid or group G which is a quotient of
the free structure F. Then the generators of this alphabet will be in
oneone correspondence with the generators (or, when G is a group, the
generators and their inverses) of F, but will not be identical to them.
This feature was necessary for technical reasons. Most of the userlevel
functions take and return words in F rather than the alphabet, but
they do this by converting from one to the other and back.
Userlevel functions have also been provided to carry out this conversion explicitly if required.
The user should also be aware of a peculiarity in the way that rewriting sytems are displayed, which is really a hangover from the GAP3 interface. They are displayed nicely as a record, which gives a useful description of the system, but it does not correspond at all to the way that they are actually stored internally!
KBMAGRewritingSystem(
G ) O
constructs and returns a rewriting system R from a finitely presented semigroup, monoid or group G. When G is a group, the alphabet members of R correspond to the generators of F together with inverses for those generators which are not obviously involutory in G.
IsKBMAGRewritingSystemRep(
rws ) R
Returns true if rws is a rewriting system created by
KBMAGRewritingSystem
. The functions IsRewritingSystem
and
IsKnuthBendixRewritingSystem
will also return true on
such a system.
IsConfluent(
rws ) A
Returns true if rws is a rewriting system that is known to be confluent.
SemigroupOfRewritingSytem(
rws ) O
FreeStructureOfSystem(
rws ) O
WordMonoidOfRewritingSystem(
rws ) O
These return, respectively, the semigroup, monoid or group G, the free structure F, and the wordmonoid of the rewriting system, as defined in the preceding section.
ExternalWordToInternalWordOfRewritingSystem(
rws,
w ) F
InternalWordToExternalWordOfRewritingSystem(
rws,
w ) F
These are the functions for converting between external words, which are those defined over the free structure F of rws, and the internal words, which are defined over the wordmonoid of rws.
Alphabet(
rws )
This is an ordered list of the generators of the wordmonoid of rws.
It will not necessarily be in the normal order of these generators,
and it can be reordered by the function
ReorderAlphabetOfKBMAGRewritingSystem
(see below).
Rules(
rws ) O
A list of the reduction rules of rws. Each rule is a twoelement list containing the left and right hand sides of the rule, which are words in the alphabet of rws.
ResetRewritingSystem(
rws ) F
This function resets the rewriting system rws back to its form as it
was before the application of KnuthBendix
or AutomaticStructure
.
However, the current ordering and values of control parameters will not
be changed. The normal form and reduction algorithms will be unavailable
after this call.
SetOrderingOfKBMAGRewritingSystem(
rws,
ordering [,
list] ) F
ReorderAlphabetOfKBMAGRewritingSystem(
rws,
p ) F
OrderingOfKBMAGRewritingSystem(
rws ) F
OrderingOfRewritingSystem(
rws ) F
SetOrderingOfKBMAGRewritingSystem
changes the ordering on the words
of the rewriting system rws to ordering.
rws is reset when the ordering is changed, so any previously
calculated results will be destroyed.
ordering must be one of the strings ``shortlex",
``recursive", ``wtlex" and ``wreathprod". The
default is ``shortlex", and this is the ordering of rewriting systems
returned by KBMAGRewritingSystem
. The orderings ``wtlex"
and ``wreathprod" require the third parameter, list, which must be a
list of positive integers in oneone correspondence with the
alphabet of rws in its current order.
They have the effect of attaching weights or
levels to the alphabet members, in the cases ``wtlex" and ``wreathprod",
respectively.
Each of these orderings depends on the order of the alphabet, The
current ordering of generators is displayed under the generatorOrder
field when rws is viewed. This ordering can be changed by the
function 'ReorderAlphabetOfKBMAGRewritingSystem'.
The second parameter p to this
function should be a permutation that moves at most ng points, where
ng is the number of generators. This permutation is applied to the
current list of generators.
OrderingOfKBMAGRewritingSystem
merely prints out a description of
the current ordering.
In the ``shortlex" ordering, shorter words come before longer ones, and, for words of equal length, the lexicographically smaller word comes first, using the ordering of the alphabet. The ``wtlex" ordering is similar, but instead of using the length of the word as the first criterion, the total weight of the word is used; this is defined as the sum of the weights of the generators in the word. So ``shortlex" is the special case of ``wtlex" in which all generators have the same nonzero weight.
The ``recursive" ordering is the special case of ``wreathprod" in which the levels of the ng generators are 1, 2, ¼, ng, in the order of the alphabet. We shall not attempt to give a complete definition of these orderings here, but refer the reader instead to pages 4650 of Sims94. The ``recursive" ordering is the one appropriate for a powerconjugate presentation of a polycyclic group, but where the generators are ordered in the reverse order from the usual convention for polycyclic groups. The confluent presentation will then be the same as the powerconjugate presentation. For example, for the Heisenberg group áx,y,z  [x,z]=[y,z]=1, [y,x]=z ñ, a good ordering is ``recursive" with the order of generators [z^{1},z,y^{1},y,x^{1},x]. This example is included in Rewriting System Examples below.
Finally, OrderingOfRewritingSystem
returns the appropriate GAP ordering
on the elements of the wordmonoid of rws. The standard GAP ordering
functions, such as IsLessThanUnder(
ord,
el1,
el2)
can then be used.
InfoRWS V
This Info
variable can be set to 0, 1, 2 or 3 to control the level
of diagnostic output.
The KnuthBendix procedure is unusually sensitive to the settings of a number of parameters that control its operation. In some examples, a small change in one of these parameters can mean the difference between obtaining a confluent rewriting system fairly quickly on the one hand, and the procedure running on until it uses all available memory on the other hand.
Unfortunately, it is almost impossible to give even very general guidelines on these settings, although the ``wreathprod" orderings appear to be more sensitive than the ``shortlex" and ``wtlex" orderings. The user can only acquire a feeling for the influence of these parameters by experimentation on a large number of examples.
The control parameters are defined by the user by setting values of certain fields of the options record of a rewriting system.
OptionsRecordOfKBMAGRewritingSystem(
rws ) F
Returns the options record OR of the rewriting system rws. The fields of OR listed below can be set by the user. Be careful to spell them correctly, because otherwise they will have no effect!
OR.maxeqns
: KnuthBendix
or AutomaticStructure
will abort.
