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FourTiTwo.m2
-- -*- coding: utf-8 -*- ---------------------------------------------------- ---------------------------------------------------- -- previous version: 0.2 30Jun08, submitted by Josephine Yu. -- author: Mike Stillman -- -- core -- author: Josephine Yu -- -- all remaining functions; documentation -- Sonja Petrovic -- -- interface for windows; edited documentation; tests -- -- latest major update: 6Jul08; -- small revision in Documentation: 6oct08; SP. ---------------------------------------------------- ---------------------------------------------------- newPackage( "FourTiTwo", Version => "1.0", Date => "February 8, 2009", Authors => { {Name => "Mike Stillman", Email => "mike@math.cornell.edu"}, {Name => "Josephine Yu", Email => "jyu@math.mit.edu"}, {Name => "Sonja Petrovic", Email => "petrovic@psu.edu"} }, Headline => "Interface to 4ti2", Keywords => {"Interfaces"}, Configuration => { "path" => "", "keep files" => true }, PackageExports => {"Polyhedra"}, -- for hilbertBasis DebuggingMode => false ) export { "toBinomial", "getMatrix", "putMatrix", "toricMarkov", "toricGroebner", "toricCircuits", "toricGraver", "toricGraverDegrees", -- "hilbertBasis", -- defined in Polyhedra "InputType", } -- for backward compatibility if not programPaths#?"4ti2" and FourTiTwo#Options#Configuration#"path" != "" then programPaths#"4ti2" = FourTiTwo#Options#Configuration#"path" fourTiTwo = null debugLimit = 5 run4ti2 = (exe, args) -> ( if fourTiTwo === null then fourTiTwo = findProgram("4ti2", "markov -h", Prefix => {(".*", "4ti2-"), -- debian (".*", "4ti2_")}, -- suse AdditionalPaths => {"/usr/lib/4ti2/bin", "/usr/lib64/4ti2/bin"}); -- fedora runProgram(fourTiTwo, exe, args) ) getFilename = () -> ( filename := temporaryFileName(); while fileExists(filename) or fileExists(filename|".mat") or fileExists(filename|".lat") do filename = temporaryFileName(); filename) putMatrix = method() putMatrix(File,Matrix) := (F,B) -> ( B = entries B; numrows := #B; numcols := #B#0; F << numrows << " " << numcols << endl; for i from 0 to numrows-1 do ( for j from 0 to numcols-1 do ( F << B#i#j << " "; ); F << endl; ); ) getMatrix = method() getMatrix String := (filename) -> ( L := lines get filename; l := toString first L; v := value("{" | replace(" +",",",l)|"}"); dimensions := select(v, vi -> vi =!= null); if dimensions#0 == 0 then ( -- matrix has no rows matrix{{}} ) else( L = drop(L,1); --L = select(L, l-> regex("^[:space:]*$",l) === null);--remove blank lines matrix select( apply(L, v -> (w := value("{" | replace(" +",",",v)|"}"); w = select(w, wi -> wi =!= null))), row -> row =!= {} ) )) toBinomial = method() toBinomial(Matrix,Ring) := (M,S) -> ( toBinom := (b) -> ( pos := 1_S; neg := 1_S; scan(#b, i -> if b_i > 0 then pos = pos*S_i^(b_i) else if b_i < 0 then neg = neg*S_i^(-b_i)); pos - neg); ideal apply(entries M, toBinom) ) toricMarkov = method(Options=> {InputType => null, Precision => 64}) toricMarkov Matrix := Matrix => o -> (A) -> ( filename := getFilename(); if debugLevel >= debugLimit then << "using temporary file name " << filename << endl; if o.InputType === "lattice" then F := openOut(filename|".lat") else F = openOut(filename|".mat"); putMatrix(F,A); close F; run4ti2("markov", "-p " | toString o.Precision | " " | rootPath | filename); getMatrix(filename|".mar") ) toricMarkov(Matrix,Ring) := o -> (A,S) -> toBinomial(toricMarkov(A,o), S) toricGroebner = method(Options=>{Weights=>null, Precision => 64}) toricGroebner Matrix := o -> (A) -> ( filename := getFilename(); if debugLevel >= debugLimit then << "using temporary file name " << filename << endl; F := openOut(filename|".mat"); putMatrix(F,A); close F; if o.Weights =!