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_truth_star(self):
ref = self.ref
a = self.a
b = self.b
c = self.c
self.assertTrue(ref.truth(PathExpressionStar(a), self.trace_1, 0, 4))
self.assertFalse(ref.truth(PathExpressionStar(a), self.trace_1, 0, 5))
self.assertTrue(
ref.truth(PathExpressionStar(Or(a, Or(b, c))), self.trace_1, 0, 5))
self.assertTrue(ref.truth(TrueFormula(), self.trace_1, 0, 1))
def test_to_nnf(self):
f1 = Not(
PathExpressionEventually(
PathExpressionSequence(PathExpressionTest(Not(self.a_and_b)),
PathExpressionStar(TrueFormula())),
self.abc))
nnf_f1 = PathExpressionAlways(
PathExpressionSequence(
PathExpressionTest(Or(Not(self.a), Not(self.b))),
# PathExpressionStar(Or(DUMMY_ATOMIC, Not(DUMMY_ATOMIC)))
PathExpressionStar(TrueFormula())),
Or(Not(self.a), Or(Not(self.b), Not(self.c))))
self.assertEqual(self.ldlf.to_nnf(f1), nnf_f1)
def test_find_atomics(self):
a = Symbol("a")
b = Symbol("b")
c = Symbol("c")
atomic_a = AtomicFormula(a)
atomic_b = AtomicFormula(b)
atomic_c = AtomicFormula(c)
self.assertEqual(
PL.find_atomics(And(atomic_a, And(atomic_b, atomic_c))),
{atomic_a, atomic_b, atomic_c})
self.assertEqual(PL.find_atomics(Or(atomic_a, Or(atomic_b, atomic_c))),
{atomic_a, atomic_b, atomic_c})
self.assertEqual(
PL.find_atomics(Equivalence(atomic_a, Or(atomic_b, atomic_c))),
{atomic_a, atomic_b, atomic_c})
def setUp(self):
"""Set up test fixtures, if any."""
self.a_sym = Symbol("a")
self.b_sym = Symbol("b")
self.c_sym = Symbol("c")
self.alphabet = Alphabet({self.a_sym, self.b_sym, self.c_sym})
# Propositions
self.a = AtomicFormula(self.a_sym)
self.b = AtomicFormula(self.b_sym)
self.c = AtomicFormula(self.c_sym)
self.not_a = Not(self.a)
self.not_a_and_b = And(self.not_a, self.b)
self.not_a_or_c = Or(self.not_a, self.c)
self.true = TrueFormula()
self.false = FalseFormula()
self.symbol2truth = {
self.a_sym: True,
self.b_sym: False,
self.c_sym: True
}
self.I = PLInterpretation(self.alphabet, self.symbol2truth)
self.PL = PL(self.alphabet)
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)
formula = f
if isinstance(formula, AtomicFormula) or isinstance(
formula, TrueFormula) or isinstance(formula, FalseFormula):
return formula
elif isinstance(formula, And) or isinstance(formula, Or):
return type(formula)(self.to_nnf(formula.f1),
self.to_nnf(formula.f2))
elif type(formula) in self.derived_formulas:
return self.to_nnf(self.derived_formulas[type(formula)](formula))
elif 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, Or):
return And(self.to_nnf(Not(subformula.f1)),
self.to_nnf((Not(subformula.f2))))
elif isinstance(subformula, AtomicFormula) or isinstance(
formula, TrueFormula) or isinstance(formula, FalseFormula):
return formula
elif type(subformula) in self.derived_formulas:
return self.to_nnf(
Not(self.derived_formulas[type(subformula)](subformula)))
else:
raise ValueError
else:
raise ValueError
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 test_to_nnf_allowed_formulas_not_normalized(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(Not(b))), b)
self.assertEqual(pl.to_nnf(Not(And(a, Not(b)))), Or(Not(a), b))
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 to_nnf(self, formula: Formula) -> Formula:
if isinstance(formula, AtomicFormula):
return formula
elif isinstance(formula, And):
return And(self.to_nnf(formula.f1), self.to_nnf(formula.f2))
elif isinstance(formula, Or):
return Or(self.to_nnf(formula.f1), self.to_nnf(formula.f2))
elif isinstance(formula, PathExpressionFormula):
return type(formula)(self.to_nnf_path(formula.p), self.to_nnf(formula.f))
elif isinstance(formula, Not):
return self._not_to_nnf(formula)
