def axisSaturate (binomials, i, degree, Verbose=False):
"""
TODO: Think about termination issue
Compute the axis saturation with respect to variable index i, terminating
at a given search degree.
Algorithm:
For each degree:
consider all pairs of monomials both divisible by the variable in index
i and check whether they can be connected, while the divided monomials
can't be connected.
"""
result = [b for b in binomials]
# This is again assumed to be a constant over all binomials, but not
# checked.
length = binomials[0].length
d = 1;
while d < degree:
if Verbose: print ("Now searching for degree " + str (d) + " moves.")
# List monomials of degree d, divisible by i.
mi = monomial(length, exponents = [1 if j==i else 0 for j in range (length)])
divisibleMonomials = listDivisibleMonomials (d+1, mi, length)
# Filter the isolated Monomials:
if Verbose: print ("Filtering isolated monomials")
divisibleMonomials = [m for m in divisibleMonomials if not isIsolated (m, result)]
if Verbose: print ("Done")
if Verbose:
n = len (divisibleMonomials)
numpairs = n*(n-1)/2
print ("There are " + str (numpairs) + " pairs to investigate.")
# Get an iterator of the pairs
pairs = combinations(divisibleMonomials, 2)
counter = 0;
for p in pairs:
counter = counter + 1
if Verbose and counter%100 == 0:
print (str(counter) + " pairs done.")
# Check if they are in the same connected component via the current result
if inSameComponent (p[0], p[1], result):
# Check if the divided monomials were already connected
if not inSameComponent(p[0].divide(mi), p[1].divide(mi), result):
# Found a new element
newbinomial = p[0].divide(p[1])
print ("Found element of the saturation: ")
print newbinomial
result.append (newbinomial)
d=d+1
return result
def findWitness(self, stoppingDegree=False):
""" Stopping degree is the last degree of potential witnesses that will be checked"""
d = 1;
variables=[createVar(i,self.nvars) for i in range(self.nvars)]
while (stoppingDegree == False or (d <= stoppingDegree)):
todo = combinations_with_replacement(variables, d)
for t in todo:
m=monomial(self.nvars)
for tt in t: m = m.multiply(tt)
if self.verbose:
print "Now testing:"
print m
mp = self.markovElement.positivePart().multiply(m)
mn = self.markovElement.negativePart().multiply(m)
if inSameComponent(mp, mn, self.moves, Verbose=self.verbose):
if self.verbose: print "Witness found!"
return m
d = d+1
print "No witness found"
raise NoWitnessFound
""" Analyzing the CI ideal of K23 """ from searching import * from monomial import * from witnessSearch import * nvars=32 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0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0]],[[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1]]] K23Binomials=[monomial(nvars, b[0]).divide(monomial(nvars, b[1])) for b in K23Edges] higherMarkovMoves=[monomial (nvars, h) for h in [ [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,-1,1,1,-1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,-1,1,-1,1,0,0,1,-1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,-1,0,1,0,0,-1,0,1,1,0,-1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,-1,0,1,-1,0,1,0,1,0,-1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,-1,1,0,0,0,0,-1,1,0,0,1,-1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,-1,1,-1,1,0,0,0,0,1,-1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,-1,0,1,0,0,-1,0,1,0,1,0,-1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,-1,0,1,-1,0,1,0,0,1,0,-1], [0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,-1,0,0,0,1,0,0,1,0,-1,0,-1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,-1,0,0,0,1,0,1,-1,0,0,-1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,1,-1,0,0,-1,1,0,0,0,0,-1,0,0,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,1,0,-1,0,-1,0,1,0,0,-1,0,0,0,1,0,0], [0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,-1,0,0], [0,0,0,0,0,0,0,0,0,0,1,-1,0,0,-1,1,-1,0,0,0,1,0,0,0,0,1,0,0,0,-1,0,0], [0,0,0,0,0,0,0,0,0,0,1,-1,0,0,-1,1,-1,1,0,0,1,-1,0,0,0,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,1,-1,0,0,-1,1,0,0,0,0,0,0,0,0,-1,1,0,0,1,-1,0,0], [0,0,0,0,0,0,0,0,0,0,1,-1,0,0,-1,1,0,1,0,0,0,-1,0,0,-1,0,0,0,1,0,0,0], [0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,-1,0,0,0,1,0,0,0,0,1,0,-1,0,-1,0,1],
from monomial import *
from witnessSearch import *
nvars=16
K22Edges=[
[[0, 0, 1, 0, 0, 0, 0, 0, 1, 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, 1, 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, 0, 0, 1, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]],
[[0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]],
[[0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]],
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0]],
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]]]
K22Binomials=[monomial(nvars, b[0]).divide(monomial(nvars, b[1])) for b in K22Edges]
# result = axisSaturate(K22Binomials, 0, 5, Verbose=True)
higherMarkovEdges=[[0,0,0,0,1,-1,-1,1,-1,1,1,-1,0,0,0,0],
[0,0,1,-1,-1,1,0,0,0,0,-1,1,1,-1,0,0],
[0,1,-1,0,0,-1,1,0,0,-1,1,0,0,1,-1,0],
[0,1,0,-1,0,-1,0,1,-1,0,1,0,1,0,-1,0],
[1,-1,-1,1,0,0,0,0,0,0,0,0,-1,1,1,-1],
[1,-1,0,0,0,0,-1,1,-1,1,0,0,0,0,1,-1],
[1,0,-1,0,-1,0,1,0,0,-1,0,1,0,1,0,-1],
[1,0,0,-1,-1,0,0,1,-1,0,0,1,1,0,0,-1]]
for h in [monomial(nvars, hh) for hh in higherMarkovEdges]:
wF = WitnessSearch(nvars, K22Binomials, h, Verbose=False)
print ("Witness for " + str(h) + " : " + str(wF.findWitness()))
def createVar (i,n):
l = [0 for j in range(n)]
l[i]=1
return monomial(n, l)