def test_is_formula_composed(self):
a_sym = Symbol("a")
alphabet = Alphabet({a_sym})
a = AtomicFormula(a_sym)
pl = PL(alphabet)
self.assertTrue(
pl.is_formula(
Implies(Not(a), And(TrueFormula(), Not(FalseFormula())))))
self.assertFalse(
pl.is_formula(
Implies(Not(a), And(TrueFormula(), Next(FalseFormula())))))
def test_to_nnf_derived_formulas(self):
fol = self.fol
john = ConstantTerm.fromString("john")
x = Variable.fromString("x")
right_equal = Equal(x, john)
true_ = TrueFormula()
false_ = FalseFormula()
or_ = Or(true_, Not(right_equal))
implies_ = Implies(or_, false_)
equivalence_ = Equivalence(implies_, false_)
forall_true_ = ForAll(x, true_)
forall_not_or_ = ForAll(x, Not(or_))
forall_equivalence_ = ForAll(x, equivalence_)
to_nnf_true_ = Equal(DUMMY_TERM, DUMMY_TERM)
to_nnf_false_ = Not(Equal(DUMMY_TERM, DUMMY_TERM))
to_nnf_or_ = Or(to_nnf_true_, Not(right_equal))
to_nnf_not_or_ = And(Not(to_nnf_true_), right_equal)
to_nnf_implies_ = Or(to_nnf_not_or_, to_nnf_false_)
not_to_nnf_implies_ = And(to_nnf_or_, to_nnf_true_)
positive_equivalence = And(to_nnf_implies_, to_nnf_false_)
negative_equivalence = And(not_to_nnf_implies_, to_nnf_true_)
to_nnf_equivalence_ = Or(positive_equivalence, negative_equivalence)
self.assertEqual(fol.to_nnf(true_), to_nnf_true_)
self.assertEqual(fol.to_nnf(false_), to_nnf_false_)
self.assertEqual(fol.to_nnf(or_), to_nnf_or_)
self.assertEqual(fol.to_nnf(implies_), to_nnf_implies_)
self.assertEqual(fol.to_nnf(equivalence_), to_nnf_equivalence_)
self.assertEqual(fol.to_nnf(forall_true_), ForAll(x, to_nnf_true_))
self.assertEqual(fol.to_nnf(forall_not_or_), ForAll(x, to_nnf_not_or_))
self.assertEqual(fol.to_nnf(forall_equivalence_),
ForAll(x, to_nnf_equivalence_))
def test_expand_formula_derived_formulas(self):
fol = self.fol
john = ConstantTerm.fromString("john")
x = Variable.fromString("x")
right_equal = Equal(x, john)
true_ = TrueFormula()
false_ = FalseFormula()
or_ = Or(true_, Not(right_equal))
implies_ = Implies(or_, false_)
equivalence_ = Equivalence(implies_, false_)
forall_ = ForAll(x, equivalence_)
expanded_true = Equal(DUMMY_TERM, DUMMY_TERM)
expanded_false = Not(Equal(DUMMY_TERM, DUMMY_TERM))
expanded_or_ = Not(And(Not(expanded_true), Not(Not(right_equal))))
expanded_implies_ = Not(And(expanded_or_, Not(expanded_false)))
positive_equivalence = And(expanded_implies_, expanded_false)
negative_equivalence = And(Not(expanded_implies_), Not(expanded_false))
expanded_equivalence_ = Not(
And(Not(positive_equivalence), Not(negative_equivalence)))
self.assertEqual(fol.expand_formula(true_), expanded_true)
self.assertEqual(fol.expand_formula(false_), expanded_false)
self.assertEqual(fol.expand_formula(or_), expanded_or_)
self.assertEqual(fol.expand_formula(implies_), expanded_implies_)
self.assertEqual(fol.expand_formula(equivalence_),
expanded_equivalence_)
self.assertEqual(fol.expand_formula(forall_),
Not(Exists(x, Not(expanded_equivalence_))))
def test_to_nnf_derived_formula(self):
a_sym = Symbol("a")
b_sym = Symbol("b")
alphabet = Alphabet({a_sym, b_sym})
a = AtomicFormula(a_sym)
b = AtomicFormula(b_sym)
pl = PL(alphabet)
