3ef53c35f09687ef4dbade911f3aea3e81433ebf
1 """Formal verification of partitioned operations
3 The approach is to take an arbitrary partition, by choosing its start point
4 and size at random. Use ``Assume`` to ensure it is a whole unbroken partition
5 (start and end points are one, with only zeros in between). Shift inputs and
6 outputs down to zero. Loop over all possible partition sizes and, if it's the
7 right size, compute the expected value, compare with the result, and assert.
9 We are turning the for-loops around (on their head), such that we start from
10 the *lengths* (and positions) and perform the ``Assume`` on the resultant
13 In other words, we have patterns as follows (assuming 32-bit words)::
20 * for 8-bit the partition bit is 1 and the previous is also 1
22 * for 16-bit the partition bit at the offset must be 0 and be surrounded by 1
24 * for 24-bit the partition bits at the offset and at offset+1 must be 0 and at
25 offset+2 and offset-1 must be 1
27 * for 32-bit all 3 bits must be 0 and be surrounded by 1 (guard bits are added
28 at each end for this purpose)
36 from nmigen
import Elaboratable
, Signal
, Module
, Const
, Repl
37 from nmigen
.asserts
import Assert
, Cover
38 from nmigen
.hdl
.ast
import Assume
40 from nmutil
.formaltest
import FHDLTestCase
41 from nmutil
.gtkw
import write_gtkw
43 from ieee754
.part_mul_add
.partpoints
import PartitionPoints
44 from ieee754
.part
.partsig
import PartitionedSignal
47 class PartitionedPattern(Elaboratable
):
48 """ Generate a unique pattern, depending on partition size.
50 * 1-byte partitions: 0x11
51 * 2-byte partitions: 0x21 0x22
52 * 3-byte partitions: 0x31 0x32 0x33
56 Useful as a test vector for testing the formal prover
59 def __init__(self
, width
, partition_points
):
61 self
.partition_points
= PartitionPoints(partition_points
)
62 self
.mwidth
= len(self
.partition_points
)+1
63 self
.output
= Signal(self
.width
, reset_less
=True)
65 def elaborate(self
, platform
):
69 # Add a guard bit at each end
70 positions
= [0] + list(self
.partition_points
.keys()) + [self
.width
]
71 gates
= [Const(1)] + list(self
.partition_points
.values()) + [Const(1)]
72 # Begin counting at one
73 last_start
= positions
[0]
74 last_end
= positions
[1]
75 last_middle
= (last_start
+last_end
)//2
76 comb
+= self
.output
[last_start
:last_middle
].eq(1)
77 # Build an incrementing cascade
78 for i
in range(1, self
.mwidth
):
81 middle
= (start
+ end
) // 2
82 # Propagate from the previous byte, adding one to it.
84 comb
+= self
.output
[start
:middle
].eq(
85 self
.output
[last_start
:last_middle
] + 1)
87 # ... unless it's a partition boundary. If so, start again.
88 comb
+= self
.output
[start
:middle
].eq(1)
91 # Mirror the nibbles on the last byte
92 last_start
= positions
[-2]
93 last_end
= positions
[-1]
94 last_middle
= (last_start
+last_end
)//2
95 comb
+= self
.output
[last_middle
:last_end
].eq(
96 self
.output
[last_start
:last_middle
])
97 for i
in range(self
.mwidth
, 0, -1):
98 start
= positions
[i
-1]
100 middle
= (start
+ end
) // 2
101 # Propagate from the previous byte.
102 with m
.If(~gates
[i
]):
103 comb
+= self
.output
[middle
:end
].eq(
104 self
.output
[last_middle
:last_end
])
106 # ... unless it's a partition boundary.
107 # If so, mirror the nibbles again.
