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22ef089 | 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 | """Independent exact arithmetic checks for the exterior kernel-feedback candidate.
These finite checks are an implementation audit; the accompanying prose review
contains the universal arguments. SymPy performs all calculations over Q.
"""
from itertools import combinations
from random import Random
import json
import sympy as s
rng = Random(202610071)
def dot(x, y):
return (x.T * y)[0]
def norm2(x):
return dot(x, x)
def wedge(x, a):
return x * a.T - a * x.T
def matrix_columns(cols, d):
return s.Matrix.hstack(*cols) if cols else s.zeros(d, 0)
def split(A, x):
d = len(x)
K = matrix_columns(A.nullspace(), d)
P = K * (K.T * K).inv() * K.T if K.cols else s.zeros(d)
w = P * x
u = x - w
h = (A + P).inv() * u
return K, P, u, w, h
def pf(A):
if A.rows == 0:
return s.Integer(1)
if A.rows == 2:
return A[0, 1]
out = s.Integer(0)
for j in range(1, A.rows):
ii = [i for i in range(1, A.rows) if i != j]
out += (-1) ** (j + 1) * A[0, j] * pf(A.extract(ii, ii))
return out
def one_step(A, x, a):
d = len(x)
assert A + A.T == s.zeros(d)
assert A * a == s.zeros(d, 1)
K, P, u, w, h = split(A, x)
B, y = A + wedge(x, a), x + a
L, Q, v, z, hh = split(B, y)
independent = matrix_columns([w, a], d).rank() == 2
if independent:
pred = matrix_columns(A.col_join(w.T).col_join(a.T).nullspace(), d)
expected_rho = norm2(u) + norm2(w + a)
assert B.rank() == A.rank() + 2
assert z == s.zeros(d, 1)
else:
pred = K - h * (a.T * K)
expected_rho = norm2(u) + norm2(h) * dot(a, w + a) ** 2 / (1 + norm2(h) * norm2(a))
assert B.rank() == A.rank()
assert B * pred == s.zeros(d, pred.cols)
assert pred.rank() == L.cols
assert norm2(v) == s.cancel(expected_rho)
assert norm2(v) >= norm2(u)
for k in range(1, d // 2 + 1):
old = [pf(A.extract(ii, ii)) for ii in combinations(range(d), 2 * k)]
new = [pf(B.extract(ii, ii)) for ii in combinations(range(d), 2 * k)]
cross = sum(p * (q - p) for p, q in zip(old, new))
assert cross == 0
assert sum(q * q for q in new) == sum(p * p for p in old) + sum((q - p) ** 2 for p, q in zip(old, new))
C, t = A + wedge(x, -w), u
KK, PP, uu, ww, hhh = split(C, t)
assert ww == s.zeros(d, 1)
assert norm2(uu) == norm2(u)
return B, y, independent
def random_state(d, rank):
while True:
C = s.Matrix(d, rank, lambda i, j: rng.randint(-2, 2))
if C.rank() == rank:
break
J = s.zeros(rank)
for i in range(0, rank, 2):
J[i, i + 1], J[i + 1, i] = 1, -1
A = C * J * C.T
x = s.Matrix([rng.randint(-2, 2) for i in range(d)])
return A, x
results = {"rational_one_step_cases": 0, "independent_cases": 0, "dependent_cases": 0, "maxrank_paths": 0}
for d in range(1, 8):
for rank in range(0, d + 1, 2):
for repeat in range(3):
A, x = random_state(d, rank)
K, P, u, w, h = split(A, x)
controls = [s.zeros(d, 1), -w]
if K.cols:
controls.append(K * s.Matrix([rng.randint(-2, 2) for _ in range(K.cols)]))
for a in controls:
B, y, independent = one_step(A, x, a)
results["rational_one_step_cases"] += 1
results["independent_cases" if independent else "dependent_cases"] += 1
# The construction also covers states with w forced to zero.
for start_x in (x, u):
AA, xx = A, start_x
KK, PP, uu, ww, hh = split(AA, xx)
m = KK.cols // 2
expected = 0 if m == 0 else 2 * m - (0 if ww == s.zeros(d, 1) else 1)
count = 0
while AA.rank() < 2 * (d // 2):
KK, PP, uu, ww, hh = split(AA, xx)
if ww == s.zeros(d, 1):
a = KK[:, 0]
else:
a = next(KK[:, i] for i in range(KK.cols) if matrix_columns([ww, KK[:, i]], d).rank() == 2)
AA, xx, independent = one_step(AA, xx, a)
count += 1
assert count == expected
results["maxrank_paths"] += 1
# Separation family, including exact signs of lambda.
for lam in map(s.Rational, [-5, -2, -1, 1, 2, 5]):
x = s.Matrix([1, 1, 0])
A = s.Matrix([[0, lam, 0], [-lam, 0, 0], [0, 0, 0]])
a = s.Matrix([0, 0, 1])
B, y, independent = one_step(A, x, a)
assert B * s.Matrix([1, -1, lam]) == s.zeros(3, 1)
K, P, u, w, h = split(B, y)
assert norm2(u) == 2 + 2 / (lam * lam + 2)
results["separation_lambdas"] = 6
# Dormant block replacement: run the same legal exact words in both systems.
for weight in (1, 2, 3):
J = s.Matrix([[0, 1], [-1, 0]])
A = s.diag(J, weight * J, s.zeros(3))
B = s.diag(J, (weight + 5) * J, s.zeros(3))
x = s.Matrix([1, 2, 0, 0, 1, -1, 0])
y = x.copy()
for step in range(5):
K, P, u, w, h = split(A, x)
L, Q, v, z, hh = split(B, y)
assert P == Q
assert x == y
a = K * s.Matrix([rng.randint(-2, 2) for _ in range(K.cols)]) if K.cols else s.zeros(7, 1)
assert a[2] == a[3] == 0
A, x = A + wedge(x, a), x + a
B, y = B + wedge(y, a), y + a
assert A[2:4, 2:4] == weight * J
assert B[2:4, 2:4] == (weight + 5) * J
results["dormant_replacement_paths"] = 3
# General three-dimensional accessibility: no projection preparation needed.
bb, cc, zz, pp, qq, rr = s.symbols("b c z p q r", real=True)
xx, kk = s.Matrix([bb, 0, zz]), s.Matrix([0, 0, cc])
for tt in (pp, qq, rr):
xx, kk = s.expand(xx + tt * kk), s.expand(kk + tt * xx.cross(kk))
expected_endpoint = s.Matrix([
bb + bb * cc**2 * pp**2 * qq * rr + bb * cc * zz * pp * qq * rr,
-bb * cc * (pp * qq + pp * rr + qq * rr),
zz + cc * (pp + qq + rr - bb**2 * pp * qq * rr),
])
assert s.simplify(xx - expected_endpoint) == s.zeros(3, 1)
jacobian_at_point = xx.jacobian((pp, qq, rr)).subs({pp: 1, qq: 1, rr: 2})
assert s.factor(jacobian_at_point.det()) == -2 * bb**2 * cc**4 * (bb**2 + 1)
results["symbolic_general_accessibility_checks"] = 1
print(json.dumps(results, indent=2))
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