else:
raise ValueError
def _not_to_nnf(self, not_formula: Not):
subformula = not_formula.f
if isinstance(subformula, AtomicFormula):
return not_formula
if isinstance(subformula, Not):
# skip two consecutive Not
new_formula = subformula.f
return self.to_nnf(new_formula)
elif isinstance(subformula, And):
return Or(self.to_nnf(Not(subformula.f1)), self.to_nnf(Not(subformula.f2)))
elif isinstance(subformula, Or):
return And(self.to_nnf(Not(subformula.f1)), self.to_nnf(Not(subformula.f2)))
elif isinstance(subformula, PathExpressionEventually):
return PathExpressionAlways(self.to_nnf_path(subformula.p), self.to_nnf(Not(subformula.f)))
elif isinstance(subformula, PathExpressionAlways):
return PathExpressionEventually(self.to_nnf_path(subformula.p), self.to_nnf(Not(subformula.f)))
else:
raise ValueError
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_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_minimal_models(self):
a = Symbol("a")
b = Symbol("b")
c = Symbol("c")
alphabet = Alphabet({a, b, c})
pl = PL(alphabet)
atomic_a = AtomicFormula(a)
atomic_b = AtomicFormula(b)
atomic_c = AtomicFormula(c)
self.assertEqual(
pl.minimal_models(TrueFormula()),
{PLInterpretation(alphabet, {
a: False,
b: False,
c: False
})})
self.assertEqual(pl.minimal_models(FalseFormula()), set())
self.assertEqual(
pl.minimal_models(atomic_a),
{PLInterpretation(alphabet, {
a: True,
b: False,
c: False
})})
self.assertEqual(
pl.minimal_models(Not(atomic_a)),
{PLInterpretation(alphabet, {
a: False,
b: False,
c: False
})})
self.assertEqual(
pl.minimal_models(And(atomic_a, atomic_b)),
{PLInterpretation(alphabet, {
a: True,
b: True,
c: False
})})
self.assertEqual(pl.minimal_models(And(atomic_a, Not(atomic_a))),
set())
self.assertEqual(
pl.minimal_models(Or(atomic_a, atomic_b)), {
PLInterpretation(alphabet, {
a: False,
b: True,
c: False
}),
PLInterpretation(alphabet, {
a: True,
b: False,
c: False
})
})
self.assertEqual(
pl.minimal_models(And.chain([atomic_a, atomic_b, atomic_c])),
{PLInterpretation(alphabet, {
a: True,
b: True,
c: True
})})
def setUp(self):
"""Set up test fixtures, if any."""
# Symbols
self.a_sym = Symbol("a")
self.b_sym = Symbol("b")
self.c_sym = Symbol("c")
# Propositions
self.a = AtomicFormula(self.a_sym)
self.b = AtomicFormula(self.b_sym)
self.c = AtomicFormula(self.c_sym)
# Propositionals
self.not_a = Not(self.a)
self.not_b = Not(self.b)
self.not_c = Not(self.c)
self.a_and_b = And(self.a, self.b)
self.a_and_c = And(self.a, self.c)
self.b_and_c = And(self.b, self.c)
self.abc = And(self.a, And(self.b, self.c))
self.b_or_c = Or(self.b, self.c)
self.a_or_b = Or(self.a, self.b)
self.not_abc = Not(And(self.a, And(self.b, self.c)))
### Path expression
# Tests
self.test_a = PathExpressionTest(self.a)
self.test_b = PathExpressionTest(self.b)
self.test_not_a = PathExpressionTest(self.not_a)
self.test_not_b = PathExpressionTest(self.not_b)
# Union
self.path_a_or_b = PathExpressionUnion(self.a, self.b)
self.path_b_or_c = PathExpressionUnion(self.b, self.c)
# Sequence
self.path_seq_a_and_b__a_and_c = PathExpressionSequence(
self.a_and_b, self.a_and_c)
self.path_a_or_b__b_or_c = PathExpressionSequence(
self.path_a_or_b, self.path_b_or_c)
# Stars
self.path_b_or_c_star = PathExpressionStar(self.path_b_or_c)
self.path_not_abc = PathExpressionStar(self.not_abc)
# Modal connective
self.eventually_propositional_a_and_b__a_and_c = PathExpressionEventually(
self.a_and_b, self.a_and_c)
self.eventually_test_a__c = PathExpressionEventually(
self.test_a, self.c)
self.eventually_test_a__b = PathExpressionEventually(
self.test_a, self.b)
