chap9.txt

  
  9 Bistellar flips
  
  
  9.1 Theory
  
  Since  two  combinatorial  manifolds are already considered distinct to each
  other  as  soon  as  they  are not combinatorially isomorphic, a topological
  PL-manifold  is  represented  by  a  whole class of combinatorial manifolds.
  Thus,  a  frequent  question  when  working  with combinatorial manifolds is
  whether  two  such  objects  are  PL-homeomorphic or not. One possibility to
  approach this problem, i. e. to find combinatorially distinct members of the
  class  of  a  PL-manifold,  is  a  heuristic  algorithm using the concept of
  bistellar moves.
  Definition (Bistellar moves [Pac87])
  Let  M  be  a combinatorial d-manifold (d-pseudomanifold), γ = ⟨ v_0 , ... ,
  v_k  ⟩ a k-face and δ = ⟨ w_0 , ... , w_d-k ⟩ a (d-k+1)-tuple of vertices of
  M  that  does  not span a (d-k)-face in M, 0 ≤ k ≤ d, such that { v_0 , ...,
  v_k  }  ∩ { w_0 , ... , w_d-k } = ∅ and { v_0 , ... , v_k, w_0 , ... w_k-d }
  spans exactly d-k+1 facets. Then the operation
  κ_(γ,δ) ( M ) = M ∖ (γ ⋆ ∂ δ) ∪ (∂ γ ⋆ δ)
  is called a bistellar (d-k)-move.
  In  other words: If there exists a bouquet D ⊂ M of d-k+1 facets on a subset
  of  vertices  W ⊂ V of order d+2 with a common k-face γ and the complement δ
  of  the  vertices  of γ in W does not span a (d-k)-face in M we can remove D
  and  replace  it  by  a bouquet of k+1 facets E ⊂ M with vertex set W with a
  common  face spanned by δ. By construction ∂ D = ∂ E and the altered complex
  is  again  a  combinatorial d-manifold (d-pseudomanifold). See Fig. 11 for a
  bistellar  1-move  of a 2-dimensional complex, see Fig. 12 for all bistellar
  moves in dimension 3.
  
  A  bistellar  0-move  is  a  stellar subdivision, i. e. the subdivision of a
  facet  δ  into d+1 new facets by introducing a new vertex at the center of δ
  (cf.  Fig. 12 on the left). In particular, the vertex set of a combinatorial
  manifold  (pseudomanifold)  is  not invariant under bistellar moves. For any
  bistellar  (d-k)-move κ_(γ,δ) we have an inverse bistellar k-move κ^-1_(γ,δ)
  =  κ_(δ,γ)  such  that  κ_(δ,γ) ( κ_(γ,δ) (M)) = M. If for two combinatorial
  manifolds  M and N there exist a sequence of bistellar moves that transforms
  one  into  the  other,  M  and  N  are called bistellarly equivalent. So far
  bistellar  moves are local operations on combinatorial manifolds that change
  its   combinatorial   type.   However,   the  strength  of  the  concept  in
  combinatorial topology is a consequence of the following
  Theorem (Bistellar moves [Pac87])
  Two combinatorial manifolds (pseudomanifolds) M and N are PL homeomorphic if
  and only if they are bistellarly equivalent.
  Unfortunately  Pachners  theorem  does  not  guarantee that the search for a
  connecting  sequence  of  bistellar moves between M and N terminates. Hence,
  using   bistellar   moves,   we   can  not  prove  that  M  and  N  are  not
  PL-homeomorphic.  However,  there  is  a  very effective simulated annealing
  approach  that  is  able  to  give  a positive answer in a lot of cases. The
  heuristic  was  first  implemented  by  Bjoerner  and  Lutz  in  [BL00]. The
  functions presented in this chapter are based on this code which can be used
  for several tasks:
  
  In many cases the heuristic reduces a given triangulation but does not reach
  a minimal triangulation after a reasonable amount of flips. Thus, we usually
  can  not  expect  the  algorithm  to  terminate.  However, in some cases the
  program normally stops after a small number of flips:
  
  Technical  note:  Since  bistellar  flips  do  not respect the combinatorial
  properties  of  a  complex,  no  attention  to the original vertex labels is
  payed,  i.  e.  the  flipped  complex  will be relabeled whenever its vertex
  labels  become different from the standard labeling (for example after every
  reverse 0-move).
  
