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93f2b5e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 | """Fail-closed executable route traces for answer-verified free-form math.
The trace is deliberately separate from the natural-language derivation and
from the boxed answer. It contains only references and operations; it never
contains model-declared intermediate results. Numeric leaves must be grounded
in the problem text, every operation is executed exactly, and every trace node
must contribute to the terminal value.
This validator establishes a checkable computational route, not a proof that
the route's modeling assumptions follow from the prose problem. Ordinary MATH
answer verification remains a separate mandatory admission condition.
"""
from __future__ import annotations
from collections import Counter
from dataclasses import dataclass
from fractions import Fraction
import json
import math
import re
from typing import Any, Mapping
import sympy
MATH_ROUTE_VERSION = "math-route-v1"
MATH_ROUTE_RPN_VERSION = "math-route-rpn-v2"
_MAX_TRACE_CHARS = 4_096
_MAX_STEPS = 20
_MAX_ARGS = 6
_MAX_INTEGER_BITS = 512
_ID_RE = re.compile(r"s[1-9][0-9]?")
_NUMBER_RE = re.compile(
r"(?<![\w.])-?\d+(?:,\d{3})*(?:\.\d+)?(?!\w)(?!\.\d)"
)
_SLASH_FRACTION_RE = re.compile(
r"(?<![\w.])(-?\d+(?:,\d{3})*)\s*/\s*(\d+(?:,\d{3})*)"
r"(?!\w)(?!\.\d)"
)
_LATEX_FRACTION_RE = re.compile(
r"\\(?:d?frac)\s*\{\s*(-?\d+(?:,\d{3})*)\s*\}"
r"\s*\{\s*(\d+(?:,\d{3})*)\s*\}"
)
_ROUTE_BLOCK_RE = re.compile(r"<route>(.*?)</route>", re.DOTALL)
_COMMUTATIVE_OPS = frozenset({"add", "mul", "gcd", "lcm", "min", "max"})
_UNARY_OPS = frozenset(
{"neg", "abs", "square", "cube", "sqrt", "factorial", "percent"}
)
_BINARY_OPS = frozenset(
{"sub", "div", "pow", "mod", "choose", "permute"}
)
_NARY_OPS = frozenset({"add", "mul", "gcd", "lcm", "min", "max", "average"})
_ALLOWED_TOP_LEVEL = frozenset({"version", "steps", "final"})
_ALLOWED_STEP_KEYS = frozenset({"id", "op", "args", "value"})
_RPN_UNARY_OPS = {
"neg": "neg",
"abs": "abs",
"square": "square",
"cube": "cube",
"sqrt": "sqrt",
"factorial": "factorial",
"percent": "percent",
}
_RPN_BINARY_OPS = {
"add": "add",
"mul": "mul",
"sub": "sub",
"div": "div",
"pow": "pow",
"mod": "mod",
"choose": "choose",
"permute": "permute",
"gcd": "gcd",
"lcm": "lcm",
"min": "min",
"max": "max",
"average": "average",
}
@dataclass(frozen=True)
class MathRouteValidation:
route_signature: str
terminal_value: sympy.Expr
operations: tuple[str, ...]
step_count: int
source_count: int
class MathRouteError(ValueError):
"""Raised for a malformed, ungrounded, or non-executable trace."""
def _fraction(text: str) -> Fraction:
compact = str(text).replace(",", "").strip()
if re.fullmatch(r"-?\d+", compact):
return Fraction(int(compact), 1)
if re.fullmatch(r"-?\d+/\d+", compact):
numerator, denominator = compact.split("/", 1)
if int(denominator) == 0:
raise MathRouteError("zero denominator")
return Fraction(int(numerator), int(denominator))
if re.fullmatch(r"-?\d+\.\d+", compact):
return Fraction(compact)
raise MathRouteError("source values must be exact integers, decimals, or fractions")
def problem_number_inventory(problem: str) -> Counter[Fraction]:
"""Return exact numeric leaves present in problem text, with multiplicity."""
