Difference between revisions of "ApCoCoA-1:CharP.MNLASolve"

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(New page: <command> <title>CharP.GBasisF2</title> <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description> <synt...)
 
m (insert version info)
 
(7 intermediate revisions by 3 users not shown)
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 +
{{Version|1}}
 
<command>
 
<command>
     <title>CharP.GBasisF2</title>
+
     <title>CharP.MNLASolve</title>
     <short_description>Computing the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description>
+
     <short_description>Computes the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description>
 
<syntax>
 
<syntax>
 
CharP.MNLASolve(F:LIST):LIST
 
CharP.MNLASolve(F:LIST):LIST
Line 9: Line 10:
  
 
<par/>
 
<par/>
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant NLA<tt>-</tt>Algorithm to find the unique zero. The Mutant NLA<tt>-</tt>Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA<tt>-</tt>Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Mutant NLA<tt>-</tt>Algorithm is the NLA<tt>-</tt>Algorithm with mutant strategy. It uses <ref>linalg.EF</ref> for gaussian elimination.
+
This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses Mutant NLA-Algorithm to find the unique zero. The Mutant NLA-Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Mutant NLA-Algorithm is the NLA-Algorithm with mutant strategy. It uses <ref>ApCoCoA-1:LinAlg.EF|LinAlg.EF</ref> for gaussian elimination.
  
  
Line 34: Line 35:
 
----------------------------------------
 
----------------------------------------
 
      
 
      
Finding Variable: x[4]
+
The size of Matrix is:
The size of Matrix is:
+
    No. of Rows=4
No. of Rows=4
+
    No. of Columns=11
No. of Columns=11
+
Applying Gaussian Elimination finding Muatants...
Appling Gaussian Elimination for finding Mutants...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Compeleted
 
Gaussian Elimination Compeleted
The size of Matrix is:
+
Finding Variable: x[4]
No. of Rows=4
+
    The size of Matrix is:
No. of Columns=11
+
    No. of Rows=11
Appling Gaussian Elimination for finding Mutants...
+
    No. of Columns=5
-- CoCoAServer: computing Cpu Time = 0
+
Applying Gaussian Elimination to check solution coordinate...
-------------------------------
 
Gaussian Elimination Compeleted
 
The size of Matrix is:
 
No. of Rows=11
 
No. of Columns=5
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
The size of Matrix is:
+
    The size of Matrix is:
No. of Rows=11
+
    No. of Rows=11
No. of Columns=5
+
    No. of Columns=5
Appling Gaussian Elimination to check solution coordinate...
+
Applying Gaussian Elimination to check solution coordinate...
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
The No. of Mutants found = 0
+
    The size of Matrix is:
The size of Matrix is:
+
    No. of Rows=4
No. of Rows=8
+
    No. of Columns=11
No. of Columns=11
+
Applying Gaussian Elimination finding Muatants...
Appling Gaussian Elimination for finding Mutants...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Compeleted
 
Gaussian Elimination Compeleted
The size of Matrix is:
+
    The No. of Mutants found = 0
No. of Rows=11
+
    The size of Matrix is:
No. of Columns=8
+
  No. of Rows=11
Appling Gaussian Elimination to check solution coordinate...
+
    No. of Columns=9
-- CoCoAServer: computing Cpu Time = 0.015
+
Applying Gaussian Elimination to check solution coordinate...
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
The size of Matrix is:
+
    The size of Matrix is:
No. of Rows=11
+
  No. of Rows=11
No. of Columns=8
+
  No. of Columns=9
Appling Gaussian Elimination to check solution coordinate...
+
Applying Gaussian Elimination to check solution coordinate...
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
The No. of Mutants found = 1
+
    The size of Matrix is:
The size of Matrix is:
+
    No. of Rows=8
No. of Rows=11
+
    No. of Columns=11
No. of Columns=11
+
Applying Gaussian Elimination finding Muatants...
Appling Gaussian Elimination for finding Mutants...
 
