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947a44b 7ed141b 947a44b 7ed141b 947a44b 7ed141b 782741e 7ed141b | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 | """Attractor computer: an energy-based threshold network.
A Boolean circuit is compiled to a quadratic pseudo-Boolean energy
E(s) = sum_i L[i] s_i + sum_{i<j} Q[i,j] s_i s_j over binary wire variables, with
per-gate gadgets that are non-negative and zero exactly on the gate truth table:
AND z=x&y : 3z + xy - 2xz - 2yz
OR z=x|y : x + y + z + xy - 2xz - 2yz
NOT z=~x : 1 - x - z + 2xz
AND/OR/NOT are functionally complete, so any circuit compiles and its consistent
assignment is the global minimum. There is no program counter or clock: the
coupling matrix Q (with linear terms L) is the program, and execution is
relaxation toward the minimum. The relaxation step is a threshold neuron over the
same integer weights, s_i <- H(-(L[i] + sum_j Q[i,j] s_j)).
The clamped wire subset selects the mode. Clamp inputs to evaluate forward (exact
via topological propagation to the energy-0 fixed point); clamp outputs to invert
(a multiplier run backward returns factors); clamp a CNF output to 1 to solve
SAT. Forward evaluation is exact and linear in gate count; inversion and
open-constraint solving are annealed ground-state search, NP-hard in general,
with a zero-energy state certifying a correct assignment.
"""
from __future__ import annotations
import math
import random
from collections import defaultdict
from typing import Dict, List, Optional, Tuple
class Circuit:
"""Wire allocator and energy accumulator. Gates append exact QUBO gadgets
and record the relation for topological forward evaluation."""
def __init__(self) -> None:
self.n = 0
self.L: Dict[int, int] = defaultdict(int)
self.Q: Dict[Tuple[int, int], int] = defaultdict(int)
self.const = 0
self.gates: List[Tuple[str, int, Tuple[int, ...]]] = []
def wire(self) -> int:
i = self.n
self.n += 1
return i
def wires(self, k: int) -> List[int]:
return [self.wire() for _ in range(k)]
def _q(self, i: int, j: int, c: int) -> None:
if i == j:
self.L[i] += c
else:
self.Q[(min(i, j), max(i, j))] += c
def AND(self, x: int, y: int) -> int:
z = self.wire()
self.L[z] += 3
self._q(x, y, 1); self._q(x, z, -2); self._q(y, z, -2)
self.gates.append(("AND", z, (x, y)))
return z
def OR(self, x: int, y: int) -> int:
z = self.wire()
self.L[x] += 1; self.L[y] += 1; self.L[z] += 1
self._q(x, y, 1); self._q(x, z, -2); self._q(y, z, -2)
self.gates.append(("OR", z, (x, y)))
return z
def NOT(self, x: int) -> int:
z = self.wire()
self.const += 1
self.L[x] += -1; self.L[z] += -1
self._q(x, z, 2)
self.gates.append(("NOT", z, (x,)))
return z
def XOR(self, x: int, y: int) -> int:
return self.OR(self.AND(x, self.NOT(y)), self.AND(self.NOT(x), y))
def full_adder(self, x: int, y: int, cin: int) -> Tuple[int, int]:
axy = self.XOR(x, y)
s = self.XOR(axy, cin)
cout = self.OR(self.AND(x, y), self.AND(cin, axy))
return s, cout
# ---- energy + couplings ------------------------------------------------
def energy(self, s: List[int]) -> int:
e = self.const
for i, c in self.L.items():
e += c * s[i]
for (i, j), c in self.Q.items():
e += c * s[i] * s[j]
return e
def neighbors(self) -> Dict[int, List[Tuple[int, int]]]:
nbr: Dict[int, List[Tuple[int, int]]] = defaultdict(list)
for (i, j), c in self.Q.items():
nbr[i].append((j, c))
nbr[j].append((i, c))
return nbr
# ---- relaxation modes --------------------------------------------------
def forward_eval(self, clamp: Dict[int, int]) -> List[int]:
"""Exact forward relaxation: propagate clamped inputs through the gate
relations in topological order onto the energy-0 fixed point."""
s = [0] * self.n
for w, v in clamp.items():
s[w] = v
for op, z, ins in self.gates:
if op == "AND":
s[z] = s[ins[0]] & s[ins[1]]
elif op == "OR":
s[z] = s[ins[0]] | s[ins[1]]
else:
s[z] = 1 - s[ins[0]]
return s
def relax_energy(self, clamp: Dict[int, int], sweeps: int = 4000,
t0: float = 4.0, t1: float = 0.02, seed: int = 0
) -> Tuple[List[int], bool]:
"""Canonical relaxation: anneal the full threshold network (every free
wire), tracking the lowest-energy state. Universal but hard; the exact
gadgets keep the target at energy 0."""