OR.tidyint
:
OR.tidyint
rules since the last
tidying.
OR.confnum
:
OR.confnum
overlaps are processed in the KnuthBendix procedure but no
new rules are found, then a fast test for confluence is
carried out. This saves a lot of time if the system really is
confluent, but usually wastes time if it is not.
OR.maxstoredlen
: KnuthBendix
with the
limits removed. (To remove the limits, unbind the field.)
OR.maxoverlaplen
:
OR.maxoverlaplen
are processed. Similar remarks apply to
those for
OR.maxstoredlen
.
OR.sorteqns
:
OR.maxoplen
:
OR.sorteqns
were true), and only those rules having left hand sides of
length up to
OR.maxoplen
are output at all. Again,
similar remarks apply to those for
OR.maxstoredlen
.
OR.maxreducelen
:
OR.maxstates
,
OR.maxwdiffs
: KnuthBendix
, and the
maximum number of worddifferences allowed when running
AutomaticStructure
, respectively. These numbers are normally increased
automatically when required, so it unusual to want to set
these flags. They can be set when either it is desired to
limit these parameters (and prevent them being increased
automatically), or (as occasionally happens), the number of
worddifferences increases too rapidly for the program to cope
 when this happens, the run is usually doomed to failure
anyway.
KnuthBendix(
rws ) O
MakeConfluent(
rws ) O
These two functions do the same thing, namely to
run the external KnuthBendix program on the rewriting system rws.
KnuthBendix
returns true if it finds a confluent rewriting system and
otherwise false. In either case, if it halts normally, then it will
update the list
of the rewriting rules of rws, and also store a finite state automaton
ReductionAutomaton(
rws)
that can be used for word reduction, and the
counting and enumeration of irreducible words.
All control parameters (as defined in the preceding section) should be
set before calling KnuthBendix
. KnuthBendix
will halt either when it
finds a finite confluent system of rewriting rules, or when one of the
control parameters (such as OR
.maxeqns
) requires it to stop. The
program can also be made to halt and output manually at any time by
hitting the interrupt key (normally ctr'C') once. (Hitting it twice
has unpredictable consequences, since GAP may intercept the
signal.)
If KnuthBendix
halts without finding a confluent system, but still manages to
output the current system and update rws, then it is possible to use
the resulting rewriting system to reduce words, and count and
enumerate the irreducible words; it cannot be guaranteed that the
irreducible words are all in normal form, however. It is also possible
to rerun KnuthBendix
on the current system, usually after altering some of
the control parameters. In fact, is some more difficult examples, this
seems to be the only means of finding a finite confluent system.
ReductionAutomaton(
rws ) F
the reduction automaton of rws. Only expert users will wish to see this explicitly. See the section on finite state automata below for general information on functions for manipulating automata.
AutomaticStructure(
rws, [
large], [
filestore], [
diff1]) O
Run the external automatic groups program on the rewriting system
rws. AutomaticStructure
returns true if successful and false otherwise. If
successful, it stores three finite state automata
FirstWordDifferenceAutomaton(
rws)
, SecondWordDifferenceAutomaton(
rws)
and WordAcceptor(
rws)
. The first two of these are used for
wordreduction, and the third for counting and enumeration of
irreducible words (i.e. words in normal form).
The three optional parameters to AutomaticStructure
are all boolean, and
false by default. Setting large true results in some of the control
parameters (such as maxeqns
and tidyint
) being set
larger than they would be otherwise. This is necessary for examples
that require a large amount of space. Setting filestore true results
in more use being made of temporary files than would be otherwise.
This makes the program run slower, but it may be necessary if you are
short of core memory. Setting diff1 to be true is a more technical
option, which is explained more fully in the documentation for the
standalone KBMAG package. It is not usually necessary or helpful,
but it enables one or two examples to complete that would otherwise
run out of space.
The ordering field of rws will usually be equal to ``shortlex"
for AutomaticStructure
to be applicable.
However, it is now possible to use some procedures written by Sarah Rees
that work when the ordering is ``wtlex" or ``wreathprod". In the latter
case, each generator must have the same level as its inverse.
The only control parameters for rws that
are likely to be relevant are maxeqns
and maxwdiffs
.
WordAcceptor(
rws ) F
FirstWordDifferenceAutomaton(
rws ) F
SecondWordDifferenceAutomaton(
rws ) F
GeneralMultiplier(
rws ) F
These functions return, respectively, the word acceptor, the first and
second worddifference automata, and the general multiplier automaton
of rws. They can only be called after a successful call of
AutomaticStructure(
rws)
. All except the wordacceptor are 2variable
automata that read pairs of words in the alphabet of rws.
Note that the general multiplier has
its states labeled, where the different labels represents the accepting
states for the different letters in the alphabet of rws.
IsReducedWord(
rws,
w ) A
IsReducedForm(
rws,
w ) A
These two functions do the same thing, namely to
test whether the word w in the generators of the freestructure
FreeStructure(
rws)
of the rewriting system system
rws is reduced or not, and return true or false.
IsReducedWord
can only be used after KnuthBendix
or AutomaticStructure
has been run successfully on rws. In the former
case, if KnuthBendix
halted without
a confluent set of rules, then irreducible words are not necessarily
in normal form (but reducible words are definitely not in normal
form). If KnuthBendix
completes with a confluent rewriting system or
AutomaticStructure
completes successfully, then it is guaranteed that
all irreducible words are in normal form.
ReducedForm(
rws,
w ) O
ReducedWord(
rws,
w ) O
Reduce the word w in the generators of the freestructure
FreeStructure(
rws)
of the rewriting system rws
(or, equivalently, in the generators of the underlying group of
rws), and return the result.
ReducedForm
can only be used after KnuthBendix
or AutomaticStructure
has been run
successfully on rws. In the former case, if KnuthBendix
halted without a
confluent set of rules, then the irreducible word returned is not
necessarily in normal form. If KnuthBendix
completes with a confluent
rewriting system or AutomaticStructure
completes successfully, then it is
guaranteed that all irreducible words are in normal form.
Size(
rws ) M
Returns the number of irreducible words in the rewriting system rws.
