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
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
|
;; Copyright (c) 2016, Regents of the University of Texas
;;
;; License: The MIT License (MIT)
;;
;; Permission is hereby granted, free of charge, to any person
;; obtaining a copy of this software and associated documentation
;; files (the "Software"), to deal in the Software without
;; restriction, including without limitation the rights to use,
;; copy, modify, merge, publish, distribute, sublicense, and/or sell
;; copies of the Software, and to permit persons to whom the
;; Software is furnished to do so, subject to the following
;; conditions:
;;
;; The above copyright notice and this permission notice shall be
;; included in all copies or substantial portions of the Software.
;;
;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
;; OTHER DEALINGS IN THE SOFTWARE.
;;
;; Original author: Nathan Wetzler <nathan.wetzler@gmail.com>
;; Last Modified: 2016-10-15 20:24
;; ============================= PACKAGE =============================
(in-package "PROOF-CHECKER-ARRAY")
;; ============================ INCLUDES =============================
(include-book "xdoc/top" :dir :system)
;; (include-book "xdoc/debug" :dir :system)
(include-book "assignment-equiv")
;; ===================================================================
;; ============================== XDOC ===============================
;; ===================================================================
(defxdoc PROOF-CHECKER-ARRAY
:parents (acl2::projects)
:short "Array-based RAT Proof Checker"
)
(xdoc::order-subtopics
PROOF-CHECKER-ARRAY
(;Background-And-Description
))
;; =========================== DESCRIPTION ===========================
(defsection Description
:extension PROOF-CHECKER-ARRAY
;; :parents (PROOF-CHECKER-ARRAY)
;; :short ""
:long
"<p>This proof is the supplemental material for Nathan Wetzler's dissertation
entitled Efficient, Mechanically-Verified Validation of Satisfiability
Solvers. Abstract and links to the dissertation are below.</p>
<h2>Links</h2>
<p>This material is further described in Nathan Wetzler's dissertation
available as a <a
href='http://www.cs.utexas.edu/~nwetzler/publications/dissertation-preprint.pdf'>preprint</a>
or on the <a href='https://repositories.lib.utexas.edu/handle/2152/30538'>UT
Digital Repository</a>.</p>
<h2>Citation</h2>
<p>Wetzler, Nathan D.: Efficient, Mechanically-Verified Validation of Satisfiability
Solvers. The University of Texas at Austin (2015).</p>" )
;; ============================ ABSTRACT =============================
(defsection Abstract
:extension PROOF-CHECKER-ARRAY
;; :parents (PROOF-CHECKER-ARRAY)
;; :short ""
:long
"<h2>Abstract From Dissertation</h2>
<p>Satisfiability (SAT) solvers are commonly used for a variety of
applications, including hardware verification, software verification, theorem
proving, debugging, and hard combinatorial problems. These applications rely
on the efficiency and correctness of SAT solvers. When a problem is determined
to be unsatisfiable, how can one be confident that a SAT solver has fully
exhausted the search space? Traditionally, unsatisfiability results have been
expressed using resolution or clausal proof systems. Resolution-based proofs
contain perfect reconstruction information, but these proofs are extremely
large and difficult to emit from a solver. Clausal proofs rely on rediscovery
of inferences using a limited number of techniques, which typically takes
several orders of magnitude longer than the solving time. Moreover, neither of
these proof systems has been able to express contemporary solving techniques
such as bounded variable addition. This combination of issues has left SAT
solver authors unmotivated to produce proofs of unsatisfiability.</p>
<p>The work from this dissertation focuses on validating satisfiability solver
output in the unsatisfiability case. We developed a new clausal proof format
called DRAT that facilitates compact proofs that are easier to emit and capable
of expressing all contemporary solving and preprocessing techniques.