self.eventually_seq_a_and_b__a_and_c__not_c = PathExpressionEventually(
self.path_seq_a_and_b__a_and_c, self.not_c)
self.eventually_seq_a_and_b__a_and_c__c = PathExpressionEventually(
self.path_seq_a_and_b__a_and_c, self.c)
self.eventually_b_or_c_star__b_and_c = PathExpressionEventually(
self.path_b_or_c_star, self.b_and_c)
self.next_a_and_c = PathExpressionEventually(TrueFormula(),
self.a_and_c)
self.liveness_b_and_c = PathExpressionEventually(
PathExpressionStar(TrueFormula()), self.b_and_c)
self.liveness_abc = PathExpressionEventually(
PathExpressionStar(TrueFormula()), self.abc)
self.always_true__a = PathExpressionAlways(
PathExpressionStar(TrueFormula()), self.a)
self.always_true__b_or_c = PathExpressionAlways(
PathExpressionStar(TrueFormula()), self.b_or_c)
self.alphabet = Alphabet({self.a_sym, self.b_sym, self.c_sym})
# Traces
self.ldlf = LDLf(self.alphabet)
self.trace_1_list = [
{self.a_sym, self.b_sym},
{self.a_sym, self.c_sym},
{self.a_sym, self.b_sym},
{self.a_sym, self.c_sym},
{self.b_sym, self.c_sym},
]
self.trace_1 = FiniteTrace(self.trace_1_list, self.alphabet)
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_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 delta(self, f: Formula, action: FrozenSet[Symbol], epsilon=False):
# TODO: should return [True|False]Formula or simply True/False?
pl, I = PL._from_set_of_propositionals(action, self.alphabet)
if pl.is_formula(f):
return self.delta(PathExpressionEventually(f, LogicalTrue()),
action, epsilon)
elif isinstance(f, LogicalTrue):
return TrueFormula()
elif isinstance(f, LogicalFalse):
return FalseFormula()
elif isinstance(f, And):
return And(self.delta(f.f1, action),
self.delta(f.f2, action, epsilon))
elif isinstance(f, Or):
return Or(self.delta(f.f1, action),
self.delta(f.f2, action, epsilon))
elif isinstance(f, PathExpressionEventually):
if pl.is_formula(f.p):
if not epsilon and pl.truth(f.p, I):
return self._expand(f.f)
else:
return FalseFormula()
elif isinstance(f.p, PathExpressionTest):
return And(self.delta(f.p.f, action, epsilon),
self.delta(f.f, action, epsilon))
elif isinstance(f.p, PathExpressionUnion):
return Or(
self.delta(PathExpressionEventually(f.p.p1, f.f), action,
epsilon),
self.delta(PathExpressionEventually(f.p.p2, f.f), action,
epsilon))
elif isinstance(f.p, PathExpressionSequence):
e2 = PathExpressionEventually(f.p.p2, f.f)
e1 = PathExpressionEventually(f.p.p1, e2)
return self.delta(e1, action, epsilon)
elif isinstance(f.p, PathExpressionStar):
o1 = self.delta(f.f, action, epsilon)
o2 = self.delta(PathExpressionEventually(f.p.p, F(f)), action,
epsilon)
return Or(o1, o2)
elif isinstance(f, PathExpressionAlways):
if pl.is_formula(f.p):
if not epsilon and pl.truth(f.p, I):
return self._expand(f.f)
else:
return TrueFormula()
elif isinstance(f.p, PathExpressionTest):
o1 = self.delta(self.to_nnf(Not(f.p.f)), action, epsilon)
o2 = self.delta(f.f, action, epsilon)
return Or(o1, o2)
elif isinstance(f.p, PathExpressionUnion):
return And(
self.delta(PathExpressionAlways(f.p.p1, f.f), action,
epsilon),
self.delta(PathExpressionAlways(f.p.p2, f.f), action,
epsilon))
elif isinstance(f.p, PathExpressionSequence):
return self.delta(
PathExpressionAlways(f.p.p1,
PathExpressionAlways(f.p.p2, f.f)),
action, epsilon)
elif isinstance(f.p, PathExpressionStar):
a1 = self.delta(f.f, action, epsilon)
a2 = self.delta(PathExpressionAlways(f.p.p, T(f)), action,
epsilon)
return And(a1, a2)
elif isinstance(f, F):
return FalseFormula()
elif isinstance(f, T):
return TrueFormula()
else:
raise ValueError
def _implies_to_or(f: Implies) -> Formula:
equivalent_formula = Or(Not(f.f1), f.f2)
return _or_to_and(equivalent_formula)