self.assertEqual(pl.to_nnf(Not(Or(b, Not(a)))), And(Not(b), a))
self.assertEqual(pl.to_nnf(Not(Implies(b, Not(a)))), And(b, a))
def test_is_formula_derived(self):
fol = self.fol
john = ConstantTerm.fromString("john")
not_a_term = ConstantTerm.fromString("NotATerm")
x = Variable.fromString("x")
y = Variable.fromString("y")
right_equal = Equal(x, john)
wrong_equal = Equal(x, not_a_term)
self.assertTrue(fol.is_formula(TrueFormula()))
self.assertTrue(fol.is_formula(FalseFormula()))
self.assertTrue(fol.is_formula(Or(right_equal, right_equal)))
self.assertFalse(fol.is_formula(Or(right_equal, wrong_equal)))
self.assertTrue(fol.is_formula(Or(right_equal, right_equal)))
self.assertFalse(fol.is_formula(Or(right_equal, wrong_equal)))
self.assertTrue(fol.is_formula(Implies(right_equal, right_equal)))
self.assertFalse(fol.is_formula(Implies(right_equal, wrong_equal)))
self.assertTrue(fol.is_formula(Equivalence(right_equal, right_equal)))
self.assertFalse(fol.is_formula(Equivalence(right_equal, wrong_equal)))
self.assertTrue(
fol.is_formula(
ForAll(x,
PredicateFormula(PredicateSymbol("Person", 2), john,
x))))
self.assertFalse(
fol.is_formula(
ForAll(
x,
PredicateFormula(PredicateSymbol("Person", 3), john, x,
john))))
self.assertFalse(
fol.is_formula(
ForAll(
x,
PredicateFormula(PredicateSymbol("Person_fake", 2), john,
x))))
def test_expand_formula_derived_formulas(self):
a_sym = Symbol("a")
b_sym = Symbol("b")
alphabet = Alphabet({a_sym, b_sym})
a = AtomicFormula(a_sym)
b = AtomicFormula(b_sym)
# T = Not(And(Not(DUMMY_ATOMIC), DUMMY_ATOMIC))
# F = And(Not(DUMMY_ATOMIC), DUMMY_ATOMIC)
T = TrueFormula()
F = FalseFormula()
pl = PL(alphabet)
self.assertEqual(pl.expand_formula(TrueFormula()), T)
self.assertEqual(pl.expand_formula(FalseFormula()), F)
self.assertEqual(pl.expand_formula(Or(a, b)), Not(And(Not(a), Not(b))))
self.assertEqual(pl.expand_formula(Implies(a, b)),
Not(And(Not(Not(a)), Not(b))))
self.assertEqual(pl.expand_formula(Implies(b, a)),
Not(And(Not(Not(b)), Not(a))))
# A === B = (A AND B) OR (NOT A AND NOT B) = NOT( NOT(A AND B) AND NOT(NOT A AND NOT B) )
self.assertEqual(pl.expand_formula(Equivalence(a, b)),
Not(And(Not(And(a, b)), Not(And(Not(a), Not(b))))))
def test_expand_formula_composed(self):
a_sym = Symbol("a")
alphabet = Alphabet({a_sym})
a = AtomicFormula(a_sym)
# T = Not(And(Not(DUMMY_ATOMIC), DUMMY_ATOMIC))
# F = And(Not(DUMMY_ATOMIC), DUMMY_ATOMIC)
T = TrueFormula()
F = FalseFormula()
pl = PL(alphabet)
self.assertEqual(pl.expand_formula(And(TrueFormula(), FalseFormula())),
And(T, F))
self.assertEqual(pl.expand_formula(Or(TrueFormula(), FalseFormula())),
Not(And(Not(T), Not(F))))
self.assertEqual(
pl.expand_formula(Implies(TrueFormula(), FalseFormula())),
Not(And(Not(Not(T)), Not(F))))
self.assertEqual(
pl.expand_formula(Equivalence(TrueFormula(), FalseFormula())),
Not(And(Not(And(T, F)), Not(And(Not(T), Not(F))))))
def test_truth(self):
w = Variable.fromString("w")
Person_x_20 = PredicateFormula(self.Person_pred_sym, self.x,
ConstantTerm.fromString("20"))
Person_x_y = PredicateFormula(self.Person_pred_sym, self.x, self.y)
not_Person_x_21 = Not(
