Package edu.jas.gb
Interface WordReduction<C extends RingElem<C>>
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- Type Parameters:
C
- coefficient type
- All Superinterfaces:
java.io.Serializable
- All Known Subinterfaces:
WordPseudoReduction<C>
- All Known Implementing Classes:
WordPseudoReductionSeq
,WordReductionAbstract
,WordReductionSeq
public interface WordReduction<C extends RingElem<C>> extends java.io.Serializable
Polynomial WordReduction interface. Defines S-Polynomial, normalform, module criterion and irreducible set.- Author:
- Heinz Kredel
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Method Summary
All Methods Instance Methods Abstract Methods Modifier and Type Method Description java.util.List<GenWordPolynomial<C>>
irreducibleSet(java.util.List<GenWordPolynomial<C>> Pp)
Irreducible set.boolean
isNormalform(java.util.List<GenWordPolynomial<C>> Pp)
Is in Normalform.boolean
isNormalform(java.util.List<GenWordPolynomial<C>> P, GenWordPolynomial<C> A)
Is in Normalform.boolean
isReducible(java.util.List<GenWordPolynomial<C>> P, GenWordPolynomial<C> A)
Is reducible.boolean
isReductionNF(java.util.List<GenWordPolynomial<C>> lrow, java.util.List<GenWordPolynomial<C>> rrow, java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap, GenWordPolynomial<C> Np)
Is reduction of normal form.boolean
isTopReducible(java.util.List<GenWordPolynomial<C>> P, GenWordPolynomial<C> A)
Is top reducible.GenWordPolynomial<C>
leftNormalform(java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Normalform with left recording.GenWordPolynomial<C>
leftNormalform(java.util.List<GenWordPolynomial<C>> lrow, java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Normalform with left recording.GenWordPolynomial<C>
normalform(java.util.List<GenWordPolynomial<C>> P, GenWordPolynomial<C> A)
Normalform.java.util.List<GenWordPolynomial<C>>
normalform(java.util.List<GenWordPolynomial<C>> Pp, java.util.List<GenWordPolynomial<C>> Ap)
Normalform Set.GenWordPolynomial<C>
normalform(java.util.List<GenWordPolynomial<C>> lrow, java.util.List<GenWordPolynomial<C>> rrow, java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Normalform with left and right recording.GenWordPolynomial<C>
rightNormalform(java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Right normalform with recording.GenWordPolynomial<C>
rightNormalform(java.util.List<GenWordPolynomial<C>> rrow, java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Right normalform with recording.GenWordPolynomial<C>
SPolynomial(C a, Word l1, GenWordPolynomial<C> A, Word r1, C b, Word l2, GenWordPolynomial<C> B, Word r2)
S-Polynomials of non-commutative polynomials.java.util.List<GenWordPolynomial<C>>
SPolynomials(GenWordPolynomial<C> Ap, GenWordPolynomial<C> Bp)
S-Polynomials of non-commutative polynomials.
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Method Detail
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SPolynomials
java.util.List<GenWordPolynomial<C>> SPolynomials(GenWordPolynomial<C> Ap, GenWordPolynomial<C> Bp)
S-Polynomials of non-commutative polynomials.- Parameters:
Ap
- word polynomial.Bp
- word polynomial.- Returns:
- list of all spol(Ap,Bp) the S-polynomials of Ap and Bp.
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SPolynomial
GenWordPolynomial<C> SPolynomial(C a, Word l1, GenWordPolynomial<C> A, Word r1, C b, Word l2, GenWordPolynomial<C> B, Word r2)
S-Polynomials of non-commutative polynomials.- Parameters:
a
- leading base coefficient of B.l1
- word.A
- word polynomial.r1
- word.b
- leading base coefficient of A.l2
- word.B
- word polynomial.r2
- word.- Returns:
- list of all spol(Ap,Bp) the S-polynomials of Ap and Bp.
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isTopReducible
boolean isTopReducible(java.util.List<GenWordPolynomial<C>> P, GenWordPolynomial<C> A)
Is top reducible. Condition is lt(B) | lt(A) for some B in F.- Parameters:
A
- polynomial.P
- polynomial list.- Returns:
- true if A is top reducible with respect to P.
