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 expand_formula(self, f: Formula):
"""Manage the case when we have a propositional formula."""
# Check first if it is a propositional
pl = PL(self.alphabet)
if pl.is_formula(f):
return super().expand_formula(
PathExpressionEventually(f, LogicalTrue()))
else:
return super().expand_formula(f)
def _is_formula(self, f: Formula):
"""Check if a formula is legal in the current formal system"""
if isinstance(f, PathExpressionUnion):
return self.is_formula(f.p1) and self.is_formula(f.p2)
elif isinstance(f, PathExpressionSequence):
return self._is_formula(f.p1) and self._is_formula(f.p2)
elif isinstance(f, PathExpressionStar):
return self._is_formula(f.p)
else:
pl = PL(self.alphabet)
return pl.is_formula(f)
def _is_path(self, p: PathExpression):
# PathExpression
if isinstance(p, PathExpressionUnion) or isinstance(p, PathExpressionSequence):
return self._is_path(p.p1) and self._is_path(p.p2)
elif isinstance(p, PathExpressionTest):
return self.is_formula(p.f)
elif isinstance(p, PathExpressionStar):
return self._is_path(p.p)
elif isinstance(p, Formula):
pl = PL(self.alphabet)
return pl.is_formula(p)
else:
raise ValueError("Argument not a valid Path")
def to_nnf(self, f: Formula):
if isinstance(f, PathExpressionUnion):
return PathExpressionUnion(self.to_nnf(f.p1), self.to_nnf(f.p2))
elif isinstance(f, PathExpressionSequence):
return PathExpressionSequence(self.to_nnf(f.p1), self.to_nnf(f.p2))
elif isinstance(f, PathExpressionStar):
return PathExpressionStar(self.to_nnf(f.p))
elif isinstance(f, Formula):
pl = PL(self.alphabet)
assert pl.is_formula(f)
return f
else:
raise ValueError
def to_nnf_path(self, path: PathExpression):
if isinstance(path, PathExpressionTest):
return PathExpressionTest(self.to_nnf(path.f))
elif isinstance(path, PathExpressionUnion):
return PathExpressionUnion(self.to_nnf_path(path.p1), self.to_nnf_path(path.p2))
elif isinstance(path, PathExpressionSequence):
return PathExpressionSequence(self.to_nnf_path(path.p1), self.to_nnf_path(path.p2))
elif isinstance(path, PathExpressionStar):
return PathExpressionStar(self.to_nnf_path(path.p))
elif isinstance(path, Formula):
pl = PL(self.alphabet)
assert pl.is_formula(path)
return pl.to_nnf(path)
else:
raise ValueError
def _is_formula(self, f: Formula):
"""Check if a formula is legal in the current formal system"""
# Check first if it is a propositional
pl = PL(self.alphabet)
if pl.is_formula(f):
return self.is_formula(PathExpressionEventually(f, LogicalTrue()))
elif isinstance(f, LogicalTrue):
return True
elif isinstance(f, Not):
return self.is_formula(f.f)
elif isinstance(f, And):
return self.is_formula(f.f1) and self.is_formula(f.f2)
elif isinstance(f, PathExpressionEventually):
return self._is_path(f.p) and self.is_formula(f.f)
else:
return False
def to_nnf(self, f: Formula) -> Formula:
formula = self.expand_formula(f)
pl = PL(self.alphabet)
if pl.is_formula(formula):
return pl.to_nnf(formula)
elif isinstance(formula, LogicalTrue):
return formula
elif isinstance(formula, And):
return And(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 _truth(self, f: Formula, trace: FiniteTrace, position: int):
assert trace.alphabet == self.alphabet
truth = self._truth
pl = PL(self.alphabet)
if pl.is_formula(f):
return self.truth(PathExpressionEventually(f, LogicalTrue()),
trace, position)
elif isinstance(f, LogicalTrue):
return True
elif isinstance(f, Not):
return not self.truth(f.f, trace, position)
elif isinstance(f, And):
return self.truth(f.f1, trace, position) and self.truth(
f.f2, trace, position)
elif isinstance(f, PathExpressionEventually):
path = f.p
assert self._is_path(path)
if isinstance(path, PathExpressionTest):
return truth(path.f, trace, position) and truth(
f.f, trace, position)
elif isinstance(path, PathExpressionUnion):
return truth(PathExpressionEventually(path.p1, f.f), trace,
position) or truth(
PathExpressionEventually(path.p2, f.f), trace,
position)
elif isinstance(path, PathExpressionSequence):
return truth(
PathExpressionEventually(