= null then ( cost := concatenate apply(o.Weights, x -> (x|" ")); (filename|".cost") << "1 " << #o.Weights << endl << cost << endl << close; ); run4ti2("groebner", "-p " | toString o.Precision | " " | rootPath | filename); getMatrix(filename|".gro") ) toricGroebner(Matrix,Ring) := o -> (A,S) -> toBinomial(toricGroebner(A,o), S) toricCircuits = method(Options => {Precision => 64}) toricCircuits Matrix := Matrix => (o -> A ->( filename := getFilename(); if debugLevel >= debugLimit then << "using temporary file name " << filename << endl; F := openOut(filename|".mat"); putMatrix(F,A); close F; run4ti2("circuits", "-p " | toString o.Precision | " " | rootPath | filename); getMatrix(filename|".cir") )) toricGraver = method(Options => {Precision => 32}) toricGraver Matrix := Matrix => (o -> A ->( filename := getFilename(); if debugLevel >= debugLimit then << "using temporary file name " << filename << endl; F := openOut(filename|".mat"); putMatrix(F,A); close F; run4ti2("graver", "-q -p " | toString o.Precision | " " | rootPath | filename); getMatrix(filename|".gra") )) toricGraver (Matrix,Ring) := Ideal => (o -> (A,S)->toBinomial(toricGraver(A),S)) -- hilbertBasis is defined in Polyhedra hilbertBasis Matrix := Matrix => { InputType => null, Precision => 32 } >> o -> A -> ( filename := getFilename(); if debugLevel >= debugLimit then << "using temporary file name " << filename << endl; if o.InputType === "lattice" then F := openOut(filename|".lat") else F = openOut(filename|".mat"); putMatrix(F,A); close F; run4ti2("hilbert", "-p " | toString o.Precision | " " | rootPath | filename); getMatrix(filename|".hil") ) rays Matrix := Matrix => { Precision => 64 } >> o -> (A ->( filename := getFilename(); if debugLevel >= debugLimit then << "using temporary file name " << filename << endl; F := openOut(filename|".mat"); putMatrix(F,A); close F; run4ti2("rays", "-p " | toString o.Precision | " " | rootPath | filename); getMatrix(filename|".ray") )) -- SP: the output command interface -- I would like to have a command that gives the list of degrees of Graver/Groebner/Circuit/Markov file; -- the way 4ti2 does this is you tell it the whatever.mar or whatever.cir file and it writes the degrees -- to the screen. -- On the other hand, it doesn't matter because you can ask M2 for those degrees directly! toricGraverDegrees = method(Options => {Precision => 32}) toricGraverDegrees Matrix := Matrix => (o -> A ->( filename := getFilename(); if debugLevel >= debugLimit then << "using temporary file name " << filename << endl; F := openOut(filename|".mat"); putMatrix(F,A); close F; run4ti2("graver", "-p " | toString o.Precision | " " | rootPath | filename); ret := run4ti2("output", "--degrees " | rootPath | filename|".gra"); print ret#"output" )) beginDocumentation() doc /// Key FourTiTwo Headline Interface for 4ti2 Description Text Interfaces most of the functionality of the software {\tt 4ti2} available at @HREF"http://www.4ti2.de/"@. (The user needs to have {\tt 4ti2} installed on his/her