Macaulay2 » Documentation
Packages » GameTheory :: intersectWithCImodel
next | previous | forward | backward | up | index | toc

intersectWithCImodel -- ideal of the intersection of a given variety with the conditional independence model

Description

intersectWithCImodel calculates the ideal of the intersection of the given variety V with the conditional independence model determined by a set of conditional probability statements or an undirected graph. More precisely, the output is the ideal of the closure of the variety given by removing the components in the coordinate hyperplanes from the intersection of the variety V and the conditional independence model.

The input for the conditional independence model can be a set of conditional probability statements or an undirected graph. A single conditional independence statement is a list consisting of three disjoint lists of indices for random variables, e.g. $\{ \{1,2\},\{4\}, \{3\} \}$ which represents the conditional independence statement ``$(X_1, X_2)$ is conditionally independent of $X_4$ given $X_3$''. In the context of game theory, the variable $X_i$ represents the strategy of player $i$.

Given an undirected graph $G$, the conditional independence statements are produced via the globalMarkov function from the GraphicalModels package. A global Markov statement for $G$ is a list $\{A, B, C\}$ of three disjoint lists of vertices of $G$, where the subset $C$ separates the subset $A$ from the subset $B$ in the graph $G$.

i1 : FF = ZZ/32003

o1 = FF

o1 : QuotientRing
i2 : d = {2,2,2};
i3 : X = randomGame(d, CoefficientRing => FF);
i4 : PR = probabilityRing(d, CoefficientRing => FF);
i5 : V = spohnIdeal(PR, X);

o5 : Ideal of PR
i6 : G1 = graph ({}, Singletons => {1,2,3});
i7 : G2 = graph ({{1,2}}, Singletons => {3});
i8 : I1 = intersectWithCImodel(V, G1)

                                                                       
o8 = ideal (p          + 3034p          - 14745p         , p          +
             {1, 0, 1}        {1, 1, 0}         {1, 1, 1}   {1, 0, 0}  
     ------------------------------------------------------------------------
                                                                             
     735p          - 10307p         , p          - 9548p          - 11660p   
         {1, 1, 0}         {1, 1, 1}   {0, 1, 1}        {1, 1, 0}         {1,
     ------------------------------------------------------------------------
                                                                      
          , p          + 6605p          - 4419p         , p          -
     1, 1}   {0, 1, 0}        {1, 1, 0}        {1, 1, 1}   {0, 0, 1}  
     ------------------------------------------------------------------------
                                                                            
     9755p          - 8595p         , p          + 7770p          + 7307p   
          {1, 1, 0}        {1, 1, 1}   {0, 0, 0}        {1, 1, 0}        {1,
     ------------------------------------------------------------------------
             2                                            2
          , p          + 15142p         p          - 9838p         )
     1, 1}   {1, 1, 0}         {1, 1, 0} {1, 1, 1}        {1, 1, 1}

o8 : Ideal of PR
i9 : I2 = intersectWithCImodel(V, G2)

                                                                           
o9 = ideal (p          + 4385p          + 10227p          + 4120p         ,
             {0, 0, 1}        {0, 1, 1}         {1, 0, 1}        {1, 1, 1} 
     ------------------------------------------------------------------------
                                                                            
     p          + 4385p          + 10227p          + 4120p         , p      
      {0, 0, 0}        {0, 1, 0}         {1, 0, 0}        {1, 1, 0}   {1, 0,
     ------------------------------------------------------------------------
                                                                        
       p          - p         p         , p         p          - p      
     1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}    {0, 1,
     ------------------------------------------------------------------------
                                                                             
       p         , p         p          + 848p         p          + 12103p   
     0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}       {0, 1, 1} {1, 0, 1}         {1,
     ------------------------------------------------------------------------
                            2                                              
          p          - 6085p          + 7894p         p          - 3300p   
     0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1, 1}        {0,
     ------------------------------------------------------------------------
                                                                            
          p          - 850p         p          - 10956p         p          +
     1, 1} {1, 1, 1}       {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}  
     ------------------------------------------------------------------------
                                     2                                
     6865p         p          + 8470p         , p         p          +
          {1, 1, 0} {1, 1, 1}        {1, 1, 1}   {0, 1, 1} {1, 0, 0}  
     ------------------------------------------------------------------------
                                                                2          
     848p         p          + 12103p         p          - 6085p          +
         {0, 1, 1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}        {1, 0, 1}  
     ------------------------------------------------------------------------
                                                                            
     7894p         p          - 3300p         p          - 850p         p   
          {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}       {1, 0, 0} {1,
     ------------------------------------------------------------------------
                                                                         2  
           - 10956p         p          + 6865p         p          + 8470p   
     1, 1}         {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {1,
     ------------------------------------------------------------------------
                                         2                                 
          , p         p          + 12103p          - 15038p         p      
     1, 1}   {0, 1, 0} {1, 0, 0}         {1, 0, 0}         {0, 1, 1} {1, 0,
     ------------------------------------------------------------------------
                                          2                                
        + 3534p         p          + 7597p          + 7894p         p      
     1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1,
     ------------------------------------------------------------------------
                                         2                                   
        - 850p         p          + 6865p          - 8785p         p         
     0}       {1, 0, 0} {1, 1, 0}        {1, 1, 0}        {0, 1, 0} {1, 1, 1}
     ------------------------------------------------------------------------
                                                                         
     + 14139p         p          + 5778p         p          + 9818p      
             {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 0,
     ------------------------------------------------------------------------
                                                      2                 
       p          + 11496p         p          - 13888p         , p      
     1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1}   {0, 1,
     ------------------------------------------------------------------------
                         2                                                   
       p          + 4731p          - 15131p         p          + 11961p      
     0} {0, 1, 1}        {0, 1, 1}         {0, 1, 1} {1, 0, 1}         {1, 0,
     ------------------------------------------------------------------------
                          2                                                  
       p          - 13703p          - 11806p         p          - 7596p      
     0} {1, 0, 1}         {1, 0, 1}         {0, 1, 0} {1, 1, 1}        {0, 1,
     ------------------------------------------------------------------------
                                                                          
       p          + 10561p         p          + 8010p         p          -
     1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}  
     ------------------------------------------------------------------------
                                      2           2                 2        
     10049p         p          + 1546p         , p          - 12264p         
           {1, 1, 0} {1, 1, 1}        {1, 1, 1}   {0, 1, 0}         {0, 1, 1}
     ------------------------------------------------------------------------
             2                                                               
     + 11961p          - 8065p         p          - 10563p         p         
             {1, 0, 0}        {0, 1, 1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}
     ------------------------------------------------------------------------
            2                                                                
     - 8689p          - 11806p         p          + 10561p         p         
            {1, 0, 1}         {0, 1, 0} {1, 1, 0}         {1, 0, 0} {1, 1, 0}
     ------------------------------------------------------------------------
             2                                                             
     - 10049p          + 10273p         p          - 10313p         p      
             {1, 1, 0}         {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1,
     ------------------------------------------------------------------------
                                                                            
