Spracherkennung für: .tst vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
#############################################################################
##
#W involutive-np.tst GAP4 package IBNP Gareth Evans & Chris Wensley
##
gap> START_TEST( "involutive-np.tst" );
gap> ibnp_infolevel_saved := InfoLevel( InfoIBNP );;
gap> SetInfoLevel( InfoIBNP, 0 );;
gap> ## Section 6.1.1
gap> A3 := Algebra3IBNP;;
gap> a:=A3.1;; b:=A3.2;; c:=A3.3;;
gap> ord := NCMonomialLeftLengthLexicographicOrdering( A3 );;
gap> M6 := [ a*b, a, b*c, a*c, c*b, c^2 ];;
gap> U6 := GM2NMList( M6 );
[ [ 1, 2 ], [ 1 ], [ 2, 3 ], [ 1, 3 ], [ 3, 2 ], [ 3, 3 ] ]
gap> LeftDivision( A3, U6, ord );
[ [ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ] ],
[ [ ], [ ], [ ], [ ], [ ], [ ] ] ]
gap> ## Section 6.1.2
gap> RightDivision( A3, U6, ord );
[ [ [ ], [ ], [ ], [ ], [ ], [ ] ],
[ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ] ] ]
gap> ## Section 6.1.3
gap> M6;
[ (1)*a*b, (1)*a, (1)*b*c, (1)*a*c, (1)*c*b, (1)*c^2 ]
gap> LeftOverlapDivision( A3, U6, ord );
[ [ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ] ],
[ [ 1, 2 ], [ 1 ], [ 1 ], [ 1 ], [ 1, 2 ], [ 1 ] ] ]
gap> ## Section 6.1.4
gap> RightOverlapDivision( A3, U6, ord );
[ [ [ 1 .. 3 ], [ 1 .. 3 ], [ 2 ], [ 1 .. 3 ], [ ], [ ] ],
[ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ] ] ]
gap> ## Section 6.1.5
gap> NoncommutativeDivision := "LeftOverlap";
"LeftOverlap"
gap> ## Section 6.1.6
gap> L3 := [ [ [ [1,2,2], [3] ], [1,-1] ],
> [ [ [2,3,3], [1] ], [1,-1] ],
> [ [ [3,1,1], [2] ], [1,-1] ] ];;
gap> PrintNPList( L3 );
ab^2 - c
bc^2 - a
ca^2 - b
gap> drec := DivisionRecord( A3, L3, ord );
rec( div := "LeftOverlap",
mvars := [ [ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ] ],
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ] ] ],
polys := [ [ [ [ 1, 2, 2 ], [ 3 ] ], [ 1, -1 ] ],
[ [ [ 2, 3, 3 ], [ 1 ] ], [ 1, -1 ] ],
[ [ [ 3, 1, 1 ], [ 2 ] ], [ 1, -1 ] ] ] )
gap> ## Section 6.1.7
gap> ## choose a polynomial to reduce
gap> p := 5*c^2*a^2*b^2 + 6*b^2*c^2*a^2 + 7*a^2*b^2*c^2;;
gap> ## convert to NP format and reduce
gap> Lp := GP2NP( p );
[ [ [ 3, 3, 1, 1, 2, 2 ], [ 2, 2, 3, 3, 1, 1 ], [ 1, 1, 2, 2, 3, 3 ] ],
[ 5, 6, 7 ] ]
gap> Lrp := IPolyReduce( A3, Lp, drec, ord );;
gap> ## convert back to a polynomial
gap> rp := NP2GP( Lrp, A3 );
(5)*c^2*a*c+(6)*b^2*c*b+(7)*a^2*b*a
gap> ## p-rp should now belong to the ideal and reduce to 0
gap> q := p - rp;;
gap> Lq := GP2NP( q );;
gap> Lrq := IPolyReduce( A3, Lq, drec, ord );
[ [ ], [ ] ]
gap> ## Section 6.1.8