machine.) A $d\times n$ integral matrix $A$ (with nonnegative entries) specifies a map from a polynomial ring in d variables to a polynomial ring with n variables by specifying exponents of the variables indexing its columns. For example, if $A$ is a matrix $$\begin{pmatrix} 3&2&1&0\\ 0&1&2&3 \end{pmatrix}$$ the map from $k[s,t]$ to $k[a,b,c,d]$ is given by $(s,t) \mapsto \ (s^3,s^2t,st^2,t^3)$. The toric ideal $I_A$ is the kernel of this map. It is minimally generated by the 2-minors of the matrix $$\begin{pmatrix} x&y&z\\ y&z&w \end{pmatrix}$$ Given the matrix $A$, one can compute its lattice basis ideal specified by the integral basis of the lattice $A$, the toric ideal $I_A$, its Groebner bases, etc. In practice, however, these are nontrivial computational tasks. The software {\tt 4ti2} is very efficient in computing these objects. For more theoretical details (and more generality), see the standard reference: B. Sturmfels, {\bf Gr\"obner bases and convex polytopes.} American Mathematical Society, University Lectures Series, No 8, Providence, Rhode Island, 1996. {\bf Note for cygwin users:} If a problem occurs during package installation and/or loading, it should be fixed by setting the path inside the file {\tt .Macaulay2/init-FourTiTwo.m2} to whatever folder {\tt 4ti2} is installed. For example, if {\tt 4ti2} has been installed in {\tt C:/cygwin/4ti2/win32}, then the line inside the {\tt init-FourTiTwo.m2} file will look like this: {\tt "path" => "C:/cygwin/4ti2/win32/"} . Alternately, the path for {\tt 4ti2} may be set when loading the package using the following command: loadPackage("FourTiTwo", Configuration=>{"path"=>"C:/cygwin/4ti2/win32/"}) assuming that 4ti2 has been installed in C:/cygwin/4ti2/win32. Caveat If the package SimpleDoc is not found when installing {\tt FourTiTwo.m2}, see questions and answers 6, 7, and 8 on the Macaulay 2 web site. ///; doc /// Key getMatrix (getMatrix, String) Headline reads a matrix from a 4ti2-formatted input file Usage getMatrix s Inputs s:String file name Outputs A:Matrix Description Text The file should contain a matrix in the format such as 2 4\break 1 1 1 1\break 1 2 3 4\break The first two numbers are the numbers of rows and columns. SeeAlso putMatrix ///; doc /// Key putMatrix (putMatrix,File,Matrix) Headline writes a matrix into a file formatted for 4ti2 Usage putMatrix(F,A) Inputs F:File A:Matrix Description Text Write the matrix {\tt A} in file {\tt F} in {\tt 4ti2} format. Example A = matrix "1,1,1,1; 1,2,3,4" s = temporaryFileName() F = openOut(s) putMatrix(F,A) close(F) getMatrix(s) SeeAlso getMatrix ///; doc /// Key toBinomial (toBinomial, Matrix, Ring) Headline creates a toric ideal from a given set of exponents of its generators Usage toBinomial(M,R) Inputs M: Matrix R: Ring ring with as least as many generators as the columns of {\tt M} Outputs I: Ideal Description Text Equivalent to "output --binomials" in 4ti2. Returns the ideal in the ring {\tt R} generated by the binomials corresponding to rows of {\tt M}. Example A = matrix "1,1,1,1; 1,2,3,4" B = syz A R = QQ[a..d] toBinomial(transpose B,R) ///; doc /// Key toricGroebner (toricGroebner, Matrix) (toricGroebner, Matrix, Ring) [toricGroebner, Weights] [toricGroebner, Precision] Headline calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" from 4ti2 Usage toricGroebner(A) or toricGroebner(A,R) Inputs