# Natural Language Toolkit: First-Order Tableau Theorem Prover
#
# Copyright (C) 2001-2013 NLTK Project
# Author: Dan Garrette <dhgarrette@gmail.com>
#
# URL: <http://nltk.org/>
# For license information, see LICENSE.TXT
"""
Module for a tableau-based First Order theorem prover.
"""
from __future__ import print_function, unicode_literals
from nltk.internals import Counter
from nltk.sem.logic import (VariableExpression, EqualityExpression,
ApplicationExpression, LogicParser,
AbstractVariableExpression, AllExpression,
NegatedExpression,
ExistsExpression, Variable, ImpExpression,
AndExpression, unique_variable,
LambdaExpression, IffExpression,
OrExpression, FunctionVariableExpression)
from nltk.inference.api import Prover, BaseProverCommand
_counter = Counter()
[docs]class ProverParseError(Exception): pass
[docs]class TableauProver(Prover):
_assume_false=False
def _prove(self, goal=None, assumptions=None, verbose=False):
if not assumptions:
assumptions = []
result = None
try:
agenda = Agenda()
if goal:
agenda.put(-goal)
agenda.put_all(assumptions)
debugger = Debug(verbose)
result = self._attempt_proof(agenda, set(), set(), debugger)
except RuntimeError as e:
if self._assume_false and str(e).startswith('maximum recursion depth exceeded'):
result = False
else:
if verbose:
print(e)
else:
raise e
return (result, '\n'.join(debugger.lines))
def _attempt_proof(self, agenda, accessible_vars, atoms, debug):
(current, context), category = agenda.pop_first()
#if there's nothing left in the agenda, and we haven't closed the path
if not current:
debug.line('AGENDA EMPTY')
return False
proof_method = { Categories.ATOM: self._attempt_proof_atom,
Categories.PROP: self._attempt_proof_prop,
Categories.N_ATOM: self._attempt_proof_n_atom,
Categories.N_PROP: self._attempt_proof_n_prop,
Categories.APP: self._attempt_proof_app,
Categories.N_APP: self._attempt_proof_n_app,
Categories.N_EQ: self._attempt_proof_n_eq,
Categories.D_NEG: self._attempt_proof_d_neg,
Categories.N_ALL: self._attempt_proof_n_all,
Categories.N_EXISTS: self._attempt_proof_n_some,
Categories.AND: self._attempt_proof_and,
Categories.N_OR: self._attempt_proof_n_or,
Categories.N_IMP: self._attempt_proof_n_imp,
Categories.OR: self._attempt_proof_or,
Categories.IMP: self._attempt_proof_imp,
Categories.N_AND: self._attempt_proof_n_and,
Categories.IFF: self._attempt_proof_iff,
Categories.N_IFF: self._attempt_proof_n_iff,
Categories.EQ: self._attempt_proof_eq,
Categories.EXISTS: self._attempt_proof_some,
Categories.ALL: self._attempt_proof_all,
}[category]
debug.line((current, context))
return proof_method(current, context, agenda, accessible_vars, atoms, debug)
def _attempt_proof_atom(self, current, context, agenda, accessible_vars, atoms, debug):
# Check if the branch is closed. Return 'True' if it is
if (current, True) in atoms:
debug.line('CLOSED', 1)
return True
if context:
if isinstance(context.term, NegatedExpression):
current = current.negate()
agenda.put(context(current).simplify())
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
else:
#mark all AllExpressions as 'not exhausted' into the agenda since we are (potentially) adding new accessible vars
agenda.mark_alls_fresh();
return self._attempt_proof(agenda, accessible_vars|set(current.args), atoms|set([(current, False)]), debug+1)
def _attempt_proof_n_atom(self, current, context, agenda, accessible_vars, atoms, debug):
# Check if the branch is closed. Return 'True' if it is
if (current.term, False) in atoms:
debug.line('CLOSED', 1)
return True
if context:
if isinstance(context.term, NegatedExpression):
current = current.negate()
agenda.put(context(current).simplify())
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
else:
#mark all AllExpressions as 'not exhausted' into the agenda since we are (potentially) adding new accessible vars
agenda.mark_alls_fresh();
return self._attempt_proof(agenda, accessible_vars|set(current.term.args), atoms|set([(current.term, True)]), debug+1)
