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
|
import z3
import re
import torch.fx as fx, torch.nn as nn
import numpy as np
import subprocess
from typing import List, Dict
__all__ = ["net", "strict_equivalence", "epsilon_equivalence", "argmax_equivalence"]
type inpla_str = str
type z3_str = str
rules: inpla_str = """
Linear(x, float q, float r) >< Add(out, b) => b ~ AddCheckLinear(out, x, q, r);
Concrete(float k) >< Add(out, b)
| k == 0 => out ~ b
| _ => b ~ AddCheckConcrete(out, k);
Linear(y, float s, float t) >< AddCheckLinear(out, x, float q, float r)
| (q == 0) && (r == 0) && (s == 0) && (t == 0) => out ~ Concrete(0), x ~ Eraser, y ~ Eraser
| (s == 0) && (t == 0) => out ~ Linear(x, q, r), y ~ Eraser
| (q == 0) && (r == 0) => out ~ (*L)Linear(y, s, t), x ~ Eraser
| _ => Linear(x, q, r) ~ Materialize(out_x), (*L)Linear(y, s, t) ~ Materialize(out_y), out ~ Linear(TermAdd(out_x, out_y), 1, 0);
Concrete(float j) >< AddCheckLinear(out, x, float q, float r) => out ~ Linear(x, q, r + j);
Linear(y, float s, float t) >< AddCheckConcrete(out, float k) => out ~ Linear(y, s, t + k);
Concrete(float j) >< AddCheckConcrete(out, float k)
| j == 0 => out ~ Concrete(k)
| _ => out ~ Concrete(k + j);
Linear(x, float q, float r) >< Mul(out, b) => b ~ MulCheckLinear(out, x, q, r);
Concrete(float k) >< Mul(out, b)
| k == 0 => b ~ Eraser, out ~ (*L)Concrete(0)
| k == 1 => out ~ b
| _ => b ~ MulCheckConcrete(out, k);
Linear(y, float s, float t) >< MulCheckLinear(out, x, float q, float r)
| ((q == 0) && (r == 0)) || ((s == 0) && (t == 0)) => out ~ Concrete(0), x ~ Eraser, y ~ Eraser
| _ => Linear(x, q, r) ~ Materialize(out_x), (*L)Linear(y, s, t) ~ Materialize(out_y), out ~ Linear(TermMul(out_x, out_y), 1, 0);
Concrete(float j) >< MulCheckLinear(out, x, float q, float r) => out ~ Linear(x, q * j, r * j);
Linear(y, float s, float t) >< MulCheckConcrete(out, float k) => out ~ Linear(y, s * k, t * k);
Concrete(float j) >< MulCheckConcrete(out, float k)
| j == 0 => out ~ Concrete(0)
| j == 1 => out ~ Concrete(k)
| _ => out ~ Concrete(k * j);
Linear(x, float q, float r) >< ReLU(out) => (*L)Linear(x, q, r) ~ Materialize(out_x), out ~ Linear(TermReLU(out_x), 1, 0);
Concrete(float k) >< ReLU(out)
| k > 0 => out ~ (*L)Concrete(k)
| _ => out ~ Concrete(0);
Linear(x, float q, float r) >< Materialize(out)
| (q == 0) => out ~ Concrete(r), x ~ Eraser
| (q == 1) && (r == 0) => out ~ x
| (q == 1) && (r != 0) => out ~ TermAdd(x, Concrete(r))
| (q != 0) && (r == 0) => out ~ TermMul(Concrete(q), x)
| _ => out ~ TermAdd(TermMul(Concrete(q), x), Concrete(r));
Concrete(float k) >< Materialize(out) => out ~ (*L)Concrete(k);
"""
class NameGen:
def __init__(self):
self.counter = 0
def next(self) -> str:
name = f"v{self.counter}"
self.counter += 1
return name
def inpla_export(model: nn.Module, input_shape: tuple) -> inpla_str:
traced = fx.symbolic_trace(model)
name_gen = NameGen()
script: List[str] = []
wire_map: Dict[str, List[str]] = {}
for node in traced.graph.nodes:
if node.op == 'placeholder':
num_inputs = int(np.prod(input_shape))
wire_map[node.name] = [f"Linear(Symbolic(X_{i}), 1.0, 0.0)" for i in range(num_inputs)]
elif node.op == 'call_module':
target_str = str(node.target)
module = dict(model.named_modules())[target_str]
input_node = node.args[0]
if not isinstance(input_node, fx.Node):
continue
input_wires = wire_map[input_node.name]
if isinstance(module, nn.Flatten):
wire_map[node.name] = input_wires
elif isinstance(module, nn.Linear):
W = (module.weight.data.detach().cpu().numpy()).astype(float)
B = (module.bias.data.detach().cpu().numpy()).astype(float)
out_dim, in_dim = W.shape
neuron_wires = [f"Concrete({B[j]})" for j in range(out_dim)]
for i in range(in_dim):
in_term = input_wires[i]
if out_dim == 1:
weight = float(W[0, i])
if weight == 0:
script.append(f"Eraser ~ {in_term};")