  
  9.2 Functions for bistellar flips
  
  9.2-1 SCBistellarOptions
  
  SCBistellarOptions global variable
  
  Record  of  global variables to adjust output an behavior of bistellar moves
  in    SCIntFunc.SCChooseMove    (9.2-4)   and   SCReduceComplexEx   (9.2-14)
  respectively.
  
  1   BaseRelaxation:  determines  the  length  of  the  relaxation  period.
        Default: 3
  
  2   BaseHeating: determines the length of the heating period. Default: 4
  
  3   Relaxation: value of the current relaxation period. Default: 0
  
  4   Heating: value of the current heating period. Default: 0
  
  5   MaxRounds:  maximal  over  all  number of bistellar flips that will be
        performed. Default: 500000
  
  6   MaxInterval:  maximal number of bistellar flips that will be performed
        without a change of the f-vector of the moved complex. Default: 100000
  
  7   Mode:  flip  mode,  0=reducing,  1=comparing, 2=reduce as sub-complex,
        3=randomize. Default: 0
  
  8   WriteLevel:  0=no output, 1=storing of every vertex minimal complex to
        user library, 2=e-mail notification. Default: 1
  
  9   MailNotifyIntervall:  (minimum)  number  of seconds between two e-mail
        notifications. Default: 24 ⋅ 60 ⋅ 60 (one day)
  
  10  MaxIntervalIsManifold:  maximal number of bistellar flips that will be
        performed  without  a  change  of  the f-vector of a vertex link while
        trying to prove that the complex is a combinatorial manifold. Default:
        5000
  
  11  MaxIntervalRandomize  :=  50:  number  of  flips performed to create a
        randomized sphere. Default: 50
  
    Example  
     gap> SCBistellarOptions.BaseRelaxation;
     3
     gap> SCBistellarOptions.BaseHeating;
     4
     gap> SCBistellarOptions.Relaxation;
     0
     gap> SCBistellarOptions.Heating;
     0
     gap> SCBistellarOptions.MaxRounds;
     500000
     gap> SCBistellarOptions.MaxInterval;
     100000
     gap> SCBistellarOptions.Mode;
     0
     gap> SCBistellarOptions.WriteLevel;
     0
     gap> SCBistellarOptions.MailNotifyInterval;
     86400
     gap> SCBistellarOptions.MaxIntervalIsManifold;
     5000
     gap> SCBistellarOptions.MaxIntervalRandomize;
     50
     
  
  
  9.2-2 SCEquivalent
  
  SCEquivalent( complex1, complex2 )  method
  Returns:  true  or  false  upon  success,  fail  or  a  list of type [ fail,
            SCSimplicialComplex, Integer, facet list] otherwise.
  
  Checks  if  the  simplicial  complex complex1 (which has to fulfill the weak
  pseudomanifold   property  with  empty  boundary)  can  be  reduced  to  the
  simplicial  complex  complex2  via  bistellar  moves,  i. e. if complex1 and
  complex2   are   PL-homeomorphic.  Note  that  in  general  the  problem  is
  undecidable. In this case fail is returned.
  
  It  is  recommended to use a minimal triangulation complex2 for the check if
  possible.
  
  Internally            calls            SCReduceComplexEx            (9.2-14)
  (complex1,complex2,1,SCIntFunc.SCChooseMove);
  
    Example  
     gap> SCBistellarOptions.WriteLevel:=0;; # do not save complexes to disk
     gap> obj:=SC([[1,2],[2,3],[3,4],[4,5],[5,6],[6,1]]);; # hexagon
     gap> refObj:=SCBdSimplex(2);; # triangle as a (minimal) reference object
     gap> SCEquivalent(obj,refObj);
     #I  SCReduceComplexEx: complexes are bistellarly equivalent.
     true
     
  
  
  9.2-3 SCExamineComplexBistellar
  
  SCExamineComplexBistellar( complex )  method
  Returns:  simplicial  complex  passed as argument with additional properties
            upon success, fail otherwise.
  