PredicateFormula(self.Person_pred_sym, self.x,
ConstantTerm.fromString("21")))
y_equal_20 = Equal(self.y, ConstantTerm.fromString("20"))
x_equal_john = Equal(self.x, ConstantTerm.fromString("john"))
x_equal_x = Equal(self.x, self.x)
x_equal_y = Equal(self.x, self.y)
x_equal_z = Equal(self.x, self.z)
x_lives_w = PredicateFormula(self.Lives_pred_sym, self.x, w)
x_lives_y = PredicateFormula(self.Lives_pred_sym, self.x, self.y)
x_lives_ny = PredicateFormula(self.Lives_pred_sym, self.x,
ConstantTerm.fromString("ny"))
x_lives_paris = PredicateFormula(self.Lives_pred_sym, self.x,
ConstantTerm.fromString("paris"))
w_lives_z = PredicateFormula(self.Lives_pred_sym, w, self.z)
exists_w__x_lives_w = Exists(w, x_lives_w)
exists_y__x_lives_y = Exists(self.y, x_lives_y)
exists_z_exists_w__w_lives_z = Exists(self.z, Exists(w, w_lives_z))
exists_x__x_equal_x = Exists(self.x, x_equal_x)
exists_x__x_equal_y = Exists(self.x, x_equal_y)
exists_x__x_equal_john_and_Lives_x_paris = Exists(
self.x, And(x_equal_john, x_lives_paris))
exists_x__x_equal_john_and_exists_x__Lives_x_ny = And(
Exists(self.x, x_equal_john), Exists(self.x, x_lives_ny))
exists_x__x_equal_x_and_exists_x__Lives_x_ny = And(
exists_x__x_equal_x, Exists(self.x, x_lives_ny))
forall_x__x_equal_x = ForAll(self.x, x_equal_x)
forall_x__x_equal_y = ForAll(self.x, x_equal_y)
not_forall_x__x_equal_x = Not(forall_x__x_equal_x)
# Person(x, 20)
# x = "john"
self.assertTrue(self.FOL.truth(Person_x_20, self.assignment))
# ~Person(x, 21)
# x = "john"
self.assertTrue(self.FOL.truth(not_Person_x_21, self.assignment))
# Equals
self.assertTrue(self.FOL.truth(y_equal_20, self.assignment))
self.assertTrue(self.FOL.truth(x_equal_x, self.assignment))
self.assertFalse(self.FOL.truth(x_equal_y, self.assignment))
self.assertFalse(self.FOL.truth(x_equal_z, self.assignment))
# y == 20 and x == "john" and Person(x, y)
self.assertTrue(
self.FOL.truth(And(y_equal_20, And(x_equal_john, Person_x_y)),
self.assignment))
self.assertTrue(
self.FOL.truth(Or(Person_x_20, x_equal_john), self.assignment))
# De Morgan on previous formula
# Not (Not y == 20 or Not x == "john" ot Not Person(x, y)
self.assertTrue(
self.FOL.truth(
Not(Or(Not(y_equal_20), Or(Not(x_equal_john),
Not(Person_x_y)))),
self.assignment))
# Or with the last formula true
self.assertTrue(
self.FOL.truth(
Or(Equal(self.x, self.y),
Or(Equal(self.x, self.y), Equal(self.z, self.z))),
self.assignment))
# (y==20 and x=="john") => Person(x,y)
self.assertTrue(
self.FOL.truth(Implies(And(y_equal_20, x_equal_john), Person_x_y),
self.assignment))
self.assertTrue(
self.FOL.truth(Implies(Not(x_equal_x), Person_x_y),
self.assignment))
# Exists
self.assertTrue(self.FOL.truth(exists_w__x_lives_w, self.assignment))
self.assertTrue(self.FOL.truth(exists_x__x_equal_x, self.assignment))
self.assertTrue(self.FOL.truth(exists_x__x_equal_y, self.assignment))
# quantified variable not present in the formula
self.assertTrue(self.FOL.truth(exists_y__x_lives_y, self.assignment))
# 2 quantified variables
self.assertTrue(
self.FOL.truth(exists_z_exists_w__w_lives_z, self.assignment))
# annidate exists
self.assertTrue(