gap> logr := LoggedIPolyReduce( A3, Lp, drec, ord );
rec( logs := [ [ [ 5, [ 3, 3, 1 ], [ ] ] ], [ [ 7, [ 1, 1, 2 ], [ ] ] ],
[ [ 6, [ 2, 2, 3 ], [ ] ] ] ],
polys := [ [ [ [ 1, 2, 2 ], [ 3 ] ], [ 1, -1 ] ],
[ [ [ 2, 3, 3 ], [ 1 ] ], [ 1, -1 ] ],
[ [ [ 3, 1, 1 ], [ 2 ] ], [ 1, -1 ] ] ],
result := [ [ [ 3, 3, 1, 3 ], [ 2, 2, 3, 2 ], [ 1, 1, 2, 1 ] ], [ 5, 6, 7 ]
] )
gap> logr.result = Lrp;
true
gap> L := logr.logs;;
gap> p1 := ScalarMulNP( BimulNP( L[1][1][2], L3[1], L[1][1][3] ), L[1][1][1] );;
gap> p2 := ScalarMulNP( BimulNP( L[2][1][2], L3[2], L[2][1][3] ), L[2][1][1] );;
gap> q := AddNP( p1, p2, 1, 1 );;
gap> p3 := ScalarMulNP( BimulNP( L[3][1][2], L3[3], L[3][1][3] ), L[3][1][1] );;
gap> q := AddNP( q, p3, 1, 1 );;
gap> Lp = AddNP( q, Lrp, 1, 1 );
true
gap> ## Section 6.1.9
gap> VerifyLoggedRecordNP( Lp, logr );
true
gap> ## Section 6.1.10
gap> L4 := Concatenation( L3, [Lp] );;
gap> R4 := IAutoreduceNP( A3, L4, ord );;
gap> PrintNPList( R4 );
5c^2ac + 6b^2cb + 7a^2ba
ca^2 - b
bc^2 - a
ab^2 - c
gap> IAutoreduceNP( A3, R4, ord );
true
gap> ## Section 6.2.1
gap> gbas := SGrobner( L3 );;
gap> Length( gbas );
64
gap> ## that's too large an example to continue with, so add a fourth poly
gap> K4 := Concatenation( L3, [ [ [ [1,1,2], [3] ], [1,-1] ] ] );;
gap> PrintNPList( K4 );
ab^2 - c
bc^2 - a
ca^2 - b
a^2b - c
gap> gbas := SGrobner( K4 );;
gap> PrintNPList( gbas );
b - a
c - a
a^3 - a
gap> ## so the only reduced elements are {1,a,a^2} with a^3=a
gap> ibasK := InvolutiveBasis( A3, K4, ord );
rec( div := "LeftOverlap",
mvars :=
[
[ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ],
[ 1 .. 3 ], [ 1 .. 3 ] ],
[ [ 2, 3 ], [ 2, 3 ], [ 2, 3 ], [ 2, 3 ], [ 2, 3 ], [ 2, 3 ], [ 2, 3 ]
] ],
polys := [ [ [ [ 3, 1, 1 ], [ 1 ] ], [ 1, -1 ] ],
[ [ [ 2, 1, 1 ], [ 1 ] ], [ 1, -1 ] ],
[ [ [ 1, 1, 1 ], [ 1 ] ], [ 1, -1 ] ],
[ [ [ 3, 1 ], [ 1, 1 ] ], [ 1, -1 ] ],
[ [ [ 2, 1 ], [ 1, 1 ] ], [ 1, -1 ] ], [ [ [ 3 ], [ 1 ] ], [ 1, -1 ] ],
[ [ [ 2 ], [ 1 ] ], [ 1, -1 ] ] ] )
gap> PrintNPList( ibasK.polys );
ca^2 - a
ba^2 - a
a^3 - a
ca - a^2
ba - a^2
c - a
b - a
gap> Lr := IPolyReduce( A3, p, ibasK, ord );;
gap> PrintNP( Lr );
18a^2
gap> NoncommutativeDivision := "RightOverlap";;
gap> ribasK := InvolutiveBasis( A3, K4, ord );;
gap> PrintNPList( ribasK.polys );
a^3 - a
c - a
b - a
gap> ## note that this different
gap> ## Section 6.3.1
gap> P4 := [ [ [ [1,2], [3] ], [1,-2] ],
> [ [ [2,1], [3] ], [1,-2] ],
> [ [ [1,3], [2] ], [1,-2] ],
> [ [ [3,1], [2] ], [1,-2] ] ];;
gap> PrintNPList( P4 );
ab - 2c
ba - 2c
ac - 2b
ca - 2b