108 comb
+= self
.output
[middle
:end
].eq(
109 self
.output
[start
:middle
])
116 def make_partitions(step
, mwidth
):
117 """Make equally spaced partition points
119 :param step: smallest partition width
120 :param mwidth: maximum number of partitions
121 :returns: partition points, and corresponding gates"""
122 gates
= Signal(mwidth
- 1)
123 points
= PartitionPoints()
124 for i
in range(mwidth
-1):
125 points
[(i
+ 1) * step
] = gates
[i
]
129 # This defines a module to drive the device under test and assert
130 # properties about its outputs
131 class Driver(Elaboratable
):
142 # Setup partition points and gates
143 step
= int(width
/mwidth
)
144 points
, gates
= make_partitions(step
, mwidth
)
145 # Instantiate the partitioned pattern producer
146 m
.submodules
.dut
= dut
= PartitionedPattern(width
, points
)
147 # Directly check some cases
148 with m
.If(gates
== 0):
149 comb
+= Assert(dut
.output
== 0x_88_87_86_85_84_83_82_81)
150 with m
.If(gates
== 0b1100101):
151 comb
+= Assert(dut
.output
== 0x_11_11_33_32_31_22_21_11)
152 with m
.If(gates
== 0b0001000):
153 comb
+= Assert(dut
.output
== 0x_44_43_42_41_44_43_42_41)
154 with m
.If(gates
== 0b0100001):
155 comb
+= Assert(dut
.output
== 0x_22_21_55_54_53_52_51_11)
156 with m
.If(gates
== 0b1000001):
157 comb
+= Assert(dut
.output
== 0x_11_66_65_64_63_62_61_11)
158 with m
.If(gates
== 0b0000001):
159 comb
+= Assert(dut
.output
== 0x_77_76_75_74_73_72_71_11)
160 # Choose a partition offset and width at random.
161 p_offset
= Signal(range(mwidth
))
162 p_width
= Signal(range(mwidth
+1))
163 p_finish
= Signal(range(mwidth
+1))
164 comb
+= p_finish
.eq(p_offset
+ p_width
)
165 # Partition must not be empty, and fit within the signal.
166 comb
+= Assume(p_width
!= 0)
167 comb
+= Assume(p_offset
+ p_width
<= mwidth
)
169 # Build the corresponding partition
170 # Use Assume to constraint the pattern to conform to the given offset
171 # and width. For each gate bit it is:
172 # 1) one, if on the partition boundary
173 # 2) zero, if it's inside the partition
174 # 3) don't care, otherwise
175 p_gates
= Signal(mwidth
+1)
176 for i
in range(mwidth
+1):
177 with m
.If(i
== p_offset
):
178 # Partitions begin with 1
179 comb
+= Assume(p_gates
[i
] == 1)
180 with m
.If((i
> p_offset
) & (i
< p_finish
)):
181 # The interior are all zeros
182 comb
+= Assume(p_gates
[i
] == 0)
183 with m
.If(i
== p_finish
):
185 comb
+= Assume(p_gates
[i
] == 1)
186 # Check some possible partitions generating a given pattern
187 with m
.If(p_gates
== 0b0100110):
188 comb
+= Assert(((p_offset
== 1) & (p_width
== 1)) |
189 ((p_offset
== 2) & (p_width
== 3)))
190 # Remove guard bits at each end and assign to the DUT gates
191 comb
+= gates
.eq(p_gates
[1:])
192 # Generate shifted down outputs:
193 p_output
= Signal(width
)
194 positions
= [0] + list(points
.keys()) + [width
]
195 for i
in range(mwidth
):
196 with m
.If(p_offset
== i
):
197 comb
+= p_output
.eq(dut
.output
[positions
[i
]:])
198 # Some checks on the shifted down output, irrespective of offset:
199 with m
.If(p_width
== 2):
200 comb
+= Assert(p_output
[:16] == 0x_22_21)
201 with m
.If(p_width
== 4):
202 comb
+= Assert(p_output
[:32] == 0x_44_43_42_41)
204 with m
.If(p_offset
== 0):
205 comb
+= Assert(p_output
== dut
.output
)
207 # Make it interesting, by having four partitions.
