def test_is_formula_exists(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(Exists(x, right_equal)))
self.assertFalse(fol.is_formula(Exists(x, wrong_equal)))
self.assertTrue(
fol.is_formula(
Exists(x,
PredicateFormula(PredicateSymbol("Person", 2), john,
x))))
self.assertFalse(
fol.is_formula(
Exists(
x,
PredicateFormula(PredicateSymbol("Person", 3), john, x,
john))))
self.assertFalse(
fol.is_formula(
Exists(
x,
PredicateFormula(PredicateSymbol("Person_fake", 2), john,
x))))
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 to_nnf(self, f: Formula):
assert self.is_formula(f)
formula = self.expand_formula(f)
if isinstance(formula, PredicateFormula) or isinstance(formula, Equal):
return formula
elif isinstance(formula, And):
return And(self.to_nnf(formula.f1), self.to_nnf(formula.f2))
if isinstance(formula, Exists):
return Exists(formula.v, self.to_nnf(formula.f))
if isinstance(formula, Not):
subformula = formula.f
if isinstance(subformula, Not):
return self.to_nnf(subformula.f)
elif isinstance(subformula, And):
return Or(self.to_nnf(Not(subformula.f1)),
self.to_nnf((Not(subformula.f2))))
elif isinstance(subformula, Exists):
return ForAll(subformula.v, self.to_nnf(Not(subformula.f)))
elif isinstance(subformula, PredicateFormula) or isinstance(
subformula, Equal):
return formula
else:
raise ValueError
else:
raise ValueError
def _expand_formula(self, f: Formula):
if isinstance(f, PredicateFormula) or isinstance(f, Equal):
return f
elif isinstance(f, And):
return And(self.expand_formula(f.f1), self.expand_formula(f.f2))
elif isinstance(f, Not):
return Not(self.expand_formula(f.f))
elif isinstance(f, Exists):
return Exists(f.v, self.expand_formula(f.f))
else:
raise ValueError("Not valid Formula to expand.")
def test_to_nnf_allowed_formulas(self):
fol = self.fol
john = ConstantTerm.fromString("john")
x = Variable.fromString("x")
right_equal = Equal(x, john)
not_ = Not(right_equal)
and_ = And(right_equal, not_)
exists_ = Exists(x, right_equal)
self.assertEqual(fol.to_nnf(right_equal), right_equal)
self.assertEqual(fol.to_nnf(not_), not_)
self.assertEqual(fol.to_nnf(and_), and_)
self.assertEqual(fol.to_nnf(exists_), exists_)
def to_equivalent_formula(self, derived_formula: Formula):
if isinstance(derived_formula, Or):
return Not(And(Not(derived_formula.f1), Not(derived_formula.f2)))
elif isinstance(derived_formula, Implies):
return Not(And(derived_formula.f1, Not(derived_formula.f2)))
elif isinstance(derived_formula, Equivalence):
positive_equivalence = And(derived_formula.f1, derived_formula.f2)
negative_equivalence = And(Not(derived_formula.f1),
Not(derived_formula.f2))
return Not(
And(Not(positive_equivalence), Not(negative_equivalence)))
elif isinstance(derived_formula, FalseFormula):
return Not(Equal(DUMMY_TERM, DUMMY_TERM))
elif isinstance(derived_formula, TrueFormula):
return Equal(DUMMY_TERM, DUMMY_TERM)
elif isinstance(derived_formula, ForAll):
return Not(Exists(derived_formula.v, Not(derived_formula.f)))
elif derived_formula in self.allowed_formulas:
return derived_formula
else:
raise ValueError("Derived formula not recognized")
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)
def test_eq(self):
# Symbols
self.assertEqual(self.a_sym, self.a_sym)
self.assertEqual(self.a_sym, Symbol("a"))
self.assertEqual(self.const_sym, ConstantSymbol("Const"))
self.assertEqual(self.fun_sym, FunctionSymbol("Fun", 3))
self.assertEqual(self.predicate_sym, PredicateSymbol("Predicate", 2))
self.assertNotEqual(self.a_sym, Symbol("c"))
self.assertNotEqual(self.const_sym, ConstantSymbol("Another_Const"))
self.assertNotEqual(self.fun_sym, FunctionSymbol("Another_Fun", 3))
self.assertNotEqual(self.fun_sym, FunctionSymbol("Fun", 2))
self.assertNotEqual(self.predicate_sym,
PredicateSymbol("Another_Predicate", 2))
self.assertNotEqual(self.predicate_sym,
PredicateSymbol("Predicate", 1))
self.assertNotEqual(self.a_sym, self.fun_sym)
self.assertNotEqual(self.const_sym, self.fun_sym)
# Terms
self.assertEqual(self.a, self.a)
self.assertEqual(self.a, Variable.fromString("a"))
self.assertEqual(self.const, ConstantTerm(ConstantSymbol("Const")))
self.assertEqual(
self.fun_abc,
FunctionTerm(FunctionSymbol("Fun", 3), self.a, self.b, self.c))
self.assertNotEqual(self.a, self.b)
self.assertNotEqual(self.a, Variable.fromString("c"))
self.assertNotEqual(self.const,
ConstantTerm(ConstantSymbol("Another_Const")))
self.assertNotEqual(
self.fun_abc,
FunctionTerm(FunctionSymbol("Another_Fun", 3), self.a, self.b,
self.c))
self.assertNotEqual(
self.fun_abc,
FunctionTerm(FunctionSymbol("Fun", 3), self.a, self.b, self.a))
self.assertNotEqual(self.a, self.fun_abc)
# Formulas
self.assertEqual(
self.predicate_ab,
PredicateFormula(PredicateSymbol("Predicate", 2),
Variable.fromString("a"),
Variable.fromString("b")))
self.assertEqual(
self.A_a,
PredicateFormula(PredicateSymbol("A", 1),
Variable.fromString("a")))
self.assertEqual(
self.b_equal_c,
Equal(Variable.fromString("b"), Variable.fromString("c")))
self.assertEqual(
self.neg_a_equal_a,
Not(Equal(Variable.fromString("a"), Variable.fromString("a"))))
self.assertEqual(
self.forall_b_exists_a_predicate_ab,
ForAll(
Variable.fromString("b"),
Exists(
Variable.fromString("a"),
PredicateFormula(PredicateSymbol("Predicate", 2),
Variable.fromString("a"),
Variable.fromString("b")))))
self.assertNotEqual(
self.predicate_ab,
PredicateFormula(PredicateSymbol("Predicate", 2),
Variable.fromString("a"),
Variable.fromString("c")))
self.assertNotEqual(
self.predicate_ab,
PredicateFormula(PredicateSymbol("Another_Predicate", 2),
Variable.fromString("a"),
Variable.fromString("c")))
self.assertNotEqual(
self.A_a,
PredicateFormula(PredicateSymbol("A", 1),
Variable.fromString("b")))
self.assertNotEqual(
self.b_equal_c,
Equal(Variable.fromString("b"), Variable.fromString("b")))
self.assertNotEqual(
self.neg_a_equal_a,
Not(Equal(Variable.fromString("b"), Variable.fromString("a"))))
self.assertNotEqual(
self.forall_b_exists_a_predicate_ab,
ForAll(
Variable.fromString("b"),
Exists(
Variable.fromString("a"),
PredicateFormula(PredicateSymbol("ANOTHER_PREDICATE", 2),
Variable.fromString("a"),
Variable.fromString("b")))))