Furthermore, we implemented a validation utility called DRAT-trim that is able
to validate proofs in a time similar to that of the discovery time. The DRAT
format has seen widespread adoption in the SAT community and the DRAT-trim
utility was used to validate the results of the 2014 SAT Competition.</p>
<p>DRAT-trim uses many advanced techniques to realize its performance gains, so
why should the results of DRAT-trim be trusted? Mechanical verification
enables users to model programs and algorithms and then prove their correctness
with a proof assistant, such as ACL2. We designed a new modeling technique for
ACL2 that combines efficient model execution with an agile and convenient
theory. Finally, we used this new technique to construct a fast,
mechanically-verified validation tool for proofs of unsatisfiability. This
research allows SAT solver authors and users to have greater confidence in
their results and applications by ensuring the validity of unsatisfiability
results.</p>" )
;; ===================================================================
(set-enforce-redundancy t)
;; ===================================================================
;; =========================== DEFINITIONS ===========================
;; ===================================================================
(defun assignmentp (x)
(declare (xargs :guard t))
(and (literal-listp x)
(unique-literalsp x)
(no-conflicting-literalsp x)))
(defun assignment-stp (st)
(declare (xargs :guard t
:guard-debug t
:stobjs (st)))
(and
;; farray
(farrayp *s* st)
;; Four fields in assignment
(equal (num-fields *s* st)
4)
;; num-vars is length 1 fieldp
(fieldp *num-vars* *s* st)
(equal (flength *num-vars* *s* st)
1)
;; let n be num-vars
(let ((n (fread *num-vars* 0 *s* st)))
(and
;; num-vars is positive
(<= 0 n)
;; stack-end is length 1 fieldp
(fieldp *stack-end* *s* st)
(equal (flength *stack-end* *s* st)
1)
;; stack is length n fieldp
(fieldp *stack* *s* st)
(equal (flength *stack* *s* st)
(1+ n))
;; lookup is length 2n+2 fieldp
(fieldp *lookup* *s* st)
(equal (flength *lookup* *s* st)
(+ 2 (* 2 n)))
;; let se be stack-end
(let ((se (fread *stack-end* 0 *s* st)))
(and
;; stack-end is field-offsetp
(field-offsetp se *stack* *s* st)
;; (or (field-offsetp (1- se) *stack* *s* st)
;; (equal se 0))
;; stack contains literals
(stack-contains-literalsp (1- se) st)
;; stack unique
(stack-uniquep (1- se) st)
;; stack not conflicting
(stack-not-conflictingp (1- se) st)
;; lookup corresponds to stack
(lookup-corresponds-with-stackp 2 st)
;;
(stack-in-rangep (1- se) st)
;; stack corresponds with lookup
(stack-corresponds-with-lookup-p (1- se) st)
;;
(equal (count-assigned 2 st)
se)
))))))
(defun assignedp (lit st)
(declare (xargs :guard (and (assignment-stp st)
(literalp lit)
(lit-in-rangep lit st))
:stobjs (st)))
(equal (fread *lookup* lit *s* st) 1))
(defun assign-lit (lit st)
(declare (xargs :guard (and (assignment-stp st)
(literalp lit)
(lit-in-rangep lit st)
(not (field-memberp lit
(1- (fread *stack-end* 0
*s* st))
0 *stack* *s* st))
(not (field-memberp (negate lit)
(1- (fread *stack-end* 0
*s* st))
0 *stack* *s* st)))
:stobjs (st)))
(let* ((stack-end (fread *stack-end* 0 *s* st))
(st (fwrite *lookup* lit 1 *s* st))
(st (fwrite *stack* stack-end lit *s* st))
(st (fwrite *stack-end* 0 (1+ stack-end) *s* st)))
st))
(defun unassign-one (st)
(declare (xargs :guard (and (assignment-stp st)
(not (equal (fread *stack-end* 0 *s* st) 0))
(sb60p (1- (fread *stack-end* 0 *s* st))))
:stobjs (st)))
(let* ((1--stack-end (1- (fread *stack-end* 0 *s* st)))
(lit (fread *stack* 1--stack-end *s* st))
(st (fwrite *lookup* lit 0 *s* st))
;; (st (fwrite *stack* 1--stack-end 0 *s* st))
(st (fwrite *stack-end* 0 1--stack-end *s* st)))
st))
(defun unassign-all (i st)
(declare (xargs :guard (and (assignment-stp st)
;; (sb60p (1- (fread *stack-end* 0 *s* st)))
(equal (fread *stack-end* 0 *s* st) i)
(or (field-offsetp i *stack* *s* st)
(equal i 0))
)
:stobjs (st)
:measure (nfix i)
))
(if (not (mbt (and (assignment-stp st)
;; (sb60p (1- (fread *stack-end* 0 *s* st)))
(equal (fread *stack-end* 0 *s* st) i)
(or (field-offsetp i *stack* *s* st)
(equal i 0))
)))
st
(if (equal i 0)
st
(let* ((st (unassign-one st))
(st (unassign-all (1- i) st)))
st))))
(defun project1 (i f start st)
(declare (xargs :guard (and (farrayp start st)
(fieldp f start st)
(or (field-offsetp i f start st)
(equal i -1)))
:stobjs (st)
:measure (nfix (1+ i))))
(if (not (mbt (and (farrayp start st)
(fieldp f start st)
(or (field-offsetp i f start st)
(equal i -1)))))
nil
(if (equal i -1)
nil
(cons (fread f i start st)
(project1 (1- i) f start st)))))
(defun project (st)
(declare (xargs :guard (assignment-stp st)
:stobjs (st)))
(project1 (1- (fread *stack-end* 0 *s* st)) *stack* *s* st))
;; ===================================================================
(defthm unit-propagation-equiv
(implies (and (assignment-stp st)
(formulap formula)
(formula-in-rangep formula st))
(equal (project (unit-propagation-st formula st))
(unit-propagation formula (project st)))))
;; ===================================================================
;; =========================== MAIN PROOF ============================
;; ===================================================================
(defthm exists-solution-implies-not-refutationp-st-contrapositive
(implies (and (assignment-stp st)
(formulap formula)
(formula-in-rangep formula st)
(clause-listp clause-list)
(clause-list-in-rangep clause-list st)
(mv-nth 0 (refutationp-st clause-list formula st)))
(not (exists-solution formula))))
(defthm main-theorem
(implies (and (formulap formula)
(refutationp clause-list formula))
(not (exists-solution formula))))
|