        + 4458p         p          - 3454p         p          + 11487p      
     1}        {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}         {1, 1,
     ------------------------------------------------------------------------
                         2
       p          + 2116p         )
     0} {1, 1, 1}        {1, 1, 1}

o9 : Ideal of PR

Here is an example where the vertices of the graph need to be relabeled.

i10 : FF = ZZ/32003;
i11 : d = {2,2,2};
i12 : X = randomGame(d, CoefficientRing => FF);
i13 : PR = probabilityRing(d, CoefficientRing => FF);
i14 : V = spohnIdeal(PR, X);

o14 : Ideal of PR
i15 : G1 = graph {{John,Matthew},{Matthew,Sarah}};
i16 : G2 = graph {{a,b},{b,c},{c,a}};
i17 : I1 = intersectWithCImodel(V, G1, {John,Matthew,Sarah})

                                                                           
o17 = ideal (p         p          - p         p         , p         p      
              {0, 1, 1} {1, 1, 0}    {0, 1, 0} {1, 1, 1}   {0, 0, 1} {1, 0,
      -----------------------------------------------------------------------
                                                                             
         - p         p         , p         p          - 7572p         p      
      0}    {0, 0, 0} {1, 0, 1}   {0, 0, 0} {1, 0, 0}        {0, 1, 0} {1, 0,
      -----------------------------------------------------------------------
                                                                            
         - 6520p         p          - 4513p         p          - 4514p      
      0}        {0, 1, 1} {1, 0, 0}        {0, 0, 0} {1, 0, 1}        {0, 0,
      -----------------------------------------------------------------------
                                                                          
        p          - 9722p         p          - 8670p         p          -
      1} {1, 0, 1}        {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}  
      -----------------------------------------------------------------------
                                                                         
      8060p         p          - 10425p         p          - 15633p      
           {0, 0, 0} {1, 1, 0}         {0, 0, 1} {1, 1, 0}         {0, 1,
      -----------------------------------------------------------------------
                                                                           
        p          + 11623p         p          + 9258p         p          -
      0} {1, 1, 0}         {0, 0, 0} {1, 1, 1}        {0, 0, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
                                                                             
      10531p         p          + 5102p         p         , p         p      
            {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}   {0, 0, 0} {0, 1,
      -----------------------------------------------------------------------
                                                                           
         - 15219p         p          - 1624p         p          + 15159p   
      0}         {0, 0, 1} {0, 1, 0}        {0, 0, 0} {0, 1, 1}         {0,
      -----------------------------------------------------------------------
                                                                             
           p          - 12583p         p          - 14208p         p         
      0, 1} {0, 1, 1}         {0, 1, 0} {1, 0, 0}         {0, 1, 1} {1, 0, 0}
      -----------------------------------------------------------------------
                                                                          
      + 11304p         p          + 9679p         p          + 9627p      
              {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}        {0, 0,
      -----------------------------------------------------------------------
                                                                          
        p          - 5593p         p          - 2957p         p          -
      0} {1, 1, 0}        {0, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 1, 0}  
      -----------------------------------------------------------------------
                                                                         
      11073p         p          - 6617p         p          + 10166p      
            {1, 0, 1} {1, 1, 0}        {0, 0, 0} {1, 1, 1}         {0, 0,
      -----------------------------------------------------------------------
                                                                          
        p          + 12802p         p          + 4686p         p         ,
      1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1} 
      -----------------------------------------------------------------------
                                                                           
      p         p          + 8469p         p          - 10163p         p   
       {0, 0, 0} {0, 0, 1}        {0, 0, 1} {0, 1, 0}         {0, 0, 0} {0,
      -----------------------------------------------------------------------
                                                                    
            - 1695p         p          - 14677p         p          +
      1, 1}        {0, 1, 0} {0, 1, 1}         {0, 1, 1} {1, 0, 0}  
      -----------------------------------------------------------------------
                                                                         
      11152p         p          - 7870p         p          + 11151p      
            {0, 0, 0} {1, 0, 1}        {0, 1, 0} {1, 0, 1}         {1, 0,
      -----------------------------------------------------------------------
                                                                          
        p          - 15920p         p          - 256p         p          +
      0} {1, 0, 1}         {0, 0, 1} {1, 1, 0}       {1, 0, 1} {1, 1, 0}  
      -----------------------------------------------------------------------
                                                                        
      12005p         p          - 5611p         p          + 7491p      
            {0, 0, 0} {1, 1, 1}        {0, 1, 0} {1, 1, 1}        {1, 0,
      -----------------------------------------------------------------------
                                                                          
        p          - 3916p         p         , p         p         p      
      0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}   {0, 1, 0} {1, 0, 0} {1, 0,
      -----------------------------------------------------------------------
                          2                                              2  
        p          - 2025p         p         p          - 8855p         p   
      1} {1, 1, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 0} {1,
      -----------------------------------------------------------------------
                                        2                                
           p          + 15298p         p         p          - 2246p      
      0, 1} {1, 1, 0}         {0, 0, 1} {1, 0, 1} {1, 1, 0}        {0, 1,
      -----------------------------------------------------------------------
         2                                    2                              
        p         p          + 5120p         p         p          + 11477p   
      0} {1, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}         {0,
      -----------------------------------------------------------------------
                      2                                    2          
           p         p          + 7638p         p         p          +
      0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1, 0, 1} {1, 1, 0}  
      -----------------------------------------------------------------------
                               2                                     2     
      1268p         p         p          - 14784p         p         p      
           {0, 1, 0} {1, 0, 1} {1, 1, 0}         {1, 0, 0} {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                2         2               2                              
         + 6758p         p          + 426p         p         p          +
      0}        {1, 0, 1} {1, 1, 0}       {0, 1, 1} {1, 0, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                     2                                     2               
      9922p         p         p          + 15161p         p         p      
           {0, 1, 0} {1, 0, 0} {1, 1, 1}         {0, 1, 1} {1, 0, 0} {1, 1,
      -----------------------------------------------------------------------
                2                                                            
         + 4579p         p         p          + 11132p         p         p   
      1}        {0, 0, 1} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {1,
      -----------------------------------------------------------------------
                             2                                          
           p          + 5195p         p         p          - 9636p      
      0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0, 0,
      -----------------------------------------------------------------------
                                                                         