nbr = self.neighbors()
rng = random.Random(seed)
s = [rng.randint(0, 1) for _ in range(self.n)]
for w, v in clamp.items():
s[w] = v
free = [i for i in range(self.n) if i not in clamp]
best, best_e = list(s), self.energy(s)
for step in range(sweeps):
T = t0 * (t1 / t0) ** (step / max(1, sweeps - 1))
for _ in range(len(free)):
i = free[rng.randrange(len(free))]
field = self.L[i] + sum(c * s[j] for j, c in nbr[i])
dE = (1 - 2 * s[i]) * field
if dE <= 0 or rng.random() < math.exp(-dE / T):
s[i] ^= 1
e = self.energy(s)
if e < best_e:
best, best_e = list(s), e
if best_e == 0:
return best, True
return best, best_e == 0
def solve(self, free_inputs: List[int], fixed: Dict[int, int],
target: Dict[int, int], sweeps: int = 3000, restarts: int = 80,
seed: int = 0) -> Optional[List[int]]:
"""Open-constraint relaxation over a chosen set of driver wires, with
the rest slaved through the circuit; anneal the output Hamming mismatch
to zero. Clamp outputs and pass the inputs here to run backward."""
rng = random.Random(seed)
def mism(vals: Dict[int, int]) -> int:
s = self.forward_eval({**fixed, **vals})
return sum(1 for w, v in target.items() if s[w] != v)
for _ in range(restarts):
vals = {w: rng.randint(0, 1) for w in free_inputs}
m = mism(vals)
if m == 0:
return self.forward_eval({**fixed, **vals})
for step in range(sweeps):
T = 2.0 * (0.02 / 2.0) ** (step / sweeps)
w = free_inputs[rng.randrange(len(free_inputs))]
vals[w] ^= 1
m2 = mism(vals)
if m2 <= m or rng.random() < math.exp(-(m2 - m) / T):
m = m2
if m == 0:
return self.forward_eval({**fixed, **vals})
else:
vals[w] ^= 1
return None
# ---------------------------------------------------------------------------
# Circuit builders
# ---------------------------------------------------------------------------
def adder(bits: int) -> Tuple[Circuit, dict]:
c = Circuit()
xs, ys = c.wires(bits), c.wires(bits)
cin = c.wire()
outs, carry = [], cin
for k in range(bits):
s, carry = c.full_adder(xs[k], ys[k], carry)
outs.append(s)
return c, {"xs": xs, "ys": ys, "cin": cin, "sum": outs + [carry]}
def multiplier(bits: int) -> Tuple[Circuit, dict]:
c = Circuit()
xs, ys = c.wires(bits), c.wires(bits)
zero = c.wire()
acc = [zero] * (2 * bits)
for i in range(bits):
carry = zero
for j in range(bits):
acc[i + j], carry = c.full_adder(acc[i + j], c.AND(xs[i], ys[j]), carry)
acc[i + bits] = carry
return c, {"xs": xs, "ys": ys, "zero": zero, "prod": acc}
_OPCODE = {"AND": 0, "OR": 1, "NOT": 2}
_OPNAME = {v: k for k, v in _OPCODE.items()}
def to_tensors(circ: Circuit, io: dict):
"""Serialize the coupling matrix (the program) and the gate list to tensors.
Q is stored sparsely as index pairs and integer values."""
import torch
qi = sorted(circ.Q)
q_idx = torch.tensor(qi if qi else [], dtype=torch.long).reshape(-1, 2)
q_val = torch.tensor([circ.Q[k] for k in qi], dtype=torch.long)
li = sorted(circ.L)
l_idx = torch.tensor(li, dtype=torch.long)
l_val = torch.tensor([circ.L[i] for i in li], dtype=torch.long)
g_op = torch.tensor([_OPCODE[op] for op, _, _ in circ.gates], dtype=torch.long)
g_out = torch.tensor([o for _, o, _ in circ.gates], dtype=torch.long)
g_in = torch.tensor([[ins[0], ins[1] if len(ins) > 1 else -1]
for _, _, ins in circ.gates], dtype=torch.long).reshape(-1, 2)
t = {"Q_idx": q_idx, "Q_val": q_val, "L_idx": l_idx, "L_val": l_val,
"gate_op": g_op, "gate_out": g_out, "gate_in": g_in}
import json
meta = {"machine": "attractor", "n": str(circ.n), "const": str(circ.const),
"io": json.dumps({k: v for k, v in io.items()})}
return t, meta
def from_tensors(t: dict, meta: dict) -> Tuple[Circuit, dict]:
import json
c = Circuit()
c.n = int(meta["n"])
c.const = int(meta["const"])
for (i, j), v in zip(t["Q_idx"].tolist(), t["Q_val"].tolist()):
c.Q[(i, j)] = v
for i, v in zip(t["L_idx"].tolist(), t["L_val"].tolist()):
c.L[i] = v
for op, out, ins in zip(t["gate_op"].tolist(), t["gate_out"].tolist(), t["gate_in"].tolist()):
c.gates.append((_OPNAME[op], out, tuple(x for x in ins if x >= 0)))
return c, json.loads(meta["io"])
def cnf(clauses: List[List[int]], n_vars: int) -> Tuple[Circuit, dict]:
"""Compile a CNF formula. Literals are +v (var v) or -v (negation), v>=1.
Returns the circuit, the variable wires, and the wire that is 1 iff the
formula is satisfied. Clamp that wire to 1 and relax to find a model."""
c = Circuit()
var = {v: c.wire() for v in range(1, n_vars + 1)}
clause_ws = []
for cl in clauses:
lits = [var[abs(l)] if l > 0 else c.NOT(var[abs(l)]) for l in cl]
acc = lits[0]
for w in lits[1:]:
acc = c.OR(acc, w)
clause_ws.append(acc)
sat = clause_ws[0]
for w in clause_ws[1:]:
sat = c.AND(sat, w)
return c, {"vars": var, "sat": sat}
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