A:Matrix whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by {\tt A}. R:Ring ring with as least as many generators as the columns of {\tt A} Precision => {ZZ, String} 32, 64, or "arbitrary", the precision of the integers used during the computation Outputs B:Matrix whose rows give binomials that form a Groebner basis of the toric ideal of {\tt A} I:Ideal whose generators form a Groebner basis for the toric ideal Description Example A = matrix "1,1,1,1; 1,2,3,4" toricGroebner(A) Text Note that the output of the command is a matrix whose rows are the exponents of the binomials that for a Groebner basis of the toric ideal $I_A$. As a shortcut, one can ask for the output to be an ideal instead: Example R = QQ[a..d] toricGroebner(A,R) Text {\tt 4ti2} offers the use of weight vectors representing term orders, as follows: Example toricGroebner(A,Weights=>{1,2,3,4}) Caveat It seems that some versions of 4ti2 do not pick up on the weight vector. It may be better to run gb computation in M2 directly with specified weights. ///; doc /// Key toricMarkov (toricMarkov, Matrix) (toricMarkov, Matrix, Ring) [toricMarkov, InputType] [toricMarkov, Precision] Headline calculates a generating set of the toric ideal I_A, given A; invokes "markov" from 4ti2 Usage toricMarkov(A) or toricMarkov(A, InputType => "lattice") or toricMarkov(A,R) Inputs A:Matrix whose columns parametrize the toric variety; the toric ideal is the kernel of the map defined by {\tt A}. Otherwise, if InputType is set to "lattice", the rows of {\tt A} are a lattice basis and the toric ideal is the saturation of the lattice basis ideal. InputType=>String which is the string "lattice" if rows of {\tt A} specify a lattice basis R:Ring polynomial ring in which the toric ideal $I_A$ should live Precision => {ZZ, String} 32, 64, or "arbitrary", the precision of the integers used during the computation Outputs B:Matrix whose rows form a Markov Basis of the lattice $\{z : z \text{ is integral and } A z = 0\}$ or the lattice spanned by the rows of {\tt A} if the option {\tt InputType => "lattice"} is used Description Text Suppose we would like to comput the toric ideal defining the variety parametrized by the following matrix: Example A = matrix"1,1,1,1;0,1,2,3" Text Since there are 4 columns, the ideal will live in the polynomial ring with 4 variables. Example R = QQ[a..d] M = toricMarkov(A) Text Note that rows of M are the exponents of minimal generators of $I_A$. To get the ideal, we can do the following: Example I = toBinomial(M,R) Text Alternately, we might wish to give a lattice basis ideal instead of the matrix A. The lattice basis will be specified by a matrix, as follows: Example B = syz A N = toricMarkov(transpose B, InputType => "lattice") J = toBinomial(N,R) -- toricMarkov(transpose B, R, InputType => "lattice") Text We can see that the two ideals are equal: Example I == J Text Also, notice that instead of the sequence of commands above, we could have used the following: Example toricMarkov(A,R) ///; doc /// Key toricGraver (toricGraver, Matrix) (toricGraver, Matrix, Ring) [toricGraver, Precision] Headline calculates the Graver basis of the toric ideal; invokes "graver" from 4ti2 Usage toricGraver(A) or toricGraver(A,R) Inputs A:Matrix whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by {\tt A} R:Ring polynomial ring in which the toric ideal $I_A$ should