208 # Make the selected partition not start at the very beginning.
209 comb
+= Cover((sum(gates
) == 3) & (p_offset
!= 0) & (p_width
== 3))
210 # Generate and check expected values for all possible partition sizes.
211 # Here, we assume partition sizes are multiple of the smaller size.
212 for w
in range(1, mwidth
+1):
213 with m
.If(p_width
== w
):
214 # calculate the expected output, for the given bit width
216 expected
= Signal(bit_width
, name
=f
"expected_{w}")
218 # lower nibble is the position
219 comb
+= expected
[b
*8:b
*8+4].eq(b
+1)
220 # upper nibble is the partition width
221 comb
+= expected
[b
*8+4:b
*8+8].eq(w
)
222 # truncate the output, compare and assert
223 comb
+= Assert(p_output
[:bit_width
] == expected
)
227 class GateGenerator(Elaboratable
):
228 """Produces partition gates at random
230 `p_offset`, `p_width` and `p_finish` describe the selected partition
232 def __init__(self
, mwidth
):
234 """Number of partitions"""
235 self
.gates
= Signal(mwidth
-1)
236 """Generated partition gates"""
237 self
.p_offset
= Signal(range(mwidth
))
238 """Generated partition start point"""
239 self
.p_width
= Signal(range(mwidth
+1))
240 """Generated partition width"""
241 self
.p_finish
= Signal(range(mwidth
+1))
242 """Generated partition end point"""
244 def elaborate(self
, _
):
249 p_offset
= self
.p_offset
250 p_width
= self
.p_width
251 p_finish
= self
.p_finish
252 comb
+= p_finish
.eq(p_offset
+ p_width
)
253 # Partition must not be empty, and fit within the signal.
254 comb
+= Assume(p_width
!= 0)
255 comb
+= Assume(p_offset
+ p_width
<= mwidth
)
257 # Build the corresponding partition
258 # Use Assume to constraint the pattern to conform to the given offset
259 # and width. For each gate bit it is:
260 # 1) one, if on the partition boundary
261 # 2) zero, if it's inside the partition
262 # 3) don't care, otherwise
263 p_gates
= Signal(mwidth
+1)
264 for i
in range(mwidth
+1):
265 with m
.If(i
== p_offset
):
266 # Partitions begin with 1
267 comb
+= Assume(p_gates
[i
] == 1)
268 with m
.If((i
> p_offset
) & (i
< p_finish
)):
269 # The interior are all zeros
270 comb
+= Assume(p_gates
[i
] == 0)
271 with m
.If(i
== p_finish
):
273 comb
+= Assume(p_gates
[i
] == 1)
274 # Remove guard bits at each end, before assigning to the output gates
275 comb
+= gates
.eq(p_gates
[1:])
279 class GeneratorDriver(Elaboratable
):
290 # Setup partition points and gates
291 step
= int(width
/mwidth
)
292 points
, gates
= make_partitions(step
, mwidth
)
293 # Instantiate the partitioned pattern producer and the DUT
294 m
.submodules
.dut
= dut
= PartitionedPattern(width
, points
)
295 m
.submodules
.gen
= gen
= GateGenerator(mwidth
)
296 comb
+= gates
.eq(gen
.gates
)
297 # Generate shifted down outputs
298 p_offset
= gen
.p_offset
299 p_width
= gen
.p_width
300 p_output
= Signal(width
)
301 for i
in range(mwidth
):
302 with m
.If(p_offset
== i
):
303 comb
+= p_output
.eq(dut
.output
[i
*step
:])
304 # Generate and check expected values for all possible partition sizes.
