src/ieee754/part/formal/proof_partition.py

   1 """Formal verification of partitioned operations
   2
   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.
   8
   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
  11 partition bits.
  12
  13 In other words, we have patterns as follows (assuming 32-bit words)::
  14
  15   8-bit offsets 0,1,2,3
  16   16-bit offsets 0,1,2
  17   24-bit offsets 0,1
  18   32-bit
  19
  20 * for 8-bit the partition bit is 1 and the previous is also 1
  21
  22 * for 16-bit the partition bit at the offset must be 0 and be surrounded by 1
  23
  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
  26
  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)
  29
  30 """
  31
  32 import os
  33 import unittest
  34
  35 from nmigen import Elaboratable, Signal, Module, Const
  36 from nmigen.asserts import Assert, Cover
  37 from nmigen.hdl.ast import Assume
  38
  39 from nmutil.formaltest import FHDLTestCase
  40 from nmutil.gtkw import write_gtkw
  41
  42 from ieee754.part_mul_add.partpoints import PartitionPoints
  43
  44
  45 class PartitionedPattern(Elaboratable):
  46     """ Generate a unique pattern, depending on partition size.
  47
  48     * 1-byte partitions: 0x11
  49     * 2-byte partitions: 0x21 0x22
  50     * 3-byte partitions: 0x31 0x32 0x33
  51
  52     And so on.
  53
  54     Useful as a test vector for testing the formal prover
  55
  56     """
  57     def __init__(self, width, partition_points):
  58         self.width = width
  59         self.partition_points = PartitionPoints(partition_points)
  60         self.mwidth = len(self.partition_points)+1
  61         self.output = Signal(self.width, reset_less=True)
  62
  63     def elaborate(self, platform):
  64         m = Module()
  65         comb = m.d.comb
  66
  67         # Add a guard bit at each end
  68         positions = [0] + list(self.partition_points.keys()) + [self.width]
  69         gates = [Const(1)] + list(self.partition_points.values()) + [Const(1)]
  70         # Begin counting at one
  71         last_start = positions[0]
  72         last_end = positions[1]
  73         last_middle = (last_start+last_end)//2
  74         comb += self.output[last_start:last_middle].eq(1)
  75         # Build an incrementing cascade
  76         for i in range(1, self.mwidth):
  77             start = positions[i]
  78             end = positions[i+1]
  79             middle = (start + end) // 2
  80             # Propagate from the previous byte, adding one to it.
  81             with m.If(~gates[i]):
  82                 comb += self.output[start:middle].eq(
  83                     self.output[last_start:last_middle] + 1)
  84             with m.Else():
  85                 # ... unless it's a partition boundary. If so, start again.
  86                 comb += self.output[start:middle].eq(1)
  87             last_start = start
  88             last_middle = middle
  89         # Mirror the nibbles on the last byte
  90         last_start = positions[-2]
  91         last_end = positions[-1]
  92         last_middle = (last_start+last_end)//2
  93         comb += self.output[last_middle:last_end].eq(
  94             self.output[last_start:last_middle])
  95         for i in range(self.mwidth, 0, -1):
  96             start = positions[i-1]
  97             end = positions[i]
  98             middle = (start + end) // 2
  99             # Propagate from the previous byte.
 100             with m.If(~gates[i]):
 101                 comb += self.output[middle:end].eq(
 102                     self.output[last_middle:last_end])
 103             with m.Else():
 104                 # ... unless it's a partition boundary.
 105                 # If so, mirror the nibbles again.
