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mm2x2x2.py
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# -*- coding: utf-8 -*-
"""
Created on Sat Oct 11 19:31:50 2014
@author: Eugene Petkevich
"""
import satmaker
# To make an input for a SAT-solver, we need to associate each variable with
# a number automatically, while keeping string reference for ourselves.
# For this we use a VariableFactory and ConstraintCollector classes.
vf = satmaker.VariableFactory();
cc = satmaker.ConstraintCollector();
# Here we define constants: size and numbers of multiplication vectors.
MATRIX_SIZE = 2
MULTIPLICATION_VECTORS = 15
# We start with basic variables, 24 variables for each of the 15
# final computed multiplications: 8 for each of elements of first matrix (A),
# 4 for second matrix (B), and 4 for third matrix(C).
a = [[[[vf.next()
for la in range(MATRIX_SIZE)]
for ja in range(MATRIX_SIZE)]
for ia in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
b = [[[[vf.next()
for lb in range(MATRIX_SIZE)]
for jb in range(MATRIX_SIZE)]
for ib in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
c = [[[[vf.next()
for lc in range(MATRIX_SIZE)]
for jc in range(MATRIX_SIZE)]
for ic in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
# For each multiplication vector m_k we define a variable
# for each combination of a, b and c variables
# that correspond to value of their product (exists or not in m_k).
m = [[[[[[[[[[vf.next()
for lc in range(MATRIX_SIZE)]
for jc in range(MATRIX_SIZE)]
for ic in range(MATRIX_SIZE)]
for lb in range(MATRIX_SIZE)]
for jb in range(MATRIX_SIZE)]
for ib in range(MATRIX_SIZE)]
for la in range(MATRIX_SIZE)]
for ja in range(MATRIX_SIZE)]
for ia in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
# Now we define first series of constraints,
# that link basic variables and multiplication vectors.
# For each k in 1..15, {ia, ja, la, ib, jb, lb, ic, jc, lc} in 1..2:
# m_k_ia_ja_la_ib_jb_lb_ic_jc_lc = a_k_ia_ja_la and b_k_ib_jb_lb and c_k_ic_jc_lc
# or in short: d = a and b and c
# which can be rewritten as
# d = (a and b) and c
# and rewritten as CNF using Tseitin transformations:
# c = a and b
# can be rewritten as
# (c or not(a) or not(b)) and (not(c) or a) and (not(c) or b).
# Additional variables n_k_ia_ja_la_ib_jb_lb are needed.
n = [[[[[[[vf.next()
for lb in range(MATRIX_SIZE)]
for jb in range(MATRIX_SIZE)]
for ib in range(MATRIX_SIZE)]
for la in range(MATRIX_SIZE)]
for ja in range(MATRIX_SIZE)]
for ia in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
for k in range(MULTIPLICATION_VECTORS):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for la in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
for lb in range(MATRIX_SIZE):
va = a[k][ia][ja][la]
vb = b[k][ib][jb][lb]
vc = n[k][ia][ja][la][ib][jb][lb]
cc.add(positive=[vc],
negative=[va, vb])
cc.add(positive=[va],
negative=[vc])
cc.add(positive=[vb],
negative=[vc])
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for lc in range(MATRIX_SIZE):
va = n[k][ia][ja][la][ib][jb][lb]
vb = c[k][ic][jc][lc]
vc = m[k][ia][ja][la][ib][jb][lb][ic][jc][lc]
cc.add(positive=[vc],
negative=[va, vb])
cc.add(positive=[va],
negative=[vc])
cc.add(positive=[vb],
negative=[vc])