-- CoCoAServer: computing Cpu Time = 0.016
 
-------------------------------
 
 
Gaussian Elimination Compeleted
 
Gaussian Elimination Compeleted
The size of Matrix is:
+
    The No. of Mutants found = 1
No. of Rows=11
+
    The size of Matrix is:
No. of Columns=10
+
    No. of Rows=11
Appling Gaussian Elimination to check solution coordinate...
+
    No. of Columns=12
-- CoCoAServer: computing Cpu Time = 0
+
Applying Gaussian Elimination to check solution coordinate...
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
The size of Matrix is:
+
    The size of Matrix is:
No. of Rows=11
+
    No. of Rows=11
No. of Columns=10
+
    No. of Columns=12
Appling Gaussian Elimination to check solution coordinate...
+
Applying Gaussian Elimination to check solution coordinate...
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
 
x[4] = 1
 
x[4] = 1
Finding Variable: x[3]
+
Finding Variable: x[3]
The size of Matrix is:
+
  The size of Matrix is:
No. of Rows=4
+
    No. of Rows=7
No. of Columns=7
+
    No. of Columns=10
Appling Gaussian Elimination for finding Mutants...
+
Applying Gaussian Elimination to check solution coordinate...
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Compeleted
 
The size of Matrix is:
 
No. of Rows=4
 
No. of Columns=7
 
Appling Gaussian Elimination for finding Mutants...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Compeleted
 
The size of Matrix is:
 
No. of Rows=7
 
No. of Columns=5
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Completed.
 
The size of Matrix is:
 
No. of Rows=7
 
No. of Columns=5
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Completed.
 
The No. of Mutants found = 0
 
The size of Matrix is:
 
No. of Rows=7
 
No. of Columns=7
 
Appling Gaussian Elimination for finding Mutants...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Compeleted
 
The size of Matrix is:
 
No. of Rows=7
 
No. of Columns=7
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
 
x[3] = 0
 
x[3] = 0
Finding Variable: x[2]
+
Finding Variable: x[2]
The size of Matrix is:
+
    The size of Matrix is:
No. of Rows=3
+
    No. of Rows=4
No. of Columns=4
+
    No. of Columns=5
Appling Gaussian Elimination for finding Mutants...
+
Applying Gaussian Elimination to check solution coordinate...
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Compeleted
 
The size of Matrix is:
 
No. of Rows=3
 
No. of Columns=4
 
Appling Gaussian Elimination for finding Mutants...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Compeleted
 
The size of Matrix is:
 
No. of Rows=4
 
No. of Columns=4
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
The size of Matrix is:
+
    The size of Matrix is:
No. of Rows=4
+
No. of Rows=4
No. of Columns=4
+
    No. of Columns=5
Appling Gaussian Elimination to check solution coordinate...
+
Applying Gaussian Elimination to check solution coordinate...
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
 
Gaussian Elimination Completed.
 
Gaussian Elimination Completed.
 
x[2] = 1
 
x[2] = 1
 +
 
[0, 1, 0, 1]
 
[0, 1, 0, 1]
 
 
</example>
 
</example>
  
Line 205: Line 125:
 
----------------------------------------
 
----------------------------------------
 
      
 