text = str(problem)
inventory: Counter[Fraction] = Counter()
def consume_latex(match: re.Match[str]) -> str:
numerator, denominator = match.groups()
inventory[Fraction(int(numerator.replace(",", "")), int(denominator.replace(",", "")))] += 1
return " "
def consume_slash(match: re.Match[str]) -> str:
numerator, denominator = match.groups()
inventory[Fraction(int(numerator.replace(",", "")), int(denominator.replace(",", "")))] += 1
return " "
text = _LATEX_FRACTION_RE.sub(consume_latex, text)
text = _SLASH_FRACTION_RE.sub(consume_slash, text)
for match in _NUMBER_RE.finditer(text):
inventory[_fraction(match.group(0))] += 1
return inventory
def extract_math_route_block(model_response: str) -> str | None:
matches = _ROUTE_BLOCK_RE.findall(str(model_response))
if len(matches) != 1:
return None
block = matches[0].strip()
if not block or len(block) > _MAX_TRACE_CHARS:
return None
return block
def _exact_integer(value: sympy.Expr, *, name: str) -> int:
simplified = sympy.simplify(value)
if not simplified.is_Integer:
raise MathRouteError(f"{name} requires an exact integer")
integer = int(simplified)
if abs(integer).bit_length() > _MAX_INTEGER_BITS:
raise MathRouteError(f"{name} integer is too large")
return integer
def _bounded(value: sympy.Expr) -> sympy.Expr:
simplified = sympy.simplify(value)
if simplified.has(
sympy.nan,
sympy.zoo,
sympy.oo,
-sympy.oo,
sympy.I,
):
raise MathRouteError("operation produced a non-finite or complex value")
if len(str(simplified)) > 1_024 or int(sympy.count_ops(simplified)) > 128:
raise MathRouteError("operation result is too large")
for atom in simplified.atoms(sympy.Integer):
if abs(int(atom)).bit_length() > _MAX_INTEGER_BITS:
raise MathRouteError("operation result integer is too large")
return simplified
def _execute_operation(op: str, arguments: tuple[sympy.Expr, ...]) -> sympy.Expr:
if op in _UNARY_OPS and len(arguments) != 1:
raise MathRouteError(f"{op} requires one argument")
if op in _BINARY_OPS and len(arguments) != 2:
raise MathRouteError(f"{op} requires two arguments")
if op in _NARY_OPS and not 2 <= len(arguments) <= _MAX_ARGS:
raise MathRouteError(f"{op} requires two to {_MAX_ARGS} arguments")
if op == "neg":
result = -arguments[0]
elif op == "abs":
result = sympy.Abs(arguments[0])
elif op == "square":
result = arguments[0] ** 2
elif op == "cube":
result = arguments[0] ** 3
elif op == "sqrt":
if arguments[0].is_nonnegative is not True:
raise MathRouteError("sqrt requires a provably nonnegative argument")
result = sympy.sqrt(arguments[0])
elif op == "factorial":
integer = _exact_integer(arguments[0], name=op)
if not 0 <= integer <= 100:
raise MathRouteError("factorial input is outside [0,100]")
result = sympy.factorial(integer)
elif op == "percent":
result = arguments[0] / 100
elif op == "add":
result = sum(arguments, sympy.Integer(0))
elif op == "mul":
result = math.prod(arguments, start=sympy.Integer(1))
elif op == "sub":
result = arguments[0] - arguments[1]
elif op == "div":
if sympy.simplify(arguments[1]) == 0:
raise MathRouteError("division by zero")
result = arguments[0] / arguments[1]
elif op == "pow":
exponent = _exact_integer(arguments[1], name=op)
if not -12 <= exponent <= 12:
raise MathRouteError("power exponent is outside [-12,12]")
if arguments[0] == 0 and exponent < 0:
raise MathRouteError("zero to a negative power")
result = arguments[0] ** exponent
elif op == "mod":
left = _exact_integer(arguments[0], name=op)
right = _exact_integer(arguments[1], name=op)
if right == 0:
raise MathRouteError("modulo by zero")
result = sympy.Integer(left % right)
elif op == "choose":
n = _exact_integer(arguments[0], name=op)
k = _exact_integer(arguments[1], name=op)
if not 0 <= k <= n <= 10_000:
raise MathRouteError("choose inputs are outside 0 <= k <= n <= 10000")