live Precision => {ZZ, String} 32, 64, or "gmp", the precision of the integers used during the computation Outputs B:Matrix whose rows give binomials that form the Graver basis of the toric ideal of {\tt A}, or I:Ideal whose generators form the Graver basis for the toric ideal Description Text The Graver basis for any toric ideal $I_A$ contains (properly) the union of all reduced Groebner basis of $I_A$. Any element in the Graver basis of the ideal is called a primitive binomial. Example A = matrix "1,1,1,1; 1,2,3,4" toricGraver(A) Text If we prefer to store the ideal instead, we may use: Example R = QQ[a..d] toricGraver(A,R) Text Note that this last ideal equals the toric ideal $I_A$ since every Graver basis element is actually in $I_A$. ///; doc /// Key toricGraverDegrees (toricGraverDegrees, Matrix) [toricGraverDegrees, Precision] Headline displays the degrees of all Graver basis elements for the toric ideal I_A Usage toricGraverDegrees(A) Inputs A:Matrix whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by {\tt A} Precision => {ZZ, String} 32, 64, or "gmp", the precision of the integers used during the computation Description Text Equivalent to "output --degrees foo.gra" in 4ti2. Very often the Graver basis consists of too many binomials, and one is only interested in their degrees. In this case, instead of looking at the Graver basis of $I_A$, we may just want to look for the degrees of binomials which show up: Example A = matrix "1,1,1,1; 1,2,3,4" toricGraver(A) -- the Graver basis toricGraverDegrees(A) -- just the degrees Text Note that these are all 1-norms of the vectors. Since $I_A$ is homogeneous, there are 3 binomials of degree 2 (norm 4) and 2 binomials of degree 3 (norm 6). Here is a more complicated example, where one may not want to see the Graver basis elements explicitly. The toric ideal I_M is the ideal of the rational normal scroll S(3,2,3): Example M = matrix "1,1,1,1,1,1,1,1,1,1,1; 1,1,1,1,0,0,0,0,0,0,0; 0,0,0,0,1,1,1,0,0,0,0; 0,0,0,0,0,0,0,1,1,1,1; 1,2,3,4,1,2,3,1,2,3,4" toricGraverDegrees(M) Text Here is another example where with many Graver basis elements. The following matrix is a design matrix for a particular statistical model for a 4-node p1 network; see Fienberg-Petrovic-Rinaldo. Example A = matrix "1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0;0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0;0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,1,0,1;0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,1,1;0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0;1,0,1,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0;0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,1;0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,1,0,1;0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1"; toricGraverDegrees(A) ///; doc /// Key (hilbertBasis, Matrix) [(hilbertBasis, Matrix), InputType] [(hilbertBasis, Matrix), Precision] Headline calculates the Hilbert basis of the cone; invokes "hilbert" from 4ti2 Usage hilbertBasis(A) or hilbertBasis(A, InputType => "lattice") Inputs A:Matrix defining the cone $\{z : Az = 0, z \ge 0 \}$ InputType => String use the string "lattice" if rows of {\tt A} specify a lattice basis Precision => {ZZ, String} 32, 64, or "gmp", the precision of the integers used during the computation Outputs B:Matrix whose rows form the Hilbert