# Now we calculate result vectors of the product matrix D (D = A*B*C).
d = []
for id in range(MATRIX_SIZE):
d.append([])
for jd in range(MATRIX_SIZE):
d[id].append([])
for ld in range(MATRIX_SIZE):
d[id][jd].append([])
for ia in range(MATRIX_SIZE):
d[id][jd][ld].append([])
for ja in range(MATRIX_SIZE):
d[id][jd][ld][ia].append([])
for la in range(MATRIX_SIZE):
d[id][jd][ld][ia][ja].append([])
for ib in range(MATRIX_SIZE):
d[id][jd][ld][ia][ja][la].append([])
for jb in range(MATRIX_SIZE):
d[id][jd][ld][ia][ja][la][ib].append([])
for lb in range(MATRIX_SIZE):
d[id][jd][ld][ia][ja][la][ib][jb].append([])
for ic in range(MATRIX_SIZE):
d[id][jd][ld][ia][ja][la][ib][jb][lb].append([])
for jc in range(MATRIX_SIZE):
d[id][jd][ld][ia][ja][la][ib][jb][lb][ic].append([])
for lc in range(MATRIX_SIZE):
d[id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc].append(False)
for id in range(MATRIX_SIZE):
for jd in range(MATRIX_SIZE):
for ld in range(MATRIX_SIZE):
for o in range(MATRIX_SIZE):
d[id][jd][ld][id][jd][o][id][o][ld][o][jd][ld] = True
# Now we define coefficients q_k_id_jd_ld, k in 1..15, {id, jd, ld} in 1..2,
# for linking multiplication and result vectors:
# <xor for all k in 1..15> m_k_ia_ja_la_ib_jb_lb_ic_jc_lc and q_k_id_jd_ld = d_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc
# for {id, jd, ld, ia, ja, la, ib, jb, lb, ic, jc, lc} in 1..2.
q = [[[[vf.next()
for ld in range(MATRIX_SIZE)]
for jd in range(MATRIX_SIZE)]
for id in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
# Now we define second series of constraints,
# that link multiplication vectors and result vectors.
# We rewrite previous constraints as a step by step computation,
# with introducing additional variables p_k_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc and t_k_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc,
# such that:
# p_k_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc = m_k_ia_ja_la_ib_jb_lb_ic_jc_lc and q_k_id_jd_ld,
# k in 1..15, {id, jd, ld, ia, ja, la, ib, jb, lb, ic, jc, lc} in 1..2;
# and
# t_k_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc = t_(k-1)_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc xor p_k_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc,
# k in 2..15, {id, jd, ld, ia, ja, la, ib, jb, lb, ic, jc, lc} in 1..2,
# t_1_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc = p_1_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc;
# and
# t_15_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc = d_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc,
# {id, jd, ld, ia, ja, la, ib, jb, lb, ic, jc, lc} in 1..2;
# The last one is rewritten as
# t_15_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc or not(t_15_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc),
# depending on d_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc value, which is known by definition.
# So we define variables p and constraints for them:
p = [[[[[[[[[[[[[vf.next()
for lc in range(MATRIX_SIZE)]
for jc in range(MATRIX_SIZE)]
for ic in range(MATRIX_SIZE)]
for lb in range(MATRIX_SIZE)]
for jb in range(MATRIX_SIZE)]
for ib in range(MATRIX_SIZE)]
for la in range(MATRIX_SIZE)]
for ja in range(MATRIX_SIZE)]
for ia in range(MATRIX_SIZE)]
for ld in range(MATRIX_SIZE)]
for jd in range(MATRIX_SIZE)]
for id in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
for k in range(MULTIPLICATION_VECTORS):
for id in range(MATRIX_SIZE):
for jd in range(MATRIX_SIZE):
for ld in range(MATRIX_SIZE):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for la in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
for lb in range(MATRIX_SIZE):
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for lc in range(MATRIX_SIZE):
va = m[k][ia][ja][la][ib][jb][lb][ic][jc][lc]
vb = q[k][id][jd][ld]
vc = p[k][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]
cc.add(positive=[vc],
negative=[va, vb])
cc.add(positive=[va],
negative=[vc])
cc.add(positive=[vb],
negative=[vc])