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isReducible
boolean isReducible(java.util.List<GenWordPolynomial<C>> P, GenWordPolynomial<C> A)
Is reducible.- Parameters:
A
- polynomial.P
- polynomial list.- Returns:
- true if A is reducible with respect to P.
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isNormalform
boolean isNormalform(java.util.List<GenWordPolynomial<C>> P, GenWordPolynomial<C> A)
Is in Normalform.- Parameters:
A
- polynomial.P
- polynomial list.- Returns:
- true if A is in normalform with respect to P.
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isNormalform
boolean isNormalform(java.util.List<GenWordPolynomial<C>> Pp)
Is in Normalform.- Parameters:
Pp
- polynomial list.- Returns:
- true if each A in Pp is in normalform with respect to Pp\{A}.
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normalform
GenWordPolynomial<C> normalform(java.util.List<GenWordPolynomial<C>> P, GenWordPolynomial<C> A)
Normalform.- Parameters:
A
- polynomial.P
- polynomial list.- Returns:
- nf(A) with respect to P.
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normalform
java.util.List<GenWordPolynomial<C>> normalform(java.util.List<GenWordPolynomial<C>> Pp, java.util.List<GenWordPolynomial<C>> Ap)
Normalform Set.- Parameters:
Ap
- polynomial list.Pp
- polynomial list.- Returns:
- list of nf(a) with respect to Pp for all a in Ap.
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normalform
GenWordPolynomial<C> normalform(java.util.List<GenWordPolynomial<C>> lrow, java.util.List<GenWordPolynomial<C>> rrow, java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Normalform with left and right recording.- Parameters:
lrow
- left recording matrix, is modified.rrow
- right recording matrix, is modified.Pp
- a polynomial list for reduction.Ap
- a polynomial.- Returns:
- nf(Pp,Ap), the normal form of Ap wrt. Pp.
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leftNormalform
GenWordPolynomial<C> leftNormalform(java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Normalform with left recording.- Parameters:
Pp
- a polynomial list for reduction.Ap
- a polynomial.- Returns:
- nf(Pp,Ap), the left normal form of Ap wrt. Pp.
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leftNormalform
GenWordPolynomial<C> leftNormalform(java.util.List<GenWordPolynomial<C>> lrow, java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Normalform with left recording.- Parameters:
lrow
- left recording matrix, is modified.Pp
- a polynomial list for reduction.Ap
- a polynomial.- Returns:
- nf(Pp,Ap), the left normal form of Ap wrt. Pp.
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irreducibleSet
java.util.List<GenWordPolynomial<C>> irreducibleSet(java.util.List<GenWordPolynomial<C>> Pp)
Irreducible set.- Parameters:
Pp
- polynomial list.- Returns:
- a list P of polynomials which are in normalform wrt. P and with ideal(Pp) = ideal(P).
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isReductionNF
boolean isReductionNF(java.util.List<GenWordPolynomial<C>> lrow, java.util.List<GenWordPolynomial<C>> rrow, java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap, GenWordPolynomial<C> Np)
Is reduction of normal form.- Parameters:
lrow
- left recording matrix.rrow
- right recording matrix.Pp
- a polynomial list for reduction.Ap
- a polynomial.Np
- nf(Pp,Ap), a normal form of Ap wrt. Pp.- Returns:
- true, if Np + sum( row[i]*Pp[i] ) == Ap, else false.
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rightNormalform
GenWordPolynomial<C> rightNormalform(java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Right normalform with recording.- Parameters:
Pp
- a polynomial list for reduction.Ap
- a polynomial.- Returns:
- nf(Pp,Ap), the right normal form of Ap wrt. Pp.
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rightNormalform
GenWordPolynomial<C> rightNormalform(java.util.List<GenWordPolynomial<C>> rrow, java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)
Right normalform with recording.- Parameters:
rrow
- right recording matrix, is modified.Pp
- a polynomial list for reduction.Ap
- a polynomial.- Returns:
- nf(Pp,Ap), the right normal form of Ap wrt. Pp.
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