Package edu.jas.gb

Class WordReductionAbstract<C extends RingElem<C>>

java.lang.Object

edu.jas.gb.WordReductionAbstract<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

WordReduction<C>, java.io.Serializable

Direct Known Subclasses:

WordPseudoReductionSeq, WordReductionSeq

public abstract class WordReductionAbstract<C extends RingElem<C>>
extends java.lang.Object
implements WordReduction<C>

Polynomial word reduction abstract class. Implements common S-Polynomial, normalform, module criterion and irreducible set.

Author:

Heinz Kredel

See Also:

Serialized Form

Constructor Summary

Constructors

Constructor	Description
WordReductionAbstract()	Constructor.

Method Summary

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>> Pp, GenWordPolynomial<C> Ap)	Is in Normalform.
boolean	isReducible(java.util.List<GenWordPolynomial<C>> Pp, GenWordPolynomial<C> Ap)	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.
java.util.List<GenWordPolynomial<C>>	normalform(java.util.List<GenWordPolynomial<C>> Pp, java.util.List<GenWordPolynomial<C>> Ap)	Normalform Set.
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.
GenWordPolynomial<C>	SPolynomial(Overlap ol, C a, GenWordPolynomial<C> A, C b, GenWordPolynomial<C> B)	S-Polynomials of non-commutative polynomials.
java.util.List<GenWordPolynomial<C>>	SPolynomials(GenWordPolynomial<C> Ap, GenWordPolynomial<C> Bp)	S-Polynomials of non-commutative polynomials.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface edu.jas.gb.WordReduction

leftNormalform, leftNormalform, normalform, normalform, rightNormalform, rightNormalform

Constructor Detail

WordReductionAbstract

public WordReductionAbstract()

Constructor.

Method Detail

SPolynomials

public java.util.List<GenWordPolynomial<C>> SPolynomials(GenWordPolynomial<C> Ap,
                                                         GenWordPolynomial<C> Bp)

S-Polynomials of non-commutative polynomials.

Specified by:

SPolynomials in interface WordReduction<C extends RingElem<C>>

Parameters:

Ap - word polynomial.

Bp - word polynomial.

Returns:

list of all spol(Ap,Bp) the S-polynomials of Ap and Bp.

SPolynomial

public 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.

Specified by:

SPolynomial in interface WordReduction<C extends RingElem<C>>

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.

SPolynomial

public GenWordPolynomial<C> SPolynomial(Overlap ol,
                                        C a,
                                        GenWordPolynomial<C> A,
                                        C b,
                                        GenWordPolynomial<C> B)

S-Polynomials of non-commutative polynomials.

Parameters:

ol - Overlap tuple.

a - leading base coefficient of B.

A - word polynomial.

b - leading base coefficient of A.

B - word polynomial.

Returns:

list of all spol(Ap,Bp) the S-polynomials of Ap and Bp.

normalform

public java.util.List<GenWordPolynomial<C>> normalform(java.util.List<GenWordPolynomial<C>> Pp,
                                                       java.util.List<GenWordPolynomial<C>> Ap)

Normalform Set.

Specified by:

normalform in interface WordReduction<C extends RingElem<C>>

Parameters:

Ap - polynomial list.

Pp - polynomial list.

Returns:

list of nf(a) with respect to Pp for all a in Ap.

isTopReducible

public boolean isTopReducible(java.util.List<GenWordPolynomial<C>> P,
                              GenWordPolynomial<C> A)

Is top reducible.

Specified by:

isTopReducible in interface WordReduction<C extends RingElem<C>>

Parameters:

A - polynomial.

P - polynomial list.

Returns:

true if A is top reducible with respect to P.

isReducible

public boolean isReducible(java.util.List<GenWordPolynomial<C>> Pp,
                           GenWordPolynomial<C> Ap)

Is reducible.

Specified by:

isReducible in interface WordReduction<C extends RingElem<C>>

Parameters:

Ap - polynomial.

Pp - polynomial list.

Returns:

true if Ap is reducible with respect to Pp.

isNormalform

public boolean isNormalform(java.util.List<GenWordPolynomial<C>> Pp,
                            GenWordPolynomial<C> Ap)

Is in Normalform.

Specified by:

isNormalform in interface WordReduction<C extends RingElem<C>>

Parameters:

Ap - polynomial.

Pp - polynomial list.

Returns:

true if Ap is in normalform with respect to Pp.

isNormalform

public boolean isNormalform(java.util.List<GenWordPolynomial<C>> Pp)

Is in Normalform.

Specified by:

isNormalform in interface WordReduction<C extends RingElem<C>>

Parameters:

Pp - polynomial list.

Returns:

true if each Ap in Pp is in normalform with respect to Pp\{Ap}.

irreducibleSet

public java.util.List<GenWordPolynomial<C>> irreducibleSet(java.util.List<GenWordPolynomial<C>> Pp)

Irreducible set.

Specified by:

irreducibleSet in interface WordReduction<C extends RingElem<C>>

Parameters:

Pp - polynomial list.

Returns:

a list P of monic polynomials which are in normalform wrt. P and with ideal(Pp) = ideal(P).

isReductionNF

public 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.

Specified by:

isReductionNF in interface WordReduction<C extends RingElem<C>>

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.