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PolyARBerNN.py
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PolyARBerNN.py
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# -*- coding: utf-8 -*-
"""
Created on Wed May 6 11:57:40 2020
@author: Wael
"""
from z3 import *
from yices import *
from scipy.optimize import minimize
from autograd import elementwise_grad as egrad
from autograd import jacobian
from autograd import grad
import itertools
import numpy as np
import polytope as pc
import multiprocessing as mp
from multiprocessing import Pool, Process, Queue
from functools import partial
import time
import keras
class PolyInequalitySolver:
# ========================================================
# Constructor
# ========================================================
def __init__(self, num_vars, boxber, pregion, orders):
self.num_vars = num_vars
#self.LipThres=LipThres
#self.bounds = bounds
self.boxber = boxber
self.pregion = pregion
self.orders = orders
self.poly_inequality_coeffs = []
self.__SMT_region_bounds = 0.01 # threshold below which we use SMT solver/CAD to compute the sign of the polynomial
self.status=False
# ========================================================
# Add A Polynomial Inequality
# ========================================================
def addPolyInequalityConstraint(self, poly):
self.poly_inequality_coeffs.append(poly)
# ========================================================
# Output s string in this form:'x0 x1...xn'
# ========================================================
def strVari(self,n):
variables=''
for i in range(0,n):
variables=variables+'x'+str(i)+' '
c= variables[:-1]
return c
# ========================================================
# Output s string in this form: '(* val xi)'
# ========================================================
def strterm(self,val,i):
valstr=str(val)
istr=str(i)
term='(*'+' '+valstr+' '+'x'+istr+')'
return term
# ========================================================
# Output s string in this form: '(* coef (^ xi i)....'
# ========================================================
def strtermpoly(self,coef,pows):
coefstr=str(coef)
termpoly='(* '+coefstr
for i in range(len(pows)):
if pows[i] !=0:
istr=str(i)
powstr=str(pows[i])
termpoly=termpoly+' '+'(^ '+'x'+istr+' '+powstr+')'
termpoly=termpoly+')'
return termpoly
# ========================================================
# 1) Output list of fmla bounds string for yices2
# ========================================================
def fmlabounds1(self,pregion):
fmlastrlist=[]
A=pregion[0]['A']
b=pregion[0]['b']
for i in range(len(A[:,1])):
fmlastr='(<=(+'
strb=str(b[i])
aux=''
for j in range(self.num_vars):
aux=aux+self.strterm(A[i,j],j)
fmlastr=fmlastr+aux+')'+''+strb+')'
fmlastrlist.append(fmlastr)
return fmlastrlist
# ========================================================
# 2) Output list of fmla bounds string for yices2
# ========================================================
def fmlabounds2(self,box):
fmlastr='(and'
for i in range(self.num_vars):
strb1=str(box[0][i][0])
strb2=str(box[1][i][0])
fmlastr=fmlastr+'( >= '+'x'+str(i)+' '+strb1+')'+''+'( <= '+'x'+str(i)+' '+strb2+')'
fmlastr=fmlastr+')'
return fmlastr
# ========================================================
# Output fmla poly string for yices2
# ========================================================
def fmlapoly(self,poly):
fmlastrpoly='(<=(+'
for monomial_counter in range(0,len(poly)):
coeff = poly[monomial_counter]['coeff']
vars = poly[monomial_counter]['vars']
pows=[]
for var_counter in range(len(vars)):
power = vars[var_counter]['power']
pows.append(power)
fmlastrpoly=fmlastrpoly+' '+ self.strtermpoly(coeff,pows)
fmlastrpoly=fmlastrpoly+')'+' '+'0)'
return fmlastrpoly
# ========================================================
# Function to output the string s in the poly structure
# for our algorithm
# ========================================================
def polyconstr(self,s):
varlist= symbols(self.strVari(self.num_vars))
numden=fraction(together(s))
s= numden[0]* numden[1]
polycs=[]
term={}
poly=sympy.poly(s,varlist)
polypowers=poly.monoms()
polycoeffs=poly.coeffs()
for j in range(len(polycoeffs)):
varspows=[]
for k in range(len(polypowers[j])):