      
Finding Variable: x[4]
+
    The size of Matrix is:
The size of Matrix is:
+
    No. of Rows=4
No. of Rows=4
+
    No. of Columns=9
No. of Columns=9
+
    Applying Gaussian Elimination finding Muatants...
Appling Gaussian Elimination for finding Mutants...
+
    Gaussian Elimination Compeleted
-- CoCoAServer: computing Cpu Time = 0
+
    Finding Variable: x[4]
-------------------------------
+
    The size of Matrix is:
Gaussian Elimination Compeleted
+
    No. of Rows=9
The size of Matrix is:
+
    No. of Columns=4
No. of Rows=3
+
    Applying Gaussian Elimination to check solution coordinate...
No. of Columns=9
+
    Gaussian Elimination Completed.
Appling Gaussian Elimination for finding Mutants...
+
    The size of Matrix is:
-- CoCoAServer: computing Cpu Time = 0
+
    No. of Rows=9
-------------------------------
+
    No. of Columns=4
Gaussian Elimination Compeleted
+
    Applying Gaussian Elimination to check solution coordinate...
The size of Matrix is:
+
    Gaussian Elimination Completed.
No. of Rows=9
+
    The size of Matrix is:
No. of Columns=4
+
    No. of Rows=3
Appling Gaussian Elimination to check solution coordinate...
+
    No. of Columns=9
-- CoCoAServer: computing Cpu Time = 0
+
    Applying Gaussian Elimination finding Muatants...
-------------------------------
+
    Gaussian Elimination Compeleted
Gaussian Elimination Completed.
+
    The No. of Mutants found = 0
The size of Matrix is:
+
    The size of Matrix is:
No. of Rows=9
+
    No. of Rows=14
No. of Columns=4
+
    No. of Columns=16
Appling Gaussian Elimination to check solution coordinate...
+
    Applying Gaussian Elimination to check solution coordinate...
-- CoCoAServer: computing Cpu Time = 0
+
    Gaussian Elimination Completed.
-------------------------------
+
    The size of Matrix is:
Gaussian Elimination Completed.
+
    No. of Rows=14
The No. of Mutants found = 0
+
    No. of Columns=16
The size of Matrix is:
+
    Applying Gaussian Elimination to check solution coordinate...
No. of Rows=15
+
    Gaussian Elimination Completed.
No. of Columns=14
+
    The size of Matrix is:
Appling Gaussian Elimination for finding Mutants...
+
    No. of Rows=15
-- CoCoAServer: computing Cpu Time = 0
+
    No. of Columns=14
-------------------------------
+
    Applying Gaussian Elimination finding Muatants...
Gaussian Elimination Compeleted
+
    Gaussian Elimination Compeleted
The size of Matrix is:
+
    The No. of Mutants found = 4
No. of Rows=14
+
    The size of Matrix is:
No. of Columns=12
+
    No. of Rows=14
Appling Gaussian Elimination to check solution coordinate...
+
    No. of Columns=28
-- CoCoAServer: computing Cpu Time = 0
+
    Applying Gaussian Elimination to check solution coordinate...
-------------------------------
+
    Gaussian Elimination Completed.
Gaussian Elimination Completed.
+
    The size of Matrix is:
The size of Matrix is:
+
    No. of Rows=14
No. of Rows=14
+
    No. of Columns=28
No. of Columns=12
+
    Applying Gaussian Elimination to check solution coordinate...
Appling Gaussian Elimination to check solution coordinate...
+
    Gaussian Elimination Completed.
-- CoCoAServer: computing Cpu Time = 0
+
    The size of Matrix is:
-------------------------------
+
    No. of Rows=27
Gaussian Elimination Completed.
+
    No. of Columns=14
The No. of Mutants found = 4
+
    Applying Gaussian Elimination finding Muatants...
The size of Matrix is:
+
    Gaussian Elimination Compeleted
No. of Rows=27
+
    The No. of Mutants found = 0
No. of Columns=14
+
    The size of Matrix is:
Appling Gaussian Elimination for finding Mutants...
+
    No. of Rows=14
-- CoCoAServer: computing Cpu Time = 0
+
    No. of Columns=13
-------------------------------
+
    Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Compeleted
+
    Gaussian Elimination Completed.
The size of Matrix is:
+
    The size of Matrix is:
No. of Rows=14
+
    No. of Rows=14
No. of Columns=13
+
    No. of Columns=13
Appling Gaussian Elimination to check solution coordinate...
+
    Applying Gaussian Elimination to check solution coordinate...
-- CoCoAServer: computing Cpu Time = 0
+
    Gaussian Elimination Completed.
-------------------------------
+
    The size of Matrix is:
Gaussian Elimination Completed.
+
    No. of Rows=12
The size of Matrix is:
+
    No. of Columns=14
No. of Rows=14
+
    Applying Gaussian Elimination finding Muatants...
No. of Columns=13
+
    Gaussian Elimination Compeleted
Appling Gaussian Elimination to check solution coordinate...
+
    The No. of Mutants found = 0
-- CoCoAServer: computing Cpu Time = 0
+
    The size of Matrix is:
-------------------------------
+
    No. of Rows=15
Gaussian Elimination Completed.
+
    No. of Columns=20
The No. of Mutants found = 0
+
    Applying Gaussian Elimination to check solution coordinate...
The size of Matrix is:
+
    Gaussian Elimination Completed.
No. of Rows=12
+
    The size of Matrix is:
No. of Columns=14
+
    No. of Rows=15
Appling Gaussian Elimination for finding Mutants...
+
    No. of Columns=20
-- CoCoAServer: computing Cpu Time = 0
+
    Applying Gaussian Elimination to check solution coordinate...
-------------------------------
+
    Gaussian Elimination Completed.
Gaussian Elimination Compeleted
+
    x[4] = NA
The size of Matrix is:
 
