def test_axiom_to_pcnf(self):
a = Symbol.Predicate('A', ['x'])
b = Symbol.Predicate('B', ['y'])
c = Symbol.Predicate('C', ['z'])
# Simple test of disjunction over conjunction
axi_one = Axiom.Axiom(Quantifier.Universal(['x', 'y', 'z'], a | b & c))
axi_one = axi_one.to_pcnf()
self.assertEqual('∀(z,y,x)[((A(z) | B(y)) & (A(z) | C(x)))]',
repr(axi_one))
# Simple sanity check, it's already FF-PCNF
axi_two = Axiom.Axiom(
Quantifier.Universal(['x', 'y', 'z'], (a | b) & c))
axi_two = axi_two.to_pcnf()
self.assertEqual('∀(z,y,x)[(C(x) & (A(z) | B(y)))]', repr(axi_two))
# Sanity check we remove functions
c = Symbol.Predicate('C', ['z', Symbol.Function('F', ['z'])])
axi_three = Axiom.Axiom(
Quantifier.Universal(['x', 'y', 'z'], a | b & c))
axi_three = axi_three.to_pcnf()
self.assertEqual(
'∀(z,y,x,w)[((A(z) | C(x,w)) & (A(z) | F(x,w)) & (A(z) | B(y)))]',
repr(axi_three))
def test_axiom_connecive_rescoping(self):
a = Symbol.Predicate('A', ['x'])
b = Symbol.Predicate('B', ['y'])
universal = Quantifier.Universal(['x'], a)
existential = Quantifier.Existential(['y'], b)
conjunction = universal & existential
disjunction = universal | existential
# Ensure we handle single quantifier case
self.assertEqual(repr((universal & b).rescope()),
'∀(x)[(A(x) & B(y))]')
self.assertEqual(repr((existential & a).rescope()),
'∃(y)[(B(y) & A(x))]')
self.assertEqual(repr((universal | b).rescope()),
'∀(x)[(A(x) | B(y))]')
self.assertEqual(repr((existential | a).rescope()),
'∃(y)[(B(y) | A(x))]')
# Ensure we catch error condition where lookahead is needed
self.assertRaises(ValueError, (existential | universal).rescope)
# Ensure that we can promote Universals when a conjunction lives above us
top = a & disjunction
self.assertEqual(repr(disjunction.rescope(top)),
'∀(x)[∃(y)[(A(x) | B(y))]]')
# Ensure that we can promote Existentials when a conjunction lives above us
top = a | conjunction
self.assertEqual(repr(conjunction.rescope(top)),
'∃(y)[∀(x)[(B(y) & A(x))]]')
def test_cnf_negation(self):
'''
Ensure we can get into conjunctive normal form
'''
alpha = Symbol.Predicate('A', ['x'])
beta = Symbol.Predicate('B', ['y'])
delta = Symbol.Predicate('D', ['z'])
s = ~(Quantifier.Universal(['x', 'y', 'z'], (~(alpha | beta) & delta)))
self.assertEqual(repr(s.push_complete()),
"∃(x,y,z)[(A(x) | B(y) | ~D(z))]")
s = ~(Quantifier.Universal(['x', 'y', 'z'], ~((alpha | beta) & delta)))
self.assertEqual(repr(s.push_complete()),
"∃(x,y,z)[((A(x) | B(y)) & D(z))]")
s = ~((~alpha | ~beta) & ~delta)
self.assertEqual(repr(s.push_complete()), "((A(x) & B(y)) | D(z))")
## Test to make sure the recursino into nested stuff actually work
s = (~~~~~~~~~alpha).push_complete()
self.assertEqual(repr(s), '~A(x)')
s = (~~~~~~~~alpha).push_complete()
self.assertEqual(repr(s), 'A(x)')
def test_cnf_quantifier_simplfy(self):
alpha = Symbol.Predicate('A', ['x'])
uni_one = Quantifier.Universal(['x'], alpha)
mixer = uni_one | alpha
uni_two = Quantifier.Universal(['y'], mixer)
self.assertEqual(repr(uni_two), "∀(y)[(∀(x)[A(x)] | A(x))]")
self.assertEqual(repr(uni_two.simplify()), "∀(y,x)[(A(x) | A(x))]")
def test_axiom_function_replacement(self):
f = Symbol.Function('f', ['x'])
t = Symbol.Function('t', ['y'])
a = Symbol.Predicate('A', [f])
b = Symbol.Predicate('B', [f, t])
axi = Axiom.Axiom(Quantifier.Universal(['x'], a | a & a))
self.assertEqual(repr(axi), '∀(x)[(A(f(x)) | (A(f(x)) & A(f(x))))]')
axi = Axiom.Axiom(Quantifier.Universal(['x', 'y'], b))
self.assertEqual(
repr(axi.substitute_functions()),
'∀(x,y)[∀(t1)[(∀(f1)[(B(f1,t1) & f(x,f1))] & t(y,t1))]]')
def substitute_function(self):
'''
Find a function that's nested and replace it by adding a new variable and term
'''