gap> NoncommutativeDivision := "LeftOverlap";;
gap> ibasP := InvolutiveBasisNP( A3, P4, ord );
rec( div := "LeftOverlap",
mvars :=
[
[ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ]\
], [ [ ], [ 2, 3 ], [ 1 ], [ 1 ], [ ], [ 2, 3 ] ] ],
polys := [ [ [ [ 3, 3 ], [ 2, 2 ] ], [ 1, -1 ] ],
[ [ [ 3, 2 ], [ 2, 3 ] ], [ 1, -1 ] ],
[ [ [ 3, 1 ], [ 2 ] ], [ 1, -2 ] ], [ [ [ 2, 1 ], [ 3 ] ], [ 1, -2 ] ],
[ [ [ 1, 3 ], [ 2 ] ], [ 1, -2 ] ], [ [ [ 1, 2 ], [ 3 ] ], [ 1, -2 ] ]
] )
gap> PrintNPList( ibasP.polys );
c^2 - b^2
cb - bc
ca - 2b
ba - 2c
ac - 2b
ab - 2c
gap> ## check that cbc reduces to b^3 and abc reduces to 2b^2
gap> IPolyReduce( A3, GP2NP( c*b*c ), ibasP, ord );
[ [ [ 2, 2, 2 ] ], [ 1 ] ]
gap> IPolyReduce( A3, GP2NP( a*b*c ), ibasP, ord );
[ [ [ 2, 2 ] ], [ 2 ] ]
gap> ## now apply the strong left overlap division - two polynomials are added
gap> NoncommutativeDivision := "StrongLeftOverlap";;
gap> sbasP := InvolutiveBasisNP( A3, P4, ord );
rec( div := "StrongLeftOverlap",
mvars :=
[
[ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ],
[ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ] ],
[ [ ], [ ], [ ], [ 2 ], [ 1 ], [ 1 ], [ ], [ 2 ] ] ],
polys := [ [ [ [ 3, 2, 3 ], [ 2, 2, 2 ] ], [ 1, -1 ] ],
[ [ [ 1, 2, 3 ], [ 2, 2 ] ], [ 1, -2 ] ],
[ [ [ 3, 3 ], [ 2, 2 ] ], [ 1, -1 ] ],
[ [ [ 3, 2 ], [ 2, 3 ] ], [ 1, -1 ] ],
[ [ [ 3, 1 ], [ 2 ] ], [ 1, -2 ] ], [ [ [ 2, 1 ], [ 3 ] ], [ 1, -2 ] ],
[ [ [ 1, 3 ], [ 2 ] ], [ 1, -2 ] ], [ [ [ 1, 2 ], [ 3 ] ], [ 1, -2 ] ]
] )
gap> PrintNPList( sbasP.polys );
cbc - b^3
abc - 2b^2
c^2 - b^2
cb - bc
ca - 2b
ba - 2c
ac - 2b
ab - 2c
gap> ## Section 6.3.2
gap> NoncommutativeDivision := "RightOverlap";;
gap> rbasP := InvolutiveBasisNP( A3, P4, ord );
rec( div := "RightOverlap",
mvars := [ [ [ 2 ], [ 2 ], [ 2 ], [ 2 ], [ 1 ], [ 1 ] ],
[ [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ], [ 1 .. 3 ],
[ 1 .. 3 ] ] ],
polys := [ [ [ [ 3, 3 ], [ 2, 2 ] ], [ 1, -1 ] ],
[ [ [ 3, 2 ], [ 2, 3 ] ], [ 1, -1 ] ],
[ [ [ 3, 1 ], [ 2 ] ], [ 1, -2 ] ], [ [ [ 2, 1 ], [ 3 ] ], [ 1, -2 ] ],
[ [ [ 1, 3 ], [ 2 ] ], [ 1, -2 ] ], [ [ [ 1, 2 ], [ 3 ] ], [ 1, -2 ] ]
] )
gap> NoncommutativeDivision := "StrongRightOverlap";;
gap> srbasP := InvolutiveBasisNP( A3, P4, ord );;
gap> ( rbasP.polys = srbasP.polys ) and ( rbasP.mvars = srbasP.mvars );
true
gap> SetInfoLevel( InfoIBNP, ibnp_infolevel_saved );;
gap> STOP_TEST( "involutive-np.tst", 10000 );
#############################################################################
##
#E involutive-np.tst . . . . . . . . . . . . . . . . . . . . . . . ends here
