Difference between revisions of "ApCoCoA-1:NC.GB"

From ApCoCoAWiki
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<title>NC.GB</title>
 
<title>NC.GB</title>
 
<short_description>
 
<short_description>
−
(Partial) Groebner basis of a finitely generated two-sided ideal over a free associative <tt>K</tt>-algebra.
+
Enumerate a (partial) Groebner basis of a finitely generated two-sided ideal over a free monoid ring (using the Buchberger procedure).
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<par/>
 
<par/>
−
Please set ring environment coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(Ordering), respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
+
Please set ring environment <em>coefficient field</em> <tt>K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NC.SetFp</ref>, <ref>NC.SetX</ref> and <ref>NC.SetOrdering</ref>, respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
 
<itemize>
 
<itemize>
−
<item>@param <em>G</em>: a LIST of non-zero polynomials generating a two-sided ideal in <tt>K&lt;X&gt;</tt>. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</item>
+
<item>@param <em>G</em>: a LIST of non-zero polynomials generating a two-sided ideal in <tt>K&lt;X&gt;</tt>. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]].</item>
 
</itemize>
 
</itemize>
−
About the optional parameters: For most cases we do not know whether there exists a finite Groebner basis. Instead of forcing computer yelling and informing nothing valuable, the function has 3 optional parameters to control the computation. Note that at the moment <em>all</em> of the following 3 additional optional parameters must be used at the same time.
+
About the optional parameters: for most cases we do not know whether or not there exists a finite Groebner basis. Thue, the function has three optional parameters to interrupt the enumerating procedure. Note that at the moment <em>all</em> of the following three optional parameters must be used at the same time.
 
<itemize>
 
<itemize>
−
<item>@param <em>DegreeBound:</em> a positive integer which gives a degree bound during Groebner basis computation. When the degree of normal remainder of some <tt>S-element</tt> reaches DegreeBound, the function stops the loop and returns a partial Groebner basis.</item>
+
<item>@param <em>DegreeBound:</em> a positive integer which gives a degree bound during the enumerating procedure. When the degree of the normal remainder of some <tt>S-polynomial</tt> reaches DegreeBound, the function stops the loop and returns a partial Groebner basis.</item>
−
<item>@param <em>LoopBound:</em> a positive integer which gives a bound for the main loop of Groebner basis computation. When it runs through the main loop LoopBound times, the function stops the loop and returns a partial Groebner basis.</item>
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<item>@param <em>LoopBound:</em> a positive integer which gives a bound for the main loop of the enumerating procedure. When it runs through the main loop LoopBound times, the function stops and returns a partial Groebner basis.</item>
−
<item>@param <em>Flag:</em> a non-negative integer which is a multi-switch for the output of ApCoCoAServer. If Flag=0, the server prints nothing on the screen. If Flag=1, the server prints basic information about computing procedure, such as number of S-elements has been checked and to be checked. If Flag=2, the server additionally prints current partial Groebner basis before each loop as well as the information when Flag=1. Note that the initial idea of Flag is to use it as a tool for debugging and tracing the computing process.</item>
+
<item>@param <em>Flag:</em> a non-negative integer which is a multi-switch for the output of ApCoCoAServer. If Flag=0, the server prints nothing on the screen. If Flag=1, the server prints basic information on the enumerating procedure, such as the number of main loop that has been proceeded, the number of elements in partial Groebner basis, the number of unselected obstructions; the total number of obstructions, the number of selected obstructions, and the number of unnecessary obstructions. If Flag=2, beside the information as FLAG=1, the server also displays explicitly the elements in paritial Groebner basis and current selected <tt>S-polynonial</tt>. Note that the initial idea of using Flag is to trace and debug the enumerating procedure.</item>
−
<item>@return: a LIST of polynomials, which is a Groebner basis if (1)finite Groebner basis exists and (2)the computation doesn't stop due to reach DegreeBound or LoopBound, and is a partial Groebner basis otherwise.</item>
+
<item>@return: a LIST of polynomials, which is a Groebner basis of the two-sided ideal generated by <tt>G</tt> if (1) there exists a finite Groebner basis and (2) the enumerating procedure doesn't terminate due to reaching <tt>DegreeBound</tt> or <tt>LoopBound</tt>, and is a partial Groebner basis otherwise.</item>
 
</itemize>
 
</itemize>
−
 
 
<example>
 
<example>
 
NC.SetX(<quotes>xyzt</quotes>);  
 
NC.SetX(<quotes>xyzt</quotes>);  
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[[1, <quotes>xx</quotes>], [-1, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]],  
 
[[1, <quotes>xx</quotes>], [-1, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]],  
 
[[1, <quotes>ttyy</quotes>], [-1, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [-1, <quotes>ttyx</quotes>]]]
 
[[1, <quotes>ttyy</quotes>], [-1, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [-1, <quotes>ttyx</quotes>]]]
 +
 
-------------------------------
 
-------------------------------
 
NC.SetFp(); -- set default Fp=F2
 
NC.SetFp(); -- set default Fp=F2
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[[1, <quotes>xx</quotes>], [1, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [1, <quotes>tyx</quotes>]],  
 
