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mm2x2.py
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mm2x2.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 = 7
# We start with basic variables, 8 variables for each of the 7
# final computed multiplications: 4 for each of elements of first matrix (A),
# and 4 for second matrix (B).
a = [[[vf.next()
for ja in range(MATRIX_SIZE)]
for ia in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
b = [[[vf.next()
for jb in range(MATRIX_SIZE)]
for ib 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 and b variables
# that correspond to value of their product (exists or not in m_k).
m = [[[[[vf.next()
for jb in range(MATRIX_SIZE)]
for ib 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..7, {ia, ja, ib, jb} in 1..2:
# m_k_ia_ja_ib_jb = a_k_ia_ja AND b_k_ib_jb
# or in short: c = a and b
# which can be rewritten as
# (c or not(a) or not(b)) and (not(c) or a) and (not(c) or b)
# This is correct and tested in test.test_and_assignment(),
# and explained also in https://en.wikipedia.org/wiki/Tseitin_transformation
for k in range(MULTIPLICATION_VECTORS):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
va = a[k][ia][ja]
vb = b[k][ib][jb]
vc = m[k][ia][ja][ib][jb]
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 C (C=A*B).
c = []
for ic in range(MATRIX_SIZE):
c.append([])
for jc in range(MATRIX_SIZE):
c[ic].append([])
for ia in range(MATRIX_SIZE):
c[ic][jc].append([])
for ja in range(MATRIX_SIZE):
c[ic][jc][ia].append([])
for ib in range(MATRIX_SIZE):
c[ic][jc][ia][ja].append([])
for jb in range(MATRIX_SIZE):
c[ic][jc][ia][ja][ib].append(False)
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for l in range(MATRIX_SIZE):
c[ic][jc][ic][l][l][jc] = True
# Now we define coefficients q_k_ic_jc, k in 1..7, {ic, jc} in 1..2,
# for linking multiplication and result verctors:
# <xor for all k in 1..7> m_k_ia_ja_ib_jb and q_k_ic_jc = c_ic_jc_ia_ja_ib_jb
# for {ic, jc, ia, ja, ib, jb} in 1..2.
q = [[[vf.next()
for jc in range(MATRIX_SIZE)]
for ic 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_ic_jc and t_k_ic_jc,
# such that:
# p_k_ic_jc_ia_ja_ib_jb = m_k_ia_ja_ib_jb and q_k_ic_jc,
# k in 1..7, {ic, jc, ia, ja, ib, jb} in 1..2;
# and
# t_k_ic_jc_ia_ja_ib_jb = t_(k-1)_ic_jc_ia_ja_ib_jb xor p_k_ic_jc_ia_ja_ib_jb,
# k in 2..7, {ic, jc, ia, ja, ib, jb} in 1..2,
# t_1_ic_jc_ia_ja_ib_jb = p_1_ic_jc_ia_ja_ib_jb;
# and
# t_7_ic_jc_ia_ja_ib_jb = c_ic_jc_ia_ja_ib_jb,
# {ic, jc, ia, ja, ib, jb} in 1..2;
# The last one is rewritten as
# t_7_ic_jc_ia_ja_ib_jb or not(t_7_ic_jc_ia_ja_ib_jb),
# depending on c_ic_jc_ia_ja_ib_jb value, which is known by definition.
# So we define variables p and constraints for them:
p = [[[[[[[vf.next()
for jb in range(MATRIX_SIZE)]
for ib in range(MATRIX_SIZE)]
for ja in range(MATRIX_SIZE)]
for ia in range(MATRIX_SIZE)]
for jc in range(MATRIX_SIZE)]
for ic in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS)]
for k in range(MULTIPLICATION_VECTORS):
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
va = m[k][ia][ja][ib][jb]
vb = q[k][ic][jc]
vc = p[k][ic][jc][ia][ja][ib][jb]
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, see https://en.wikipedia.org/wiki/Tseitin_transformation
t = [[[[[[[vf.next()
for jb in range(MATRIX_SIZE)]
for ib in range(MATRIX_SIZE)]
for ja in range(MATRIX_SIZE)]
for ia in range(MATRIX_SIZE)]
for jc in range(MATRIX_SIZE)]
for ic in range(MATRIX_SIZE)]
for k in range(MULTIPLICATION_VECTORS-1)]
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
va = p[0][ic][jc][ia][ja][ib][jb]
vb = p[1][ic][jc][ia][ja][ib][jb]
vc = t[0][ic][jc][ia][ja][ib][jb]
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 ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
va = t[k-1][ic][jc][ia][ja][ib][jb]
vb = p[k+1][ic][jc][ia][ja][ib][jb]
vc = t[k][ic][jc][ia][ja][ib][jb]
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_6_ic_jc_ia_ja_ib_jb = c_ic_jc_ia_ja_ib_jb,
# {ic, jc, ia, ja, ib, jb} in 1..2.
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
if c[ic][jc][ia][ja][ib][jb]:
cc.add(positive=[t[MULTIPLICATION_VECTORS-2][ic][jc][ia][ja][ib][jb]], negative=[])
else:
cc.add(positive=[], negative=[t[MULTIPLICATION_VECTORS-2][ic][jc][ia][ja][ib][jb]])
# We have in the end 1028 variables and 3280 constraints.
# 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-2x2.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-2x2.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(MULTIPLICATION_VECTORS):
file.write('M_' + str(k+1) + ' = (')
members = []
for ia in range(MATRIX_SIZE):
for ja in range(MATRIX_SIZE):
if (a[k][ia][ja]['value']):
members.append('A' + str(ia+1) + '' + str(ja+1))
file.write(' + '.join(members) + ')(')
members = []
for ib in range(MATRIX_SIZE):
for jb in range(MATRIX_SIZE):
if (b[k][ib][jb]['value']):
members.append('B' + str(ib+1) + '' + str(jb+1))
file.write(' + '.join(members) + ')\n')
for ic in range(MATRIX_SIZE):
for jc in range(MATRIX_SIZE):
file.write('C' + str(ic+1) + str(jc+1) + ' = ')
members = []
for k in range(MULTIPLICATION_VECTORS):
if q[k][ic][jc]['value']:
members.append('M' + str(k+1))
file.write(' + '.join(members) + '\n')
file.close()
"""