305 for w
in range(1, mwidth
+1):
306 with m
.If(p_width
== w
):
307 # calculate the expected output, for the given bit width
309 expected
= Signal(bit_width
, name
=f
"expected_{w}")
311 # lower nibble is the position
312 comb
+= expected
[b
*8:b
*8+4].eq(b
+1)
313 # upper nibble is the partition width
314 comb
+= expected
[b
*8+4:b
*8+8].eq(w
)
315 # truncate the output, compare and assert
316 comb
+= Assert(p_output
[:bit_width
] == expected
)
318 # Make it interesting, by having four partitions.
319 # Make the selected partition not start at the very beginning.
320 comb
+= Cover((sum(gates
) == 3) & (p_offset
!= 0) & (p_width
== 3))
324 class OpDriver(Elaboratable
):
325 """Checks operations on partitioned signals"""
326 def __init__(self
, op
, width
=64, mwidth
=8, nops
=2, part_out
=True):
328 """Operation to perform"""
330 """Partition full width"""
332 """Maximum number of equally sized partitions"""
334 """Number of input operands"""
335 self
.part_out
= part_out
336 """True if output is partition-like"""
337 def elaborate(self
, _
):
343 part_out
= self
.part_out
344 # setup partition points and gates
345 step
= int(width
/mwidth
)
346 points
, gates
= make_partitions(step
, mwidth
)
347 # setup inputs and outputs
349 for i
in range(nops
):
350 inp
= PartitionedSignal(points
, width
, name
=f
"i_{i+1}")
359 output
= Signal(out_width
)
360 # perform the operation on the partitioned signals
361 comb
+= output
.eq(self
.op(*operands
))
362 # instantiate the partitioned gate generator and connect the gates
363 m
.submodules
.gen
= gen
= GateGenerator(mwidth
)
364 comb
+= gates
.eq(gen
.gates
)
365 p_offset
= gen
.p_offset
366 p_width
= gen
.p_width
367 # generate shifted down inputs and outputs
369 for i
in range(nops
):
370 p_i
= Signal(width
, name
=f
"p_{i+1}")
371 p_operands
.append(p_i
)
372 for pos
in range(mwidth
):
373 with m
.If(p_offset
== pos
):
374 comb
+= p_i
.eq(operands
[i
].sig
[pos
* step
:])
375 p_output
= Signal(out_width
)
376 for pos
in range(mwidth
):
377 with m
.If(p_offset
== pos
):
378 comb
+= p_output
.eq(output
[pos
* out_step
:])
379 # generate and check expected values for all possible partition sizes
380 all_operands_non_zero
= Signal()
381 for w
in range(1, mwidth
+1):
382 with m
.If(p_width
== w
):
383 # calculate the expected output, for the given bit width,
384 # truncating the inputs to the partition size
385 input_bit_width
= w
* step
386 output_bit_width
= w
* out_step
387 expected
= Signal(output_bit_width
, name
=f
"expected_{w}")
388 trunc_operands
= list()
389 for i
in range(nops
):
390 t_i
= Signal(input_bit_width
, name
=f
"t{w}_{i+1}")
391 trunc_operands
.append(t_i
)
392 comb
+= t_i
.eq(p_operands
[i
][:input_bit_width
])
394 # for partition-like outputs, calculate the LSB
395 # and replicate it on the partition
396 lsb
= Signal(name
=f
"lsb_{w}")
397 comb
+= lsb
.eq(self
.op(*trunc_operands
))
398 comb
+= expected
.eq(Repl(lsb