varspows.append({'power':polypowers[j][k]})
term={'coeff':polycoeffs[j],'vars':varspows}
polycs.append(term)
return polycs
# ========================================================
# Compute the approx Lipchtz constant L of multivar poly
# in region
# ========================================================
def Lipchtz(self,poly,region,num_samples):
all_list=[]
for i in range(self.num_vars):
X=np.linspace(region[i]['min'], region[i]['max'], num_samples, endpoint=True)
all_list.append(X)
all_coords=list(itertools.product(*all_list))
all_coordsarray=[]
for i in range(len(all_coords)):
all_coordsarray.append(list(all_coords[i]))
all_coordsarray=np.array(all_coordsarray)
poly_vals=[]
for i in range(len(all_coordsarray)):
poly_vals.append(self.evaluate_multivar_poly(poly,all_coordsarray[i]))
poly_vals=np.array(poly_vals)
poly_vals_diff=np.diff(poly_vals)
all_coordsarray_diff=np.diff(all_coordsarray,axis=0)
all_coordsarray_diff=np.linalg.norm(all_coordsarray_diff,axis=1)
L=max(abs(poly_vals_diff)/all_coordsarray_diff)
return L
# ========================================================
# Partition region into subregions around the components
# that have higher rate change threshold
# ========================================================
def Partition_ratechange(self,poly,polype,num_samples):
ambiguous_regions=[]
# Areg=pregion[0]['A']
# breg=pregion[0]['b']
# polype=pc.Polytope(Areg, breg)
boundingbox=pc.bounding_box(polype)
all_list=[]
for i in range(self.num_vars):
X=np.linspace(boundingbox[0][i][0], boundingbox[1][i][0], num_samples, endpoint=True)
all_list.append(X)
sampldist=all_list[0][1]- all_list[0][0]
all_coords=list(itertools.product(*all_list))
all_coordsarray=[]
for i in range(len(all_coords)):
all_coordsarray.append(list(all_coords[i]))
all_coordsarray=np.array(all_coordsarray)
ratechange=[]
for i in range(len(all_coordsarray)):
ratechange.append(np.linalg.norm(self.Gradient(poly,all_coordsarray[i])))
res = []
for idx in range(0, len(ratechange)) :
if ratechange[idx] > 50000000:
res.append(idx)
if len(res)==0:
return 0, ambiguous_regions
highratechangev=all_coordsarray[res,:]
ps=[]
for i in range(len(highratechangev)):
box=[]
for j in range(self.num_vars):
box.append([highratechangev[i][j]-sampldist,highratechangev[i][j]+sampldist])
box=np.array(box)
p=pc.box2poly(box)
p=p.intersect(polype)
ps.append(p)
psreg=pc.Region(ps)
pambig=(polype.diff(psreg)).union(psreg)
for polytope in pambig:
ambiguous_regions.append([{'A':polytope.A,'b':polytope.b}])
return 1,ambiguous_regions
#return 1,ratechange
# ========================================================
# Partition polype to 2 polypes along the long dimension
# ========================================================
def Part_polype(self,polype):
box=pc.bounding_box(polype)
boxn=np.append(box[0], box[1], axis=1)
indexmax=np.argmax(box[1]-box[0])
mid=0.5*(boxn[indexmax][0]+boxn[indexmax][1])
box1=boxn
box2=boxn
box1=np.delete(box1, indexmax, axis=0)
box1=np.insert(box1, indexmax, np.array([boxn[indexmax][0],mid]), axis=0)
box2=np.delete(box2, indexmax, axis=0)
box2=np.insert(box2, indexmax, np.array([mid,boxn[indexmax][1]]), axis=0)
p1=pc.box2poly(box1)
p2=pc.box2poly(box2)
p1=polype.intersect(p1)
p2=polype.intersect(p2)
return p1,p2
# ========================================================
# Compute the Hessian Matrix of multivar poly at point x
# ========================================================
def Hessian(self,poly,x):
def multivar_poly(x):
result = 0
for monomial_counter in range(0,len(poly)):
coeff = poly[monomial_counter]['coeff']
vars = poly[monomial_counter]['vars']
product = coeff
for var_counter in range(len(vars)):
power = vars[var_counter]['power']
var = x[var_counter]
product = product * (var**power)
result = result + product
return result
H_f = jacobian(egrad(multivar_poly))
return H_f(x)
# ========================================================
# Compute the Gradient Vector of multivar poly at point x
# ========================================================
def Gradient(self,poly,x):
def multivar_poly(x):
result = 0
for monomial_counter in range(0,len(poly)):
coeff = poly[monomial_counter]['coeff']
vars = poly[monomial_counter]['vars']
product = coeff