def _attempt_proof_prop(self, current, context, agenda, accessible_vars, atoms, debug):
# Check if the branch is closed. Return 'True' if it is
if (current, True) in atoms:
debug.line('CLOSED', 1)
return True
#mark all AllExpressions as 'not exhausted' into the agenda since we are (potentially) adding new accessible vars
agenda.mark_alls_fresh();
return self._attempt_proof(agenda, accessible_vars, atoms|set([(current, False)]), debug+1)
def _attempt_proof_n_prop(self, current, context, agenda, accessible_vars, atoms, debug):
# Check if the branch is closed. Return 'True' if it is
if (current.term, False) in atoms:
debug.line('CLOSED', 1)
return True
#mark all AllExpressions as 'not exhausted' into the agenda since we are (potentially) adding new accessible vars
agenda.mark_alls_fresh();
return self._attempt_proof(agenda, accessible_vars, atoms|set([(current.term, True)]), debug+1)
def _attempt_proof_app(self, current, context, agenda, accessible_vars, atoms, debug):
f, args = current.uncurry()
for i, arg in enumerate(args):
if not TableauProver.is_atom(arg):
ctx = f
nv = Variable('X%s' % _counter.get())
for j,a in enumerate(args):
ctx = (ctx(VariableExpression(nv)) if i == j else ctx(a))
if context:
ctx = context(ctx).simplify()
ctx = LambdaExpression(nv, ctx)
agenda.put(arg, ctx)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
raise Exception('If this method is called, there must be a non-atomic argument')
def _attempt_proof_n_app(self, current, context, agenda, accessible_vars, atoms, debug):
f, args = current.term.uncurry()
for i, arg in enumerate(args):
if not TableauProver.is_atom(arg):
ctx = f
nv = Variable('X%s' % _counter.get())
for j,a in enumerate(args):
ctx = (ctx(VariableExpression(nv)) if i == j else ctx(a))
if context:
#combine new context with existing
ctx = context(ctx).simplify()
ctx = LambdaExpression(nv, -ctx)
agenda.put(-arg, ctx)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
raise Exception('If this method is called, there must be a non-atomic argument')
def _attempt_proof_n_eq(self, current, context, agenda, accessible_vars, atoms, debug):
###########################################################################
# Since 'current' is of type '~(a=b)', the path is closed if 'a' == 'b'
###########################################################################
if current.term.first == current.term.second:
debug.line('CLOSED', 1)
return True
agenda[Categories.N_EQ].add((current,context))
current._exhausted = True
return self._attempt_proof(agenda, accessible_vars|set([current.term.first, current.term.second]), atoms, debug+1)
def _attempt_proof_d_neg(self, current, context, agenda, accessible_vars, atoms, debug):
agenda.put(current.term.term, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_n_all(self, current, context, agenda, accessible_vars, atoms, debug):
agenda[Categories.EXISTS].add((ExistsExpression(current.term.variable, -current.term.term), context))
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_n_some(self, current, context, agenda, accessible_vars, atoms, debug):
agenda[Categories.ALL].add((AllExpression(current.term.variable, -current.term.term), context))
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_and(self, current, context, agenda, accessible_vars, atoms, debug):
agenda.put(current.first, context)
agenda.put(current.second, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_n_or(self, current, context, agenda, accessible_vars, atoms, debug):
agenda.put(-current.term.first, context)
agenda.put(-current.term.second, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_n_imp(self, current, context, agenda, accessible_vars, atoms, debug):
agenda.put(current.term.first, context)
agenda.put(-current.term.second, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_or(self, current, context, agenda, accessible_vars, atoms, debug):