basis of the cone $\{z : Az = 0, z \ge 0 \}$ or the cone $\{z A : z \text{ is an integral vector and } z A \ge 0 \}$ if {\tt InputType => "lattice"} is used Description Example A = matrix "1,1,1,1; 1,2,3,4" B = syz A hilbertBasis(transpose B) hilbertBasis(A, InputType => "lattice") ///; doc /// Key (rays, Matrix) Headline calculates the extreme rays of the cone; invokes "rays" from 4ti2 Usage rays(A) Inputs A:Matrix defining the cone $\{z : Az = 0, z \ge 0 \}$ Precision => {ZZ, String} 32, 64, or "arbitrary", the precision of the integers used during the computation Outputs B:Matrix whose rows are the extreme rays of the cone $\{z : Az = 0, z \ge 0 \}$ Description Example A = matrix "1,1,1,1; 1,2,3,4" B = syz A rays(transpose B) ///; doc /// Key toricCircuits (toricCircuits, Matrix) [toricCircuits, Precision] Headline calculates the circuits of the toric ideal; invokes "circuits" from 4ti2 Usage toricCircuits(A) Inputs A:Matrix whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by {\tt A} Precision => {ZZ, String} 32, 64, or "arbitrary", the precision of the integers used during the computation Outputs B:Matrix whose rows form the circuits of A Description Text The circuits are contained in the Graver basis of $I_A$. In fact, they are precisely the primitive binomials in the ideal with minimal support. Example A = matrix "1,1,1,1; 1,2,3,4" C = toricCircuits A Text The ideal generated by the circuits of A in general differs from the toric ideal of A. For example: Example R = QQ[a..d] Icircuit = toBinomial(toricCircuits(A), R) -- this is the circuit ideal of A I = toBinomial(toricMarkov(A), R) I==Icircuit Text The two ideals are not the same. There is a minimal generator of I which is not a circuit: Example a*d-b*c % I -- this binomial is in I: a*d-b*c % Icircuit -- but not in Icircuit: ///; doc /// Key InputType Description Text Put {\tt InputType => "lattice"} as an argument in the functions toricMarkov and hilbertBasis SeeAlso toricMarkov hilbertBasis ///; TEST/// A = matrix "1,1,1,1; 1,2,3,4" M = toricMarkov(A) R = QQ[x_0,x_1,x_2,x_3] I = toBinomial(M,R) Irnc3 = ideal(x_0*x_2-x_1^2,x_1*x_3-x_2^2,x_0*x_3-x_1*x_2) assert(I==Irnc3) /// TEST /// B = matrix "1,-2,1,0; 0,1,-2,1" M = toricMarkov(B, InputType => "lattice") R = QQ[x_0,x_1,x_2,x_3] I = toBinomial(M,R) Irnc3 = ideal(x_0*x_2-x_1^2,x_1*x_3-x_2^2,x_0*x_3-x_1*x_2) assert(I== Irnc3) /// TEST /// R=CC[x_0,x_1,x_2,x_3] A = matrix "1,1,1,1; 1,2,3,4" C = toricCircuits(A) --circuits returned by 4ti2 Icir = toBinomial(C,R) -- circuit ideal returned by 4ti2 Ctrue = matrix{{0,1,-2,1},{1,-2,1,0},{1,0,-3,2},{2,-3,0,1}} --known: all circuits IcirTrue = toBinomial(Ctrue,R) --known: circuit ideal Irnc3 = ideal(x_0*x_2-x_1^2,x_1*x_3-x_2^2,x_0*x_3-x_1*x_2) assert(Icir==IcirTrue) shouldBeFalse = (Icir==Irnc3) assert(shouldBeFalse==false) assert(target C == target Ctrue) assert(source C == source Ctrue) /// TEST /// B = matrix "1,-2,1,0; 0,1,-2,1" R = QQ[a..d] I = toBinomial(B,R) assert(a*c-b^2 % I == 0) assert(a*c-d^2 %I =!