path.p1, PathExpressionEventually(path.p2, f.f)),
trace, position)
elif isinstance(path, PathExpressionStar):
return truth(f.f, trace,
position) or (position < trace.last() and truth(
PathExpressionEventually(
path.p, PathExpressionEventually(
path, f.f)), trace, position)
and not self._is_testonly(path))
# path should be a Propositional Formula
else:
pl, I = PL._from_set_of_propositionals(trace.get(position),
trace.alphabet)
return position < trace.length() and pl.truth(
path, I) and truth(f.f, trace, position + 1)
else:
raise ValueError("Argument not a valid Formula")
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_is_formula_error(self):
a_sym = Symbol("a")
alphabet = Alphabet({a_sym})
a = Next(AtomicFormula(a_sym))
pl = PL(alphabet)
self.assertFalse(pl.is_formula(a))
def test_is_formula_atomic(self):
a_sym = Symbol("a")
alphabet = Alphabet({a_sym})
a = AtomicFormula(a_sym)
pl = PL(alphabet)
self.assertTrue(pl.is_formula(a))
class LDLf_EmptyTraces(FormalSystem):
def __init__(self, alphabet: Alphabet):
super().__init__(alphabet)
self.pl = PL(self.alphabet)
allowed_formulas = {
LogicalTrue, Not, And, PathExpressionEventually, TrueFormula,
FalseFormula
}
derived_formulas = {
LogicalFalse, Or, Next, Until, End, PathExpressionAlways, LDLfLast,
AtomicFormula
}
def _is_formula(self, f: Formula):
"""Check if a formula is legal in the current formal system"""
# Check first if it is a propositional
pl = PL(self.alphabet)
if pl.is_formula(f):
return self.is_formula(PathExpressionEventually(f, LogicalTrue()))
elif isinstance(f, LogicalTrue):
return True
elif isinstance(f, Not):
return self.is_formula(f.f)
elif isinstance(f, And):
return self.is_formula(f.f1) and self.is_formula(f.f2)
elif isinstance(f, PathExpressionEventually):
return self._is_path(f.p) and self.is_formula(f.f)
else:
return False
def _is_path(self, p: PathExpression):
# PathExpression
if isinstance(p, PathExpressionUnion) or isinstance(
p, PathExpressionSequence):
return self._is_path(p.p1) and self._is_path(p.p2)
elif isinstance(p, PathExpressionTest):
if self.pl.is_formula(p.f):
return True
else:
return self.is_formula(p.f)
elif isinstance(p, PathExpressionStar):
return self._is_path(p.p)
elif isinstance(p, Formula):
pl = PL(self.alphabet)
return pl.is_formula(p)
else:
raise ValueError("Argument not a valid Path")
def expand_formula(self, f: Formula):
"""Manage the case when we have a propositional formula."""
# Check first if it is a propositional
pl = PL(self.alphabet)
if pl.is_formula(f):
return super().expand_formula(
PathExpressionEventually(f, LogicalTrue()))
else:
return super().expand_formula(f)
def _expand_formula(self, f: Formula):
if isinstance(f, LogicalTrue):
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, PathExpressionEventually):
return PathExpressionEventually(self._expand_path(f.p),
self.expand_formula(f.f))
else:
raise ValueError("Not valid Formula to expand")
def _expand_path(self, p: PathExpression) -> PathExpression:
if isinstance(p, PathExpressionUnion) or isinstance(
p, PathExpressionSequence):
return type(p)(self._expand_path(p.p1), self._expand_path(p.p2))
elif isinstance(p, PathExpressionTest):
return PathExpressionTest(self.expand_formula(p.f))
elif isinstance(p, PathExpressionStar):
return PathExpressionStar(self._expand_path(p.p))
elif isinstance(p, Formula):
pl = PL(self.alphabet)
return pl.expand_formula(p)
else:
raise ValueError
@staticmethod
def _is_testonly(p: PathExpression):
if isinstance(p, PathExpressionUnion) or isinstance(
p, PathExpressionSequence):
return LDLf_EmptyTraces._is_testonly(
p.p1) and LDLf_EmptyTraces._is_testonly(p.p2)
elif isinstance(p, PathExpressionStar):
return LDLf_EmptyTraces._is_testonly(p.p)
else:
return isinstance(p, PathExpressionTest)
def _truth(self, f: Formula, trace: FiniteTrace, position: int):
assert trace.alphabet == self.alphabet
truth = self._truth
pl = PL(self.alphabet)
if pl.is_formula(f):
return self.truth(PathExpressionEventually(f, LogicalTrue()),
trace, position)
elif isinstance(f, LogicalTrue):
return True