new_agenda = agenda.clone()
agenda.put(current.first, context)
new_agenda.put(current.second, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1) and \
self._attempt_proof(new_agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_imp(self, current, context, agenda, accessible_vars, atoms, debug):
new_agenda = agenda.clone()
agenda.put(-current.first, context)
new_agenda.put(current.second, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1) and \
self._attempt_proof(new_agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_n_and(self, current, context, agenda, accessible_vars, atoms, debug):
new_agenda = agenda.clone()
agenda.put(-current.term.first, context)
new_agenda.put(-current.term.second, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1) and \
self._attempt_proof(new_agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_iff(self, current, context, agenda, accessible_vars, atoms, debug):
new_agenda = agenda.clone()
agenda.put(current.first, context)
agenda.put(current.second, context)
new_agenda.put(-current.first, context)
new_agenda.put(-current.second, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1) and \
self._attempt_proof(new_agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_n_iff(self, current, context, agenda, accessible_vars, atoms, debug):
new_agenda = agenda.clone()
agenda.put(current.term.first, context)
agenda.put(-current.term.second, context)
new_agenda.put(-current.term.first, context)
new_agenda.put(current.term.second, context)
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1) and \
self._attempt_proof(new_agenda, accessible_vars, atoms, debug+1)
def _attempt_proof_eq(self, current, context, agenda, accessible_vars, atoms, debug):
#########################################################################
# Since 'current' is of the form '(a = b)', replace ALL free instances
# of 'a' with 'b'
#########################################################################
agenda.put_atoms(atoms)
agenda.replace_all(current.first, current.second)
accessible_vars.discard(current.first)
agenda.mark_neqs_fresh();
return self._attempt_proof(agenda, accessible_vars, set(), debug+1)
def _attempt_proof_some(self, current, context, agenda, accessible_vars, atoms, debug):
new_unique_variable = VariableExpression(unique_variable())
agenda.put(current.term.replace(current.variable, new_unique_variable), context)
agenda.mark_alls_fresh()
return self._attempt_proof(agenda, accessible_vars|set([new_unique_variable]), atoms, debug+1)
def _attempt_proof_all(self, current, context, agenda, accessible_vars, atoms, debug):
try:
current._used_vars
except AttributeError:
current._used_vars = set()
#if there are accessible_vars on the path
if accessible_vars:
# get the set of bound variables that have not be used by this AllExpression
bv_available = accessible_vars - current._used_vars
if bv_available:
variable_to_use = list(bv_available)[0]
debug.line('--> Using \'%s\'' % variable_to_use, 2)
current._used_vars |= set([variable_to_use])
agenda.put(current.term.replace(current.variable, variable_to_use), context)
agenda[Categories.ALL].add((current,context))
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
else:
#no more available variables to substitute
debug.line('--> Variables Exhausted', 2)
current._exhausted = True
agenda[Categories.ALL].add((current,context))
return self._attempt_proof(agenda, accessible_vars, atoms, debug+1)
else:
new_unique_variable = VariableExpression(unique_variable())
debug.line('--> Using \'%s\'' % new_unique_variable, 2)
current._used_vars |= set([new_unique_variable])
agenda.put(current.term.replace(current.variable, new_unique_variable), context)
agenda[Categories.ALL].add((current,context))
agenda.mark_alls_fresh()
return self._attempt_proof(agenda, accessible_vars|set([new_unique_variable]), atoms, debug+1)
@staticmethod
[docs] def is_atom(e):
if isinstance(e, NegatedExpression):