        p         p         p          - 10604p         p         p      
      0} {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {1, 0,
      -----------------------------------------------------------------------
                                                                          2  
        p          + 7191p         p         p         p          - 14257p   
      1} {1, 1, 1}        {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}         {0,
      -----------------------------------------------------------------------
                                                                            
           p         p          - 13121p         p         p         p      
      1, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                                                                2            
         + 12659p         p         p         p          + 7266p         p   
      1}         {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1,
      -----------------------------------------------------------------------
                                       2                                
           p          - 8855p         p         p          + 3518p      
      0, 1} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 0,
      -----------------------------------------------------------------------
         2                                    2                              
        p         p          - 4926p         p         p          - 15176p   
      1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}         {0,
      -----------------------------------------------------------------------
            2                                    2                    
           p         p          - 1233p         p         p          -
      1, 1} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
            2                                                               
      13140p         p         p          - 14057p         p         p      
            {0, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                          2                                                 
        p          + 9623p         p         p          + 7952p         p   
      0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1,
      -----------------------------------------------------------------------
                                       2                              
           p         p          + 3226p         p         p          +
      0, 0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                                                                            
      12897p         p         p         p          - 13896p         p      
            {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {1, 0,
      -----------------------------------------------------------------------
                                                                            
        p         p          - 694p         p         p         p          +
      1} {1, 1, 0} {1, 1, 1}       {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                                                           2               
      11611p         p         p         p          - 6750p         p      
            {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                                    2                                    2  
        p          + 9008p         p         p          - 8064p         p   
      0} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1,
      -----------------------------------------------------------------------
                                       2                                
           p          - 4313p         p         p          - 5828p      
      1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}        {1, 0,
      -----------------------------------------------------------------------
         2                           2         2                       
        p         p          - 10710p         p          + 14163p      
      1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {1, 1, 1}         {0, 0,
      -----------------------------------------------------------------------
                   2               2         2                             
        p         p          + 790p         p          + 8184p         p   
      1} {0, 1, 0} {1, 1, 1}       {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0,
      -----------------------------------------------------------------------
            2                                   2                 2     
           p          - 228p         p         p          - 13861p      
      1, 1} {1, 1, 1}       {0, 1, 0} {0, 1, 1} {1, 1, 1}         {0, 1,
      -----------------------------------------------------------------------
         2                                    2                             
        p          - 3536p         p         p          - 2039p         p   
      1} {1, 1, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 1}        {0, 1, 1} {1,
      -----------------------------------------------------------------------
            2                2         2                                
           p          + 5602p         p          + 1594p         p      
      0, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {0, 0, 0} {1, 0,
      -----------------------------------------------------------------------
         2                                    2                              
        p          - 1643p         p         p          + 13400p         p   
      1} {1, 1, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1,
      -----------------------------------------------------------------------
            2                                    2                      
           p          - 1626p         p         p          + 7660p      
      0, 1} {1, 1, 1}        {0, 1, 1} {1, 0, 1} {1, 1, 1}        {1, 0,
      -----------------------------------------------------------------------
                   2                2         2                              
        p         p          - 6267p         p          + 11901p         p   
      0} {1, 0, 1} {1, 1, 1}        {1, 0, 1} {1, 1, 1}         {0, 0, 1} {1,
      -----------------------------------------------------------------------
            2                                    2                      
           p          - 1563p         p         p          - 2002p      
      1, 0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}        {1, 0,
      -----------------------------------------------------------------------
                   2                                     2                2  
        p         p          + 13912p         p         p          - 1383p   
      0} {1, 1, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1,
      -----------------------------------------------------------------------
            2                          3                           3        
           p          - 5666p         p          - 11175p         p         
      1, 0} {1, 1, 1}        {0, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 1, 1}
      -----------------------------------------------------------------------
                        3                          3                      
      + 12875p         p          + 5080p         p          - 7775p      
              {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 0,
      -----------------------------------------------------------------------
         3                          3                                     
        p          + 9606p         p         , p         p         p      
      1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}   {0, 0, 1} {0, 1, 0} {0, 1,
      -----------------------------------------------------------------------
                           2                                           
        p          - 14243p         p         p          - 15958p      
      1} {1, 0, 1}         {0, 1, 0} {0, 1, 1} {1, 0, 1}         {0, 0,
      -----------------------------------------------------------------------
         2                                    2                             
        p         p          + 6166p         p         p          - 5634p   
      0} {0, 1, 1} {1, 0, 1}        {0, 0, 1} {0, 1, 1} {1, 0, 1}        {0,
      -----------------------------------------------------------------------
            2                           2                              
           p         p          + 14492p         p         p          +
      1, 0} {0, 1, 1} {1, 0, 1}         {0, 1, 1} {1, 0, 0} {1, 0, 1}  
      -----------------------------------------------------------------------
                     2                           2         2          
      2501p         p         p          - 15165p         p          -
           {0, 1, 1} {1, 0, 0} {1, 0, 1}         {0, 0, 1} {1, 0, 1}  
      -----------------------------------------------------------------------
                                2               2         2          
      15250p         p         p          - 406p         p          +
            {0, 0, 1} {0, 1, 0} {1, 0, 1}       {0, 1, 0} {1, 0, 1}  
      -----------------------------------------------------------------------
                               2                                     2     
      3752p         p         p          + 12382p         p         p      
           {0, 0, 0} {0, 1, 1} {1, 0, 1}         {0, 0, 1} {0, 1, 1} {1, 0,
      -----------------------------------------------------------------------
                                     2                2         2          
         - 11217p         p         p          + 6170p         p          -
      1}         {0, 1, 0} {0, 1, 1} {1, 0, 1}        {0, 1, 1} {1, 0, 1}  
      -----------------------------------------------------------------------
                               2                                    2        
      7630p         p         p          + 7038p         p         p         
           {0, 1, 0} {1, 0, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 0} {1, 0, 1}
      -----------------------------------------------------------------------
              2         2                           3                      
      + 15402p         p          - 12723p         p          + 2012p      
              {1, 0, 0} {1, 0, 1}         {0, 0, 0} {1, 0, 1}        {0, 0,
      -----------------------------------------------------------------------
         3                          3                          3          
        p          - 6736p         p          + 8283p         p          -
      1} {1, 0, 1}        {0, 1, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}  
      -----------------------------------------------------------------------
                     3                                                       
      3584p         p          - 3794p         p         p         p         
           {1, 0, 0} {1, 0, 1}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 0}
      -----------------------------------------------------------------------
             2                                     2                         
      - 3223p         p         p          - 13454p         p         p      
             {0, 1, 0} {1, 0, 1} {1, 1, 0}         {1, 0, 0} {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                           2                                    2            
         + 10279p         p         p          - 7844p         p         p   
      0}         {0, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1, 0, 1} {1,
      -----------------------------------------------------------------------
                             2                                    2     
            - 3516p         p         p          + 8958p         p      
      1, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0}        {1, 0, 0} {1, 0,
      -----------------------------------------------------------------------
                          3                           2         2          
        p          - 8268p         p          + 11120p         p          -
      1} {1, 1, 0}        {1, 0, 1} {1, 1, 0}         {0, 0, 1} {1, 1, 0}  
      -----------------------------------------------------------------------
                              2                                    2        
      117p         p         p          + 7933p         p         p         
          {0, 0, 1} {0, 1, 0} {1, 1, 0}        {0, 0, 0} {1, 0, 1} {1, 1, 0}
      -----------------------------------------------------------------------
                                 2                                    2     
      - 4540p         p         p          + 4523p         p         p      
             {0, 0, 1} {1, 0, 1} {1, 1, 0}        {0, 1, 0} {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                                    2                 2         2          
         + 6609p         p         p          + 11413p         p          +
      0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}         {1, 0, 1} {1, 1, 0}  
      -----------------------------------------------------------------------
                      3                          3                      
      14051p         p          + 3378p         p          - 2516p      
            {0, 0, 1} {1, 1, 0}        {1, 0, 1} {1, 1, 0}        {0, 0,
      -----------------------------------------------------------------------
         2                          3                                       
        p         p          + 3051p         p          - 6094p         p   
      1} {0, 1, 0} {1, 1, 1}        {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0,
      -----------------------------------------------------------------------
                                       2                              
           p         p          + 4590p         p         p          +
      1, 0} {0, 1, 1} {1, 1, 1}        {0, 1, 0} {0, 1, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
                      2                           2                         
      10665p         p         p          - 13935p         p         p      
            {0, 1, 0} {0, 1, 1} {1, 1, 1}         {0, 1, 1} {1, 0, 0} {1, 1,
      -----------------------------------------------------------------------
                          2                                    2            
         + 4295p         p         p          - 6110p         p         p   
      1}        {0, 1, 0} {1, 0, 0} {1, 1, 1}        {0, 1, 1} {1, 0, 0} {1,
      -----------------------------------------------------------------------
                   2                                                    
            - 9317p         p         p          + 8038p         p      
      1, 1}        {0, 0, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1,
      -----------------------------------------------------------------------
                                    2                                       
        p         p          - 8809p         p         p          - 5623p   
      0} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0,
      -----------------------------------------------------------------------
                                                                           