Size
can only be used after KnuthBendix
or
AutomaticStructure
has been run
successfully on rws. In the former case, if KnuthBendix
halted without a
confluent set of rules, then the number of irreducible words may be
greater than the number of words in normal form (which is equal to the
order of the underlying group, monoid or semigroup G of rws). If
KnuthBendix
completes with a confluent rewriting system or
AutomaticStructure
completes successfully, then it is guaranteed that
Size
will return the correct order of G.
Order(
rws,
w ) M
The order of the element w of the free structure FreeStructure(
rws)
of
rws as an element of the group or monoid from which rws was defined.
Order
can only be used after KnuthBendix
or
AutomaticStructure
has been run successfully on rws.
It is not guaranteed to terminate in the case of infinite order, but it
usually seems to do so in practice!
EnumerateReducedWords(
rws,
min,
max) O
Enumerate all irreducible words in the rewriting system rws that have lengths between min and max (inclusive), which should be nonnegative integers. The result is returned as a list of words. The enumeration is by depthfirst search of a finite state automaton, and so the words in the list returned are ordered lexicographically (not by shortlex).
EnumerateReducedWords
can only be used after KnuthBendix
or AutomaticStructure
has been run
successfully on rws. In the former case, if KnuthBendix
halted without a
confluent set of rules, then not all irreducible words in the list
returned will necessarily be in normal form. If KnuthBendix
completes with a
confluent rewriting system or AutomaticStructure
completes successfully, then
it is guaranteed that all words in the list will be in normal form.
GrowthFunction(
rws ) F
Returns the growth function of the set of irreducible words in the rewriting system rws. This is a rational function, of which the coefficient of x^{n} in its Taylor expansion is equal to the number of irreducible words of length n.
If the coefficients in this rational function are larger than about 16000 then strange error messages will appear and fail will be returned.
GrowthFunction
can only be used after KnuthBendix
or AutomaticStructure
has been run
successfully on rws. In the former case, if KnuthBendix
halted without a
confluent set of rules, then not all irreducible words in the list
returned will necessarily be in normal form. If KnuthBendix
completes with a
confluent rewriting system or AutomaticStructure
completes successfully,
then it is guaranteed that all words in the list will be in normal form.
We start with a easy example  the alternating group A_{4}.
gap> F:=FreeGroup("a","b");; gap> a:=F.1;; b:=F.2;; gap> G:=F/[a^2, b^3, (a*b)^3];; gap> R:=KBMAGRewritingSystem(G); rec( isRWS := true, generatorOrder := [_g1,_g2,_g3], inverses := [_g1,_g3,_g2], ordering := "shortlex", equations := [ [_g2^2,_g3], [_g1*_g2*_g1,_g3*_g1*_g3] ] ) #Notice that monoid generators printed as _g1, _g2, _g3 are used #internally. These correspond to the group generators a, b, b^1. gap> KnuthBendix(R); true gap> R; rec( isRWS := true, isConfluent := true, generatorOrder := [_g1,_g2,_g3], inverses := [_g1,_g3,_g2], ordering := "shortlex", equations := [ [_g1^2,IdWord], [_g2*_g3,IdWord], [_g3*_g2,IdWord], [_g2^2,_g3], [_g3*_g1*_g3,_g1*_g2*_g1], [_g3^2,_g2], [_g2*_g1*_g2,_g1*_g3*_g1], [_g3*_g1*_g2*_g1,_g2*_g1*_g3], [_g1*_g2*_g1*_g3,_g3*_g1*_g2], [_g2*_g1*_g3*_g1,_g3*_g1*_g2], [_g1*_g3*_g1*_g2,_g2*_g1*_g3] ] ) #The `equations' field of <R> is now a complete system of rewriting rules gap> Size(R); 12 gap> EnumerateReducedWords(R,0,12); [ <identity ...>, a, a*b, a*b*a, a*b^1, a*b^1*a, b, b*a, b*a*b^1, b^1, b^1*a, b^1*a*b ] #We have enumerated all of the elements of the group  note that they #are returned as words in the free group F.
Example 2
The Fibonacci group F(2,5) defined by a semigroup rather than a group presentation. Interestingly this defines the same structure (although ir would not do so for F(2,r) with r even). beginexample gap> S:=FreeSemigroup(5);; a:=S.1;; b:=S.2;; c:=S.3;; d:=S.4;; e:=S.5;; gap> Q := S/[ [ab,c], [bc,d], [cd,e], [de,a], [ea,b] ]; fp semigroup on the generators [ s1, s2, s3, s4, s5 ] gap> R:=KBMAGRewritingSystem(Q); rec( isRWS := true, silent := true, generatorOrder := [_s1,_s2,_s3,_s4,_s5], inverses := [,,,,], ordering := shortlex, equations := [ [_s1_s2,_s3], [_s2_s3,_s4], [_s3_s4,_s5], [_s4_s5,_s1], [_s5_s1,_s2] ] ) gap> KnuthBendix(R); true gap> Size(R); 11 gap> EnumerateReducedWords(R,0,4); [ s1, s1^2, s1^2s4, s1s3, s1s4, s2, s2^2, s2s5, s3, s4, s5 ] #Let's do the same thing using the recursive ordering. gap> SetOrderingOfKBMAGRewritingSystem(R,recursive); gap> KnuthBendix(R); true gap> Size(R); 11 gap> EnumerateReducedWords(R,0,11); [ s1, s1^2, s1^3, s1^4, s1^5, s1^6, s1^7, s1^8, s1^9, s1^10, s1^11 ] endexample
Example 3
The Heisenberg group  that is, the free 2generator nilpotent group of class 2. For this to complete, we need to use the recursive ordering, and reverse our initial order of generators. (Alternatively, we could avoid this reversal, by using a ``wreathprod" ordering, and setting the levels of the generators to be 6,5,4,3,2,1.)