result = sympy.binomial(n, k)
elif op == "permute":
n = _exact_integer(arguments[0], name=op)
k = _exact_integer(arguments[1], name=op)
if not 0 <= k <= n <= 1_000:
raise MathRouteError("permute inputs are outside 0 <= k <= n <= 1000")
result = sympy.factorial(n) / sympy.factorial(n - k)
elif op in {"gcd", "lcm"}:
integers = [_exact_integer(argument, name=op) for argument in arguments]
function = math.gcd if op == "gcd" else math.lcm
result = sympy.Integer(function(*integers))
elif op == "min":
if not all(argument.is_real is True for argument in arguments):
raise MathRouteError("min requires real arguments")
result = sympy.Min(*arguments)
elif op == "max":
if not all(argument.is_real is True for argument in arguments):
raise MathRouteError("max requires real arguments")
result = sympy.Max(*arguments)
elif op == "average":
result = sum(arguments, sympy.Integer(0)) / len(arguments)
else:
raise MathRouteError(f"unsupported route operation {op!r}")
return _bounded(result)
def _signature(op: str, argument_signatures: tuple[str, ...]) -> str:
children = (
tuple(sorted(argument_signatures))
if op in _COMMUTATIVE_OPS
else argument_signatures
)
return f"{op}({','.join(children)})"
def validate_math_route_trace(
trace: str | Mapping[str, Any],
problem: str,
) -> MathRouteValidation | None:
"""Parse and execute a grounded route trace; return ``None`` on any failure."""
try:
parsed = json.loads(trace) if isinstance(trace, str) else dict(trace)
if not isinstance(parsed, dict) or set(parsed) != _ALLOWED_TOP_LEVEL:
raise MathRouteError("route object has missing or extra fields")
if parsed["version"] != MATH_ROUTE_VERSION:
raise MathRouteError("unsupported route version")
raw_steps = parsed["steps"]
if not isinstance(raw_steps, list) or not 2 <= len(raw_steps) <= _MAX_STEPS:
raise MathRouteError("route has an invalid number of steps")
final_id = str(parsed["final"])
if _ID_RE.fullmatch(final_id) is None:
raise MathRouteError("invalid final node ID")
inventory = problem_number_inventory(problem)
values: dict[str, sympy.Expr] = {}
signatures: dict[str, str] = {}
dependencies: dict[str, tuple[str, ...]] = {}
operations: list[str] = []
source_count = 0
non_source_count = 0
for expected_index, raw_step in enumerate(raw_steps, start=1):
if not isinstance(raw_step, dict):
raise MathRouteError("route step must be an object")
if not set(raw_step).issubset(_ALLOWED_STEP_KEYS):
raise MathRouteError("route step has extra fields")
step_id = str(raw_step.get("id", ""))
if step_id != f"s{expected_index}" or step_id in values:
raise MathRouteError("route node IDs must be contiguous and ordered")
op = str(raw_step.get("op", ""))
if op == "source":
if set(raw_step) != {"id", "op", "value"}:
raise MathRouteError("source step schema is invalid")
value = _fraction(str(raw_step["value"]))
if inventory[value] <= 0:
raise MathRouteError("source value is not available in the problem")
inventory[value] -= 1
values[step_id] = sympy.Rational(value.numerator, value.denominator)
signatures[step_id] = "input"
dependencies[step_id] = ()
source_count += 1
operations.append(op)
continue
if set(raw_step) != {"id", "op", "args"}:
raise MathRouteError("operation step schema is invalid")
raw_args = raw_step["args"]
if not isinstance(raw_args, list):
raise MathRouteError("operation args must be a list")
argument_ids = tuple(str(argument) for argument in raw_args)
if (
not argument_ids
or len(argument_ids) > _MAX_ARGS
or len(set(argument_ids)) != len(argument_ids)
or any(argument_id not in values for argument_id in argument_ids)
):
raise MathRouteError("operation dependencies are invalid")
arguments = tuple(values[argument_id] for argument_id in argument_ids)
values[step_id] = _execute_operation(op, arguments)
