-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathmm2x2.py
264 lines (232 loc) · 9.88 KB
/
mm2x2.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
# -*- 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()
"""