for var_counter in range(len(vars)):
power = vars[var_counter]['power']
var = x[var_counter]
product = product * (var**power)
result = result + product
return result
grad_f = grad(multivar_poly)
return grad_f(x)
# ========================================================
# Evaluate the multivar poly at point x
# ========================================================
def evaluate_multivar_poly(self,poly, x):
result = 0
for monomial_counter in range(0,len(poly)):
coeff = poly[monomial_counter]['coeff']
vars = poly[monomial_counter]['vars']
product = coeff
for var_counter in range(len(vars)):
power = vars[var_counter]['power']
var = x[var_counter]
product = product * (var**power)
result = result + product
return result
# ========================================================
# Check if a matrix M is positive semidefinite or not
# ========================================================
def is_pos_sem_def(self,M):
return np.all(np.linalg.eigvals(M) >= 0)
# ========================================================
# Check if a matrix M is positive definite or not
# ========================================================
def is_pos_def(self,M):
return np.all(np.linalg.eigvals(M) > 1e-8)
# ========================================================
# Check if a matrix M is negative definite or not
# ========================================================
def is_neg_def(self,M):
return np.all(np.linalg.eigvals(M) < 0)
# ========================================================
# Number of positive eigenvalues: Lambda_i>0
# ========================================================
def num_pos_eig(self,M):
w=np.linalg.eigvals(M)
return np.sum(w > 1e-2)
# ========================================================
# Compute the upper bound of 1st ord Remin of Taylor app
# ========================================================
def remainder1cst(self,poly,pregion,mid_point,Gradi):
cons=[{'type':'ineq','fun':lambda x:pregion[0]['b']-(pregion[0]['A']).dot(x)}]
objectiveFunction = lambda x:-abs(self.evaluate_multivar_poly(poly,x)-(self.evaluate_multivar_poly(poly,mid_point)+Gradi.dot((x-mid_point))))
res = minimize(objectiveFunction, np.ones(self.num_vars)/(self.num_vars), constraints=cons, options={'disp': False})
# np.random.uniform(low=lowvals, high=highvals, size=self.num_vars)
regneg=res.fun
# print(regneg)
# ccc=abs(self.evaluate_multivar_poly(poly,regneg)-(self.evaluate_multivar_poly(poly,mid_point)+Gradi.dot((regneg-mid_point))))
# print(ccc)
stat=res.status
# print(stat)
if stat==0:
return -regneg
else:
b=[]
return b
# ========================================================
# Compute the upper bound of 2nd ord Remin of Taylor app
# ========================================================
def remainder2cst(self,poly,pregion,mid_point,Gradi,Hess):
cons=[{'type':'ineq','fun':lambda x:pregion[0]['b']-(pregion[0]['A']).dot(x)}]
objectiveFunction = lambda x:-abs(self.evaluate_multivar_poly(poly,x)-(self.evaluate_multivar_poly(poly,mid_point)+Gradi.dot((x-mid_point))+0.5*(x-mid_point).dot(Hess.dot((x-mid_point)))))
res = minimize(objectiveFunction, np.ones(self.num_vars)/(self.num_vars), constraints=cons, options={'disp': False})
# np.random.uniform(low=lowvals, high=highvals, size=self.num_vars)
regneg=res.fun
#print( regneg)
# ccc=abs(self.evaluate_multivar_poly(poly,regneg)-(self.evaluate_multivar_poly(poly,mid_point)+Gradi.dot((regneg-mid_point))+0.5*(regneg-mid_point).dot(Hess.dot((regneg-mid_point)))))
# print(ccc)
stat=res.status
# print(stat)
if stat==0:
return -regneg
else:
b=[]
return b
# ========================================================
# Output:the rectangle incribed in polytope: A,b
# ========================================================
def RecinPolytope(self,A,b):
Ap=A.clip(0)
Am=(-A).clip(0)
cons=[{'type':'ineq','fun':lambda x:b-(Ap.dot(x[0:self.num_vars]))+(Am.dot(x[self.num_vars:]))},{'type':'ineq','fun':lambda x:x[0:self.num_vars]-x[self.num_vars:]-0.001}]
objectiveFunction = lambda x: -np.prod(x[0:self.num_vars]-x[self.num_vars:])
#objectiveFunction = lambda x: -np.sum(np.log(x[0:self.num_vars]-x[self.num_vars:3]))
#objectiveFunction = lambda x: -(x[0]-x[2])*(x[1]-x[3])
res = minimize(objectiveFunction, np.ones(2*(self.num_vars))/(2*(self.num_vars)), constraints=cons, options={'disp': False})