  Computes  the  face lattice, the f-vector, the AS-determinant, the dimension
  and the maximal vertex label of complex.
  
    Example  
     gap> obj:=SC([[1,2],[2,3],[3,4],[4,5],[5,6],[6,1]]);
     [SimplicialComplex
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="unnamed complex 7"
      Dim=1
     
     /SimplicialComplex]
     gap> SCExamineComplexBistellar(obj);
     [SimplicialComplex
     
      Properties known: AltshulerSteinberg, BoundaryEx, Dim, FVector, 
                        FacetsEx, HasBoundary, IsPseudoManifold, IsPure, 
                        Name, NumFaces[], SkelExs[], Vertices.
     
      Name="unnamed complex 7"
      Dim=1
      AltshulerSteinberg=0
      FVector=[ 6, 6 ]
      HasBoundary=false
      IsPseudoManifold=true
      IsPure=true
     
     /SimplicialComplex]
     
  
  
  9.2-4 SCIntFunc.SCChooseMove
  
  SCIntFunc.SCChooseMove( dim, moves )  function
  Returns:  a  bistellar  move,  i.  e.  a  pair  of  lists upon success, fail
            otherwise.
  
  Since  the  problem  of  finding  a  bistellar  flip sequence that reduces a
  simplicial  complex  is undecidable, we have to use an heuristic approach to
  choose the next move.
  
  The  implemented  strategy  SCIntFunc.SCChooseMove  first  tries to directly
  remove  vertices, edges, i-faces in increasing dimension etc. If this is not
  possible it inserts high dimensional faces in decreasing co-dimension. To do
  this  in an efficient way a number of parameters have to be adjusted, namely
  SCBistellarOptions.BaseHeating  and  SCBistellarOptions.BaseRelaxation.  See
  SCBistellarOptions (9.2-1) for further options.
  
  If this strategy does not work for you, just implement a customized strategy
  and pass it to SCReduceComplexEx (9.2-14).
  
  See SCRMoves (9.2-10) for further information.
  
  9.2-5 SCIsKStackedSphere
  
  SCIsKStackedSphere( complex, k )  method
  Returns:  a list upon success, fail otherwise.
  
  Checks,  whether  the  given  simplicial  complex  complex that must be a PL
  d-sphere  is  a  k-stacked  sphere  with 1≤ k≤ ⌊fracd+22⌋ using a randomized
  algorithm based on bistellar moves (see [Eff11b], [Eff11a]). Note that it is
  not checked whether complex is a PL sphere -- if not, the algorithm will not
  succeed.  Returns  a  list upon success: the first entry is a boolean, where
  true  means  that  the complex is k-stacked and false means that the complex
  cannot  be  k-stacked.  A  value  of -1 means that the question could not be
  decided.  The second argument contains a simplicial complex that, in case of
  success,   contains   the   trigangulated   (d+1)-ball  B  with  ∂  B=S  and
  operatornameskel_d-k(B)=operatornameskel_d-k(S),   where   S   denotes   the
  simplicial complex passed in complex.
  
  Internally calls SCReduceComplexEx (9.2-14).
  
    Example  
     gap> SCLib.SearchByName("S^4~S^1");
     [ [ 463, "S^4~S^1 (VT)" ], [ 1473, "S^4~S^1 (VT)" ], [ 1474, "S^4~S^1 (VT)" ],
       [ 2477, "S^4~S^1 (VT)" ], [ 4395, "S^4~S^1 (VT)" ], 
       [ 4396, "S^4~S^1 (VT)" ], [ 4397, "S^4~S^1 (VT)" ], 
       [ 4398, "S^4~S^1 (VT)" ], [ 4399, "S^4~S^1 (VT)" ], 
       [ 4402, "S^4~S^1 (VT)" ], [ 4403, "S^4~S^1 (VT)" ], 
       [ 4404, "S^4~S^1 (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> l:=SCLink(c,1);
     [SimplicialComplex
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="lk([ 1 ]) in S^4~S^1 (VT)"
      Dim=4
     
     /SimplicialComplex]
     gap> SCIsKStackedSphere(l,1);
     #I  SCIsKStackedSphere: checking if complex is a 1-stacked sphere...
     #I  SCIsKStackedSphere: try 1/1
     #I  SCIsKStackedSphere: complex is a 1-stacked sphere.
     [ true, [SimplicialComplex
         
          Properties known: Dim, FacetsEx, Name, Vertices.
         