           p         p         p          + 3537p         p         p      
      0, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 0,
      -----------------------------------------------------------------------
                                                                          2  
        p          - 14684p         p         p         p          - 8102p   
      1} {1, 1, 1}         {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0,
      -----------------------------------------------------------------------
                                                                            
           p         p          + 15472p         p         p         p      
      1, 1} {1, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                                                                2            
         - 13852p         p         p         p          - 5666p         p   
      1}         {0, 1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1,
      -----------------------------------------------------------------------
                                       2                                 
           p          - 2863p         p         p          - 14919p      
      0, 1} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}         {0, 0,
      -----------------------------------------------------------------------
         2                                    2                            
        p         p          + 1694p         p         p          - 360p   
      1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}       {0,
      -----------------------------------------------------------------------
            2                                    2                    
           p         p          - 9597p         p         p          +
      1, 1} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
            3                           2                              
      11132p         p          + 10999p         p         p          -
            {1, 0, 1} {1, 1, 1}         {0, 0, 1} {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                                                          2               
      13375p         p         p         p          + 121p         p      
            {0, 0, 1} {0, 1, 0} {1, 1, 0} {1, 1, 1}       {0, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                                                                         2  
        p          - 7302p         p         p         p          - 6621p   
      0} {1, 1, 1}        {0, 1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1}        {1,
      -----------------------------------------------------------------------
                                                                            
           p         p          + 16p         p         p         p         
      0, 0} {1, 1, 0} {1, 1, 1}      {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}
      -----------------------------------------------------------------------
                                                                           
      - 14517p         p         p         p          - 14891p         p   
              {0, 0, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 1, 0} {1,
      -----------------------------------------------------------------------
                                                                            