[[1, <quotes>xx</quotes>], [1, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [1, <quotes>tyx</quotes>]],  
 
[[1, <quotes>ttyy</quotes>], [1, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [1, <quotes>ttyx</quotes>]]]
 
[[1, <quotes>ttyy</quotes>], [1, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [1, <quotes>ttyx</quotes>]]]
 +
 
-------------------------------
 
-------------------------------
 
NC.SetFp(3);
 
NC.SetFp(3);
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[[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [2, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [2, <quotes>tyx</quotes>]],  
 
[[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [2, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [2, <quotes>tyx</quotes>]],  
 
[[1, <quotes>ttyy</quotes>], [2, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [2, <quotes>ttyx</quotes>]]]
 
[[1, <quotes>ttyy</quotes>], [2, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [2, <quotes>ttyx</quotes>]]]
 +
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>

Revision as of 11:44, 7 June 2012

NC.GB

Enumerate a (partial) Groebner basis of a finitely generated two-sided ideal over a free monoid ring (using the Buchberger procedure).

Syntax

NC.GB(G:LIST):LIST
NC.GB(G:LIST, DegreeBound:INT, LoopBound:INT, Flag:INT):LIST

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Please set ring environment coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NC.SetFp, NC.SetX and NC.SetOrdering, respectively, before calling the function. Default coefficient field is Q. Default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

  • @param G: a LIST of non-zero polynomials generating a two-sided ideal in K<X>. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial F=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].

About the optional parameters: for most cases we do not know whether or not there exists a finite Groebner basis. Thue, the function has three optional parameters to interrupt the enumerating procedure. Note that at the moment all of the following three optional parameters must be used at the same time.

  • @param DegreeBound: a positive integer which gives a degree bound during the enumerating procedure. When the degree of the normal remainder of some S-polynomial reaches DegreeBound, the function stops the loop and returns a partial Groebner basis.

  • @param LoopBound: a positive integer which gives a bound for the main loop of the enumerating procedure. When it runs through the main loop LoopBound times, the function stops and returns a partial Groebner basis.

  • @param Flag: a non-negative integer which is a multi-switch for the output of ApCoCoAServer. If Flag=0, the server prints nothing on the screen. If Flag=1, the server prints basic information on the enumerating procedure, such as the number of main loop that has been proceeded, the number of elements in partial Groebner basis, the number of unselected obstructions; the total number of obstructions, the number of selected obstructions, and the number of unnecessary obstructions. If Flag=2, beside the information as FLAG=1, the server also displays explicitly the elements in paritial Groebner basis and current selected S-polynonial. Note that the initial idea of using Flag is to trace and debug the enumerating procedure.

  • @return: a LIST of polynomials, which is a Groebner basis of the two-sided ideal generated by G if (1) there exists a finite Groebner basis and (2) the enumerating procedure doesn't terminate due to reaching DegreeBound or LoopBound, and is a partial Groebner basis otherwise.

Example

NC.SetX(<quotes>xyzt</quotes>); 
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]];
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]];
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]];
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]];
G := [F1, F2,F3,F4]; 
NC.GB(G); -- over Q (default field), LLEX ordering (default ordering)
[[[1, <quotes>yt</quotes>], [-1, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [-1, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [-1, <quotes>ty</quotes>]], 
[[1, <quotes>xx</quotes>], [-1, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]], 
[[1, <quotes>ttyy</quotes>], [-1, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [-1, <quotes>ttyx</quotes>]]]

-------------------------------
NC.SetFp(); -- set default Fp=F2
NC.GB(G); -- over F2, LLEX ordering
[[[1, <quotes>yt</quotes>], [1, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [1, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [1, <quotes>ty</quotes>]], 
[[1, <quotes>xx</quotes>], [1, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [1, <quotes>tyx</quotes>]], 
[[1, <quotes>ttyy</quotes>], [1, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [1, <quotes>ttyx</quotes>]]]

-------------------------------
NC.SetFp(3);
NC.GB(G); -- over F3, LLEX ordering
[[[1, <quotes>yt</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [2, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [2, <quotes>ty</quotes>]], 
[[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], [[1, <quotes>tyy</quotes>], [2, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [2, <quotes>tyx</quotes>]], 
[[1, <quotes>ttyy</quotes>], [2, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [2, <quotes>ttyx</quotes>]]]

-------------------------------

See also

NC.Add

NC.Deg

NC.FindPolynomials

NC.GB

NC.HF

NC.Intersection

NC.IsGB

NC.KernelOfHomomorphism

NC.LC

NC.LT

NC.LTIdeal

NC.MinimalPolynomial

NC.Multiply

NC.NR

NC.ReducedGB

NC.SetFp

NC.SetOrdering

NC.SetRelations

NC.SetRules

NC.SetX

NC.Subtract

NC.UnsetFp

NC.UnsetOrdering

NC.UnsetRelations

NC.UnsetRules

NC.UnsetX

Introduction to CoCoAServer




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