          Name="Filled 1-stacked sphere (lk([ 1 ]) in S^4~S^1 (VT))"
          Dim=5
         
         /SimplicialComplex] ]
     
  
  
  9.2-6 SCBistellarIsManifold
  
  SCBistellarIsManifold( complex )  method
  Returns:  true or false upon success, fail otherwise.
  
  Tries  to prove that a closed simplicial d-pseudomanifold is a combinatorial
  manifold by reducing its vertex links to the boundary of the d-simplex.
  
  false  is returned if it can be proven that there exists a vertex link which
  is  not  PL-homeomorphic  to the standard PL-sphere, true is returned if all
  vertex links are bistellarly equivalent to the boundary of the simplex, fail
  is  returned  if the algorithm does not terminate after the number of rounds
  indicated by SCBistellarOptions.MaxIntervallIsManifold.
  
  Internally            calls            SCReduceComplexEx            (9.2-14)
  (link,SCEmpty(),0,SCIntFunc.SCChooseMove);  for  every link of complex. Note
  that false is returned in case of a bounded manifold.
  
  See  SCIsManifoldEx  (12.1-18)  and  SCIsManifold  (12.1-17) for alternative
  methods for manifold verification.
  
    Example  
     gap> c:=SCBdCrossPolytope(3);;
     gap> SCBistellarIsManifold(c);
     true
     
  
  
  9.2-7 SCIsMovableComplex
  
  SCIsMovableComplex( complex )  method
  Returns:  true or false upon success, fail otherwise.
  
  Checks  if  a simplicial complex complex can be modified by bistellar moves,
  i.  e.  if  it  is  a  pure  simplicial  complex  which  fulfills  the  weak
  pseudomanifold property with empty boundary.
  
    Example  
     gap> c:=SCBdCrossPolytope(3);;
     gap> SCIsMovableComplex(c);
     true
     
  
  
  Complex with non-empty boundary:
  
    Example  
     gap> c:=SC([[1,2],[2,3],[3,4],[3,1]]);;
     gap> SCIsMovableComplex(c);
     false
     
  
  
  9.2-8 SCMove
  
  SCMove( c, move )  method
  Returns:  simplicial  complex of type SCSimplicialComplex upon success, fail
            otherwise.
  
  Applies  the bistellar move move to a simplicial complex c. move is given as
  a  (r+1)-tuple  together with a (d+1-r)-tuple if d is the dimension of c and
  if  move  is  a r-move. See SCRMoves (9.2-10) for detailed information about
  bistellar r-moves.
  
  Note:  move  and  c should be given in standard labeling to ensure a correct
  result.
  
    Example  
     gap> obj:=SC([[1,2],[2,3],[3,4],[4,1]]);
     [SimplicialComplex
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="unnamed complex 5"
      Dim=1
     
     /SimplicialComplex]
     gap> moves:=SCMoves(obj);
     [ [ [ [ 1, 2 ], [  ] ], [ [ 1, 4 ], [  ] ], [ [ 2, 3 ], [  ] ], 
           [ [ 3, 4 ], [  ] ] ], 
       [ [ [ 1 ], [ 2, 4 ] ], [ [ 2 ], [ 1, 3 ] ], [ [ 3 ], [ 2, 4 ] ], 
           [ [ 4 ], [ 1, 3 ] ] ] ]
     gap> obj:=SCMove(obj,last[2][1]);
     [SimplicialComplex
     
      Properties known: Dim, FacetsEx, Name, NumFaces[], SkelExs[], 
                        Vertices.
     
      Name="unnamed complex 6"
      Dim=1
     
     /SimplicialComplex]
     
  
  
  9.2-9 SCMoves
  
  SCMoves( complex )  method
  Returns:  a list of list of pairs of lists upon success, fail otherwise.
  