gap> F:=FreeGroup("x","y","z");; gap> x:=F.1;; y:=F.2;; z:=F.3;; gap> G:=F/[Comm(y,x)*z^1, Comm(z,x), Comm(z,y)];; gap> R:=KBMAGRewritingSystem(G); rec( isRWS := true, generatorOrder := [_g1,_g2,_g3,_g4,_g5,_g6], inverses := [_g2,_g1,_g4,_g3,_g6,_g5], ordering := "shortlex", equations := [ [_g4*_g2*_g3,_g5*_g2], [_g6*_g2,_g2*_g6], [_g6*_g4,_g4*_g6] ] ) gap> SetOrderingOfKBMAGRewritingSystem(R,"recursive"); gap> ReorderAlphabetOfKBMAGRewritingSystem(R,(1,6)(2,5)(3,4)); gap> R; rec( isRWS := true, generatorOrder := [_g6,_g5,_g4,_g3,_g2,_g1], inverses := [_g5,_g6,_g3,_g4,_g1,_g2], ordering := "recursive", equations := [ [_g4*_g2*_g3,_g5*_g2], [_g6*_g2,_g2*_g6], [_g6*_g4,_g4*_g6] ] ) gap> SetInfoLevel(InfoRWS,1); gap> KnuthBendix(R); #I Calling external KnuthBendix program. #System is confluent. #Halting with 18 equations. #I External KnuthBendix program complete. #I System computed is confluent. true gap> R; rec( isRWS := true, isConfluent := true, generatorOrder := [_g6,_g5,_g4,_g3,_g2,_g1], inverses := [_g5,_g6,_g3,_g4,_g1,_g2], ordering := "recursive", equations := [ [_g6*_g5,IdWord], [_g5*_g6,IdWord], [_g4*_g3,IdWord], [_g3*_g4,IdWord], [_g2*_g1,IdWord], [_g1*_g2,IdWord], [_g6*_g2,_g2*_g6], [_g6*_g4,_g4*_g6], [_g4*_g2,_g2*_g4*_g5], [_g5*_g2,_g2*_g5], [_g6*_g1,_g1*_g6], [_g5*_g4,_g4*_g5], [_g6*_g3,_g3*_g6], [_g3*_g1,_g1*_g3*_g5], [_g4*_g1,_g1*_g4*_g6], [_g3*_g2,_g2*_g3*_g6], [_g5*_g1,_g1*_g5], [_g5*_g3,_g3*_g5] ] ) gap> Size(R); infinity gap> IsReducedWord(R,z*y*x); false gap> ReducedForm(R,z*y*x); x*y*z^2 gap> IsReducedForm(R,x*y*z^2); true
Example 4
This is an example of the use of the KnuthBendix algorithm to prove
the nilpotence of a finitely presented group. (The method is due to
Sims, and is described in Chapter 11.8 of Sims94.) This example
is of intermediate difficulty, and demonstrates the necessity of using
the maxstoredlen
control parameter.
The group is

NilpotentQuotient
command, from the package
``nq". We find that there is a maximal such quotient, and it has
class 7, and the layers going down the lower central series have the
abelian structures [0,0], [0], [0], [0], [0], [2], [2].
By using the standalone C
nilpotent quotient program, it is
possible to find a powercommutator presentation of this maximal
quotient. We now construct a new presentation of the same group, by
introducing the generators in this powercommutator presentation,
together with their definitions as powers or commutators of earlier
generators. It is this new presentation that we use as input for the
KnuthBendix program. Again we use the recursive ordering, but this
time we will be careful to introduce the generators in the correct
order in the first place!
gap> F:=FreeGroup("h","g","f","e","d","c","b","a");; gap> h:=F.1;;g:=F.2;;f:=F.3;;e:=F.4;;d:=F.5;;c:=F.6;;b:=F.7;;a:=F.8;; gap> G:=F/[Comm(b,a)*c^1, Comm(c,a)*d^1, Comm(d,a)*e^1, > Comm(e,b)*f^1, Comm(f,a)*g^1, Comm(g,b)*h^1, > Comm(g,a), Comm(c,b), Comm(e,a)];; gap> R:=KBMAGRewritingSystem(G); rec( isRWS := true, generatorOrder := [_g1,_g2,_g3,_g4,_g5,_g6,_g7,_g8,_g9,_g10, _g11,_g12,_g13,_g14,_g15,_g16], inverses := [_g2,_g1,_g4,_g3,_g6,_g5,_g8,_g7,_g10,_g9, _g12,_g11,_g14,_g13,_g16,_g15], ordering := "shortlex", equations := [ [_g14*_g16*_g13,_g11*_g16], [_g12*_g16*_g11,_g9*_g16], [_g10*_g16*_g9,_g7*_g16], [_g8*_g14*_g7,_g5*_g14], [_g6*_g16*_g5,_g3*_g16], [_g4*_g14*_g3,_g1*_g14], [_g4*_g16,_g16*_g4], [_g12*_g14,_g14*_g12], [_g8*_g16,_g16*_g8] ] ) gap> SetOrderingOfKBMAGRewritingSystem(R,"recursive");A little experimentation reveals that this example works best when only those equations with left and right hand sides of lengths at most 10 are kept.
gap> O:=OptionsRecordOfKBMAGRewritingSystem(R); gap> O.maxstoredlen:=[10,10];; gap> SetInfoLevel(InfoRWS,2); gap> KnuthBendix(R); # 60 eqns; total len: lhs, rhs = 129, 143; 25 states; 0 secs. # 68 eqns; total len: lhs, rhs = 364, 326; 28 states; 0 secs. # 77 eqns; total len: lhs, rhs = 918, 486; 45 states; 0 secs. # 91 eqns; total len: lhs, rhs = 728, 683; 58 states; 0 secs. # 102 eqns; total len: lhs, rhs = 1385, 1479; 89 states; 0 secs. . . . . # 310 eqns; total len: lhs, rhs = 4095, 4313; 489 states; 1 secs. # 200 eqns; total len: lhs, rhs = 2214, 2433; 292 states; 1 secs. # 194 eqns; total len: lhs, rhs = 835, 922; 204 states; 1 secs. # 157 eqns; total len: lhs, rhs = 702, 723; 126 states; 1 secs. # 151 eqns; total len: lhs, rhs = 553, 444; 107 states; 1 secs. # 101 eqns; total len: lhs, rhs = 204, 236; 19 states; 1 secs. #No new eqns for some time  testing for confluence #System is not confluent. # 172 eqns; total len: lhs, rhs = 616, 473; 156 states; 1 secs. # 171 eqns; total len: lhs, rhs = 606, 472; 156 states; 1 secs. #No new eqns for some time  testing for confluence #System is not confluent. # 151 eqns; total len: lhs, rhs = 452, 453; 92 states; 1 secs. # 151 eqns; total len: lhs, rhs = 452, 453; 92 states; 1 secs. #No new eqns for some time  testing for confluence #System is not confluent. # 101 eqns; total len: lhs, rhs = 200, 239; 15 states; 1 secs. # 101 eqns; total len: lhs, rhs = 200, 239; 15 states; 1 secs. #No new eqns for some time  testing for confluence #System is confluent. #Halting with 101 equations. WARNING: The monoid defined by the presentation may have changed, since equations have been discarded. If you rerun, include the original equations. #Exit status is 0 #I External KnuthBendix program complete. #WARNING: Because of the control parameters you set, the system may # not be confluent. Unbind the parameters and rerun KnuthBendix # to check! #I System computed is NOT confluent. false #Now it is essential to rerun with the `maxstoredlen' limit removed #to check that the system really is confluent. gap> Unbind(O.maxstoredlen); gap> KnuthBendix(R); # 101 eqns; total len: lhs, rhs = 200, 239; 15 states; 0 secs. #No new eqns for some time  testing for confluence #System is confluent. #Halting with 101 equations. #Exit status is 0 #I External KnuthBendix program complete. #I System computed is confluent. true #In fact, in this case, we did have a confluent set already.Inspection of the confluent set now reveals it to be precisely a powercommutator presentation of a nilpotent group, and so we have proved that the group we started with really is nilpotent. Of course, this means also that it is equal to its largest nilpotent quotient, of which we already know the structure.
