edu.jas.gb

Interface WordReduction<C extends RingElem<C>>

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

Method Summary

Modifier and Type	Method and 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.

Method Detail

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.

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.

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.

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.

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.

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}.

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.

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.

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.

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.

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.

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).

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.

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.

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.