           p         p          + 10497p         p         p         p      
      0, 1} {1, 1, 0} {1, 1, 1}         {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                 2                                               2     
         - 13928p         p         p          - 14933p         p      
      1}         {1, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                                    2                                    2  
        p          + 9570p         p         p          - 2423p         p   
      0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1,
      -----------------------------------------------------------------------
                                        2                           3     
           p          - 11148p         p         p          + 10669p      
      1, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 0} {1, 1, 1}         {1, 1,
      -----------------------------------------------------------------------
                           2         2                                    2  
        p          + 14252p         p          + 4352p         p         p   
      0} {1, 1, 1}         {0, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1, 0} {1,
      -----------------------------------------------------------------------
                   2         2                                    2          
            - 5405p         p          - 2494p         p         p          -
      1, 1}        {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
                               2               2         2                   
      2458p         p         p          + 221p         p          - 8810p   
           {0, 1, 0} {0, 1, 1} {1, 1, 1}       {0, 1, 1} {1, 1, 1}        {0,
      -----------------------------------------------------------------------
                      2                                     2          
           p         p          - 13968p         p         p          +
      1, 0} {1, 0, 0} {1, 1, 1}         {0, 1, 1} {1, 0, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
           2         2                                    2          
      8726p         p          - 7769p         p         p          -
           {1, 0, 0} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
                               2                                    2        
      2338p         p         p          + 3358p         p         p         
           {0, 0, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}
      -----------------------------------------------------------------------
                                  2                                    2     
      - 11821p         p         p          + 2532p         p         p      
              {0, 1, 1} {1, 0, 1} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                 2         2                                    2          
         - 12316p         p          + 5849p         p         p          +
      1}         {1, 0, 1} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                                2                                    2     
      15371p         p         p          + 9917p         p         p      
            {0, 1, 0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                                    2               2         2          
         + 8415p         p         p          - 784p         p          -
      1}        {1, 0, 1} {1, 1, 0} {1, 1, 1}       {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                     3                          3                       
      6420p         p          + 3362p         p          + 13997p      
           {0, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 1, 1}         {0, 1,
      -----------------------------------------------------------------------
         3                          3                           3          
        p          - 9431p         p          + 12750p         p          +
      1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
                      3           2                                    2     
      11941p         p         , p         p         p          + 8301p      
            {1, 1, 0} {1, 1, 1}   {0, 0, 1} {0, 1, 1} {1, 0, 1}        {0, 1,
      -----------------------------------------------------------------------
                                              2                             
        p         p          - 2872p         p         p          + 1676p   
      0} {0, 1, 1} {1, 0, 1}        {0, 0, 0} {0, 1, 1} {1, 0, 1}        {0,
      -----------------------------------------------------------------------
            2                                    2                    
           p         p          + 8028p         p         p          -
      0, 1} {0, 1, 1} {1, 0, 1}        {0, 1, 0} {0, 1, 1} {1, 0, 1}  
      -----------------------------------------------------------------------
           2                                              2                  
      9021p         p         p          - 1907p         p         p         
           {0, 1, 1} {1, 0, 0} {1, 0, 1}        {0, 1, 1} {1, 0, 0} {1, 0, 1}
      -----------------------------------------------------------------------
             2         2                                    2          
      + 5805p         p          + 4603p         p         p          +
             {0, 0, 1} {1, 0, 1}        {0, 0, 1} {0, 1, 0} {1, 0, 1}  
      -----------------------------------------------------------------------
            2         2                                    2          
      14590p         p          + 6650p         p         p          +
            {0, 1, 0} {1, 0, 1}        {0, 0, 0} {0, 1, 1} {1, 0, 1}  
      -----------------------------------------------------------------------
                              2                                     2        
      101p         p         p          + 11496p         p         p         
          {0, 0, 1} {0, 1, 1} {1, 0, 1}         {0, 1, 0} {0, 1, 1} {1, 0, 1}
      -----------------------------------------------------------------------
             2         2                                   2          
      + 9211p         p          - 593p         p         p          +
             {0, 1, 1} {1, 0, 1}       {0, 1, 0} {1, 0, 0} {1, 0, 1}  
      -----------------------------------------------------------------------
                                2               2         2          
      11858p         p         p          + 182p         p          -
            {0, 1, 1} {1, 0, 0} {1, 0, 1}       {1, 0, 0} {1, 0, 1}  
      -----------------------------------------------------------------------
                     3                          3                      
      7855p         p          - 6563p         p          - 1505p      
           {0, 0, 0} {1, 0, 1}        {0, 0, 1} {1, 0, 1}        {0, 1,
      -----------------------------------------------------------------------
         3                          3                           3          
        p          - 2221p         p          + 11818p         p          -
      0} {1, 0, 1}        {0, 1, 1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}  
      -----------------------------------------------------------------------
                                                            2               
      13742p         p         p         p          - 12821p         p      
            {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1, 0}         {0, 1, 0} {1, 0,
      -----------------------------------------------------------------------
                          2                                              2  
        p          - 5391p         p         p          + 7556p         p   
      1} {1, 1, 0}        {1, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 0} {1,
      -----------------------------------------------------------------------
                                       2                                
           p          + 9935p         p         p          + 6379p      
      0, 1} {1, 1, 0}        {0, 0, 1} {1, 0, 1} {1, 1, 0}        {0, 1,
      -----------------------------------------------------------------------
         2                                     2                          3  
        p         p          - 11501p         p         p          - 5097p   
      0} {1, 0, 1} {1, 1, 0}         {1, 0, 0} {1, 0, 1} {1, 1, 0}        {1,
      -----------------------------------------------------------------------
                             2         2                               
           p          + 4765p         p          - 525p         p      
      0, 1} {1, 1, 0}        {0, 0, 1} {1, 1, 0}       {0, 0, 1} {0, 1,
      -----------------------------------------------------------------------
         2                                     2                             
        p          - 12818p         p         p          - 3960p         p   
      0} {1, 1, 0}         {0, 0, 0} {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1,
      -----------------------------------------------------------------------
            2                                    2                      
           p          - 2681p         p         p          - 4809p      
      0, 1} {1, 1, 0}        {0, 1, 0} {1, 0, 1} {1, 1, 0}        {1, 0,
      -----------------------------------------------------------------------
                   2                 2         2                          3  
        p         p          - 14435p         p          - 4239p         p   
      0} {1, 0, 1} {1, 1, 0}         {1, 0, 1} {1, 1, 0}        {0, 0, 1} {1,
      -----------------------------------------------------------------------
                             3                         2                    
            - 9460p         p          + 830p         p         p          -
      1, 0}        {1, 0, 1} {1, 1, 0}       {0, 0, 1} {0, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
           3                          2                              
      3542p         p          + 9178p         p         p          +
           {0, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
                                                           2               
      13401p         p         p         p          + 8812p         p      
            {0, 0, 1} {0, 1, 0} {0, 1, 1} {1, 1, 1}        {0, 1, 0} {0, 1,
      -----------------------------------------------------------------------
                                    2                                   2  
        p          - 3781p         p         p          + 754p         p   
      1} {1, 1, 1}        {0, 0, 1} {0, 1, 1} {1, 1, 1}       {0, 1, 0} {0,
      -----------------------------------------------------------------------
                            2                                          
           p          - 939p         p         p          + 6143p      
      1, 1} {1, 1, 1}       {0, 1, 1} {1, 0, 0} {1, 1, 1}        {0, 1,
      -----------------------------------------------------------------------
         2                                    2                          2  
        p         p          - 9628p         p         p          + 1145p   
      0} {1, 0, 0} {1, 1, 1}        {0, 1, 1} {1, 0, 0} {1, 1, 1}        {0,
      -----------------------------------------------------------------------
                                                                           
           p         p          - 6841p         p         p         p      
      0, 1} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {0, 1, 0} {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                 2                                                           
         - 12922p         p         p          + 5777p         p         p   
      1}         {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 0} {0, 1, 1} {1,
      -----------------------------------------------------------------------
                                                                       
           p          - 13310p         p         p         p          -
      0, 1} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {1, 0, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
                                                           2               
      11476p         p         p         p          - 9476p         p      
            {0, 1, 0} {0, 1, 1} {1, 0, 1} {1, 1, 1}        {0, 1, 1} {1, 0,
      -----------------------------------------------------------------------
                                                                             
        p          + 10793p         p         p         p          - 3645p   
      1} {1, 1, 1}         {0, 1, 0} {1, 0, 0} {1, 0, 1} {1, 1, 1}        {0,
      -----------------------------------------------------------------------
                                                  2                         
           p         p         p          + 14904p         p         p      
      1, 1} {1, 0, 0} {1, 0, 1} {1, 1, 1}         {1, 0, 0} {1, 0, 1} {1, 1,
      -----------------------------------------------------------------------
                          2                                    2            
         + 1585p         p         p          + 7652p         p         p   
      1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {1, 0, 1} {1,
      -----------------------------------------------------------------------
                             2                                     2     
            + 3086p         p         p          + 14919p         p      
      1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}         {0, 1, 1} {1, 0,
      -----------------------------------------------------------------------
                                    2                          3            
        p          - 6168p         p         p          - 7490p         p   
      1} {1, 1, 1}        {1, 0, 0} {1, 0, 1} {1, 1, 1}        {1, 0, 1} {1,
      -----------------------------------------------------------------------
                   2                                                    
            + 3678p         p         p          - 9667p         p      
      1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {0, 1,
      -----------------------------------------------------------------------
                                    2                                        
        p         p          - 3560p         p         p          - 15533p   
      0} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1, 1, 0} {1, 1, 1}         {0,
      -----------------------------------------------------------------------
                                                  2                         
           p         p         p          - 14939p         p         p      
      1, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1}         {1, 0, 0} {1, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                                                                             
         + 12326p         p         p         p          - 8128p         p   
      1}         {0, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 0, 1} {1,
      -----------------------------------------------------------------------
                                                                           