Example 5
Our final example illustrates the use of the AutomaticStructure
command, which
runs the automatic groups programs. The group has a balanced
symmetrical presentation with 3 generators and 3 relators, and was
originally proposed by Heineken as a possible example of a finite
group with such a presentation. In fact, the AutomaticStructure
command proves
it to be infinite.
This example is of intermediate difficulty, but there is no need to
use any special options. It takes a few minutes to run on a
WorkStation. It works better with the optional large parameter of
AutomaticStructure
set to true
.
We will not attempt to explain all of the output in detail here; the interested user should consult the documentation for the standalone KBMAG package. Roughly speaking, it first runs the KnuthBendix program, which does not halt with a confluent rewriting system, but is used instead to construct a worddifference finite state automaton. This in turn is used to construct the wordacceptor and multiplier automata for the group. Sometimes the initial constructions are incorrect, and part of the procedure consists in checking for this, and making corrections. In fact, in this example, the correct automata are considerably smaller than the ones first constructed. The final stage is to run an axiomchecking program, which essentially checks that the automata satisfy the group relations. If this completes successfully, then the correctness of the automata has been proved, and they can be used for correct wordreduction and enumeration in the group.
gap> F:=FreeGroup("a","b","c");; gap> a:=F.1;;b:=F.2;;c:=F.3;; gap> G:=F/[Comm(a,Comm(a,b))*c^1, Comm(b,Comm(b,c))*a^1, > Comm(c,Comm(c,a))*b^1];; gap> R:=KBMAGRewritingSystem(G); rec( isRWS := true, verbose := true, generatorOrder := [_g1,_g2,_g3,_g4,_g5,_g6], inverses := [_g2,_g1,_g4,_g3,_g6,_g5], ordering := "shortlex", equations := [ [_g2*_g4*_g2*_g3*_g1,_g5*_g4*_g2*_g3], [_g4*_g6*_g4*_g5*_g3,_g1*_g6*_g4*_g5], [_g6*_g2*_g6*_g1*_g5,_g3*_g2*_g6*_g1] ] ) gap> SetInfoLevel(InfoRWS,1); gap> AutomaticStructure(R,true); #I Calling external automatic groups program. #Running KnuthBendix Program (pathname)/kbprog mt 20 hf 100 cn 0 wd me 262144 t 500 (filename) #Halting with 42317 equations. #First worddifference machine with 271 states computed. #Second worddifference machine with 271 states computed. #System is confluent, or halting factor condition holds. #Running program to construct wordacceptor and multiplier automata (pathname)/gpmakefsa l (filename) #Wordacceptor with 1106 states computed. #General multiplier with 2428 states computed. #Validity test on general multiplier succeeded. #Running program to verify axioms on the automatic structure (pathname)/gpaxioms l (filename) #General length2 multiplier with 2820 states computed. #Checking inverse and short relations. #Checking relation: _g2*_g4*_g2*_g3*_g1 = _g5*_g4*_g2*_g3 #Checking relation: _g4*_g6*_g4*_g5*_g3 = _g1*_g6*_g4*_g5 #Checking relation: _g6*_g2*_g6*_g1*_g5 = _g3*_g2*_g6*_g1 #Axiom checking succeeded. #I Computation was successful  automatic structure computed. #Minimal reducible word acceptor with 1058 states computed. #Minimal KnuthBendix equation fsa with 1891 states computed. #Correct diff1 fsa with 271 states computed. #Correct diff2 fsa with 271 states computed. true gap> Size(R); infinity gap> Order(R,a); infinity gap> Order(R,Comm(a,b)); infinity
This functions in this section are currently only applicable when the rewriting system is defined from a group G.
It is possible to use the KnuthBendix and Automatic groups program on cosets of a specified subgroup H of G. Most of the functions in the preceding sections have analogues for cosets rather than for elements. It is also possible sometimes to compute a complete rewriting system or a subgroup presentation of H.
SubgroupOfKBMAGRewritingSystem(
rws,
H ) F
The subgroup H of the group G (= SemigroupOfRewritingSystem(
rws)
)
from which rws is defined can be specified either as a subgroup of
G or as a list of elements of G that generate H, or as a subgroup of
F = FreeStructureOfRewritingSystem(
rws)
that maps onto H, or as
a list of elements of F that generate a subgroup mapping onto H.
SubgroupOfKBMAGRewritingSystem
returns a rewriting system subrws for H,
but subrws has no rules, and is only intended to be used as a parameter
in the functions that follow.
ResetRewritingSystemOnCosets(
rws,
subrws ) F
This function resets subrws back to its form as it
was before the application of KnuthBendixOnCosets
or
AutomaticStructureOnCosets
.
The normal form and reduction algorithms on cosets will be
unavailable after this call.
Any optional control parameters set for rws will automatically be used when applying the KnuthBendix and Automatic Structure functions on cosets, that are now to be described.
