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33f388b | 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 | """Exact reciprocal repair, over GF(2). Python 3.10+, standard library only.
Square matrices use one Python integer per row. Reported work is a specified
word-RAM model, NOT FLOPs or a claimed hardware-neutral speedup. n <= 64.
Every matrix traversal, word xor, set-bit scan, comparison, RNG draw and
search bookkeeping has a deterministic charge; elapsed time is also reported.
"""
from __future__ import annotations
from collections import Counter
from dataclasses import dataclass, field
import random
Mat = tuple[int, ...]
class BudgetExhausted(RuntimeError):
pass
@dataclass
class Meter:
limit: int | None = None
used: int = 0
counts: Counter = field(default_factory=Counter)
def use(self, amount: int, kind: str = "misc") -> None:
if self.limit is not None and self.used + amount > self.limit:
remaining = self.limit - self.used
self.counts[kind] += remaining
self.used = self.limit
raise BudgetExhausted
self.used += amount
self.counts[kind] += amount
@dataclass(frozen=True)
class Problem:
d: Mat
Q: Mat
P: Mat
h0: Mat
grades: tuple[int, ...]
@property
def n(self) -> int:
return len(self.d)
def zero(n: int) -> Mat:
return (0,) * n
def eye(n: int) -> Mat:
return tuple(1 << i for i in range(n))
def add(A: Mat, B: Mat, m: Meter) -> Mat:
m.use(2 * len(A), "xor")
return tuple(a ^ b for a, b in zip(A, B))
def mul(A: Mat, B: Mat, m: Meter) -> Mat:
out = []
for row in A:
m.use(2, "multiply")
value = 0
while row:
m.use(6, "multiply")
low = row & -row
value ^= B[low.bit_length() - 1]
row ^= low
out.append(value)
return tuple(out)
def nnz(A: Mat, m: Meter) -> int:
m.use(2 * len(A), "score")
return sum(row.bit_count() for row in A)
def equal(A: Mat, B: Mat, m: Meter) -> bool:
m.use(len(A), "compare")
return A == B
def differential(p: Problem, f: Mat, m: Meter) -> Mat:
# Signs coincide in characteristic two, for every degree.
return add(mul(p.d, f, m), mul(f, p.d, m), m)
def K(p: Problem, h: Mat, f: Mat, m: Meter, cross: bool = True) -> Mat:
hf = mul(h, f, m)
if not cross:
return hf
return add(hf, mul(mul(p.Q, f, m), h, m), m)
def valid(p: Problem, h: Mat, m: Meter) -> bool:
z = zero(p.n)
# Check all identities; do not short-circuit invalid candidates.
checks = [equal(differential(p, h, m), p.P, m),
equal(mul(h, h, m), z, m),
equal(mul(p.Q, h, m), z, m),
equal(mul(h, p.Q, m), z, m)]
m.use(p.n * p.n, "degree_check")
graded = all(not (row >> j & 1) or p.grades[i] == p.grades[j] - 1
for i, row in enumerate(h) for j in range(p.n))
return all(checks) and graded
def normalize(p: Problem, g: Mat, m: Meter) -> Mat:
return mul(mul(g, p.d, m), g, m)
def repaired_candidate(p: Problem, current: Mat, generator: Mat,
a: Mat, m: Meter, protect: bool = False) -> Mat:
# z = P (a - K_generator(partial a)) P; minus = plus over GF(2).
z = add(a, K(p, generator, differential(p, a, m), m), m)
if protect:
z = mul(mul(p.P, z, m), p.P, m)
return normalize(p, add(current, z, m), m)
def boundary_candidate(p: Problem, current: Mat, B: Mat, m: Meter) -> Mat:
# Strong simple reduction: any fixed-Q closed variation is partial B.
z = differential(p, B, m)
# gdg automatically removes Q mixing; explicit P sandwiches are redundant.
return normalize(p, add(current, z, m), m)
def positions(p: Problem, degree: int, m: Meter) -> tuple[tuple[int, int], ...]:
m.use(p.n * p.n, "setup")
return tuple((i, j) for i in range(p.n) for j in range(p.n)
if p.grades[i] == p.grades[j] + degree)
def proposal(p: Problem, available: tuple[tuple[int, int], ...],
rng: random.Random, m: Meter, width: int = 1) -> Mat:
m.use(p.n + 7 * width, "proposal")
rows = [0] * p.n
for _ in range(width):
i, j = available[rng.randrange(len(available))]
rows[i] ^= 1 << j
return tuple(rows)
def kernel_work(p: Problem, h: Mat, tasks: list[Mat],
charge: Meter | None = None) -> int:
"""Execute, rather than estimate, the kernel on fresh repair problems."""
local = Meter()
for task in tasks:
K(p, h, task, local)
if charge is not None:
charge.use(local.used, "kernel_evaluation")
return local.used
def semantic_certificate(p: Problem, h: Mat, f: Mat, m: Meter,
cross: bool = True) -> bool:
"""Check partial K + K partial = f - Q f Q for an arbitrary End task."""
lhs = add(differential(p, K(p, h, f, m, cross), m),
K(p, h, differential(p, f, m), m, cross), m)
rhs = add(f, mul(mul(p.Q, f, m), p.Q, m), m)
return equal(lhs, rhs, m)
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