# Natural Language Toolkit: Interface to the Prover9 Theorem Prover
#
# Copyright (C) 2001-2023 NLTK Project
# Author: Dan Garrette <dhgarrette@gmail.com>
# Ewan Klein <ewan@inf.ed.ac.uk>
#
# URL: <https://www.nltk.org/>
# For license information, see LICENSE.TXT
"""
A theorem prover that makes use of the external 'Prover9' package.
"""
import os
import subprocess
import nltk
from nltk.inference.api import BaseProverCommand, Prover
from nltk.sem.logic import (
AllExpression,
AndExpression,
EqualityExpression,
ExistsExpression,
Expression,
IffExpression,
ImpExpression,
NegatedExpression,
OrExpression,
)
#
# Following is not yet used. Return code for 2 actually realized as 512.
#
p9_return_codes = {
0: True,
1: "(FATAL)", # A fatal error occurred (user's syntax error).
2: False, # (SOS_EMPTY) Prover9 ran out of things to do
# (sos list exhausted).
3: "(MAX_MEGS)", # The max_megs (memory limit) parameter was exceeded.
4: "(MAX_SECONDS)", # The max_seconds parameter was exceeded.
5: "(MAX_GIVEN)", # The max_given parameter was exceeded.
6: "(MAX_KEPT)", # The max_kept parameter was exceeded.
7: "(ACTION)", # A Prover9 action terminated the search.
101: "(SIGSEGV)", # Prover9 crashed, most probably due to a bug.
}
[docs]class Prover9CommandParent:
"""
A common base class used by both ``Prover9Command`` and ``MaceCommand``,
which is responsible for maintaining a goal and a set of assumptions,
and generating prover9-style input files from them.
"""
[docs] def print_assumptions(self, output_format="nltk"):
"""
Print the list of the current assumptions.
"""
if output_format.lower() == "nltk":
for a in self.assumptions():
print(a)
elif output_format.lower() == "prover9":
for a in convert_to_prover9(self.assumptions()):
print(a)
else:
raise NameError(
"Unrecognized value for 'output_format': %s" % output_format
)
[docs]class Prover9Command(Prover9CommandParent, BaseProverCommand):
"""
A ``ProverCommand`` specific to the ``Prover9`` prover. It contains
the a print_assumptions() method that is used to print the list
of assumptions in multiple formats.
"""
[docs] def __init__(self, goal=None, assumptions=None, timeout=60, prover=None):
"""
:param goal: Input expression to prove
:type goal: sem.Expression
:param assumptions: Input expressions to use as assumptions in
the proof.
:type assumptions: list(sem.Expression)
:param timeout: number of seconds before timeout; set to 0 for
no timeout.
:type timeout: int
:param prover: a prover. If not set, one will be created.
:type prover: Prover9
"""
if not assumptions:
assumptions = []
if prover is not None:
assert isinstance(prover, Prover9)
else:
prover = Prover9(timeout)
BaseProverCommand.__init__(self, prover, goal, assumptions)
[docs] def decorate_proof(self, proof_string, simplify=True):
"""
:see BaseProverCommand.decorate_proof()
"""
if simplify:
return self._prover._call_prooftrans(proof_string, ["striplabels"])[
0
].rstrip()
else:
return proof_string.rstrip()
[docs]class Prover9Parent:
"""
A common class extended by both ``Prover9`` and ``Mace <mace.Mace>``.
It contains the functionality required to convert NLTK-style
expressions into Prover9-style expressions.
"""
_binary_location = None
[docs] def config_prover9(self, binary_location, verbose=False):
if binary_location is None:
self._binary_location = None
self._prover9_bin = None
else:
name = "prover9"
self._prover9_bin = nltk.internals.find_binary(
name,
path_to_bin=binary_location,
env_vars=["PROVER9"],
url="https://www.cs.unm.edu/~mccune/prover9/",
binary_names=[name, name + ".exe"],
verbose=verbose,
)
self._binary_location = self._prover9_bin.rsplit(os.path.sep, 1)
[docs] def binary_locations(self):
"""
A list of directories that should be searched for the prover9
executables. This list is used by ``config_prover9`` when searching
for the prover9 executables.
