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 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, PathExpressionAlways):
return Not(PathExpressionEventually(derived_formula.p, Not(derived_formula.f)))
elif isinstance(derived_formula, FalseFormula):
return And(Not(DUMMY_ATOMIC), DUMMY_ATOMIC)
elif isinstance(derived_formula, TrueFormula):
return Not(FalseFormula())
elif isinstance(derived_formula, LDLfLast):
return PathExpressionAlways(TrueFormula(), FalseFormula())
else:
raise ValueError("Derived formula not recognized")
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, Always):
return Not(Eventually(Not(derived_formula.f)))
elif isinstance(derived_formula, Eventually):
return Until(TrueFormula(), derived_formula.f)
elif isinstance(derived_formula, FalseFormula):
return And(Not(DUMMY_ATOMIC), DUMMY_ATOMIC)
elif isinstance(derived_formula, TrueFormula):
return Not(FalseFormula())
elif isinstance(derived_formula, LDLfLast):
return Next(TrueFormula(), FalseFormula())
else:
raise ValueError("Derived formula not recognized")
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_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_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_truth_propositional(self):
ref = self.ref
a = self.a
b = self.b
c = self.c
self.assertTrue(ref.truth(a, self.trace_1, 0, 1))
self.assertTrue(ref.truth(And(Not(b), And(a, c)), self.trace_1, 1, 2))
self.assertFalse(ref.truth(And(b, And(a, c)), self.trace_1, 1, 2))
self.assertTrue(ref.truth(TrueFormula(), self.trace_1, 0, 1))
self.assertFalse(ref.truth(FalseFormula(), self.trace_1, 0, 1))
self.assertFalse(ref.truth(TrueFormula(), self.trace_1, 0, 5))
self.assertFalse(ref.truth(TrueFormula(), self.trace_1, 0, 0))
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_equivalent_formula(self, derived_formula: Formula):
# make lines shorter
ef = self.to_equivalent_formula
if isinstance(derived_formula, AtomicFormula):
return PathExpressionEventually(derived_formula, LogicalTrue())
elif isinstance(derived_formula, LogicalFalse):
return Not(LogicalTrue())
elif isinstance(derived_formula, Or):
return Not(And(Not(derived_formula.f1), Not(derived_formula.f2)))
elif isinstance(derived_formula, PathExpressionAlways):
return Not(
PathExpressionEventually(derived_formula.p,
Not(derived_formula.f)))
elif isinstance(derived_formula, Next):
return PathExpressionEventually(
TrueFormula(), And(derived_formula.f, Not(ef(End()))))
elif isinstance(derived_formula, End):
return ef(PathExpressionAlways(TrueFormula(), ef(LogicalFalse())))
elif isinstance(derived_formula, Until):
return PathExpressionEventually(
PathExpressionStar(
PathExpressionSequence(
PathExpressionTest(derived_formula.f1),
ef(TrueFormula()))),
And(derived_formula.f2, Not(ef(End()))))
elif isinstance(derived_formula, FalseFormula):
return FalseFormula()
elif isinstance(derived_formula, TrueFormula):
return TrueFormula()
elif isinstance(derived_formula, LDLfLast):
return PathExpressionEventually(ef(TrueFormula()), ef(End()))
# propositional
elif isinstance(derived_formula, Formula):
pl = PL(self.alphabet)
assert pl.is_formula(derived_formula)
f = pl.to_nnf(derived_formula)
return PathExpressionEventually(f, LogicalTrue())
else:
raise ValueError("Derived formula not recognized")
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 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 to_nfa(self, f: Formula):
# TODO: optimize!!!
assert self.is_formula(f)
nnf_f = self.to_nnf(f)
alphabet = powerset(self.alphabet.symbols)
initial_states = {frozenset([nnf_f])}
final_states = {frozenset()}
delta = set()
pl, I = PL._from_set_of_propositionals(set(), Alphabet(set()))
d = self.delta(nnf_f, frozenset(), epsilon=True)
if pl.truth(d, I):
final_states.add(frozenset([nnf_f]))
states = {frozenset(), frozenset([nnf_f])}
states_changed, delta_changed = True, True
while states_changed or delta_changed:
states_changed, delta_changed = False, False
for actions_set in alphabet:
states_list = list(states)
for q in states_list:
delta_formulas = [
self.delta(subf, actions_set) for subf in q
]
atomics = [
s for subf in delta_formulas
for s in PL.find_atomics(subf)
]
symbol2formula = {
Symbol(str(f)): f
for f in atomics
if f != TrueFormula() and f != FalseFormula()
}
formula2atomic_formulas = {
f: AtomicFormula.fromName(str(f))
if f != TrueFormula() and f != FalseFormula() else f
for f in atomics
}
transformed_delta_formulas = [
self._tranform_delta(f, formula2atomic_formulas)
for f in delta_formulas
]
conjunctions = And.chain(transformed_delta_formulas)
models = frozenset(
PL(Alphabet(
set(symbol2formula))).minimal_models(conjunctions))
if len(models) == 0:
continue
for min_model in models:
q_prime = frozenset({
symbol2formula[s]
for s in min_model.symbol2truth
if min_model.symbol2truth[s]
})
len_before = len(states)
states.add(q_prime)
if len(states) == len_before + 1:
states_list.append(q_prime)
states_changed = True
len_before = len(delta)
delta.add((q, actions_set, q_prime))
if len(delta) == len_before + 1:
delta_changed = True
# check if q_prime should be added as final state
if len(q_prime) == 0:
final_states.add(q_prime)
else:
q_prime_delta_conjunction = And.chain([
self.delta(subf, frozenset(), epsilon=True)
for subf in q_prime
])
pl, I = PL._from_set_of_propositionals(
set(), Alphabet(set()))
if pl.truth(q_prime_delta_conjunction, I):
final_states.add(q_prime)
return {
"alphabet": alphabet,
"states": frozenset(states),
"initial_states": frozenset(initial_states),
"transitions": delta,
"accepting_states": frozenset(final_states)
}