signatures[step_id] = _signature(
op,
tuple(signatures[argument_id] for argument_id in argument_ids),
)
dependencies[step_id] = argument_ids
operations.append(op)
non_source_count += 1
if final_id not in values or non_source_count < 1 or source_count < 1:
raise MathRouteError("route lacks a computed terminal value")
reachable = {final_id}
frontier = [final_id]
while frontier:
node = frontier.pop()
for dependency in dependencies[node]:
if dependency not in reachable:
reachable.add(dependency)
frontier.append(dependency)
if reachable != set(values):
raise MathRouteError("every route node must contribute to the final value")
return MathRouteValidation(
route_signature=(
f"math-route:{MATH_ROUTE_VERSION}:{signatures[final_id]}"
),
terminal_value=values[final_id],
operations=tuple(operations),
step_count=len(values),
source_count=source_count,
)
except Exception:
return None
def validate_math_route_rpn(
trace: str,
problem: str,
) -> MathRouteValidation | None:
"""Execute the compact v2 reverse-Polish route language.
Numeric tokens are grounded problem leaves. Unary and binary operator
tokens consume the stack, so a successful one-item terminal stack also
proves that every emitted token contributes to the final value. The
language has no result literals, variable names, prose, or code execution.
"""
try:
compact = str(trace).strip()
if not compact or len(compact) > 1_024:
raise MathRouteError("RPN route is empty or too long")
tokens = compact.split()
if not tokens or tokens[0] not in {
"v2",
MATH_ROUTE_RPN_VERSION,
}:
raise MathRouteError("unsupported RPN route version")
tokens = tokens[1:]
if not 2 <= len(tokens) <= 40:
raise MathRouteError("RPN route has an invalid token count")
inventory = problem_number_inventory(problem)
stack: list[tuple[sympy.Expr, str]] = []
operations: list[str] = []
source_count = 0
operation_count = 0
for token in tokens:
if token in _RPN_UNARY_OPS:
if len(stack) < 1:
raise MathRouteError("RPN unary operation underflow")
argument_value, argument_signature = stack.pop()
op = _RPN_UNARY_OPS[token]
stack.append(
(
_execute_operation(op, (argument_value,)),
_signature(op, (argument_signature,)),
)
)
operations.append(op)
operation_count += 1
continue
if token in _RPN_BINARY_OPS:
if len(stack) < 2:
raise MathRouteError("RPN binary operation underflow")
right_value, right_signature = stack.pop()
left_value, left_signature = stack.pop()
op = _RPN_BINARY_OPS[token]
stack.append(
(
_execute_operation(op, (left_value, right_value)),
_signature(op, (left_signature, right_signature)),
)
)
operations.append(op)
operation_count += 1
continue
value = _fraction(token)
if inventory[value] <= 0:
raise MathRouteError("RPN source value is not available in the problem")
inventory[value] -= 1
stack.append(
(
sympy.Rational(value.numerator, value.denominator),
"input",
)
)
operations.append("source")
source_count += 1
if len(stack) != 1 or source_count < 1 or operation_count < 1:
raise MathRouteError("RPN route lacks one computed terminal value")
terminal_value, terminal_signature = stack[0]
return MathRouteValidation(
route_signature=(
f"math-route:{MATH_ROUTE_RPN_VERSION}:{terminal_signature}"
),
terminal_value=terminal_value,
operations=tuple(operations),
step_count=len(tokens),
source_count=source_count,
)
except Exception:
return None
def validate_math_route_response(
model_response: str,
problem: str,
) -> MathRouteValidation | None:
"""Extract the response's unique route block and execute it."""
block = extract_math_route_block(model_response)
if block is None:
return None
if block.lstrip().startswith("{"):
return validate_math_route_trace(block, problem)
return validate_math_route_rpn(block, problem)
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