# TODO This a dirty hack because cyclic imports are painful
import macleod.logical.Quantifier as Quantifier
if not self.has_functions():
return self
for idx, var in enumerate(self.variables):
if isinstance(var, Function):
function = var
pos = idx
# TODO Get a global variable singleton
n_var = function.name.lower()[0] + '1'
f_vars = copy.deepcopy(function.variables)
f_vars.append(n_var)
n_pred = Predicate(function.name, f_vars)
e_vars = copy.deepcopy(self.variables)
e_vars[pos] = n_var
e_pred = Predicate(self.name, e_vars)
return Quantifier.Universal([n_var], [e_pred & n_pred]), None
def test_axiom_variable_standardize(self):
a = Symbol.Predicate('A', ['x'])
b = Symbol.Predicate('B', ['y', 'x'])
c = Symbol.Predicate('C',
['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'])
axi = Axiom.Axiom(Quantifier.Universal(['x'], a | a & a))
self.assertEqual(repr(axi.standardize_variables()),
'∀(z)[(A(z) | (A(z) & A(z)))]')
axi = Axiom.Axiom(Quantifier.Universal(['x', 'y'], b))
self.assertEqual(repr(axi.standardize_variables()), '∀(z,y)[B(y,z)]')
axi = Axiom.Axiom(
Quantifier.Existential(
['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'], c))
self.assertEqual(repr(axi.standardize_variables()),
'∃(z,y,x,w,v,u,t,s,r)[C(z,y,x,w,v,u,t,s,r)]')
def test_quantifiers(self):
alpha = Symbol.Predicate('A', ['x'])
beta = Symbol.Predicate('B', ['y'])
delta = Symbol.Predicate('D', ['z'])
uni = Quantifier.Universal(['x', 'y', 'z'], alpha | beta | delta)
exi = Quantifier.Existential(['x', 'y', 'z'], alpha & beta & delta)
self.assertEqual(repr(uni), "∀(x,y,z)[(A(x) | B(y) | D(z))]")
self.assertEqual(repr(exi), "∃(x,y,z)[(A(x) & B(y) & D(z))]")
self.assertEqual(repr(~uni), "~∀(x,y,z)[(A(x) | B(y) | D(z))]")
self.assertEqual(repr(~exi), "~∃(x,y,z)[(A(x) & B(y) & D(z))]")
self.assertEqual(repr((~uni).push()),
"∃(x,y,z)[~(A(x) | B(y) | D(z))]")
self.assertEqual(repr((~exi).push()),
"∀(x,y,z)[~(A(x) & B(y) & D(z))]")
def test_axiom_quantifier_coalesence(self):
a = Symbol.Predicate('A', ['x'])
b = Symbol.Predicate('B', ['y'])
universal = Quantifier.Universal(['x'], a)
universal_two = Quantifier.Universal(['y'], b)
existential = Quantifier.Existential(['y'], b)
existential_two = Quantifier.Existential(['x'], a)
# Coalescence over conjunction should merge Universals
conjunction = universal & universal_two & existential & existential_two
self.assertEqual(repr(conjunction.coalesce()),
'(∃(y)[B(y)] & ∃(x)[A(x)] & ∀(x)[(B(x) & A(x))])')
# Coalescence over disjunction should merge Existentials
disjunction = universal | universal_two | existential | existential_two
self.assertEqual(repr(disjunction.coalesce()),
'(∀(x)[A(x)] | ∀(y)[B(y)] | ∃(y)[(A(y) | B(y))])')
def push(self):
'''
Push negation inwards and apply to all children
'''
# Can be a conjunction or disjunction
# Can be a single predicate
# Can be a quantifier
if isinstance(self.term(), Connective.Conjunction):
ret = Connective.Disjunction(
[Negation(x) for x in self.term().get_term()])
elif isinstance(self.term(), Connective.Disjunction):
ret = Connective.Conjunction(
[Negation(x) for x in self.term().get_term()])
elif isinstance(self.term(), Symbol.Predicate):
ret = self
elif isinstance(self.term(), Quantifier.Existential):
ret = Quantifier.Universal(self.term().variables,
Negation(self.term().get_term()))
elif isinstance(self.term(), Quantifier.Universal):
ret = Quantifier.Existential(self.term().variables,
Negation(self.term().get_term()))
elif isinstance(self.term(), Negation):
ret = self.term().term()
else:
raise ValueError("Negation onto unknown type!", self.term)
return copy.deepcopy(ret)
def coalesce(self):
'''
Coalesce or merge any like quantifiers held as terms of this
connective. Disjunctions will coalesce existentials and will merge
universals.