#
regneg=res.x
stat=res.status
#print(res)
if stat==0:
b=(np.array([regneg[self.num_vars:]]).T,np.array([regneg[0:self.num_vars]]).T)
return b
else:
b=[]
return b
# ========================================================
# Output:Vertex v of under-approx polytope tangent to tem (one sheet case)
# ========================================================
def Ver_tang_one_sheet(self,poly,pregion,mid_point,Hess,Gradi,Rem2,Tem,sign):
if sign=='N':
cons=[{'type':'ineq','fun':lambda x:-(self.evaluate_multivar_poly(poly,mid_point)+Gradi.dot((x-mid_point))+0.5*(x-mid_point).dot(Hess.dot((x-mid_point)))+Rem2+0.01)},{'type':'ineq','fun':lambda x:pregion[0]['b']-(pregion[0]['A']).dot(x)}]
# cons=[{'type':'ineq','fun':lambda x:-(self.evaluate_multivar_poly(poly,x))}]
objectiveFunction = lambda x: -x.dot(Tem)
res = minimize(objectiveFunction, np.ones(self.num_vars)/(self.num_vars), constraints=cons, options={'disp': False})
# np.random.uniform(low=lowvals, high=highvals, size=self.num_vars)
regneg=res.x
stat=res.status
# print(res)
# print(stat)
if stat==0:
return regneg
else:
b=[]
return b
else:
cons=[{'type':'ineq','fun':lambda x:(self.evaluate_multivar_poly(poly,mid_point)+Gradi.dot((x-mid_point))+0.5*(x-mid_point).dot(Hess.dot((x-mid_point)))-Rem2)},{'type':'ineq','fun':lambda x:-pregion[0]['b']+(pregion[0]['A']).dot(x)}]
# cons=[{'type':'ineq','fun':lambda x:-(self.evaluate_multivar_poly(poly,x))}]
objectiveFunction = lambda x: -x.dot(Tem)
res = minimize(objectiveFunction, np.ones(self.num_vars)/(self.num_vars), constraints=cons, options={'disp': False})
# np.random.uniform(low=lowvals, high=highvals, size=self.num_vars)
regneg=res.x
stat=res.status
# print(stat)
if stat==0:
return regneg
else:
b=[]
return b
# ========================================================
# Output:Vertex v of under-approx polytope tangent to tem
# (two sheets case): The left side of the hyperplane:
# Ax<=b
# ========================================================
def Ver_tang_two_sheet(self,poly,pregion,mid_point,Hess,Gradi,Rem2,Tem,A,b,sign):
if sign=='N':
cons=[{'type':'ineq','fun':lambda x:-(self.evaluate_multivar_poly(poly,mid_point)+Gradi.dot((x-mid_point))+0.5*(x-mid_point).dot(Hess.dot((x-mid_point)))+Rem2)},{'type':'ineq','fun':lambda x:b-A.dot(x)},{'type':'ineq','fun':lambda x:pregion[0]['b']-(pregion[0]['A']).dot(x)}]
# cons=[{'type':'ineq','fun':lambda x:-(self.evaluate_multivar_poly(poly,x))}]
objectiveFunction = lambda x: -x.dot(Tem)
res = minimize(objectiveFunction, np.ones(self.num_vars)/(self.num_vars), constraints=cons, options={'disp': False})
# np.random.uniform(low=lowvals, high=highvals, size=self.num_vars)
regneg=res.x
stat=res.status
# print(stat)
if stat==0:
return regneg
else:
b=[]
return b
else:
cons=[{'type':'ineq','fun':lambda x:(self.evaluate_multivar_poly(poly,mid_point)+Gradi.dot((x-mid_point))+0.5*(x-mid_point).dot(Hess.dot((x-mid_point)))-Rem2)},{'type':'ineq','fun':lambda x:-b+A.dot(x)},{'type':'ineq','fun':lambda x:pregion[0]['b']-(pregion[0]['A']).dot(x)}]
# cons=[{'type':'ineq','fun':lambda x:-(self.evaluate_multivar_poly(poly,x))}]
objectiveFunction = lambda x: -x.dot(Tem)
res = minimize(objectiveFunction, np.ones(self.num_vars)/(self.num_vars), constraints=cons, options={'disp': False})
# np.random.uniform(low=lowvals, high=highvals, size=self.num_vars)
regneg=res.x
stat=res.status
# print(stat)
if stat==0:
return regneg
else:
b=[]
return b
def quitfc(self,result):
if result:
self.p.terminate()
# ========================================================
# Compute the binomial of n choose k
# ========================================================
def binom(self, n, k):
return math.factorial(n) // math.factorial(k) // math.factorial(n - k)
# ========================================================
# compute min bernstein coeffs of the univariate monomial
# x_{i}^{k} in the interval [xmin_i, xmax_i],
# where k is the power and l_i is the order
# inputs:
# k: the power
# l_i: the order
# interval = [xmin_i, xmax_i]
# output:
# bern, list of the bernstein coefficients
# ========================================================
def min_univ_monom_bernst(self, k, l_i, interval):
if k == 0:
sum = 1