elif weight == 1:
new_s = name_gen.next()
script.append(f"Add({new_s}, {in_term}) ~ {neuron_wires[0]};")
neuron_wires[0] = new_s
else:
mul_out = name_gen.next()
new_s = name_gen.next()
script.append(f"Mul({mul_out}, Concrete({weight})) ~ {in_term};")
script.append(f"Add({new_s}, {mul_out}) ~ {neuron_wires[0]};")
neuron_wires[0] = new_s
else:
branch_wires = [name_gen.next() for _ in range(out_dim)]
def nest_dups(names: List[str]) -> str:
if len(names) == 1: return names[0]
if len(names) == 2: return f"Dup({names[0]}, {names[1]})"
return f"Dup({names[0]}, {nest_dups(names[1:])})"
script.append(f"{nest_dups(branch_wires)} ~ {in_term};")
for j in range(out_dim):
weight = float(W[j, i])
if weight == 0:
script.append(f"Eraser ~ {branch_wires[j]};")
elif weight == 1:
new_s = name_gen.next()
script.append(f"Add({new_s}, {branch_wires[j]}) ~ {neuron_wires[j]};")
neuron_wires[j] = new_s
else:
mul_out = name_gen.next()
new_s = name_gen.next()
script.append(f"Mul({mul_out}, Concrete({weight})) ~ {branch_wires[j]};")
script.append(f"Add({new_s}, {mul_out}) ~ {neuron_wires[j]};")
neuron_wires[j] = new_s
wire_map[node.name] = neuron_wires
elif isinstance(module, nn.ReLU):
output_wires = []
for i, w in enumerate(input_wires):
r_out = name_gen.next()
script.append(f"ReLU({r_out}) ~ {w};")
output_wires.append(r_out)
wire_map[node.name] = output_wires
elif node.op == 'output':
output_node = node.args[0]
if isinstance(output_node, fx.Node):
final_wires = wire_map[output_node.name]
for i, w in enumerate(final_wires):
res_name = f"result{i}"
script.append(f"Materialize({res_name}) ~ {w};")
script.append(f"{res_name};")
return "\n".join(script)
def inpla_run(model: inpla_str) -> z3_str:
return subprocess.run(["./inpla"], input=f"{rules}\n{model}", capture_output=True, text=True).stdout
syms = {}
def Symbolic(id):
if id not in syms:
syms[id] = z3.Real(id)
return syms[id]
def Concrete(val): return z3.RealVal(val)
def TermAdd(a, b): return a + b
def TermMul(a, b): return a * b
def TermReLU(x): return z3.If(x > 0, x, 0)
context = {
'Concrete': Concrete,
'Symbolic': Symbolic,
'TermAdd': TermAdd,
'TermMul': TermMul,
'TermReLU': TermReLU
}
wrap = re.compile(r"Symbolic\((.*?)\)")
def z3_evaluate(model: z3_str):
model = wrap.sub(r'Symbolic("\1")', model);
return eval(model, context)
def net(model: nn.Module, input_shape: tuple):
return z3_evaluate(inpla_run(inpla_export(model, input_shape)))
def strict_equivalence(net_a, net_b):
solver = z3.Solver()
for sym in syms.values():
solver.add(z3.Or(sym == 0, sym == 1))
solver.add(net_a != net_b)
result = solver.check()
print("Strict Equivalence")
if result == z3.unsat:
print("VERIFIED: The networks are strictly equivalent.")
elif result == z3.sat:
print("FAILED: The networks are different.")
print("Counter-example input:")
print(solver.model())
else:
print("UNKNOWN: Solver could not decide.")
def epsilon_equivalence(net_a, net_b, epsilon):
solver = z3.Solver()
for sym in syms.values():
solver.add(z3.Or(sym == 0, sym == 1))
solver.add(z3.Abs(net_a - net_b) > epsilon)
result = solver.check()
print(f"Epsilon-Equivalence | Epsilon={epsilon}.")
if result == z3.unsat:
print("VERIFIED: The networks are epsilon-equivalent.")
elif result == z3.sat:
print("FAILED: The networks are different.")
print("Counter-example input:")
print(solver.model())
else:
print("UNKNOWN: Solver could not decide.")
def argmax_equivalence(net_a, net_b):
solver = z3.Solver()
for sym in syms.values():
solver.add(z3.Or(sym == 0, sym == 1))
solver.add((net_a > 0.5) != (net_b > 0.5))
result = solver.check()
print("ARGMAX Equivalence")
if result == z3.unsat:
print("VERIFIED: The networks are ARGMAX equivalent.")
elif result == z3.sat:
print("FAILED: The networks are different.")
print("Counter-example input:")
print(solver.model())
else:
print("UNKNOWN: Solver could not decide.")
|