 106                 comb += self.output[middle:end].eq(
 107                     self.output[start:middle])
 108             last_middle = middle
 109             last_end = end
 110
 111         return m
 112
 113
 114 # This defines a module to drive the device under test and assert
 115 # properties about its outputs
 116 class Driver(Elaboratable):
 117     def __init__(self):
 118         # inputs and outputs
 119         pass
 120
 121     @staticmethod
 122     def elaborate(_):
 123         m = Module()
 124         comb = m.d.comb
 125         width = 64
 126         mwidth = 8
 127         # Setup partition points and gates
 128         points = PartitionPoints()
 129         gates = Signal(mwidth-1)
 130         step = int(width/mwidth)
 131         for i in range(mwidth-1):
 132             points[(i+1)*step] = gates[i]
 133         # Instantiate the partitioned pattern producer
 134         m.submodules.dut = dut = PartitionedPattern(width, points)
 135         # Directly check some cases
 136         with m.If(gates == 0):
 137             comb += Assert(dut.output == 0x_88_87_86_85_84_83_82_81)
 138         with m.If(gates == 0b1100101):
 139             comb += Assert(dut.output == 0x_11_11_33_32_31_22_21_11)
 140         with m.If(gates == 0b0001000):
 141             comb += Assert(dut.output == 0x_44_43_42_41_44_43_42_41)
 142         with m.If(gates == 0b0100001):
 143             comb += Assert(dut.output == 0x_22_21_55_54_53_52_51_11)
 144         with m.If(gates == 0b1000001):
 145             comb += Assert(dut.output == 0x_11_66_65_64_63_62_61_11)
 146         with m.If(gates == 0b0000001):
 147             comb += Assert(dut.output == 0x_77_76_75_74_73_72_71_11)
 148         # Make it interesting, by having four partitions.
 149         comb += Cover(sum(gates) == 3)
 150         # Choose a partition offset and width at random.
 151         p_offset = Signal(range(mwidth))
 152         p_width = Signal(range(mwidth+1))
 153         p_finish = Signal(range(mwidth+1))
 154         comb += p_finish.eq(p_offset + p_width)
 155         # Partition must not be empty, and fit within the signal.
 156         comb += Assume(p_width != 0)
 157         comb += Assume(p_offset + p_width <= mwidth)
 158
 159         # Build the corresponding partition
 160         # Use Assume to constraint the pattern to conform to the given offset
 161         # and width. For each gate bit it is:
 162         # 1) one, if on the partition boundary
 163         # 2) zero, if it's inside the partition
 164         # 3) don't care, otherwise
 165         p_gates = Signal(mwidth+1)
 166         for i in range(mwidth+1):
 167             with m.If(i == p_offset):
 168                 # Partitions begin with 1
 169                 comb += Assume(p_gates[i] == 1)
 170             with m.If((i > p_offset) & (i < p_finish)):
 171                 # The interior are all zeros
 172                 comb += Assume(p_gates[i] == 0)
 173             with m.If(i == p_finish):
 174                 # End with 1 again
 175                 comb += Assume(p_gates[i] == 1)
 176         # Check some possible partitions generating a given pattern
 177         with m.If(p_gates == 0b0100110):
 178             comb += Assert(((p_offset == 1) & (p_width == 1)) |
 179                            ((p_offset == 2) & (p_width == 3)))
 180         # Remove guard bits at each end and assign to the DUT gates
 181         comb += gates.eq(p_gates[1:])
 182         return m
 183
 184
 185 class PartitionTestCase(FHDLTestCase):
 186     def test_formal(self):
 187         traces = [
 188             ('dut', {'submodule': 'dut'}, [
 189                 'output[63:0]',
 190                 ('gates[6:0]', {'base': 'bin'})]),
 191             ('p_offset[2:0]', {'base': 'dec'}),
 192             ('p_width[3:0]', {'base': 'dec'}),
 193             ('p_finish[3:0]', {'base': 'dec'}),
 194             ('p_gates[8:0]', {'base': 'bin'})]
 195         write_gtkw(
 196             'proof_partition_cover.gtkw',
 197             os.path.dirname(__file__) +
 198             '/proof_partition_formal/engine_0/trace0.vcd',
 199             traces,
 200             module='top',
 201             zoom=-3
 202         )
 203         write_gtkw(
 204             'proof_partition_bmc.gtkw',
 205             os.path.dirname(__file__) +
 206             '/proof_partition_formal/engine_0/trace.vcd',
 207             traces,
 208             module='top',
 209             zoom=-3
 210         )
 211         module = Driver()
 212         self.assertFormal(module, mode="bmc", depth=1)
 213         self.assertFormal(module, mode="cover", depth=1)
 214
 215
 216 if __name__ == '__main__':
 217     unittest.main()