No. of Rows=14
 
No. of Columns=13
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0.016
 
-------------------------------
 
Gaussian Elimination Completed.
 
The size of Matrix is:
 
No. of Rows=14
 
No. of Columns=13
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Completed.
 
The No. of Mutants found = 0
 
The size of Matrix is:
 
No. of Rows=19
 
No. of Columns=15
 
Appling Gaussian Elimination for finding Mutants...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Compeleted
 
The size of Matrix is:
 
No. of Rows=15
 
No. of Columns=14
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Completed.
 
The size of Matrix is:
 
No. of Rows=15
 
No. of Columns=14
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Completed.
 
The No. of Mutants found = 0
 
The size of Matrix is:
 
No. of Rows=14
 
No. of Columns=15
 
Appling Gaussian Elimination for finding Mutants...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Compeleted
 
The size of Matrix is:
 
No. of Rows=15
 
No. of Columns=14
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Completed.
 
The size of Matrix is:
 
No. of Rows=15
 
No. of Columns=14
 
Appling Gaussian Elimination to check solution coordinate...
 
-- CoCoAServer: computing Cpu Time = 0
 
-------------------------------
 
Gaussian Elimination Completed.
 
x[4] = NA
 
 
Please Check the uniqueness of solution.
 
Please Check the uniqueness of solution.
 
The Given system of polynomials does not
 
The Given system of polynomials does not
 
seem to have a unique solution or it has
 
seem to have a unique solution or it has
 
no solution over the finite field F2.
 
no solution over the finite field F2.
 
  
 
</example>
 
</example>
Line 356: Line 216:
 
     </description>
 
     </description>
 
     <seealso>
 
     <seealso>
       <see>CharP.MXLSolve</see>
+
       <see>ApCoCoA-1:CharP.MXLSolve|CharP.MXLSolve</see>
     <see>Introduction to CoCoAServer</see>
+
     <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
     <see>Introduction to Groebner Basis in CoCoA</see>
+
     <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see>
     <see>CharP.GBasisF2</see>
+
     <see>ApCoCoA-1:CharP.GBasisF2|CharP.GBasisF2</see>
     <see>CharP.XLSolve</see>
+
     <see>ApCoCoA-1:CharP.XLSolve|CharP.XLSolve</see>
     <see>CharP.IMXLSolve</see>
+
     <see>ApCoCoA-1:CharP.IMXLSolve|CharP.IMXLSolve</see>
     <see>CharP.IMNLASolve</see>
+
     <see>ApCoCoA-1:CharP.IMNLASolve|CharP.IMNLASolve</see>
    <see>CharP.MNLASolve</see>
 
 
 
 
 
 
   </seealso>
 
   </seealso>
  
 
     <types>
 
     <types>
 
       <type>apcocoaserver</type>
 
       <type>apcocoaserver</type>
       <type>ideal</type>
+
       <type>poly_system</type>
      <type>groebner</type>
 
 
     </types>
 
     </types>
  
     <key>charP.GBasisF2</key>
+
     <key>charP.mnlasolve</key>
     <key>GBasisF2</key>
+
     <key>mnlasolve</key>
 
     <key>finite field</key>
 
     <key>finite field</key>
     <wiki-category>Package_charP</wiki-category>
+
     <wiki-category>ApCoCoA-1:Package_charP</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 09:56, 7 October 2020

This article is about a function from ApCoCoA-1.

CharP.MNLASolve

Computes the unique F_2-rational zero of a given polynomial system over F_2.

Syntax

CharP.MNLASolve(F:LIST):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.