           p         p          - 4053p         p         p         p      
      0, 1} {1, 1, 0} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                                                                2            
         + 14913p         p         p         p          + 2800p         p   
      1}         {1, 0, 0} {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, 0, 1} {1,
      -----------------------------------------------------------------------
                                       2                                
           p          - 5138p         p         p          + 3554p      
      1, 0} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}        {0, 1,
      -----------------------------------------------------------------------
         2                                    2                              
        p         p          + 2257p         p         p          - 10618p   
      0} {1, 1, 0} {1, 1, 1}        {1, 0, 0} {1, 1, 0} {1, 1, 1}         {1,
      -----------------------------------------------------------------------
            2                          3                          2     
           p         p          + 9306p         p          - 3609p      
      0, 1} {1, 1, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}        {0, 0,
      -----------------------------------------------------------------------
         2                                     2                 2     
        p          - 12092p         p         p          - 12825p      
      1} {1, 1, 1}         {0, 0, 1} {0, 1, 0} {1, 1, 1}         {0, 1,
      -----------------------------------------------------------------------
         2                                     2                       
        p          + 14746p         p         p          + 12149p      
      0} {1, 1, 1}         {0, 0, 1} {0, 1, 1} {1, 1, 1}         {0, 1,
      -----------------------------------------------------------------------
                   2               2         2                              
        p         p          - 615p         p          + 13952p         p   
      0} {0, 1, 1} {1, 1, 1}       {0, 1, 1} {1, 1, 1}         {0, 1, 0} {1,
      -----------------------------------------------------------------------
            2                                    2                2     
           p          - 2536p         p         p          + 2032p      
      0, 0} {1, 1, 1}        {0, 1, 1} {1, 0, 0} {1, 1, 1}        {1, 0,
      -----------------------------------------------------------------------
         2                                    2                             
        p          + 4764p         p         p          - 6395p         p   
      0} {1, 1, 1}        {0, 0, 0} {1, 0, 1} {1, 1, 1}        {0, 0, 1} {1,
      -----------------------------------------------------------------------
            2                                    2                      
           p          - 9950p         p         p          - 6007p      
      0, 1} {1, 1, 1}        {0, 1, 0} {1, 0, 1} {1, 1, 1}        {0, 1,
      -----------------------------------------------------------------------
                   2                                     2                2  
        p         p          + 11585p         p         p          + 8863p   
      1} {1, 0, 1} {1, 1, 1}         {1, 0, 0} {1, 0, 1} {1, 1, 1}        {1,
      -----------------------------------------------------------------------
            2                                    2                       
           p          + 6661p         p         p          + 14790p      
      0, 1} {1, 1, 1}        {0, 0, 1} {1, 1, 0} {1, 1, 1}         {0, 1,
      -----------------------------------------------------------------------
                   2                                     2          
        p         p          - 13623p         p         p          -
      0} {1, 1, 0} {1, 1, 1}         {1, 0, 0} {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                                2                2         2          
      11140p         p         p          - 6313p         p          +
            {1, 0, 1} {1, 1, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                     3                           3                      
      7416p         p          + 15703p         p          + 4551p      
           {0, 0, 1} {1, 1, 1}         {0, 1, 0} {1, 1, 1}        {0, 1,
      -----------------------------------------------------------------------
         3                           3                          3          
        p          + 12129p         p          - 6987p         p          +
      1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}  
      -----------------------------------------------------------------------
                     3
      1926p         p         )
           {1, 1, 0} {1, 1, 1}

o17 : Ideal of PR
i18 : I2 = intersectWithCImodel(V, G2, {a,b,c})

o18 = ideal (p         p          - 2364p         p          - 7572p      
              {0, 0, 0} {1, 0, 0}        {0, 0, 1} {1, 0, 0}        {0, 1,
      -----------------------------------------------------------------------
        p          - 6520p         p          - 2149p         p          -
      0} {1, 0, 0}        {0, 1, 1} {1, 0, 0}        {0, 0, 0} {1, 0, 1}  
      -----------------------------------------------------------------------
      4514p         p          - 9722p         p          - 8670p      
           {0, 0, 1} {1, 0, 1}        {0, 1, 0} {1, 0, 1}        {0, 1,
      -----------------------------------------------------------------------
        p          - 8060p         p          - 10425p         p          -
      1} {1, 0, 1}        {0, 0, 0} {1, 1, 0}         {0, 0, 1} {1, 1, 0}  
      -----------------------------------------------------------------------
      15633p         p          - 14581p         p          + 11623p      
            {0, 1, 0} {1, 1, 0}         {0, 1, 1} {1, 1, 0}         {0, 0,
      -----------------------------------------------------------------------
        p          + 9258p         p          + 4050p         p          +
      0} {1, 1, 1}        {0, 0, 1} {1, 1, 1}        {0, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
      5102p         p         , p         p          - 15219p         p      
           {0, 1, 1} {1, 1, 1}   {0, 0, 0} {0, 1, 0}         {0, 0, 1} {0, 1,
      -----------------------------------------------------------------------
         - 1624p         p          + 15159p         p          - 12583p   
      0}        {0, 0, 0} {0, 1, 1}         {0, 0, 1} {0, 1, 1}         {0,
      -----------------------------------------------------------------------
           p          - 14208p         p          + 11304p         p         
      1, 0} {1, 0, 0}         {0, 1, 1} {1, 0, 0}         {0, 1, 0} {1, 0, 1}
      -----------------------------------------------------------------------
      + 9679p         p          + 9627p         p          - 5593p      
             {0, 1, 1} {1, 0, 1}        {0, 0, 0} {1, 1, 0}        {0, 0,
      -----------------------------------------------------------------------
        p          - 2957p         p          - 11073p         p          -
      1} {1, 1, 0}        {1, 0, 0} {1, 1, 0}         {1, 0, 1} {1, 1, 0}  
      -----------------------------------------------------------------------
      6617p         p          + 10166p         p          + 12802p      
           {0, 0, 0} {1, 1, 1}         {0, 0, 1} {1, 1, 1}         {1, 0,
      -----------------------------------------------------------------------
        p          + 4686p         p         , p         p          +
      0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}   {0, 0, 0} {0, 0, 1}  
      -----------------------------------------------------------------------
      8469p         p          - 10163p         p          - 1695p      
           {0, 0, 1} {0, 1, 0}         {0, 0, 0} {0, 1, 1}        {0, 1,
      -----------------------------------------------------------------------
        p          - 4513p         p          - 14677p         p          +
      0} {0, 1, 1}        {0, 0, 1} {1, 0, 0}         {0, 1, 1} {1, 0, 0}  
      -----------------------------------------------------------------------
      15665p         p          - 7870p         p          + 11151p      
            {0, 0, 0} {1, 0, 1}        {0, 1, 0} {1, 0, 1}         {1, 0,
      -----------------------------------------------------------------------
        p          - 15920p         p          + 5919p         p          -
      0} {1, 0, 1}         {0, 0, 1} {1, 1, 0}        {0, 1, 1} {1, 1, 0}  
      -----------------------------------------------------------------------
      256p         p          + 12005p         p          - 11530p      
          {1, 0, 1} {1, 1, 0}         {0, 0, 0} {1, 1, 1}         {0, 1,
      -----------------------------------------------------------------------
        p          + 7491p         p          - 3916p         p         )
      0} {1, 1, 1}        {1, 0, 0} {1, 1, 1}        {1, 1, 0} {1, 1, 1}