e = e.term
if isinstance(e, ApplicationExpression):
for arg in e.args:
if not TableauProver.is_atom(arg):
return False
return True
elif isinstance(e, AbstractVariableExpression) or \
isinstance(e, LambdaExpression):
return True
else:
return False
[docs]class TableauProverCommand(BaseProverCommand):
def __init__(self, goal=None, assumptions=None, 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)
"""
if prover is not None:
assert isinstance(prover, TableauProver)
else:
prover = TableauProver()
BaseProverCommand.__init__(self, prover, goal, assumptions)
[docs]class Agenda(object):
def __init__(self):
self.sets = tuple(set() for i in range(21))
[docs] def clone(self):
new_agenda = Agenda()
set_list = [s.copy() for s in self.sets]
new_allExs = set()
for allEx,_ in set_list[Categories.ALL]:
new_allEx = AllExpression(allEx.variable, allEx.term)
try:
new_allEx._used_vars = set(used for used in allEx._used_vars)
except AttributeError:
new_allEx._used_vars = set()
new_allExs.add((new_allEx,None))
set_list[Categories.ALL] = new_allExs
set_list[Categories.N_EQ] = set((NegatedExpression(n_eq.term),ctx)
for (n_eq,ctx) in set_list[Categories.N_EQ])
new_agenda.sets = tuple(set_list)
return new_agenda
def __getitem__(self, index):
return self.sets[index]
[docs] def put(self, expression, context=None):
if isinstance(expression, AllExpression):
ex_to_add = AllExpression(expression.variable, expression.term)
try:
ex_to_add._used_vars = set(used for used in expression._used_vars)
except AttributeError:
ex_to_add._used_vars = set()
else:
ex_to_add = expression
self.sets[self._categorize_expression(ex_to_add)].add((ex_to_add, context))
[docs] def put_all(self, expressions):
for expression in expressions:
self.put(expression)
[docs] def put_atoms(self, atoms):
for atom, neg in atoms:
if neg:
self[Categories.N_ATOM].add((-atom,None))
else:
self[Categories.ATOM].add((atom,None))
[docs] def pop_first(self):
""" Pop the first expression that appears in the agenda """
for i,s in enumerate(self.sets):
if s:
if i in [Categories.N_EQ, Categories.ALL]:
for ex in s:
try:
if not ex[0]._exhausted:
s.remove(ex)
return (ex, i)
except AttributeError:
s.remove(ex)
return (ex, i)
else:
return (s.pop(), i)
return ((None, None), None)
[docs] def replace_all(self, old, new):
for s in self.sets:
for ex,ctx in s:
ex.replace(old.variable, new)
if ctx is not None:
ctx.replace(old.variable, new)
[docs] def mark_alls_fresh(self):
for u,_ in self.sets[Categories.ALL]:
u._exhausted = False
[docs] def mark_neqs_fresh(self):
for neq,_ in self.sets[Categories.N_EQ]:
neq._exhausted = False
def _categorize_expression(self, current):
if isinstance(current, NegatedExpression):
return self._categorize_NegatedExpression(current)
elif isinstance(current, FunctionVariableExpression):
return Categories.PROP
elif TableauProver.is_atom(current):
return Categories.ATOM
elif isinstance(current, AllExpression):
return Categories.ALL
elif isinstance(current, AndExpression):
return Categories.AND
elif isinstance(current, OrExpression):
return Categories.OR
elif isinstance(current, ImpExpression):
return Categories.IMP
elif isinstance(current, IffExpression):
return Categories.IFF
elif isinstance(current, EqualityExpression):
return Categories.EQ
elif isinstance(current, ExistsExpression):
return Categories.EXISTS
elif isinstance(current, ApplicationExpression):
return Categories.APP
else:
raise ProverParseError("cannot categorize %s" % \
current.__class__.__name__)
def _categorize_NegatedExpression(self, current):
negated = current.term
if isinstance(negated, NegatedExpression):
return Categories.D_NEG
elif isinstance(negated, FunctionVariableExpression):
return Categories.N_PROP
elif TableauProver.is_atom(negated):
return Categories.N_ATOM
elif isinstance(negated, AllExpression):
return Categories.N_ALL
elif isinstance(negated, AndExpression):
return Categories.N_AND
elif isinstance(negated, OrExpression):
