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June 14, 2020 05:11
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| # The following program checks if a functions symmetric block multilinear representation can be not bounded by 1. | |
| # We check this over all functions on n variables | |
| # A function is a counterexample if its block multilinear rep is not bounded | |
| from sys import argv | |
| import math | |
| import itertools | |
| from mip import * | |
| #from random import seed | |
| #from random import randint | |
| print("Give the number of variables") | |
| n=int(input()) # number of variables | |
| size_tt = 2**n # size of the truth table of f | |
| t=0 # degree of the polynomials | |
| n_bm = t * (n + 1) | |
| count = 0 # global variable to count number of counterexamples | |
| def countSetBits(n): | |
| count = 0 | |
| while (n): | |
| count += n & 1 | |
| n >>= 1 | |
| return count | |
| def binary(l,dig): #binary representation of l (0 replaced with -1) with dig number of digits | |
| rep = [] | |
| m=int(l) | |
| for i in range(0,dig): | |
| k = m%2 | |
| #print(m,k) | |
| if (k == 0): | |
| rep.append(-1) | |
| else: | |
| rep.append(1) | |
| m = int(m / 2) | |
| return rep | |
| #print(binary(63)) | |
| def is_func(func): # is this function a counterexample, needs input in block multilinear representation | |
| m=Model() | |
| global t | |
| print("t: "+str(t)) | |
| permutations_t=list(itertools.permutations(list(range(t)))) | |
| print("permutation") | |
| print(permutations_t) | |
| x_b=[] | |
| x_ext=[] | |
| x_sum=[] | |
| t_fact=math.factorial(t) | |
| for x in range(n+1): | |
| x_b.append( [m.add_var(name="x_b_"+str(x)+"_"+str(i),var_type=BINARY) for i in range(t)] ) | |
| for x in range(size_tt): | |
| x_ext.append([m.add_var(name="x_ext_"+str(x)+"_"+str(i),var_type=BINARY) for i in range(t_fact)]) | |
| x_sum.append([m.add_var(name="x_sum_"+str(x)+"_"+str(i),var_type=INTEGER) for i in range(t_fact)]) | |
| vars=[] | |
| temp=x | |
| cnt=0 | |
| for i in range(n): | |
| if 1&temp==1: | |
| vars.append(i+1) | |
| cnt+=1 | |
| temp//=2 | |
| vars+=[0]*(t-cnt) | |
| for y in range(t_fact): | |
| m+=(x_ext[x][y]+2*x_sum[x][y]==xsum(x_b[vars[z]][permutations_t[y][z]] for z in range(t))) | |
| m.objective=xsum( func[x][y]*(1-2*x_ext[x][y]) for x in range(size_tt) for y in range(t_fact) ) | |
| status = m.optimize(max_seconds=500) | |
| if status == OptimizationStatus.OPTIMAL: | |
| print('Optimal value is'+str(m.objective_value)) | |
| if m.objective_value<-1-1e-5: | |
| print('Counter eample found with {} as its minimum value'.format(m.objective_value)) | |
| for k in x_ext: | |
| # if abs(v.x) > 1e-6: # only printing non-zeros | |
| for v in k: | |
| print('{} : {}'.format(v.name, v.x)) | |
| print(func) | |
| exit() | |
| # exit() | |
| def chi(i,j): # Calculates \chi_i(j) , where i should be viewed as a set (-1 means index not present) and j should be viewed as an assignment. | |
| value = 1 | |
| bin_j = binary(j,n) | |
| bin_i = binary(i,n) | |
| for k in range(0,n): | |
| if (bin_i[k] == 1): | |
| value = value * bin_j[k] | |
| #print(value, bin_i, bin_j) | |
| return value | |
| def poly(func_tt): # give the block multilinear rep of f from its truth table representation, the first entry of truth table is -1,-1,..., -1 | |
| global t | |
| t=0 | |
| func_poly = [0] * size_tt | |
| for i in range(0,size_tt): | |
| func_poly[i] = 0 | |
| for j in range(0,size_tt): | |
| func_poly[i] = func_poly[i] + func_tt[j] * chi(i,j) | |
| func_poly[i] = (1.0 / size_tt) * func_poly[i] # ith Fourier coefficient | |
| if abs(func_poly[i])>1e-6: | |
| t=max(countSetBits(i),t) | |
| #print("the poly rep is ", func_poly) | |
| t_fact=math.factorial(t) | |
| h = 1/(t_fact) | |
| func_bm=[] | |
| for i in range(size_tt): | |
| func_bm.append([0]*t_fact) | |
| for j in range(t_fact): | |
| func_bm[i][j]=func_poly[i]*h | |
| print(func_bm) | |
| return func_bm | |
| for i in range(0,2**(size_tt)): | |
| f_tt = binary(i, size_tt) | |
| print("The truth table of the function is ", f_tt) | |
| f_bm = poly(f_tt) | |
| #print("Its bm poly representation is ", f_bm) | |
| is_func(f_bm) | |
| print("count is ", count) |
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