, output_bit_width
))
400 # otherwise, just take the operation result
401 comb
+= expected
.eq(self
.op(*trunc_operands
))
402 # truncate the output, compare and assert
403 comb
+= Assert(p_output
[:output_bit_width
] == expected
)
404 # ensure a test case with all non-zero operands
405 non_zero_op
= Signal(nops
)
406 for i
in range(nops
):
407 comb
+= non_zero_op
[i
].eq(trunc_operands
[i
].any())
408 comb
+= all_operands_non_zero
.eq(non_zero_op
.all())
410 comb
+= Cover((p_offset
!= 0) & (p_width
== 3) & (sum(output
) > 1) &
411 (p_output
!= 0) & all_operands_non_zero
)
415 class PartitionTestCase(FHDLTestCase
):
416 def test_formal(self
):
418 'dec': {'base': 'dec'},
419 'bin': {'base': 'bin'}
422 ('p_offset[2:0]', 'dec'),
423 ('p_width[3:0]', 'dec'),
424 ('p_finish[3:0]', 'dec'),
425 ('p_gates[8:0]', 'bin'),
426 ('dut', {'submodule': 'dut'}, [
427 ('gates[6:0]', 'bin'),
429 'p_output[63:0]', 'expected_3[21:0]']
431 'proof_partition_cover.gtkw',
432 os
.path
.dirname(__file__
) +
433 '/proof_partition_formal/engine_0/trace0.vcd',
439 'proof_partition_bmc.gtkw',
440 os
.path
.dirname(__file__
) +
441 '/proof_partition_formal/engine_0/trace.vcd',
447 self
.assertFormal(module
, mode
="bmc", depth
=1)
448 self
.assertFormal(module
, mode
="cover", depth
=1)
450 def test_generator(self
):
452 'dec': {'base': 'dec'},
453 'bin': {'base': 'bin'}
456 ('p_offset[2:0]', 'dec'),
457 ('p_width[3:0]', 'dec'),
458 ('p_finish[3:0]', 'dec'),
459 ('p_gates[8:0]', 'bin'),
460 ('dut', {'submodule': 'dut'}, [
461 ('gates[6:0]', 'bin'),
463 'p_output[63:0]', 'expected_3[21:0]',
464 'a_3[23:0]', 'b_3[32:0]', 'expected_3[2:0]']
466 'proof_partition_generator_cover.gtkw',
467 os
.path
.dirname(__file__
) +
468 '/proof_partition_generator/engine_0/trace0.vcd',
474 'proof_partition_generator_bmc.gtkw',
475 os
.path
.dirname(__file__
) +
476 '/proof_partition_generator/engine_0/trace.vcd',
481 module
= GeneratorDriver()
482 self
.assertFormal(module
, mode
="bmc", depth
=1)
483 self
.assertFormal(module
, mode
="cover", depth
=1)
485 def test_partsig_eq(self
):
487 'dec': {'base': 'dec'},
488 'bin': {'base': 'bin'}
491 ('p_offset[2:0]', 'dec'),
492 ('p_width[3:0]', 'dec'),
493 ('p_gates[8:0]', 'bin'),
494 'i_1[63:0]', 'i_2[63:0]',
495 ('eq_1', {'submodule': 'eq_1'}, [
496 ('gates[6:0]', 'bin'),
497 'a[63:0]', 'b[63:0]',
498 ('output[7:0]', 'bin')]),
499 'p_1[63:0]', 'p_2[63:0]',
500 ('p_output[7:0]', 'bin'),
501 't3_1[23:0]', 't3_2[23:0]', 'lsb_3',
502 ('expected_3[2:0]', 'bin')]
504 'proof_partsig_eq_cover.gtkw',
505 os
.path
.dirname(__file__
) +
506 '/proof_partition_partsig_eq/engine_0/trace0.vcd',
512 'proof_partsig_eq_bmc.gtkw',
513 os
.path
.dirname(__file__
) +
514 '/proof_partition_partsig_eq/engine_0/trace.vcd',
519 module
= OpDriver(operator
.eq
)
520 self
.assertFormal(module
, mode
="bmc", depth
=1)
521 self
.assertFormal(module
, mode
="cover", depth
=1)
523 def test_partsig_ne(self