return Categories.N_OR
elif isinstance(negated, ImpExpression):
return Categories.N_IMP
elif isinstance(negated, IffExpression):
return Categories.N_IFF
elif isinstance(negated, EqualityExpression):
return Categories.N_EQ
elif isinstance(negated, ExistsExpression):
return Categories.N_EXISTS
elif isinstance(negated, ApplicationExpression):
return Categories.N_APP
else:
raise ProverParseError("cannot categorize %s" % \
negated.__class__.__name__)
[docs]class Debug(object):
def __init__(self, verbose, indent=0, lines=None):
self.verbose = verbose
self.indent = indent
if not lines: lines = []
self.lines = lines
def __add__(self, increment):
return Debug(self.verbose, self.indent+1, self.lines)
[docs] def line(self, data, indent=0):
if isinstance(data, tuple):
ex, ctx = data
if ctx:
data = '%s, %s' % (ex, ctx)
else:
data = '%s' % ex
if isinstance(ex, AllExpression):
try:
used_vars = "[%s]" % (",".join("%s" % ve.variable.name for ve in ex._used_vars))
data += ': %s' % used_vars
except AttributeError:
data += ': []'
newline = '%s%s' % (' '*(self.indent+indent), data)
self.lines.append(newline)
if self.verbose:
print(newline)
[docs]class Categories(object):
ATOM = 0
PROP = 1
N_ATOM = 2
N_PROP = 3
APP = 4
N_APP = 5
N_EQ = 6
D_NEG = 7
N_ALL = 8
N_EXISTS = 9
AND = 10
N_OR = 11
N_IMP = 12
OR = 13
IMP = 14
N_AND = 15
IFF = 16
N_IFF = 17
EQ = 18
EXISTS = 19
ALL = 20
[docs]def testTableauProver():
tableau_test('P | -P')
tableau_test('P & -P')
tableau_test('Q', ['P', '(P -> Q)'])
tableau_test('man(x)')
tableau_test('(man(x) -> man(x))')
tableau_test('(man(x) -> --man(x))')
tableau_test('-(man(x) and -man(x))')
tableau_test('(man(x) or -man(x))')
tableau_test('(man(x) -> man(x))')
tableau_test('-(man(x) and -man(x))')
tableau_test('(man(x) or -man(x))')
tableau_test('(man(x) -> man(x))')
tableau_test('(man(x) iff man(x))')
tableau_test('-(man(x) iff -man(x))')
tableau_test('all x.man(x)')
tableau_test('all x.all y.((x = y) -> (y = x))')
tableau_test('all x.all y.all z.(((x = y) & (y = z)) -> (x = z))')
# tableau_test('-all x.some y.F(x,y) & some x.all y.(-F(x,y))')
# tableau_test('some x.all y.sees(x,y)')
parse = LogicParser().parse
p1 = 'all x.(man(x) -> mortal(x))'
p2 = 'man(Socrates)'
c = 'mortal(Socrates)'
tableau_test(c, [p1, p2])
p1 = 'all x.(man(x) -> walks(x))'
p2 = 'man(John)'
c = 'some y.walks(y)'
tableau_test(c, [p1, p2])
p = '((x = y) & walks(y))'
c = 'walks(x)'
tableau_test(c, [p])
p = '((x = y) & ((y = z) & (z = w)))'
c = '(x = w)'
tableau_test(c, [p])
p = 'some e1.some e2.(believe(e1,john,e2) & walk(e2,mary))'
c = 'some e0.walk(e0,mary)'
tableau_test(c, [p])
c = '(exists x.exists z3.((x = Mary) & ((z3 = John) & sees(z3,x))) <-> exists x.exists z4.((x = John) & ((z4 = Mary) & sees(x,z4))))'
tableau_test(c)
# p = 'some e1.some e2.((believe e1 john e2) and (walk e2 mary))'
# c = 'some x.some e3.some e4.((believe e3 x e4) and (walk e4 mary))'
# tableau_test(c, [p])
[docs]def testHigherOrderTableauProver():
tableau_test('believe(j, -lie(b))', ['believe(j, -lie(b) & -cheat(b))'])
tableau_test('believe(j, lie(b) & cheat(b))', ['believe(j, lie(b))'])
tableau_test('believe(j, lie(b))', ['lie(b)']) #how do we capture that John believes all things that are true
tableau_test('believe(j, know(b, cheat(b)))', ['believe(j, know(b, lie(b)) & know(b, steals(b) & cheat(b)))'])
tableau_test('P(Q(y), R(y) & R(z))', ['P(Q(x) & Q(y), R(y) & R(z))'])
tableau_test('believe(j, cheat(b) & lie(b))', ['believe(j, lie(b) & cheat(b))'])
tableau_test('believe(j, -cheat(b) & -lie(b))', ['believe(j, -lie(b) & -cheat(b))'])
[docs]def tableau_test(c, ps=None, verbose=False):
lp = LogicParser()
pc = lp.parse(c)
pps = ([lp.parse(p) for p in ps] if ps else [])
if not ps:
ps = []
print('%s |- %s: %s' % (', '.join(ps), pc, TableauProver().prove(pc, pps, verbose=verbose)))
[docs]def demo():
testTableauProver()
testHigherOrderTableauProver()
if __name__ == '__main__':
demo()