  See SCRMoves (9.2-10) for further information.
  
    Example  
     gap> c:=SCBdCrossPolytope(3);;
     gap> moves:=SCMoves(c);
     [ [ [ [ 1, 3, 5 ], [  ] ], [ [ 1, 3, 6 ], [  ] ], [ [ 1, 4, 5 ], [  ] ], 
           [ [ 1, 4, 6 ], [  ] ], [ [ 2, 3, 5 ], [  ] ], [ [ 2, 3, 6 ], [  ] ], 
           [ [ 2, 4, 5 ], [  ] ], [ [ 2, 4, 6 ], [  ] ] ], 
       [ [ [ 1, 3 ], [ 5, 6 ] ], [ [ 1, 4 ], [ 5, 6 ] ], [ [ 1, 5 ], [ 3, 4 ] ], 
           [ [ 1, 6 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 5, 6 ] ], [ [ 2, 4 ], [ 5, 6 ] ], 
           [ [ 2, 5 ], [ 3, 4 ] ], [ [ 2, 6 ], [ 3, 4 ] ], [ [ 3, 5 ], [ 1, 2 ] ], 
           [ [ 3, 6 ], [ 1, 2 ] ], [ [ 4, 5 ], [ 1, 2 ] ], [ [ 4, 6 ], [ 1, 2 ] ] ]
         , [  ] ]
     
  
  
  9.2-10 SCRMoves
  
  SCRMoves( complex, r )  method
  Returns:  a list of pairs of the form [ list, list ], fail otherwise.
  
  A  bistellar  r-move  of a d-dimensional combinatorial manifold complex is a
  r-face  m_1  together  with  a  d-r-tuple  m_2 where m_1 is a common face of
  exactly (d+1-r) facets and m_2 is not a face of complex.
  
  The  r-move removes all facets containing m_1 and replaces them by the (r+1)
  faces obtained by uniting m_2 with any subset of m_1 of order r.
  
  The resulting complex is PL-homeomorphic to complex.
  
    Example  
     gap> c:=SCBdCrossPolytope(3);;
     gap> moves:=SCRMoves(c,1);
     [ [ [ 1, 3 ], [ 5, 6 ] ], [ [ 1, 4 ], [ 5, 6 ] ], [ [ 1, 5 ], [ 3, 4 ] ], 
       [ [ 1, 6 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 5, 6 ] ], [ [ 2, 4 ], [ 5, 6 ] ], 
       [ [ 2, 5 ], [ 3, 4 ] ], [ [ 2, 6 ], [ 3, 4 ] ], [ [ 3, 5 ], [ 1, 2 ] ], 
       [ [ 3, 6 ], [ 1, 2 ] ], [ [ 4, 5 ], [ 1, 2 ] ], [ [ 4, 6 ], [ 1, 2 ] ] ]
     
  
  
  9.2-11 SCRandomize
  
  SCRandomize( complex[[, rounds][, allowedmoves]] )  function
  Returns:  a simplicial complex upon success, fail otherwise.
  
  Randomizes  the  given simplicial complex complex via bistellar moves chosen
  at  random.  By  passing  the optional array allowedmoves, which has to be a
  dense array of integer values of length SCDim(complex), certain moves can be
  allowed  or  forbidden  in  the  proccess. An entry allowedmoves[i]=1 allows
  (i-1)-moves  and  an  entry  allowedmoves[i]=0  forbids  (i-1)-moves  in the
  randomization process.
  
  With  optional positive integer argument rounds, the amount of randomization
  can  be controlled. The higher the value of rounds, the more bistellar moves
  will  be  randomly  performed  on  complex.  Note  that  the argument rounds
  overrides  the  global setting SCBistellarOptions.MaxIntervalRandomize (this
  value   is   used,   if   rounds   is   not   specified).  Internally  calls
  SCReduceComplexEx (9.2-14).
  