KnuthBendixOnCosets(
rws,
subrws ) O
KnuthBendixOnCosetsWithSubgroupRewritingSystem(
rws,
subrws ) O
Run the external KnuthBendix program on the rewriting system rws
with respect to the cosets of the subgroup corresponding to subrws.
KnuthBendixOnCosets
returns true if it finds a confluent rewriting
system on coset representatives, and otherwise false.
If KnuthBendixOnCosets
halts without finding a confluent system, but still
manages to output the current system and update rws, then it is possible to
use the resulting rewriting system to reduce coset representatives, and
count and enumerate the irreducible coset representatives;
it cannot be guaranteed that the
irreducible coset representatives are all in normal form, however.
KnuthBendixOnCosetsWithSubgroupRewritingSystem
does the same and, in
addition, tries to compute a confluent rewriting system for the subgroup H.
RewritingSystemOfSubgroupOfKBMAGRewritingSystem(
rws,
subrws ) F
This can only be used after a successful call of
KnuthBendixOnCosetsWithSubgroupRewritingSystem
.
It returns a confluent rewriting system for H on a generating set
corresponding to the generators of H that were used to define subrws.
AutomaticStructureOnCosets(
rws,
subrws, [
large], [
filestore], [
diff1]) O
AutomaticStructureOnCosetsWithSubgroupPresentation(
rws,
subrws, [
large], [
filestore], [
diff1]) O
Run the external automatic cosets program on the rewriting system
rws with respect to the cosets of the subgroup H from which subrws
was defined.
AutomaticStructureOnCosets
returns true if successful and false otherwise.
The optional parameters to AutomaticStructureOnCosets
are the same as for
AutomaticStructure
.
The ordering of rws must be ``shortlex".
AutomaticStructureOnCosetsWithSubgroupPresentation
does the same and, in
addition, tries to compute a presentation of the subgroup H.
PresentationOfSubgroupOfKBMAGRewritingSystem(
rws,
subrws ) F
This can only be used after a successful call of
AutomaticStructureOnCosetsWithSubgroupPresentation
.
It returns a presentation for H, but this is not on the generators
used to define H.
IsReducedCosetRepresentative(
rws,
subrws,
w ) A
Test whether the word w in the generators of the freestructure
FreeStructure(
rws)
of the rewriting system system
rws is reduced or not as a coset representative of the subgroup H
of G, and return true or false.
IsReducedCosetRepresentative
can only be used after
KnuthBendixOnCosets
or AutomaticStructureOnCosets
has been run
successfully on rws and subrws. In the former case, if
KnuthBendixOnCosets
halted without a confluent set of rules, then
irreducible words are not necessarily in normal form (but reducible words
are definitely not in normal form). If KnuthBendixOnCosets
completes
with a confluent rewriting system or AutomaticStructureOnCosets
completes successfully, then it is guaranteed that all irreducible words
are in normal form.
ReducedFormOfCosetRepresentative(
rws,
subrws,
w ) O
ReducedCosetRepresentative(
rws,
subrws,
w ) O
Reduce the word w in the generators of the free structure
FreeStructure(
rws)
of the rewriting system rws
as a coset representative of the subgroup H from which subrws
was defined, and return the result.
ReducedFormOfCosetRepresentative
can only be used after
KnuthBendixOnCosets
or AutomaticStructureOnCosets
has been run
successfully on rws and subrws.
In the former case, if KnuthBendixOnCosets
halted
without a confluent set of rules, then the irreducible word returned is
not necessarily in normal form. If KnuthBendixOnCosets
completes
with a confluent rewriting system or AutomaticStructureOnCosets
completes successfully, then it is guaranteed that all irreducible words are
in normal form.
Index(
rws,
subrws ) M
Returns the number of irreducible words for coset represenatitives of the subgroup H of G corresponding to subrws.
Index
can only be used after KnuthBendixOnCosets
or
AutomaticStructureOnCosets
has been run
successfully on rws and subrws.
In the former case, if KnuthBendixOnCosets
halted without a
confluent set of rules, then the number of irreducible words may be
greater than the number of words in normal form (which is equal to the
index of H in G). If KnuthBendixOnCosets
completes with a
confluent rewriting system or AutomaticStructureOnCosets
completes
successfully, then
it is guaranteed that Index
will return the correct index of H
in G.
EnumerateReducedCosetRepresentatives(
rws,
subrws,
min,
max) O
Enumerate all irreducible words for coset representatives of H in G, that have lengths between min and max (inclusive), which should be nonnegative integers. The result is returned as a list of words. The enumeration is by depthfirst search of a finite state automaton, and so the words in the list returned are ordered lexicographically (not by shortlex).
EnumerateReducedCosetRepresentatives
can only be used after
KnuthBendixOnCosets
or AutomaticStructureOnCosets
has been run
successfully on rws and subrws.
In the former case, if KnuthBendixOnCosets
halted without a
confluent set of rules, then not all irreducible words in the list
returned will necessarily be in normal form.
If KnuthBendixOnCosets
completes with a
confluent rewriting system or AutomaticStructureOnCosets
completes successfully, then
it is guaranteed that all words in the list will be in normal form.
GrowthFunctionOfCosetRepresentatives(
rws,
subrws ) F
Returns the growth function of the set of irreducible words for coset representatives of H in G, where subrws and rws are the rewriting systems for H and G. This is a rational function, of which the coefficient of x^{n} in its Taylor expansion is equal to the number of coset representatives words of length n.
If the coefficients in this rational function are larger than about 16000 then strange error messages will appear and fail will be returned.
GrowthFunctionOfCosetRepresentatives
can only be used after
KnuthBendixOnCosets
or AutomaticStructureOnCosets
has been run
successfully on rws and subrws.
In the former case, if KnuthBendixOnCosets
halted without a
confluent set of rules, then not all irreducible words in the list
returned will necessarily be in normal form.