= 0) assert(degree I == 4) M = hilbertBasis B assert(numrows M == 3) assert(numcols M == 4) M1 = rays B assert(numrows M1 == 2) assert(numcols M1 == 4) /// TEST/// A = matrix "1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0;0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0;0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,1,0,1;0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,1,1;0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0;1,0,1,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0;0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,1;0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,1,0,1;0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1" M = toricGroebner(A); --note this matrix is the design matrix for the p1 statistical model on 4 nodes using a constant rho. (see fienberg/rinaldo/petrovic; in prep-missing reference). assert(numrows M == 137) assert(numcols M == 18) R = QQ[x_1..x_18] I = toBinomial(M,R); assert(degree I == 192) A1 = matrix "3,2,1,0;0,1,2,3" --one more example R=QQ[x_0..x_3] G4ti2=gens toricGroebner(A1,R) GM2 =gens gb toricMarkov(A1,R) Gtrue=toList flatten entries GM2 G = toList flatten entries G4ti2 apply(0..#Gtrue-1, j-> (isSubset({Gtrue_j},G) )) --checking 4ti2's gb against M2's gb assert(numrows GM2 == numrows G4ti2) assert(numcols GM2 == numcols G4ti2) Rwt=QQ[x_0..x_3,Weights=>{3,2,4,1}] --with wt vector G4ti2=gens toBinomial(toricGroebner(A1,Weights=>{3,2,4,1}),Rwt) GM2=gens gb toricMarkov(A1,Rwt) Gtrue=toList flatten entries GM2 G = toList flatten entries G4ti2 assert( numrows GM2 == numrows G4ti2 ) assert( numcols GM2 == numcols G4ti2 ) apply(0..#Gtrue-1, j-> assert(isSubset({Gtrue_j},G)) ) --checking 4ti2's gb against M2's gb /// TEST/// needsPackage "FourTiTwo" --testing graver A1 = matrix "3,2,1,0;0,1,2,3" R=QQ[x_0..x_3] G = toricGraver(A1) assert( numrows G==5) assert(numcols G==4) Gtrue = toBinomial(matrix{{1,-2,1,0},{0,1,-2,1},{1,-1,-1,1},{2,-3,0,1},{1,0,-3,2}},R) --known: Graver basis Gtrue=toList flatten entries gens Gtrue G = toList flatten entries gens toBinomial(G,R) apply(0..#Gtrue-1, j-> assert(isSubset({Gtrue_j},G)) ) --testing 4ti2 output against by-hand calculation! A = matrix "1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0;0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0;0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,1,0,1;0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,1,1;0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0;1,0,1,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0;0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,1;0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,1,0,1;0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1" M = toricGraver(A); --note this matrix is the design matrix for the p1 statistical model on 4 nodes using a constant rho. (see fienberg/rinaldo/petrovic; in prep-missing reference). assert(numrows M == 7462) assert(numcols M == 18) AS=matrix"1,1,1,1,1,1,1,1;1,1,1,0,0,0,0,0;1,2,3,1,2,3,4,5"--scroll S(2,4) R=QQ[x_1..x_8] G4ti2 = toList flatten entries gens toBinomial(toricGraver(AS),R) assert(#G4ti2 == 82) Gtrue=toList flatten entries gens toBinomial(matrix" 1,-2,1,0,0,0,0,0;1,-1,0,-1,1,0,0,0;2,-2,0,-1,0,1,0,0;3,-3,0,-1,0,0,1,0;4,-4,0,-1,0,0,0,1;3,-3,0,0,-1,0,0,1;2,-2,0,0,0,-1,0,1;1,-1,0,0,0,0,-1,1;2,-2,0,0,-1,0,1,0;1,-1,0,0,0,-1,1,0;1,-1,0,0,-1,1,0,0;0,1,-1,-1,1,0,0,0;0,1,-1,0,0,-1,1,0;0,1,-1,0,0,0,-1,1;0,1,-1,0,-1,1,0,0;1,0,-1,-1,0,1,0,0;1,0,-1,0,-1,0,1,0;1,0,-1,0,0,-1,0,1;2,-1,-1,-1,0,0,1,0;2,-1,-1,0,-1,0,0,1;3,-2,-1,-1,0,0,0,1;0,2,-2,0,-1,0,1,0;1,1,-2,-1,0,0,1,0;0,2,-2,-1,0,1,0,0;2,0,-2,-1,0,0,0,1;0,2,-2,0,0,-1,0,1;1,1,-2,0,-1,0,0,1;0,0,0,1,-1,-1,1,0;0,0,0,1,-1,0,-1,1;0,0,0,1,-2,1,0,0;1,-