    Example  
     gap> c:=SCRandomize(SCBdSimplex(4));
     [SimplicialComplex
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="Randomized S^3_5"
      Dim=3
     
     /SimplicialComplex]
     gap> c.F;
     [ 16, 65, 98, 49 ]
     
  
  
  9.2-12 SCReduceAsSubcomplex
  
  SCReduceAsSubcomplex( complex1, complex2 )  method
  Returns:  SCBistellarOptions.WriteLevel=0:  a  triple of the form [ boolean,
            simplicial  complex,  rounds  performed  ] upon termination of the
            algorithm.
  
            SCBistellarOptions.WriteLevel=1: A library of simplicial complexes
            with  a  number  of  complexes from the reducing process and (upon
            termination)  a  triple of the form [ boolean, simplicial complex,
            rounds performed ].
  
            SCBistellarOptions.WriteLevel=2:  A mail in case a smaller version
            of  complex1  was  found, a library of simplicial complexes with a
            number   of   complexes   from  the  reducing  process  and  (upon
            termination)  a  triple of the form [ boolean, simplicial complex,
            rounds performed ].
  
            Returns fail upon an error.
  
  Reduces  a  simplicial  complex complex1 (satisfying the weak pseudomanifold
  property  with  empty  boundary)  as a sub-complex of the simplicial complex
  complex2.
  
  Main  application:  Reduce  a  sub-complex  of  the  cross  polytope without
  introducing diagonals.
  
  Internally            calls            SCReduceComplexEx            (9.2-14)
  (complex1,complex2,2,SCIntFunc.SCChooseMove);
  
    Example  
     gap> c:=SCFromFacets([[1,3],[3,5],[4,5],[4,1]]);;
     gap> SCBistellarOptions.WriteLevel:=0;; # do not save complexes                      
     gap> SCReduceAsSubcomplex(c,SCBdCrossPolytope(3));
     [ true, [SimplicialComplex
         
          Properties known: Dim, FacetsEx, Name, Vertices.
         
          Name="unnamed complex 36"
          Dim=1
         
         /SimplicialComplex], 1 ]
  
  
  9.2-13 SCReduceComplex
  
  SCReduceComplex( complex )  method
  Returns:  SCBistellarOptions.WriteLevel=0:  a  triple of the form [ boolean,
            simplicial  complex,  rounds  performed  ] upon termination of the
            algorithm.
  
            SCBistellarOptions.WriteLevel=1: A library of simplicial complexes
            with  a  number  of  complexes from the reducing process and (upon
            termination)  a  triple of the form [ boolean, simplicial complex,
            rounds performed ].
  
            SCBistellarOptions.WriteLevel=2:  A mail in case a smaller version
            of  complex1  was  found, a library of simplicial complexes with a
            number   of   complexes   from  the  reducing  process  and  (upon
            termination)  a  triple of the form [ boolean, simplicial complex,
            rounds performed ].
  
            Returns fail upon an error..
  
  Reduces a pure simplicial complex complex satisfying the weak pseudomanifold
  property  via  bistellar  moves. Internally calls SCReduceComplexEx (9.2-14)
  (complex,SCEmpty(),0,SCIntFunc.SCChooseMove);
  
    Example  
     gap> obj:=SC([[1,2],[2,3],[3,4],[4,5],[5,6],[6,1]]);; # hexagon
     gap> SCBistellarOptions.WriteLevel:=0;; # do not save complexes                      
     gap> tmp := SCReduceComplex(obj);
     [ true, [SimplicialComplex
         
          Properties known: Dim, FacetsEx, Name, Vertices.
         
          Name="unnamed complex 27"
          Dim=1
         
         /SimplicialComplex], 3 ]
     
  
  
  9.2-14 SCReduceComplexEx
  
  SCReduceComplexEx( complex, refComplex, mode, choosemove )  function
  Returns:  SCBistellarOptions.WriteLevel=0:  a  triple of the form [ boolean,
            simplicial complex, rounds ] upon termination of the algorithm.
  
            SCBistellarOptions.WriteLevel=1: A library of simplicial complexes
            with  a  number  of  complexes from the reducing process and (upon
            termination)  a  triple of the form [ boolean, simplicial complex,
            rounds ].
  
            SCBistellarOptions.WriteLevel=2:  A mail in case a smaller version
            of  complex1  was  found, a library of simplicial complexes with a
            number   of   complexes   from  the  reducing  process  and  (upon
            termination)  a  triple of the form [ boolean, simplicial complex,
            rounds ].
  