o18 : Ideal of PR

Here is an example where the conditional independence relations are given with a List.

i19 : FF = ZZ/32003;
i20 : d = {2,2,2};
i21 : X = randomGame(d, CoefficientRing => FF);
i22 : PR = probabilityRing(d, CoefficientRing => FF);
i23 : V = spohnIdeal(PR, X);

o23 : Ideal of PR
i24 : G = graph ({{1,2}},Singletons => {3});
i25 : L = {{{1,2},{3},{}}};
i26 : I1 = intersectWithCImodel(V, G)

                                                                            
o26 = ideal (p          - 11083p          - 13237p          - 771p         ,
              {0, 0, 1}         {0, 1, 1}         {1, 0, 1}       {1, 1, 1} 
      -----------------------------------------------------------------------
                                                                             
      p          - 11083p          - 13237p          - 771p         , p      
       {0, 0, 0}         {0, 1, 0}         {1, 0, 0}       {1, 1, 0}   {1, 0,
      -----------------------------------------------------------------------
                                                                         
        p          - p         p         , p         p          - p      
      1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}    {0, 1,
      -----------------------------------------------------------------------
                                                                             
        p         , p         p          + 705p         p          + 9353p   
      0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}       {0, 1, 1} {1, 0, 1}        {1,
      -----------------------------------------------------------------------
                             2                                              
           p          - 8959p          + 5062p         p          - 1880p   
      0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1, 1}        {0,
      -----------------------------------------------------------------------
                                                                             
           p          + 15763p         p          + 11084p         p         
      1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}
      -----------------------------------------------------------------------
                                          2                                
      - 12581p         p          - 13990p         , p         p          +
              {1, 1, 0} {1, 1, 1}         {1, 1, 1}   {0, 1, 1} {1, 0, 0}  
      -----------------------------------------------------------------------
                                                                2          
      705p         p          + 9353p         p          - 8959p          +
          {0, 1, 1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}  
      -----------------------------------------------------------------------
                                                                        
      5062p         p          - 1880p         p          + 15763p      
           {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}         {1, 0,
      -----------------------------------------------------------------------
                                                                            
        p          + 11084p         p          - 12581p         p          -
      0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
            2                                      2                       
      13990p         , p         p          + 9353p          + 15023p      
            {1, 1, 1}   {0, 1, 0} {1, 0, 0}        {1, 0, 0}         {0, 1,
      -----------------------------------------------------------------------
                                                       2                   
        p          - 10206p         p          + 11504p          + 5062p   
      1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}         {1, 0, 1}        {0,
      -----------------------------------------------------------------------
                                                          2          
           p          + 15763p         p          - 12581p          +
      1, 0} {1, 1, 0}         {1, 0, 0} {1, 1, 0}         {1, 1, 0}  
      -----------------------------------------------------------------------
                                                                         
      13746p         p          + 13277p         p          + 3210p      
            {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0,
      -----------------------------------------------------------------------
                                                                          
        p          - 5488p         p          - 9216p         p          +
      0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
           2                                      2                       
      6026p         , p         p          - 7547p          + 13612p      
           {1, 1, 1}   {0, 1, 0} {0, 1, 1}        {0, 1, 1}         {0, 1,
      -----------------------------------------------------------------------
                                                      2                      
        p          + 11912p         p          + 2824p          + 5387p      
      1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1,
      -----------------------------------------------------------------------
                                                                           
        p          - 13671p         p          - 4486p         p          -
      0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                                                                  2         
      8494p         p          + 15671p         p          - 6785p         ,
           {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}        {1, 1, 1} 
      -----------------------------------------------------------------------
       2                2                 2                                
      p          + 8131p          + 11912p          + 4574p         p      
       {0, 1, 0}        {0, 1, 1}         {1, 0, 0}        {0, 1, 1} {1, 0,
      -----------------------------------------------------------------------
                                            2                                
         + 1159p         p          - 14795p          + 5387p         p      
      1}        {1, 0, 0} {1, 0, 1}         {1, 0, 1}        {0, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                                            2                                
         - 4486p         p          + 15671p          - 3277p         p      
      0}        {1, 0, 0} {1, 1, 0}         {1, 1, 0}        {0, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                                                                             
         - 9205p         p          + 8997p         p          - 15475p      
      1}        {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}         {1, 0,
      -----------------------------------------------------------------------
                                                       2
        p          + 15686p         p          + 12435p         )
      1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1}

o26 : Ideal of PR
i27 : I2 = intersectWithCImodel(V, L)