return sum
else:
i = 0
if k == l_i:
sum = interval[0]**(k - i) * interval[1]**i
return sum
else:
sum = 0
min_i_k = min(i, k)
for j in range(min_i_k + 1):
sum = sum + (self.binom(i, j)/self.binom(l_i, j)) * (interval[1] - interval[0])**j * self.binom(k, j) * interval[0]**(k - j)
return sum
# ========================================================
# compute max bernstein coeffs of the univariate monomial
# x_{i}^{k} in the interval [xmin_i, xmax_i],
# where k is the power and l_i is the order
# inputs:
# k: the power
# l_i: the order
# interval = [xmin_i, xmax_i]
# output:
# bern, list of the bernstein coefficients
# ========================================================
def max_univ_monom_bernst(self, k, l_i, interval):
if k == 0:
sum = 1
return sum
else:
i = l_i
if k == l_i:
sum = interval[0]**(k - i) * interval[1]**i
return sum
else:
sum = 0
min_i_k = min(i, k)
for j in range(min_i_k + 1):
sum = sum + (self.binom(i, j)/self.binom(l_i, j)) * (interval[1] - interval[0])**j * self.binom(k, j) * interval[0]**(k - j)
# print(sum)
return sum
# ========================================================
# compute min bernstein coeffs of the multivariate polynomial
# poly in the box using implicit berns representation:
# inputs:
# poly: polynomial into consideration
# box: region into consideration
# Lpoly: orders of the polynomial
# output:
# bern: matrix that represents the bernstein coefficients
# of poly over box
# ========================================================
def min_poly_implicit_bernst(self, poly, box, Lpoly):
min_bern = 0
for term in poly:
min_bern_term = 1
a = term['coeff']
i = 0
for x in term['vars']:
if a > 0:
min_res = self.min_univ_monom_bernst(x['power'], Lpoly[i], box[i])
min_bern_term = min_bern_term * (min_res)
elif (a < 0):
min_res = self.max_univ_monom_bernst(x['power'], Lpoly[i], box[i])
min_bern_term = min_bern_term * (min_res)
i = i + 1
min_bern = min_bern + a * min_bern_term
return min_bern
# ========================================================
# compute max bernstein coeffs of the multivariate polynomial
# poly in the box using implicit berns representation:
# inputs:
# poly: polynomial into consideration
# box: region into consideration
# Lpoly: orders of the polynomial
# output:
# bern: matrix that represents the bernstein coefficients
# of poly over box
# ========================================================
def max_poly_implicit_bernst(self, poly, box, Lpoly):
max_bern = 0
for term in poly:
max_bern_term = 1
a = term['coeff']
i = 0
for x in term['vars']:
if a > 0:
max_res = self.max_univ_monom_bernst(x['power'], Lpoly[i], box[i])
max_bern_term = max_bern_term * (max_res)
elif (a < 0):
max_res = self.min_univ_monom_bernst(x['power'], Lpoly[i], box[i])
max_bern_term = max_bern_term * (max_res)
i = i + 1
max_bern = max_bern + a * max_bern_term
return max_bern
def UNSAT_Remov_Berns(self, poly_list):
i = 0
# for poly, order in zip([poly_list[0]], [self.orders[0]]):
for poly, order in zip(poly_list[-2:], self.orders[-2:]):
# while (len(poly_list) != 0) :
# print('i', i)
# print(len(poly_list))
min_bern = self.min_poly_implicit_bernst(poly, self.boxber, order)
max_bern = self.max_poly_implicit_bernst(poly, self.boxber, order)
# print('max_bern', max(min_bern, max_bern))
# print('min_bern', min(min_bern, max_bern))
if min_bern > 0:
return 'UNSAT'
if max_bern <= 0:
poly_list.remove(poly)
self.orders.remove(order)
# else:
# i = i + 1
return 'SAT_UNSAT'
# ========================================================
# The main solver that solve the multivar constraints
# ========================================================
def solve(self):
if self.poly_inequality_coeffs == []:
print('ERROR: At least one polynomial constraint is needed')
return self.bounds[0]['min']
poly_list=list(range(0,len(self.poly_inequality_coeffs)))
result = self.UNSAT_Remov_Berns(self.poly_inequality_coeffs)
if (result == 'UNSAT') or (len(poly_list) == 0):
return 'UNSAT'
negative_regions=self.pregion
aux=[]
aux2=[]
while poly_list:
aux=negative_regions
poly_index=poly_list.pop(0)
# compute min/max Bernstein coeffs and gradients coefficients
minb, maxb = self.min_max_Ber(aux, poly_index)
minb_grad, maxb_grad = self.min_max_Ber_grad(aux, poly_index)