"""
return [
"/usr/local/bin/prover9",
"/usr/local/bin/prover9/bin",
"/usr/local/bin",
"/usr/bin",
"/usr/local/prover9",
"/usr/local/share/prover9",
]
def _find_binary(self, name, verbose=False):
binary_locations = self.binary_locations()
if self._binary_location is not None:
binary_locations += [self._binary_location]
return nltk.internals.find_binary(
name,
searchpath=binary_locations,
env_vars=["PROVER9"],
url="https://www.cs.unm.edu/~mccune/prover9/",
binary_names=[name, name + ".exe"],
verbose=verbose,
)
def _call(self, input_str, binary, args=[], verbose=False):
"""
Call the binary with the given input.
:param input_str: A string whose contents are used as stdin.
:param binary: The location of the binary to call
:param args: A list of command-line arguments.
:return: A tuple (stdout, returncode)
:see: ``config_prover9``
"""
if verbose:
print("Calling:", binary)
print("Args:", args)
print("Input:\n", input_str, "\n")
# Call prover9 via a subprocess
cmd = [binary] + args
try:
input_str = input_str.encode("utf8")
except AttributeError:
pass
p = subprocess.Popen(
cmd, stdout=subprocess.PIPE, stderr=subprocess.STDOUT, stdin=subprocess.PIPE
)
(stdout, stderr) = p.communicate(input=input_str)
if verbose:
print("Return code:", p.returncode)
if stdout:
print("stdout:\n", stdout, "\n")
if stderr:
print("stderr:\n", stderr, "\n")
return (stdout.decode("utf-8"), p.returncode)
[docs]def convert_to_prover9(input):
"""
Convert a ``logic.Expression`` to Prover9 format.
"""
if isinstance(input, list):
result = []
for s in input:
try:
result.append(_convert_to_prover9(s.simplify()))
except:
print("input %s cannot be converted to Prover9 input syntax" % input)
raise
return result
else:
try:
return _convert_to_prover9(input.simplify())
except:
print("input %s cannot be converted to Prover9 input syntax" % input)
raise
def _convert_to_prover9(expression):
"""
Convert ``logic.Expression`` to Prover9 formatted string.
"""
if isinstance(expression, ExistsExpression):
return (
"exists "
+ str(expression.variable)
+ " "
+ _convert_to_prover9(expression.term)
)
elif isinstance(expression, AllExpression):
return (
"all "
+ str(expression.variable)
+ " "
+ _convert_to_prover9(expression.term)
)
elif isinstance(expression, NegatedExpression):
return "-(" + _convert_to_prover9(expression.term) + ")"
elif isinstance(expression, AndExpression):
return (
"("
+ _convert_to_prover9(expression.first)
+ " & "
+ _convert_to_prover9(expression.second)
+ ")"
)
elif isinstance(expression, OrExpression):
return (
"("
+ _convert_to_prover9(expression.first)
+ " | "
+ _convert_to_prover9(expression.second)
+ ")"
)
elif isinstance(expression, ImpExpression):
return (
"("
+ _convert_to_prover9(expression.first)
+ " -> "
+ _convert_to_prover9(expression.second)
+ ")"
)
elif isinstance(expression, IffExpression):
return (
"("
+ _convert_to_prover9(expression.first)
+ " <-> "
+ _convert_to_prover9(expression.second)
+ ")"
)
elif isinstance(expression, EqualityExpression):
return (
"("
+ _convert_to_prover9(expression.first)
+ " = "
+ _convert_to_prover9(expression.second)
+ ")"
)
else:
return str(expression)
[docs]class Prover9(Prover9Parent, Prover):
_prover9_bin = None
_prooftrans_bin = None
[docs] def __init__(self, timeout=60):
self._timeout = timeout
"""The timeout value for prover9. If a proof can not be found
in this amount of time, then prover9 will return false.
(Use 0 for no timeout.)"""
def _prove(self, goal=None, assumptions=None, verbose=False):
"""
Use Prover9 to prove a theorem.
:return: A pair whose first element is a boolean indicating if the
proof was successful (i.e. returns value of 0) and whose second element
is the output of the prover.
"""
if not assumptions:
assumptions = []
stdout, returncode = self._call_prover9(
self.prover9_input(goal, assumptions), verbose=verbose
)
return (returncode == 0, stdout)
def _call_prover9(self, input_str, args=[], verbose=False):
"""
Call the ``prover9`` binary with the given input.
:param input_str: A string whose contents are used as stdin.
:param args: A list of command-line arguments.