            Returns fail upon an error.
  
  Reduces a pure simplicial complex complex satisfying the weak pseudomanifold
  property via bistellar moves mode = 0, compares it to the simplicial complex
  refComplex  (mode  = 1) or reduces it as a sub-complex of refComplex (mode =
  2).
  
  choosemove   is   a   function   containing   a   flip  strategy,  see  also
  SCIntFunc.SCChooseMove (9.2-4).
  
  The  currently  smallest  complex  is stored to the variable minComplex, the
  currently smallest f-vector to minF. Note that in general the algorithm will
  not  stop  until the maximum number of rounds is reached. You can adjust the
  maximum  number  of  rounds via the property SCBistellarOptions (9.2-1). The
  number  of  rounds  performed  is  returned in the third entry of the triple
  returned by this function.
  
  This function is called by
  
  1   SCReduceComplex (9.2-13),
  
  2   SCEquivalent (9.2-2),
  
  3   SCReduceAsSubcomplex (9.2-12),
  
  4   SCBistellarIsManifold (9.2-6).
  
  5   SCRandomize (9.2-11).
  
  Please  see SCMailIsPending (15.2-3) for further information about the email
  notification system in case SCBistellarOptions.WriteLevel is set to 2.
  
    Example  
     gap> c:=SCBdCrossPolytope(4);;
     gap> SCBistellarOptions.WriteLevel:=0;; # do not save complexes                      
     gap> SCReduceComplexEx(c,SCEmpty(),0,SCIntFunc.SCChooseMove);
     [ true, [SimplicialComplex
         
          Properties known: Dim, FacetsEx, Name, Vertices.
         
          Name="unnamed complex 13"
          Dim=3
         
         /SimplicialComplex], 7 ]
     gap> SCReduceComplexEx(c,SCEmpty(),0,SCIntFunc.SCChooseMove);
     [ true, [SimplicialComplex
         
          Properties known: Dim, FacetsEx, Name, Vertices.
         
          Name="unnamed complex 18"
          Dim=3
         
         /SimplicialComplex], 9 ]
     gap> SCMailSetAddress("johndoe@somehost");   
     true
     gap> SCMailIsEnabled();                     
     true
     gap> SCReduceComplexEx(c,SCEmpty(),0,SCIntFunc.SCChooseMove);
     [ true, [SimplicialComplex
         
          Properties known: Dim, FacetsEx, Name, Vertices.
         
          Name="unnamed complex 23"
          Dim=3
         
         /SimplicialComplex], 7 ]
     
  
  
  Content of sent mail:
  
    Example  
     Greetings master,
     
     this is simpcomp 2.1.7 running on igt215.mathematik.uni-stuttgart.de
     (Linux igt215 2.6.26-2-amd64 #1 SMP Thu Nov 5 02:23:12 UTC 2009 x86_64
     GNU/Linux), GAP 4.4.12.
     
     I have been working hard for 0 seconds and have a message for you, see below.
     
     #### START MESSAGE ####
     
     SCReduceComplex:
     
     Computed locally minimal complex after 7 rounds:
     
     [SimplicialComplex
     
      Properties known: Boundary, Chi, Date, Dim, F, Faces, Facets, G, H,
      HasBoundary, Homology, IsConnected, IsManifold, IsPM, Name, SCVertices,
      Vertices.
     
      Name="ReducedComplex_5_vertices_7"
      Dim=3
      Chi=0
      F=[ 5, 10, 10, 5 ]
      G=[ 0, 0 ]
      H=[ 1, 1, 1, 1 ]
      HasBoundary=false
      Homology=[ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ]
      IsConnected=true
      IsPM=true
     
     /SimplicialComplex]
     
     ##### END MESSAGE #####
     
     That's all, I hope this is good news! Have a nice day.
     
  
  
  9.2-15 SCReduceComplexFast
  
  SCReduceComplexFast( complex )  function
  Returns:  a simplicial complex upon success, fail otherwise.
  
  Same as SCReduceComplex (9.2-13), but calls an external binary provided with
  the simpcomp package.