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"""Satisfiability thresholds of random $k$-XORSAT -- numberdb.org/T272.
For k >= 3, alpha_k is the threshold density m/n for satisfiability of a
random k-XORSAT formula with n variables and m equations, each equation
choosing k distinct variables and a uniform right-hand side. Pittel and
Sorkin give
alpha_k = xi_k / (k (1 - exp(-xi_k))^(k - 1)),
where xi_k is the positive solution of
k = xi (1 - exp(-xi)) / (1 - exp(-xi) - xi exp(-xi)).
This table stores alpha_k for 3 <= k <= 12 at 100 digits in ball arithmetic.
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # fill the draft, with NUMBERDB_API_KEY set
The root xi_k is enclosed by bisection on psi(x) - k, with the signs at the
two ends checked in ball arithmetic. The bracket has half-width
10^-(digits+6), and evaluating alpha_k on the ball around that bracket
encloses the value written to the table.
"""
import os
import sys
import numberdb.sage as numberdb
from numberdb._write import to_text
from sage.rings.real_arb import RealBallField
from sage.rings.real_mpfr import RealField
TABLE = os.environ.get("NUMBERDB_TABLE", "T272")
# Measured before filling the draft: k = 3..12 gives 10 entries, the longest
# entry is under 260 characters, and the entries block is under 4 KB.
K_MAX = 12
# Bits of working precision beyond what the written digits need. Measured at
# 100 digits: every result ball has radius below 3e-106.
WORKING_GUARD = 64
# Decimal places by which the bracket around xi_k is narrower than the values
# written. The derivative of alpha at xi_k is harmless on this range.
BRACKET_GUARD = 6
COMMENT_DIGITS = 12
PUBLISHED_TEN_DIGITS = {
3: "0.9179352767",
4: "0.9767701649",
5: "0.9924383913",
6: "0.9973795528",
7: "0.9990637588",
}
def _key_from_stdin():
if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
return
token = sys.stdin.read().strip()
if "=" in token and token.split("=", 1)[0].isupper():
token = token.split("=", 1)[1].strip().strip("'\"")
if token:
os.environ["NUMBERDB_API_KEY"] = token
def psi(x):
"""x(1-e^-x)/(1-e^-x-xe^-x), the root equation's left side."""
e = (-x).exp()
return x * (1 - e) / (1 - e - x * e)
def root_function(x, k):
"""The sign-changing function whose positive zero is xi_k."""
return psi(x) - k
def alpha_from_xi(x, k):
"""The XORSAT satisfiability threshold from the root xi_k."""
return x / (k * (1 - (-x).exp()) ** (k - 1))
def root(k, digits):
"""A ball enclosing xi_k, found by bisection with ball sign checks."""
if k < 3:
raise ValueError("the XORSAT threshold here is defined for k >= 3")
bits = numberdb.bits(digits, losing=WORKING_GUARD)
RBF, RR = RealBallField(bits), RealField(bits)
delta = RR(10) ** (-(digits + BRACKET_GUARD))
lo, hi = RR(1), RR(4 * k + 10)
if not (root_function(RBF(lo), k) < 0 and root_function(RBF(hi), k) > 0):
raise ArithmeticError("root was not bracketed for k = %s" % k)
while hi - lo > delta / 4:
mid = (lo + hi) / 2
v = root_function(RBF(mid), k)
if v.contains_zero():
raise ArithmeticError("could not decide root sign at x = %s for k = %s" % (mid, k))
if v < 0:
lo = mid
else:
hi = mid
x0 = (lo + hi) / 2
left = root_function(RBF(x0 - delta), k)
right = root_function(RBF(x0 + delta), k)
if not (left < 0 and right > 0):
raise ArithmeticError("no sign change across xi_k bracket for k = %s" % k)
return RBF(x0).add_error(delta)
def threshold(k, digits):
"""(alpha_k, xi_k) as real balls."""
xi = root(k, digits)
alpha = alpha_from_xi(xi, k)
if not (0 < alpha and alpha < 1):
raise ArithmeticError("threshold came out as %s at xi = %s for k = %s" % (alpha, xi, k))
return alpha, xi
class XORSATSatisfiabilityThresholds(numberdb.Generator):
table = TABLE
parameters = ("k",)
type = "R"
digits = 100
rigour = "proven"
def enumerate(self):
for k in range(3, K_MAX + 1):
yield {"k": str(k)}
def value(self, params, digits):
k = int(params["k"])
alpha, xi = threshold(k, digits)
if k in PUBLISHED_TEN_DIGITS:
comment = (
r"The root is $\xi_{%d}=%s$; Dietzfelbinger, Goerdt, "
r"Mitzenmacher, Montanari, Pagh and Rink tabulate this "
r"threshold as $%s$ to ten decimal places CITE{DGMMPR}."
% (k, to_text(xi, COMMENT_DIGITS), PUBLISHED_TEN_DIGITS[k])
)
else:
comment = r"The root is $\xi_{%d}=%s$." % (k, to_text(xi, COMMENT_DIGITS))
return {"number": alpha, "comment": comment}
def fill_draft_once(generator, message):
"""Fill a fresh draft without the client's empty upsert probe."""
from numberdb._generate import (
_check_precision,
_check_rigour,
_producer,
_run_name,
_source_files,
)
from numberdb._write import Entries, attach, submit_entries, to_text
table = generator.table
run = _run_name(generator)
entries = Entries(*generator.parameters)
for params in generator.enumerate():
params = dict(params)
wanted = generator.digits_for(params)
entry = generator._entry(params, wanted)
value = entry["number"]
identity = ",".join(str(params[name]) for name in generator.parameters)
_check_rigour(generator, table, identity, value)
written = to_text(value, wanted, generator.format)
_check_precision(table, identity, written, wanted, lowering=False)
record = dict(entry)
record.pop("digits", None)
entries.add(**params, **record, digits=wanted)
answer = submit_entries(
table,
entries,
message=message,
produced_by=_producer(generator, os.environ.get("NUMBERDB_ASSISTED_BY", "")),
upsert=False,
run=run,
rigour=generator.rigour,
)
for name, body in sorted(_source_files(generator).items()):
attach(table, name, body, run=run, message=message, rigour=generator.rigour)
return answer
if __name__ == "__main__":
_key_from_stdin()
generator = XORSATSatisfiabilityThresholds()
if "--publish" in sys.argv or os.environ.get("NUMBERDB_PUBLISH") == "1":
print(fill_draft_once(
generator,
message=(
"random k-XORSAT satisfiability thresholds alpha_k for 3 <= k <= 12, "
"with the defining root enclosed in ball arithmetic"
),
))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)