# So we define variables t and constraints for them. First, a transformation:
# c = a xor b
# can be rewritten as
# c = (a or b) and (not(a) or not(b))
# which in turn is
# (c or not((a or b)) or not((not(a) or not(b)))) and (not(c) or (a or b)) and (not(c) or (not(a) or not(b)))
# which is equal to
# (c or (not(a) and not(b)) or (a and b)) and (not(c) or a or b) and (not(c) or not(a) or not(b))
# which is equal to
# (c or not(a) or b) and (c or a or not(b)) and (not(c) or a or b) and (not(c) or not(a) or not(b))
# This is correct, Tseitin transformations
t = [[[[[[[[[[[[[vf.next()
for lc in range(MATRIX_SIZE)]
for jc in range(MATRIX_SIZE)]
for ic in range(MATRIX_SIZE)]
for lb in range(MATRIX_SIZE)]
for jb in range(MATRIX_SIZE)]
for ib in range(MATRIX_SIZE)]
for la in range(MATRIX_SIZE)]
for ja in range(MATRIX_SIZE)]
for ia in range(MATRIX_SIZE)]
for ld in range(MATRIX_SIZE)]
for jd in range(MATRIX_SIZE)]
for id in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS-1)]
for id in range(MATRIX_SIZE):
for jd in range(MATRIX_SIZE):
for ld in range(MATRIX_SIZE):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for la in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
for lb in range(MATRIX_SIZE):
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for lc in range(MATRIX_SIZE):
va = p[0][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]
vb = p[1][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]
vc = t[0][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]
cc.add(positive=[vc, va],
negative=[vb])
cc.add(positive=[vc, vb],
negative=[va])
cc.add(positive=[va, vb],
negative=[vc])
cc.add(positive=[],
negative=[vc, va, vb])
for k in range(1, MULTIPLICATION_VECTORS-1):
for id in range(MATRIX_SIZE):
for jd in range(MATRIX_SIZE):
for ld in range(MATRIX_SIZE):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for la in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
for lb in range(MATRIX_SIZE):
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for lc in range(MATRIX_SIZE):
va = t[k-1][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]
vb = p[k+1][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]
vc = t[k][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]
cc.add(positive=[vc, va],
negative=[vb])
cc.add(positive=[vc, vb],
negative=[va])
cc.add(positive=[va, vb],
negative=[vc])
cc.add(positive=[],
negative=[vc, va, vb])
# And last constraints:
# t_14_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc = c_id_jd_ld_ia_ja_la_ib_jb_lb_ic_jc_lc,
# {id, jd, ld, ia, ja, la, ib, jb, lb, ic, jc, lc} in 1..2.
for id in range(MATRIX_SIZE):
for jd in range(MATRIX_SIZE):
for ld in range(MATRIX_SIZE):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for la in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
for lb in range(MATRIX_SIZE):
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for lc in range(MATRIX_SIZE):
if d[id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]:
cc.add(positive=[t[MULTIPLICATION_VECTORS-2][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]], negative=[])
else:
cc.add(positive=[], negative=[t[MULTIPLICATION_VECTORS-2][id][jd][ld][ia][ja][la][ib][jb][lb][ic][jc][lc]])
# We have in the end 127904 variables, 443712 clauses, 1182784 literals.
# Now we will output all the constraints to a file that will be an input to
# a SAT solver.
# For printing we will use a SatPrinter class.
sp = satmaker.SatPrinter(vf, cc);
file = open('input-2x2x2-v1.txt', 'wt')
sp.print(file)
file.close()
'''
# Now a SAT solver should be executed and store its output in output.txt file.
# We get back data from the output to variables.
file = open('output.txt', 'rt')
sp.decode_output(file)
file.close()
# We print the important variables into a text file in easily readable form.
file = open('solution.txt', 'wt')
for k in range(7):
file.write('M_' + str(k+1) + ' = (')
members = []
for ia in range(2):
for ja in range(2):
if (a[k][ia][ja]['value']):
members.append('A' + str(ia+1) + '' + str(ja+1))
file.write(' + '.join(members) + ')(')
members = []
for ib in range(2):
for jb in range(2):
if (b[k][ib][jb]['value']):
members.append('B' + str(ib+1) + '' + str(jb+1))
file.write(' + '.join(members) + ')\n')
for ic in range(2):
for jc in range(2):
file.write('C' + str(ic+1) + str(jc+1) + ' = ')
members = []
for k in range(7):
if q[k][ic][jc]['value']:
members.append('M' + str(k+1))
file.write(' + '.join(members) + '\n')
file.close()
'''