self.FOL.truth(exists_x__x_equal_john_and_exists_x__Lives_x_ny,
self.assignment))
self.assertTrue(
self.FOL.truth(exists_x__x_equal_x_and_exists_x__Lives_x_ny,
self.assignment))
self.assertFalse(
self.FOL.truth(exists_x__x_equal_john_and_Lives_x_paris,
self.assignment))
# ForAll
self.assertTrue(self.FOL.truth(forall_x__x_equal_x, self.assignment))
self.assertFalse(self.FOL.truth(forall_x__x_equal_y, self.assignment))
self.assertFalse(
self.FOL.truth(not_forall_x__x_equal_x, self.assignment))
def setUp(self):
"""Set up symbols, terms and formulas. Both legal and illegal, according to some FOL system"""
# Symbols
self.a_sym = Symbol('a')
self.b_sym = Symbol('b')
self.const_sym = ConstantSymbol("Const")
self.fun_sym = FunctionSymbol("Fun", 3)
self.predicate_sym = PredicateSymbol("Predicate", 2)
self.A = PredicateSymbol("A", 1)
# Terms
self.a = Variable(self.a_sym)
self.b = Variable(self.b_sym)
self.c = Variable.fromString("c")
self.const = ConstantTerm(self.const_sym)
self.fun_abc = FunctionTerm(self.fun_sym, self.a, self.b, self.c)
# Formulas
self.predicate_ab = PredicateFormula(self.predicate_sym, self.a,
self.b)
self.predicate_ac = PredicateFormula(self.predicate_sym, self.a,
self.c)
self.A_a = PredicateFormula(self.A, self.a)
self.a_equal_a = Equal(self.a, self.a)
self.b_equal_c = Equal(self.b, self.c)
self.neg_a_equal_a = Not(self.a_equal_a)
self.neg_Aa = Not(self.A_a)
self.Aa_and_b_equal_c = And(self.A_a, self.b_equal_c)
self.Aa_or_b_equal_c = Or(self.A_a, self.b_equal_c)
self.Aa_implies_b_equal_c = Implies(self.A_a, self.b_equal_c)
self.exists_a_predicate_ab = Exists(self.a, self.predicate_ab)
self.forall_b_exists_a_predicate_ab = ForAll(
self.b, self.exists_a_predicate_ab)
# FOL
self.vars = {self.a, self.b, self.c}
self.functions = {self.const_sym, self.fun_sym}
self.predicates = {self.predicate_sym, self.A}
self.alphabet = FOLAlphabet(self.functions, self.predicates)
self.myFOL = FOL(self.alphabet)
# define dummy stuff
# does not belong to myFOL. They are used for test membership to myFOL
self.dummy_variable = Variable.fromString(
"ThisVariableDoesNotBelongToFOLSystem")
self.dummy_fun_sym = FunctionSymbol(
"ThisFunctionDoesNotBelongToFOLSystem", 3)
self.dummy_constant_sym = ConstantSymbol(
"ThisConstDoesNotBelongToFOLSystem")
self.dummy_predicate_sym = PredicateSymbol(
"ThisPredicateDoesNotBelongToFOLSystem", 2)
self.dummy_fun = FunctionTerm(self.dummy_fun_sym, self.a, self.b,
self.dummy_variable)
self.dummy_constant = ConstantTerm(self.dummy_constant_sym)
self.dummy_predicate = PredicateFormula(self.dummy_predicate_sym,
self.a, self.b)
self.dummy_predicate_only_one_symbol_false = PredicateFormula(
self.predicate_sym, self.dummy_variable, self.dummy_constant)
self.dummy_equal = Equal(self.c, self.dummy_constant)
self.dummy_neg = Not(self.dummy_predicate_only_one_symbol_false)
self.dummy_and = And(self.dummy_predicate, self.predicate_ab)
self.dummy_or = Or(self.dummy_predicate_only_one_symbol_false,
self.predicate_ac)
self.dummy_exists = Exists(self.dummy_variable,
self.dummy_predicate_only_one_symbol_false)
self.dummy_forall = ForAll(self.b, self.dummy_predicate)