1,0,1,-2,0,1,0;1,-1,0,1,-1,-1,0,1;2,-2,0,1,-2,0,0,1;0,0,0,0,0,1,-2,1;0,0,0,0,1,-2,1,0;1,-1,0,-1,0,2,-1,0;0,0,0,0,1,-1,-1,1;1,-1,0,-1,0,1,1,-1;1,-1,0,0,-1,0,2,-1;2,-2,0,-1,0,0,2,-1;1,-1,0,0,1,-2,0,1;0,0,0,1,0,-2,0,1;0,1,-1,0,1,-2,0,1;0,1,-1,-1,0,1,1,-1;0,1,-1,-1,0,2,-1,0;1,0,-1,-1,0,0,2,-1;0,1,-1,0,-1,0,2,-1;0,1,-1,1,-1,-1,0,1;0,1,-1,1,-2,0,1,0;1,0,-1,1,-2,0,0,1;0,3,-3,-1,0,0,1,0;1,2,-3,-1,0,0,0,1;0,3,-3,0,-1,0,0,1;0,2,-2,1,-2,0,0,1;0,2,-2,-1,0,0,2,-1;0,4,-4,-1,0,0,0,1;0,0,0,1,-2,0,2,-1;0,0,0,2,-2,-1,0,1;0,0,0,2,-3,0,1,0;1,-1,0,2,-3,0,0,1;0,0,0,1,0,-3,2,0;0,0,0,1,0,-1,-2,2;1,-1,0,-1,0,0,3,-2;0,0,0,0,1,0,-3,2;0,0,0,0,2,-3,0,1;0,1,-1,-1,0,0,3,-2;0,1,-1,2,-3,0,0,1;0,0,0,3,-4,0,0,1;0,0,0,1,0,0,-4,3;1,0,-1,0,-1,1,-1,1;1,0,-1,0,-2,2,0,0;1,0,-1,-1,1,-1,1,0;1,0,-1,-1,1,0,-1,1;1,0,-1,-2,2,0,0,0;1,0,-1,0,0,-2,2,0;1,0,-1,0,0,0,-2,2;2,0,-2,0,-2,1,0,1;2,0,-2,-2,1,0,1,0;2,0,-2,-1,0,-1,2,0;2,0,-2,0,-1,0,-1,2;3,0,-3,-2,0,0,2,0;3,0,-3,0,-2,0,0,2" ,R) apply(0..#Gtrue-1, j-> assert(isSubset({Gtrue_j},G4ti2)) ) -- checking 4ti2 output against by hand input!! /// end restart --load "4ti2.m2" installPackage ("FourTiTwo", RemakeAllDocumentation => true, UserMode=>true) installPackage("FourTiTwo",UserMode=>true,DebuggingMode => true) viewHelp FourTiTwo check FourTiTwo debug FourTiTwo A = matrix{{1,1,1,1},{0,1,2,3}} A = matrix{{1,1,1,1},{0,1,3,4}} B = syz A time toricMarkov A A toricMarkov(A, InputType => "lattice") R = QQ[a..d] time toricGroebner(A) toBinomial(transpose B, R) toricCircuits(A) H = hilbertBasis(A) hilbertBasis(transpose B) toBinomial(H,QQ[x,y]) toricGraver(A) A toricMarkov(A) 7 9 A = matrix" 1,1,1,-1,-1,-1, 0, 0, 0; 1,1,1, 0, 0, 0,-1,-1,-1; 0,1,1,-1, 0, 0,-1, 0, 0; 1,0,1, 0,-1, 0, 0,-1, 0; 1,1,0, 0, 0,-1, 0, 0,-1; 0,1,1, 0,-1, 0, 0, 0,-1; 1,1,0, 0,-1, 0,-1, 0, 0" transpose A toricMarkov transpose A hilbertBasis transpose A toricGraver transpose A toricCircuits transpose A 27 27 A = matrix" 1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0; 0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1; 1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1; 1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1" toricMarkov A R = QQ[x_1..x_27] toricMarkov(A,R) toricGroebner(A,R) gens gb oo I = toBinomial(matrix{{}}, QQ[x]) gens I gens gb I -- Notes from Mike talking with Peter Malkin, 4/21/09 in 1.3.2 (and in 1.3.1): These routines use the structure below: groebner, markov, hilbert, graver, zsolve rays, circuits, qsolve also: minimise, walk, normalform a.mat: m by n a.rel: 1 by m: symbols: >, =, < (means: >= 0, == 0, <= 0) a.sign: 1 by n matrix: 0,1,-1,2 a.sign: 0: x_i unrestricted in sign 1: x_i >= 0 -1: x_i <= 0. 2: x_i is a Graver component a.mat, a.rel, a.sign. groebner, markov, zsolve: also can give a.rhs (1 x n matrix). groebner does this: Ax >= 0, x>=0. Ax-Iy = 0, x >= 0, y >= 0. doc on main page of 4ti2 web site: manual, and the slides. .rhs doesn't work for qsolve though, possibly. for hilbert, graver, zsolve, have the following filter: .ub, .lb can be used to provide upper and lower bounds (lower only for Graver components, or for <= vars). install glpk first, and make sure gmp is visible. ./configure --with-gmp=.... --with-glpk=.... call the different routines as: -p32, -p64, -parb tests are in 4ti2-1.3.2/test email Peter if I have more questions 4ti2 google group: joined. glpk:open source linear programming
Simpan