'''
obj = copy.deepcopy(self)
existentials = [
x for x in obj.terms if isinstance(x, Quantifier.Existential)
]
universals = [
x for x in obj.terms if isinstance(x, Quantifier.Universal)
]
if len(existentials) != 0:
LOGGER.debug("Coalescing existentials")
new_existential = functools.reduce(lambda x, y: x.coalesce(y),
existentials)
if len(existentials) == len(obj.terms):
return new_existential
else:
for term in existentials:
obj.remove_term(term)
obj.set_term(new_existential)
return obj
elif len(universals) != 0 and len(existentials) == 0:
LOGGER.debug("Merging nested Universals")
variables = []
terms = []
for quant in universals:
variables += quant.variables
terms.append(quant.terms[0])
for term in obj.terms:
LOGGER.debug("Adding term: " + repr(term))
if term not in universals:
terms.append(term)
terms = [type(self)(terms)]
return Quantifier.Universal(variables, terms)
else:
return obj
def test_onf_detection(self):
alpha = Symbol.Predicate('A', ['x'])
beta = Symbol.Predicate('B', ['y'])
delta = Symbol.Predicate('D', ['z'])
uni = Quantifier.Universal(['x', 'y', 'z'], alpha | beta | delta)
exi = Quantifier.Existential(['x', 'y', 'z'], alpha & beta | delta)
self.assertEqual(alpha.is_onf(), True)
self.assertEqual((alpha | beta).is_onf(), True)
self.assertEqual((alpha & beta).is_onf(), True)
self.assertEqual((alpha | (beta & delta)).is_onf(), False)
self.assertEqual((alpha & (beta | delta)).is_onf(), True)
self.assertEqual((~(alpha | beta)).is_onf(), False)
self.assertEqual((~(alpha & beta)).is_onf(), False)
self.assertEqual(uni.is_onf(), True)
self.assertEqual(exi.is_onf(), False)
# Note that is_onf() is not a recursive call, it's a top level feature
# If will actually if you need an ONF axiom then create a Logical.Axiom and to_onf()
self.assertEqual((alpha & (alpha | (beta & delta)) & delta).is_onf(),
True)
def test_cnf_quantifier_scoping(self):
alpha = Symbol.Predicate('A', ['x'])
beta = Symbol.Predicate('B', ['y'])
delta = Symbol.Predicate('D', ['z'])
uni_one = Quantifier.Universal(['x', 'y', 'z'], alpha | beta | delta)
exi_one = Quantifier.Existential(['x', 'y', 'z'], alpha & beta | delta)
term = exi_one | beta
term_two = exi_one | beta | uni_one
self.assertEqual(
repr(Quantifier.Universal(['x', 'y', 'z'], uni_one).simplify()),
"∀(x,y,z)[(A(x) | B(y) | D(z))]")
self.assertEqual(
repr(Quantifier.Universal(['x', 'y', 'z'], term_two).simplify()),
"∀(x,y,z)[(∃(x,y,z)[((A(x) & B(y)) | D(z))] | B(y) | (A(x) | B(y) | D(z)))]"
)
self.assertEqual(
repr(
Quantifier.Universal(['x', 'y', 'z'],
term_two).simplify().rescope()),
"∀(x,y,z)[∃(x,y,z)[(B(y) | (A(x) | B(y) | D(z)) | ((A(x) & B(y)) | D(z)))]]"
)
def coalesce(self):
'''
Coalesce or merge any like quantifiers held as terms of this
connective. Conjunctions will coalesce universals and will merge
existentials.
precondition: Must not have both universals and existentials
'''
obj = copy.deepcopy(self)
LOGGER.debug("Attempting to coalesce: " + repr(obj))
universals = [
x for x in obj.terms if isinstance(x, Quantifier.Universal)
]
existentials = [
x for x in obj.terms if isinstance(x, Quantifier.Existential)
]
if len(universals) != 0:
LOGGER.debug("Coalescing Universals")
new_universal = functools.reduce(lambda x, y: x.coalesce(y),
universals)
LOGGER.debug("Coalesced Universal: " + repr(new_universal))
if len(universals) == len(obj.terms):
LOGGER.debug("Returning plain: " + repr(new_universal))
return new_universal
else:
for term in universals:
obj.remove_term(term)
obj.set_term(new_universal)
LOGGER.debug("Returning added: " + repr(obj))
return obj
elif len(existentials) != 0 and len(universals) == 0:
LOGGER.debug("Merging nested existentials")
variables = []
terms = []
for quant in existentials:
variables += quant.variables
terms.append(quant.terms[0])
for term in obj.terms:
if term not in existentials:
terms.append(term)
terms = [type(self)(terms)]
return Quantifier.Existential(variables, terms)
else:
return obj
def p_universal(p):
"""
universal : LPAREN FORALL LPAREN nonlogicals RPAREN axiom RPAREN
"""
p[0] = Quantifier.Universal(p[4], p[6])
def p_existential(p):
"""
existential : LPAREN EXISTS LPAREN nonlogicals RPAREN axiom RPAREN
"""
p[0] = Quantifier.Existential(p[4], p[6])