# load the trained NN.
NN_model = keras.models.load_model("model")
# predict the right action
input_NN = np.array([[minb, maxb, minb_grad, maxb_grad]])
actions = NN_model(input_NN)
# print(actions)
# compute the index of the right action, could be 0, 1, or 3
max_action = max(list(actions[0]))
action_index =list(actions[0]).index(max_action)
if action_index == 0:
ambiguous_regions,negative_regions = self.iterat(aux, poly_index,'N')
elif action_index == 1:
ambiguous_regions,positive_regions = self.iterat(aux, poly_index,'P')
else:
ambiguous_region1, ambiguous_region2 = self.split(aux)
ambiguous_regions = ambiguous_region1 + ambiguous_region2
aux2.insert(0,ambiguous_regions)
if not negative_regions:
for i in range(len(aux2)):
aux3=aux2.pop(0)
for ccc in range(len(aux3)):
regionf=aux3.pop(0)
Areg=regionf[0]['A']
breg=regionf[0]['b']
polype=pc.Polytope(Areg, breg)
boxxpolype=pc.bounding_box(polype)
boxxpolype=np.append(boxxpolype[0], boxxpolype[1], axis=1)
polype=pc.box2poly(boxxpolype)
###########################################################################
p1,p2=self.Part_polype(polype)
plist=[[p1,p2]]
kk=0
while kk<1:
plistsub=[]
for j in range(len(plist[kk])):
p1,p2=self.Part_polype(plist[kk][j])
plistsub=plistsub+[p1,p2]
plist.append(plistsub)
kk=kk+1
##############################2)Parallel###################################
iterable=[]
for kkk in range(len(plist[0])):
boxx=pc.bounding_box(plist[0][kkk])
iterable.append(boxx)
procs=[]
q=Queue()
for box in iterable:
proc= Process(target=self.Yicesmany_multivars, args=(self.poly_inequality_coeffs,box,q))
# proc= Process(target=self.solveZ3_many_multivars, args=(self.poly_inequality_coeffs,box,q))
proc.start()
procs.append(proc)
is_done = True
counter=0
while is_done:
time.sleep(0.01)
for process in procs:
if (not process.is_alive()):
if (not q.empty()):
is_done = False
self.status=True
break
else:
counter=counter+1
if counter==len(procs):
break
counter=0
if self.status:
for process in procs:
process.terminate()
return 'SAT', q.get()
else:
for process in procs:
process.terminate()
return 'UNSAT'
aux=[]
if negative_regions :
if (len(negative_regions)==0):
region = negative_regions
polytope = pc.Polytope(region[0]['A'], region[0]['b'])
r,sol=pc.cheby_ball(polytope)
else:
region = negative_regions[0]
polytope = pc.Polytope(region[0]['A'], region[0]['b'])
r,sol=pc.cheby_ball(polytope)
return 'SAT'
else:
return 'UNSAT'
# ========================================================
# Compute the gradient of a polynomial poly
# ========================================================
def grad(self, poly):
gradient = []
occ_pow_0 = 0
for i in range(self.num_vars):
grad_poly_i = poly
for monomial_counter in range(0,len(grad_poly_i)):
coeff = grad_poly_i[monomial_counter]['coeff']
vars = grad_poly_i[monomial_counter]['vars']
product = coeff
for var_counter in range(len(vars)):
power = vars[var_counter]['power']
var = x[var_counter]
product = product * (var**power)
power = vars[i]['power']
if power == 0:
occ_pow_0 = occ_pow_0 + 1
vars[var_counter]['power'] == power
else:
vars[var_counter]['power'] == power - 1
gradient.append(grad_poly_i)
if occ_pow_0 == self.num_vars:
for i in range(self.num_vars):
gradient[i] = gradient[i][:-1]
return gradient