:return: A tuple (stdout, returncode)
:see: ``config_prover9``
"""
if self._prover9_bin is None:
self._prover9_bin = self._find_binary("prover9", verbose)
updated_input_str = ""
if self._timeout > 0:
updated_input_str += "assign(max_seconds, %d).\n\n" % self._timeout
updated_input_str += input_str
stdout, returncode = self._call(
updated_input_str, self._prover9_bin, args, verbose
)
if returncode not in [0, 2]:
errormsgprefix = "%%ERROR:"
if errormsgprefix in stdout:
msgstart = stdout.index(errormsgprefix)
errormsg = stdout[msgstart:].strip()
else:
errormsg = None
if returncode in [3, 4, 5, 6]:
raise Prover9LimitExceededException(returncode, errormsg)
else:
raise Prover9FatalException(returncode, errormsg)
return stdout, returncode
def _call_prooftrans(self, input_str, args=[], verbose=False):
"""
Call the ``prooftrans`` binary with the given input.
:param input_str: A string whose contents are used as stdin.
:param args: A list of command-line arguments.
:return: A tuple (stdout, returncode)
:see: ``config_prover9``
"""
if self._prooftrans_bin is None:
self._prooftrans_bin = self._find_binary("prooftrans", verbose)
return self._call(input_str, self._prooftrans_bin, args, verbose)
[docs]class Prover9Exception(Exception):
[docs] def __init__(self, returncode, message):
msg = p9_return_codes[returncode]
if message:
msg += "\n%s" % message
Exception.__init__(self, msg)
[docs]class Prover9FatalException(Prover9Exception):
pass
[docs]class Prover9LimitExceededException(Prover9Exception):
pass
######################################################################
# { Tests and Demos
######################################################################
[docs]def test_config():
a = Expression.fromstring("(walk(j) & sing(j))")
g = Expression.fromstring("walk(j)")
p = Prover9Command(g, assumptions=[a])
p._executable_path = None
p.prover9_search = []
p.prove()
# config_prover9('/usr/local/bin')
print(p.prove())
print(p.proof())
[docs]def test_convert_to_prover9(expr):
"""
Test that parsing works OK.
"""
for t in expr:
e = Expression.fromstring(t)
print(convert_to_prover9(e))
[docs]def test_prove(arguments):
"""
Try some proofs and exhibit the results.
"""
for (goal, assumptions) in arguments:
g = Expression.fromstring(goal)
alist = [Expression.fromstring(a) for a in assumptions]
p = Prover9Command(g, assumptions=alist).prove()
for a in alist:
print(" %s" % a)
print(f"|- {g}: {p}\n")
arguments = [
("(man(x) <-> (not (not man(x))))", []),
("(not (man(x) & (not man(x))))", []),
("(man(x) | (not man(x)))", []),
("(man(x) & (not man(x)))", []),
("(man(x) -> man(x))", []),
("(not (man(x) & (not man(x))))", []),
("(man(x) | (not man(x)))", []),
("(man(x) -> man(x))", []),
("(man(x) <-> man(x))", []),
("(not (man(x) <-> (not man(x))))", []),
("mortal(Socrates)", ["all x.(man(x) -> mortal(x))", "man(Socrates)"]),
("((all x.(man(x) -> walks(x)) & man(Socrates)) -> some y.walks(y))", []),
("(all x.man(x) -> all x.man(x))", []),
("some x.all y.sees(x,y)", []),
(
"some e3.(walk(e3) & subj(e3, mary))",
[
"some e1.(see(e1) & subj(e1, john) & some e2.(pred(e1, e2) & walk(e2) & subj(e2, mary)))"
],
),
(
"some x e1.(see(e1) & subj(e1, x) & some e2.(pred(e1, e2) & walk(e2) & subj(e2, mary)))",
[
"some e1.(see(e1) & subj(e1, john) & some e2.(pred(e1, e2) & walk(e2) & subj(e2, mary)))"
],
),
]
expressions = [
r"some x y.sees(x,y)",
r"some x.(man(x) & walks(x))",
r"\x.(man(x) & walks(x))",
r"\x y.sees(x,y)",
r"walks(john)",
r"\x.big(x, \y.mouse(y))",
r"(walks(x) & (runs(x) & (threes(x) & fours(x))))",
r"(walks(x) -> runs(x))",
r"some x.(PRO(x) & sees(John, x))",
r"some x.(man(x) & (not walks(x)))",
r"all x.(man(x) -> walks(x))",
]
[docs]def spacer(num=45):
print("-" * num)
[docs]def demo():
print("Testing configuration")
spacer()
test_config()
print()
print("Testing conversion to Prover9 format")
spacer()
test_convert_to_prover9(expressions)
print()
print("Testing proofs")
spacer()
test_prove(arguments)
if __name__ == "__main__":
demo()