If KnuthBendixOnCosets
completes with a confluent rewriting system or
AutomaticStructureOnCosets
completes successfully, then
it is guaranteed that all words in the list will be in normal form.
gap> F:=FreeGroup("a","b","c");; gap> a:=F.1;;b:=F.2;;c:=F.3;; gap> G := F/[b^3,c^3,(b*c)^4,(b*c^1)^5,a^1*b^1*c*b*c*b^1*c*b*c^1]; <fp group on the generators [ a, b, c ]> gap> R:=KBMAGRewritingSystem(G); rec( isRWS := true, silent := true, generatorOrder := [_g1,_g2,_g3,_g4,_g5,_g6], inverses := [_g2,_g1,_g4,_g3,_g6,_g5], ordering := "shortlex", equations := [ [_g3^2,_g4], [_g5^2,_g6], [_g3*_g5*_g3*_g5,_g6*_g4*_g6*_g4], [_g3*_g6*_g3*_g6*_g3,_g5*_g4*_g5*_g4*_g5], [_g2*_g4*_g5*_g3*_g5,_g5*_g4*_g6*_g3] ] ) gap> S:=SubgroupOfKBMAGRewritingSystem(R,[a^3,c*a^2]); rec( isRWS := true, silent := true, generatorOrder := [_x1,_X1,_x2,_X2], inverses := [_X1,_x1,_X2,_x2], ordering := "shortlex", equations := [ ] ) gap> KnuthBendixOnCosetsWithSubgroupRewritingSystem(R,S); true gap> Index(R,S); 18 gap> IsReducedCosetRepresentative(R,S,b*a*b*a); false gap> ReducedFormOfCosetRepresentative(R,S,b*a*b*a); b^1*a^1 gap> EnumerateReducedCosetRepresentatives(R,S,0,4); [ <identity ...>, a, a*b, a*b*c, a*b^1, a^1, a^1*b, a^1*b*c, a^1*b^1, b, b*c, b*c*a, b*c*a^1, b*c^1, b^1, b^1*a, b^1*a^1, b^1*a^1*b ] gap> SS:=RewritingSystemOfSubgroupOfKBMAGRewritingSystem(R,S);; gap> Size(SS); 60
Example 2
We find a free subgroup of the Fibonacci group F(2,8). This example may take about 20 minutes to run on a typical WorkStation.
gap> F:=FreeGroup(8);; gap> a:=F.1; b:=F.2; c:=F.3; d:=F.4; e:=F.5; f:=F.6; g:=F.7; h:=F.8; gap> G := F/[a*b*c^1, b*c*d^1, c*d*e^1, d*e*f^1, > e*f*g^1, f*g*h^1, g*h*a^1, h*a*b^1]; gap> R:=KBMAGRewritingSystem(G);; gap> S:=SubgroupOfKBMAGRewritingSystem(R,[a,e]);; gap> AutomaticStructureOnCosetsWithSubgroupPresentation(R,S); gap> P:=PresentationOfSubgroupOfKBMAGRewritingSystem(R,S); <fp group on the generators [ f1, f3 ]> gap> RelatorsOfFpGroup(P); [ ] gap> Index(R,S); infinity
The KBMAG package contains GAP interfaces to many of the functions
for manipulating finite state automata (fsa) that are available in the
standalone. We shall list these here, without giving much detail.
For more detail, the user could try looking in the source file
gap/fsa4.g
.
fsa are currently implemented as GAP records, as they were
previously in GAP3. This interface may be updated to the style of
GAP4 at some stage. (Note that the abbreviation fsa will be used for
both singular and the plural.)
The alphabet of an fsa is itself a record that must contain at least
the two components type
and size
, where type
is a string, and
size
a positive integer. The easiest possibility is to use the
type ``simple", and then no other record components are necessary.
There are several more complicated possibilities, which are used by
the other rewriting system functions. For example, there is the type
``identifiers", for which fields ``format" and ``names" are necessary.
For example
gap> M:=FreeMonoid(3);; gap> alph := rec(type:="identifiers", size:=3, format:="dense", names:=[M.1,M.2,M.3] );;defines a valid alphabet for an fsa. The members of the alphabet are referred to as
letters
, and can be
represented either by a positive integer or by their name (usually
a generator of a free group or monoid) if they have one.
The functions ReductionAutomaton(
rws)
, WordAcceptor(
rws)
,
FirstWordDifferenceAutomaton(
rws)
, SecondWordDifferenceAutomaton(
rws)
and GeneralMultiplie(
rws)
mentioned in earlier sections all
return a fsa. The other possibilities for the user to construct
a fsa are to use the function FSA
or to read one in from a file.
In the latter case, the user must immediately call InitializeFSA
on the record that has been read in. For example, running GAP from the
package directory:
gap> Read("standalone/fsa_data/fsa_2"); gap> InitializeFSA(fsa_2);
IsInitializedFSA(
fsa ) F
Tests whether fsa is a record describing a valid initialized fsa.
InitializeFSA(
fsa ) F
Initializes a record representing a fsa that has been read in from a file.
FSA(
alph ) F
Returns an initialized fsa with alphabet alph having one state that is an initial and final state, and no transitions (edges).
The arguments of the following functions must be initialized fsa.
WriteFSA(
fsa )
Displays fsa nicely. (In a proper implementation, this would be the
ViewObj
function.)
IsDeterministicFSA(
fsa ) F
Tests whether fsa is a deterministic fsa.
Many of the functions below work only for deterministic fsa.
A deterministic fsa with the same language as fsa can
be constructed with the function DeterminizeFSA
.
AlphabetFSA(
fsa ) F
StatesFSA(
fsa ) F
Return, respectively, the records representing the alphabet and stateset of fsa.
NumberOfStatesFSA(
fsa ) F
Returns the number of states of fsa.
NumberOfLettersFSA(
fsa ) F
SizeOfAlphabetFSA(
fsa ) F
Returns the size of the alphabet of fsa.
AcceptingStatesFSA(
fsa ) F
Returns the list of accepting states of fsa.
InitialStatesFSA(
fsa ) F
Returns the list of initial states of fsa.
DenseDTableFSA(
fsa ) F
fsa must be deterministic. The transition table is returned as a list
of lists, where the eth entry in the sth inner list is
TargetDFA(
fsa,
e,
s)
(see below).
SparseTableFSA(
fsa ) F
The transition table is returned as a list of lists,
where each entry in the sth inner list is
is a twoelement list [
e,
t]
of integers that represents a transition
from state number s to state number t under letter number e.
We can also have e=0, representing transitions with no label
(e transitions).
TargetDFA(
fsa,
e,
s ) F
fsa must be a deterministic fsa, e should be a number or name of a letter of the alphabet, and s a number of a state of fsa. The target of s under e is returned, or 0 if there is no target.