                                                                            
o27 = ideal (p          - 11083p          - 13237p          - 771p         ,
              {0, 0, 1}         {0, 1, 1}         {1, 0, 1}       {1, 1, 1} 
      -----------------------------------------------------------------------
                                                                             
      p          - 11083p          - 13237p          - 771p         , p      
       {0, 0, 0}         {0, 1, 0}         {1, 0, 0}       {1, 1, 0}   {1, 0,
      -----------------------------------------------------------------------
                                                                         
        p          - p         p         , p         p          - p      
      1} {1, 1, 0}    {1, 0, 0} {1, 1, 1}   {0, 1, 1} {1, 1, 0}    {0, 1,
      -----------------------------------------------------------------------
                                                                             
        p         , p         p          + 705p         p          + 9353p   
      0} {1, 1, 1}   {0, 1, 0} {1, 0, 1}       {0, 1, 1} {1, 0, 1}        {1,
      -----------------------------------------------------------------------
                             2                                              
           p          - 8959p          + 5062p         p          - 1880p   
      0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1, 0} {1, 1, 1}        {0,
      -----------------------------------------------------------------------
                                                                             
           p          + 15763p         p          + 11084p         p         
      1, 1} {1, 1, 1}         {1, 0, 0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}
      -----------------------------------------------------------------------
                                          2                                
      - 12581p         p          - 13990p         , p         p          +
              {1, 1, 0} {1, 1, 1}         {1, 1, 1}   {0, 1, 1} {1, 0, 0}  
      -----------------------------------------------------------------------
                                                                2          
      705p         p          + 9353p         p          - 8959p          +
          {0, 1, 1} {1, 0, 1}        {1, 0, 0} {1, 0, 1}        {1, 0, 1}  
      -----------------------------------------------------------------------
                                                                        
      5062p         p          - 1880p         p          + 15763p      
           {0, 1, 0} {1, 1, 1}        {0, 1, 1} {1, 1, 1}         {1, 0,
      -----------------------------------------------------------------------
                                                                            
        p          + 11084p         p          - 12581p         p          -
      0} {1, 1, 1}         {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
            2                                      2                       
      13990p         , p         p          + 9353p          + 15023p      
            {1, 1, 1}   {0, 1, 0} {1, 0, 0}        {1, 0, 0}         {0, 1,
      -----------------------------------------------------------------------
                                                       2                   
        p          - 10206p         p          + 11504p          + 5062p   
      1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}         {1, 0, 1}        {0,
      -----------------------------------------------------------------------
                                                          2          
           p          + 15763p         p          - 12581p          +
      1, 0} {1, 1, 0}         {1, 0, 0} {1, 1, 0}         {1, 1, 0}  
      -----------------------------------------------------------------------
                                                                         
      13746p         p          + 13277p         p          + 3210p      
            {0, 1, 0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0,
      -----------------------------------------------------------------------
                                                                          
        p          - 5488p         p          - 9216p         p          +
      0} {1, 1, 1}        {1, 0, 1} {1, 1, 1}        {1, 1, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
           2                                      2                       
      6026p         , p         p          - 7547p          + 13612p      
           {1, 1, 1}   {0, 1, 0} {0, 1, 1}        {0, 1, 1}         {0, 1,
      -----------------------------------------------------------------------
                                                      2                      
        p          + 11912p         p          + 2824p          + 5387p      
      1} {1, 0, 1}         {1, 0, 0} {1, 0, 1}        {1, 0, 1}        {0, 1,
      -----------------------------------------------------------------------
                                                                           
        p          - 13671p         p          - 4486p         p          -
      0} {1, 1, 1}         {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}  
      -----------------------------------------------------------------------
                                                                  2         
      8494p         p          + 15671p         p          - 6785p         ,
           {1, 0, 1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}        {1, 1, 1} 
      -----------------------------------------------------------------------
       2                2                 2                                
      p          + 8131p          + 11912p          + 4574p         p      
       {0, 1, 0}        {0, 1, 1}         {1, 0, 0}        {0, 1, 1} {1, 0,
      -----------------------------------------------------------------------
                                            2                                
         + 1159p         p          - 14795p          + 5387p         p      
      1}        {1, 0, 0} {1, 0, 1}         {1, 0, 1}        {0, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                                            2                                
         - 4486p         p          + 15671p          - 3277p         p      
      0}        {1, 0, 0} {1, 1, 0}         {1, 1, 0}        {0, 1, 0} {1, 1,
      -----------------------------------------------------------------------
                                                                             
         - 9205p         p          + 8997p         p          - 15475p      
      1}        {0, 1, 1} {1, 1, 1}        {1, 0, 0} {1, 1, 1}         {1, 0,
      -----------------------------------------------------------------------
                                                       2
        p          + 15686p         p          + 12435p         )
      1} {1, 1, 1}         {1, 1, 0} {1, 1, 1}         {1, 1, 1}

o27 : Ideal of PR
i28 : I1 == I2

o28 = true

The Verbose=>true option prints the progress of each step in the saturation process - a message is printed after saturating the ideal $V$, the conditional independence ideal $I$, and the sum $V + I$ with respect to each hyperplane of the probability simplex.

i29 : FF = ZZ/32003;
i30 : d = {2,3,2};
i31 : X = randomGame(d, CoefficientRing => FF);
i32 : PR = probabilityRing(d, CoefficientRing => FF);
i33 : V = spohnIdeal(PR, X);

o33 : Ideal of PR
i34 : L = {{{1,2},{3},{}}};
i35 : I = intersectWithCImodel(V, L, Verbose=>true);
Completed step 1 of saturating CI ideal
Completed step 1 of saturating input ideal
Completed step 2 of saturating CI ideal
Completed step 2 of saturating input ideal
Completed step 3 of saturating CI ideal
Completed step 3 of saturating input ideal
Completed step 4 of saturating CI ideal
Completed step 4 of saturating input ideal
Completed step 5 of saturating CI ideal
Completed step 5 of saturating input ideal
Completed step 6 of saturating CI ideal
Completed step 6 of saturating input ideal
Completed step 7 of saturating CI ideal
Completed step 7 of saturating input ideal
Completed step 8 of saturating CI ideal
Completed step 8 of saturating input ideal
Completed step 9 of saturating CI ideal
Completed step 9 of saturating input ideal
Completed step 10 of saturating CI ideal
Completed step 10 of saturating input ideal
Completed step 11 of saturating CI ideal
Completed step 11 of saturating input ideal
Completed step 12 of saturating CI ideal
Completed step 12 of saturating input ideal
Completed step 13 of saturating CI ideal
Completed step 13 of saturating input ideal
Completed step 1 of saturating sum
Completed step 2 of saturating sum
Completed step 3 of saturating sum
Completed step 4 of saturating sum
Completed step 5 of saturating sum
Completed step 6 of saturating sum
Completed step 7 of saturating sum
Completed step 8 of saturating sum
Completed step 9 of saturating sum
Completed step 10 of saturating sum
Completed step 11 of saturating sum
Completed step 12 of saturating sum

o35 : Ideal of PR

See also

Ways to use intersectWithCImodel:

  • intersectWithCImodel(Ideal,Graph)
  • intersectWithCImodel(Ideal,Graph,List)
  • intersectWithCImodel(Ideal,List)
  • intersectWithCImodel(Ideal,List,List)

For the programmer

The object intersectWithCImodel is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/GameTheory.m2:2061:0.