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_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))
# Test recursive distribution
#axi_one = Axiom.Axiom(Quantifier.Universal(['x','y','z'], a | (b & (a | (c & b)))))
#print(repr(axi_one))
#self.assertEqual('', repr(axi_one.to_pcnf()))
# 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,v)[((A(z) | ~C(w,v) | F(w,v)) & (A(z) | B(y)))]', repr(axi_three))
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))
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_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_quantifier_simplfy(self):
alpha = Symbol.Predicate('A', ['x'])
beta = Symbol.Predicate('B', ['y'])
uni_one = Quantifier.Universal(['x'], alpha)
mixer = uni_one | beta
uni_two = Quantifier.Universal(['y'], mixer)
uni_nested = Quantifier.Universal(['z'],
alpha & (beta | (alpha & uni_one)))
self.assertEqual('∀(z)[(A(x) & (B(y) | (A(x) & ∀(x)[A(x)])))]',
repr(uni_nested))
self.assertEqual('∀(z,x)[(A(x) & (B(y) | (A(x) & A(x))))]',
repr(uni_nested.simplify()))
self.assertEqual(repr(uni_two), "∀(y)[(∀(x)[A(x)] | B(y))]")
self.assertEqual(repr(uni_two.simplify()), "∀(y,x)[(B(y) | A(x))]")
def test_axiom_simple_function_replacement(self):
f = Symbol.Function('f', ['x'])
t = Symbol.Function('t', ['y'])
p = Symbol.Function('p', ['z'])
a = Symbol.Predicate('A', [f, t, p])
b = Symbol.Predicate('B', [f, t])
c = Symbol.Predicate('C', [f])
axi = Axiom.Axiom(Quantifier.Universal(['x', 'y', 'z'], a ))
self.assertEqual(repr(axi.substitute_functions()), '∀(x,y,z)[∀(f2,t3,p4)[(~A(f2,t3,p4) | (f(x,f2) & t(y,t3) & p(z,p4)))]]')
axi = Axiom.Axiom(Quantifier.Universal(['x',], ~c ))
self.assertEqual(repr(axi.substitute_functions()), '∀(x)[~~∀(f5)[(C(f5) | f(x,f5))]]')
c = Symbol.Predicate('C', [Symbol.Function('f', [Symbol.Function('g', [Symbol.Function('h', ['x'])])])])
axi = Axiom.Axiom(Quantifier.Universal(['x'], c))
self.assertEqual(repr(axi.substitute_functions()), '∀(x)[∀(f5,g6,h7)[(~C(f5) | (h(x,h7) & g(h7,g6) & f(g6,f5)))]]')
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 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_cnf_quantifier_scoping(self):
a = Symbol.Predicate('A', ['x'])
b = Symbol.Predicate('B', ['y'])
c = Symbol.Predicate('C', ['z'])
e = Quantifier.Existential(['x'], a)
u = Quantifier.Universal(['y'], b)
# Test the effect over an OR
self.assertEqual('∃(x)[(A(x) | B(y))]', repr((e | b).rescope()))
self.assertEqual('∀(y)[(B(y) | A(x))]', repr((u | a).rescope()))
# Test the effect over an AND
self.assertEqual('∃(x)[(A(x) & B(y))]', repr((e & b).rescope()))
self.assertEqual('∀(y)[(B(y) & A(x))]', repr((u & a).rescope()))
# Test with more than two to make sure things aren't dropped
self.assertEqual('∀(y)[(B(y) & (A(x) | C(z) | B(y)))]',
repr((u & (a | c | b)).rescope()))
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 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(), False)
def substitute_function(self, negated = False):
'''
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 logical.Quantifier as Quantifier
import logical.Negation as Negation
import logical.Connective as Connective
global gen
def sub_functions(term, predicates, variables):
'''
Does three things: (1) returns a variable that serves as the placeholder
for the function. (2) adds a minted predicate to the accumulator (3) adds
any the newly introduced variable for each function to the accumulator
'''
# Singleton used to access an increasing integer sequence
global gen
# Base Case 0: I'm not event a function, I'm a variable
if isinstance(term, str):
# I'm a variable so I go in the variable accumulator
variables.append(term)
return term
# Base Case 1: I'm a function with no nested functions!
if isinstance(term, Function) and term.has_functions() == False:
# Mint a new variable special
new_variable = term.name.lower()[0] + gen()
variables.append(new_variable)
# Mint a new predicate with our original variables + the new one
predicate_variables = term.variables
predicate_variables.append(new_variable)
predicate = Predicate(term.name, predicate_variables)
predicates.append(predicate)
# Return our fresh variable
return new_variable
# Recursive Case: I'm a function with nested functions!
if isinstance(term, Function): #and term.has_functions() == True:
# Still mint a new variable cuz I'm still a function
new_variable = term.name.lower()[0] + gen()
variables.append(new_variable)
# Assemble my variables by a DFS down on my function / atom children
predicate_variables = [sub_functions(x, predicates, variables) for x in term.variables]
predicate_variables.append(new_variable)
predicate = Predicate(term.name, predicate_variables)
predicates.append(predicate)
return new_variable
predicate_accumulator = []
variable_accumulator = []
variables = [sub_functions(x, predicate_accumulator, variable_accumulator) for x in self.variables]
if len(predicate_accumulator) > 1:
term = Connective.Conjunction(predicate_accumulator)
else:
term = predicate_accumulator.pop()
predicate = Predicate(self.name, variables)
if negated:
# The negation cancels out the normal conditional breakdown
universal = Quantifier.Universal(variable_accumulator, predicate | term)
else:
universal = Quantifier.Universal(variable_accumulator, ~predicate | term)
return universal, predicate_accumulator
def p_universal(p):
"""
universal : LPAREN FORALL LPAREN nonlogicals RPAREN axiom RPAREN
"""
p[0] = Quantifier.Universal(p[4], p[6])