TargetsFSA(
fsa,
e,
s ) F
Similar to TargetDFA
, but fsa need not be deterministic, and a list
of targets is returned.
SourcesFSA(
fsa,
e,
s ) F
Similar to TargetsFSA
, but the list of sources of transitions to s
under e is returned. e can also be zero here.
WordTargetDFA(
fsa,
w ) F
fsa must be a deterministic fsa, and w should be a list of integers or a word in the alphabet (in the case when the alphabet is defined from a free group or monoid). The target of the initial state of fsa under w is returned, or 0 if there is no such target.
IsAcceptedWordDFA(
fsa,
w ) F
fsa must be a deterministic fsa, and w should be a list of integers or a word in the alphabet (in the case when the alphabet is defined from a free group or monoid). This function tests whether w is in the language of fsa.
AddStateFSA(
fsa ) F
Adds an extra nonaccepting state to fsa with no transitions to or from it.
DeleteStateFSA(
fsa ) F
Deletes the highest numbered state together with all transitions to and from
it from fsa. Use PermuteStatesFSA
first to delete a different state.
PermuteStatesFSA(
fsa,
p )
p should be a permutation of [1..ns]
, where fsa has ns states.
The states are permuted, where the original state number n becomes
the new state number n^{p}.
AddLetterFSA(
fsa [,
name] ) F
Adds an extra letter to the alphabet of fsa with no associated transitions. If the alphabet of fsa is defined over a free group or monoid, then name specifies the element of this free structure corresponding to the new letter.
DeleteLetterFSA(
fsa ) F
Deletes the highest numbered letter together with all associated transitions
from the alphabet of fsa. Use PermuteLettersFSA
first to delete a
different letter.
PermuteLettersFSA(
fsa,
p )
p should be a permutation of [1..na]
, where the alphabet of
fsa has na letters.
The letters are permuted, where the original letter number n becomes
the new letter number n^{p}.
AddEdgeFSA(
fsa,
e,
s,
t ) F
Adds an extra transition to fsa with source s, target t and letter e. If there is already such a transition, then this function does nothing.
DeleteEdgeFSA(
fsa,
e,
s,
t ) F
Deletes the transition from fsa with source s, target t and letter e if there is one.
SetAcceptingFSA(
fsa,
s,
flag ) F
flag should be true or false, and state number s is made accepting or nonaccepting, respectively.
SetInitialFSA(
fsa,
s,
flag ) F
flag should be true or false, and state number s is made initial or noninitial, respectively.
IsAccessibleFSA(
fsa ) F
Tests whether fsa is accessible; that is, whether all states can be reached from the initial states.
AccessibleFSA(
fsa ) F
Removes all inaccessible states from fsa thus rendering it accessible without altering its language.
IsTrimFSA(
fsa ) F
Tests whether fsa is trim; that is, whether all states can be reached from the initial states, and a final state can be reached from all states.
TrimFSA(
fsa ) F
Removes all inaccessible states from fsa and all states from which an accepting state cannot be reached, thus rendering it trim without altering its language.
IsBFSFSA(
fsa ) F
Tests whether fsa is in bfs
(breadthfirstsearch) form; that is,
whether the initial states come first and the other states
appear in ascending order if one scans the transition table first
by state number and then by alphabet number. Note that fsa must
be accessible for it to be bfs
.
BFSFSA(
fsa ) F
Replaces fsa by one with the same language but in bfs
form.
This can be useful for comparing the languages of two fsa.
In fact MinimizeFSA
and BFSFSA
are applied in turn to a
deteministic fsa, then the resulting transition table is
uniquely determined by the language of the fsa.
LSizeDFA(
fsa ) F
The size of the acceted language of fsa, which must be deterministic.
This only works if fsa is trim. If not, then TrimFSA
must be called
first.
LEnumerateDFA(
fsa,
min,
max ) F
The words in the language of fsa of length l satisfying min £ l £ max are returned in a list. The words will either be lists of integers, for ``simple" type alphabets, of lists of words in the underlying free group or monoid for alphabets of type ``identifiers".
The remaining fsa functions all call external programs from the
standalone. All of them except DeterminizeFSA
take only
deterministic fsa as input, and all of them return deterministic fsa
as output.
DeterminizeFSA(
fsa ) F
Returns a deterministic fsa with the same language as fsa.
MinimizeFSA(
fsa ) F
Returns a fsa with the same language as fsa and a minimal number of states.
NotFSA(
fsa ) F
Returns a fsa with the same alphabet as fsa whose language is the complement of that of fsa.
StarFSA(
fsa ) F
Returns a fsa whose language is L^{*}, where L is the langauge of fsa.
ReverseFSA(
fsa ) F
Returns a fsa whose language consists of the reversed words in the language of fsa.
ExistsFSA(
fsa ) F
fsa should be twovariable fsa on an alphabet A. An fsa is returned that accepts a word w_{1} Î A^{*} if and only if there exists a words w_{2} Î A^{*} with (w_{1},w_{2}) in the language of fsa.
SwapCoordsFSA(
fsa ) F
fsa should be twovariable fsa on an alphabet A. A twovariable fsa on A is returned that accepts (w_{1},w_{2}) if and only if (w_{2},w_{1}) is accepted by fsa.
AndFSA(
fsa1,
fsa2 ) F
fsa1 and fsa2 must have the same alphabet. The returned fsa has language equal to the interssection of those of fsa1 and fsa2.
OrFSA(
fsa1,
fsa2 ) F
fsa1 and fsa2 must have the same alphabet. The returned fsa has language equal to the union of those of fsa1 and fsa2.
ConcatFSA(
fsa1,
fsa2 ) F
fsa1 and fsa2 must have the same alphabet. The returned fsa accepts words of the form w_{1}w_{2}, where w_{1} and w_{2} are in the languages of fsa1 and fsa2, respectively.
LanguagesEqualFSA(
fsa1,
fsa2 ) F
fsa1 and fsa2 must have the same alphabet. This function tests whether the languages of fsa1 and fsa2 are equal, and returns True or false.
GrowthFSA(
fsa ) F
Returns the growth function of fsa. This is a rational function, of which the the coefficient of x^{n} in its Taylor expansion is equal to the number of words of length n in the accepted language of fsa.
If the coefficients in this rational function are larger than about 16000 then strange error messages will appear and fail will be returned.
kbmag manual