This function computes the unique zero in F_2^n of a polynomial system over F_2 . It uses Mutant NLA-Algorithm to find the unique zero. The Mutant NLA-Algorithm generates a sequence of linear systems to solve the given system. The Mutant NLA-Algorithm can find the unique zero only. If the given polynomial system has more than one zeros in F_2^n then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. In fact Mutant NLA-Algorithm is the NLA-Algorithm with mutant strategy. It uses LinAlg.EF for gaussian elimination.


  • @param F: List of polynomials of given system.

  • @return The unique solution of the given system in F_2^n.

Example

Use Z/(2)[x[1..4]];
F:=[
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, 
    x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, 
    x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, 
    x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1
    ];


-- Then we compute the solution with
CharP.MNLASolve(F);

-- And we achieve the following information on the screen together with the solution at the end.
----------------------------------------
    
The size of Matrix is:
     No. of Rows=4
     No. of Columns=11
Applying Gaussian Elimination finding Muatants...
Gaussian Elimination Compeleted
Finding Variable: x[4]
    The size of Matrix is:
    	No. of Rows=11
    	No. of Columns=5
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=11
    	No. of Columns=5
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=4
    	No. of Columns=11
Applying Gaussian Elimination finding Muatants...
Gaussian Elimination Compeleted
    The No. of Mutants found = 0
    The size of Matrix is:
   	No. of Rows=11
    	No. of Columns=9
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
    The size of Matrix is:
  	No. of Rows=11
   	No. of Columns=9
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=8
    	No. of Columns=11
Applying Gaussian Elimination finding Muatants...
Gaussian Elimination Compeleted
    The No. of Mutants found = 1
    The size of Matrix is:
    	No. of Rows=11
    	No. of Columns=12
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=11
    	No. of Columns=12
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
x[4] = 1
Finding Variable: x[3]
   The size of Matrix is:
    	No. of Rows=7
    	No. of Columns=10
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
x[3] = 0
Finding Variable: x[2]
    The size of Matrix is:
    	No. of Rows=4
    	No. of Columns=5
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
    The size of Matrix is:
	No. of Rows=4
    	No. of Columns=5
Applying Gaussian Elimination to check solution coordinate...
Gaussian Elimination Completed.
x[2] = 1

[0, 1, 0, 1]


Example

Use Z/(2)[x[1..4]];
F:=[ 
    x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], 
    x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4],  
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2],  
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]
   ];

-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions 

-- Then we compute the solution with
CharP.MNLASolve(F);

-- And we achieve the following information on the screen.
----------------------------------------
    
     The size of Matrix is:
    	No. of Rows=4
    	No. of Columns=9
    Applying Gaussian Elimination finding Muatants...
    Gaussian Elimination Compeleted
    Finding Variable: x[4]
    The size of Matrix is:
    	No. of Rows=9
    	No. of Columns=4
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=9
    	No. of Columns=4
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=3
    	No. of Columns=9
    Applying Gaussian Elimination finding Muatants...
    Gaussian Elimination Compeleted
    The No. of Mutants found = 0
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=16
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=16
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=15
    	No. of Columns=14
    Applying Gaussian Elimination finding Muatants...
    Gaussian Elimination Compeleted
    The No. of Mutants found = 4
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=28
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=28
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=27
    	No. of Columns=14
    Applying Gaussian Elimination finding Muatants...
    Gaussian Elimination Compeleted
    The No. of Mutants found = 0
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=13
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=13
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=12
    	No. of Columns=14
    Applying Gaussian Elimination finding Muatants...
    Gaussian Elimination Compeleted
    The No. of Mutants found = 0
    The size of Matrix is:
    	No. of Rows=15
    	No. of Columns=20
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    The size of Matrix is:
    	No. of Rows=15
    	No. of Columns=20
    Applying Gaussian Elimination to check solution coordinate...
    Gaussian Elimination Completed.
    x[4] = NA
	Please Check the uniqueness of solution.
	The Given system of polynomials does not
	seem to have a unique solution or it has
	no solution over the finite field F2.


See also

CharP.MXLSolve

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.GBasisF2

CharP.XLSolve

CharP.IMXLSolve

CharP.IMNLASolve