):
524 module
= OpDriver(operator
.ne
)
525 self
.assertFormal(module
, mode
="bmc", depth
=1)
527 def test_partsig_gt(self
):
528 module
= OpDriver(operator
.gt
)
529 self
.assertFormal(module
, mode
="bmc", depth
=1)
531 def test_partsig_ge(self
):
532 module
= OpDriver(operator
.ge
)
533 self
.assertFormal(module
, mode
="bmc", depth
=1)
535 def test_partsig_lt(self
):
536 module
= OpDriver(operator
.lt
)
537 self
.assertFormal(module
, mode
="bmc", depth
=1)
539 def test_partsig_le(self
):
540 module
= OpDriver(operator
.le
)
541 self
.assertFormal(module
, mode
="bmc", depth
=1)
543 def test_partsig_all(self
):
545 'dec': {'base': 'dec'},
546 'bin': {'base': 'bin'}
549 ('p_offset[2:0]', 'dec'),
550 ('p_width[3:0]', 'dec'),
551 ('p_gates[8:0]', 'bin'),
553 ('eq_1', {'submodule': 'eq_1'}, [
554 ('gates[6:0]', 'bin'),
555 'a[63:0]', 'b[63:0]',
556 ('output[7:0]', 'bin')]),
558 ('p_output[7:0]', 'bin'),
559 't3_1[23:0]', 'lsb_3',
560 ('expected_3[2:0]', 'bin')]
562 'proof_partsig_all_cover.gtkw',
563 os
.path
.dirname(__file__
) +
564 '/proof_partition_partsig_all/engine_0/trace0.vcd',
573 module
= OpDriver(op_all
, nops
=1)
574 self
.assertFormal(module
, mode
="bmc", depth
=1)
575 self
.assertFormal(module
, mode
="cover", depth
=1)
577 def test_partsig_any(self
):
582 module
= OpDriver(op_any
, nops
=1)
583 self
.assertFormal(module
, mode
="bmc", depth
=1)
585 def test_partsig_xor(self
):
590 # 8-bit partitions take a long time, for some reason
591 module
= OpDriver(op_xor
, nops
=1, width
=32, mwidth
=4)
592 self
.assertFormal(module
, mode
="bmc", depth
=1)
594 def test_partsig_add(self
):
596 'dec': {'base': 'dec'},
597 'bin': {'base': 'bin'}
600 ('p_offset[2:0]', 'dec'),
601 ('p_width[3:0]', 'dec'),
602 ('p_gates[8:0]', 'bin'),
603 'i_1[63:0]', 'i_2[63:0]',
604 ('add_1', {'submodule': 'add_1'}, [
605 ('gates[6:0]', 'bin'),
606 'a[63:0]', 'b[63:0]',
608 'p_1[63:0]', 'p_2[63:0]',
610 't3_1[23:0]', 't3_2[23:0]',
613 'proof_partsig_add_cover.gtkw',
614 os
.path
.dirname(__file__
) +
615 '/proof_partition_partsig_add/engine_0/trace0.vcd',
621 module
= OpDriver(operator
.add
, part_out
=False)
622 self
.assertFormal(module
, mode
="bmc", depth
=1)
623 self
.assertFormal(module
, mode
="cover", depth
=1)
625 def test_partsig_sub(self
):
626 module
= OpDriver(operator
.sub
, part_out
=False)
627 self
.assertFormal(module
, mode
="bmc", depth
=1)
629 def test_partsig_neg(self
):
631 'dec': {'base': 'dec'},
632 'bin': {'base': 'bin'}
635 ('p_offset[2:0]', 'dec'),
636 ('p_width[3:0]', 'dec'),
637 ('p_gates[8:0]', 'bin'),
639 ('add_1', {'submodule': 'add_1'}, [
640 ('gates[6:0]', 'bin'),
641 'a[63:0]', 'b[63:0]',
648 'proof_partsig_neg_bmc.gtkw',
649 os
.path
.dirname(__file__
) +
650 '/proof_partition_partsig_neg/engine_0/trace.vcd',
656 module
= OpDriver(operator
.neg
, nops
=1, part_out
=False)
657 self
.assertFormal(module
, mode
="bmc", depth
=1)
660 if __name__
== '__main__':