  
  [1X1 [33X[0;0YThe Deep Thought algorithm[133X[101X
  
  [33X[0;0YPolycyclic  and,  especially,  finitely generated nilpotent groups exhibit a
  rich  structure  allowing  a  special  approach towards multiplication using
  polynomials.  The  so-called  Deep  Thought  algorithm introduced in [LGS98]
  computes  these  polynomials  in  practice  for a suitable presentation of a
  finitely  generated nilpotent group. Such a presentation is of the following
  form[133X
  
  
  [24X[33X[0;6Y\langle  a_1,  \ldots,  a_n  \mid a_i^{s_i} = a_{i+1}^{c_{i, i, i+1}} \cdots
  a_n^{c_{i,  i,  n}},  1  \leq  i \leq n, a_j a_i = a_i a_j a_{j+1}^{c_{i, j,
  j+1}} \cdots a_n^{c_{i, j, n}}, 1 \leq i < j \leq n \rangle[133X
  
  [124X
  
  [33X[0;0Ywith [23Xs_i \in \mathbb{N} \cup \{ \infty \}[123X and [23Xc_{i, j, k} \in \mathbb{Z}[123X. If
  [23Xs_i = \infty[123X, then the power relation [23Xa_i^{s_i}[123X is skipped.[133X
  
  [33X[0;0YLet  [23XG[123X  denote  the  group  presented  by the above presentation. Then every
  element in [23XG[123X can be written uniquely in a so-called normal form. That is, if
  [23XG_i  := \langle a_i, \ldots, a_n \rangle[123X and [23Xr_i := | G_i : G_{i+1}|[123X, [23X1 \leq
  i  \leq n[123X, are the relative orders, then for certain [23Xe_i \in \mathbb{Z}[123X each
  element can be written as[133X
  
  
  [24X[33X[0;6Ya_1^{e_1} \cdots a_n^{e_n}[133X
  
  [124X
  
  [33X[0;0Ywith [23X0 \leq e_i < r_i[123X if [23Xr_i < \infty[123X. A presentation is called confluent if
  and  only  if  [23Xs_i  =  r_i[123X for all [23X1 \leq i \leq n[123X. If a presentation is not
  confluent,  not  all  functions provided in this package are applicable, see
  function  [10XDTP_DTapplicability[110X.  For  more  theoretical  background  see  for
  example the documentation of the [5XGAP[105X package [5XPolycyclic[105X.[133X
  
  [33X[0;0YThe Deep Thought algorithm computes rational polynomials [23Xf_1, \ldots, f_n[123X in
  [23X2n[123X  indeterminates  such  that if [23X x := a_1^{x_1} \cdots a_n^{x_n} [123X and [23Xy :=
  a_1^{y_1}  \cdots  a_n^{y_n}  [123X  are two elements (in normal form or not with
  [23Xx_1, \ldots, x_n, y_1, \ldots, y_n \in \mathbb{Z}[123X), then their product [23Xxy[123X is
  given by[133X
  
  
  [24X[33X[0;6Ya_1^{f_1(x_1,  \ldots,  x_n, y_1, \ldots, y_n)} \cdots a_n^{f_n(x_1, \ldots,
  x_n, y_1, \ldots, y_n)}.[133X
  
  [124X
  
  [33X[0;0YIf  the  collector  is confluent, also the normal form of the product can be
  computed.  Otherwise  this  is not possible. Moreover, there exists a second
  version  of  the  Deep  Thought  algorithms  which  computes [23Xn^2[123X polynomials
  [23Xf_{rs}[123X, [23X1 \leq r, s \leq n[123X, suitable for multiplications of the form[133X
  
  
  [24X[33X[0;6Y(a_1^{x_1} \cdots a_n^{x_n}) \cdot a_s^{y_s} = a_1^{f_{1s}(x_1, \ldots, x_n,
  y_s)} \cdots a_n^{f_{ns}(x_1, \ldots, x_n, y_s)}[133X
  
  [124X
  
  [33X[0;0Yfor  [23X1  \leq  s  \leq  n[123X. Each general multiplication can be expressed using
  these special multiplications. Depending on the presentation, polynomials of
  one  version  may be more efficient for computations than the polynomials of
  the  other version. For a suggestion on which polynomials to use for a given
  presentation,  see  [10XDTP_DTapplicability[110X. In the following, Deep Thought type
  [23Xf_r[123X  refers  to the Deep Thought algorithm which uses [23Xn[123X polynomials and type
  [23Xf_{rs}[123X refers to the Deep Thought algorithm using [23Xn^2[123X polynomials.[133X
  
  [33X[0;0YIn  order to work with the Deep Thought functions, the group presentation is
  expected  to  be  given  as  a collector [10Xcoll[110X, as defined in the [5XGAP[105X package
  [5XPolycyclic[105X.  Moreover,  the  [5XPolycyclic[105X  package introduces the structure of
  exponent  vectors  [10Xexpvec[110X,  which  will  be  used  also  in  this package to
  represent  group  elements. In the following text, a group element [23Xa_1^{x_1}
  \cdots  a_n^{x_n}[123X is identified with its exponent vector in form of the list
  [10X[x_1,  ...,  x_n][110X.  For example, if [10Xexpvec1, expvec2[110X are exponent vectors of
  elements   in   the  same  group,  then  [10Xexpvec1  *  expvec2[110X  describes  the
  multiplication  of  the  corresponding  group elements, and so on. Note that
  generally exponent vectors are not assumed to represent normal forms.[133X
  
  
  [1X1.1 [33X[0;0YCategory DTObj[133X[101X
  
  [33X[0;0YThis package uses the category [10XDTObj[110X. A [10XDTObj[110X is a [10XIsFromTheLeftCollectorRep[110X
  with certain further list entries to store the Deep Thought polynomials of a
  collector  together with some additional information. That is, the functions
  [10XDTP_DTpols_r[110X   and  [10XDTP_DTpols_rs[110X  return  a  [10XDTObj[110X  which  has  entries  as
  [10XIsFromTheLeftCollectorRep[110X and additionally:[133X
  
  [30X    [33X[0;6Y[10XDTObj![PC_DTPPolynomials][110X:   Deep   Thought  polynomials  in  form  of
        (nested) lists[133X
  
  [30X    [33X[0;6Y[10XDTObj![PC_DTPOrders][110X:  list  containing  orders of group generators if
        the collector is confluent[133X
  
  [30X    [33X[0;6Y[10XDTObj![PC_DTPConfluent][110X:   boolean   value   indicating   whether  the
        collector is confluent or not[133X
  