elif isinstance(f, Not):
return not self.truth(f.f, trace, position)
elif isinstance(f, And):
return self.truth(f.f1, trace, position) and self.truth(
f.f2, trace, position)
elif isinstance(f, PathExpressionEventually):
path = f.p
assert self._is_path(path)
if isinstance(path, PathExpressionTest):
return truth(path.f, trace, position) and truth(
f.f, trace, position)
elif isinstance(path, PathExpressionUnion):
return truth(PathExpressionEventually(path.p1, f.f), trace,
position) or truth(
PathExpressionEventually(path.p2, f.f), trace,
position)
elif isinstance(path, PathExpressionSequence):
return truth(
PathExpressionEventually(
path.p1, PathExpressionEventually(path.p2, f.f)),
trace, position)
elif isinstance(path, PathExpressionStar):
return truth(f.f, trace,
position) or (position < trace.last() and truth(
PathExpressionEventually(
path.p, PathExpressionEventually(
path, f.f)), trace, position)
and not self._is_testonly(path))
# path should be a Propositional Formula
else:
pl, I = PL._from_set_of_propositionals(trace.get(position),
trace.alphabet)
return position < trace.length() and pl.truth(
path, I) and truth(f.f, trace, position + 1)
else:
raise ValueError("Argument not a valid Formula")
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 to_nnf(self, f: Formula) -> Formula:
formula = self.expand_formula(f)
pl = PL(self.alphabet)
if pl.is_formula(formula):
return pl.to_nnf(formula)
elif isinstance(formula, LogicalTrue):
return formula
elif isinstance(formula, And):
return And(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, Not):
# skip two consecutive Not
new_formula = subformula.f
return self.to_nnf(new_formula)
elif isinstance(subformula, LogicalTrue):
return LogicalFalse()
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 to_nnf_path(self, path: PathExpression):
if isinstance(path, PathExpressionTest):
return PathExpressionTest(self.to_nnf(path.f))
elif isinstance(path, PathExpressionUnion):
return PathExpressionUnion(self.to_nnf_path(path.p1),
self.to_nnf_path(path.p2))
elif isinstance(path, PathExpressionSequence):
return PathExpressionSequence(self.to_nnf_path(path.p1),
self.to_nnf_path(path.p2))
elif isinstance(path, PathExpressionStar):
return PathExpressionStar(self.to_nnf_path(path.p))
elif isinstance(path, Formula):
pl = PL(self.alphabet)
if pl.is_formula(path):
return pl.to_nnf(path)
else:
raise ValueError
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)
}
def _tranform_delta(self, f: Formula, formula2AtomicFormula):
if isinstance(f, Not):
return Not(self._tranform_delta(f, formula2AtomicFormula))
elif isinstance(f, And) or isinstance(f, Or):
return type(f)(self._tranform_delta(f.f1, formula2AtomicFormula),
self._tranform_delta(f.f2, formula2AtomicFormula))
elif isinstance(f, TrueFormula) or isinstance(f, FalseFormula):
return f
else:
return formula2AtomicFormula[f]
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 _expand(self, f: Formula):
if isinstance(f, F) or isinstance(f, T):
return self._expand(f.f)
elif isinstance(f, PathExpressionEventually) or isinstance(
f, PathExpressionAlways):
return type(f)(f.p, self._expand(f.f))
else:
return f
class REf(FormalSystem):
def __init__(self, alphabet: Alphabet):
super().__init__(alphabet)
self.pl = PL(self.alphabet)
allowed_formulas = {
PathExpressionUnion, PathExpressionSequence, PathExpressionStar,
AtomicFormula, And, Not
}
derived_formulas = {Or, Implies, Equivalence, TrueFormula, FalseFormula}
def _is_formula(self, f: Formula):
"""Check if a formula is legal in the current formal system"""
if isinstance(f, PathExpressionUnion):
return self.is_formula(f.p1) and self.is_formula(f.p2)
elif isinstance(f, PathExpressionSequence):
return self._is_formula(f.p1) and self._is_formula(f.p2)
elif isinstance(f, PathExpressionStar):
return self._is_formula(f.p)
else:
pl = PL(self.alphabet)
return pl.is_formula(f)
def _truth(self, f: Formula, trace: FiniteTrace, start: int, end: int):
assert self._is_formula(f)
assert trace.alphabet == self.alphabet
truth = self.truth
if isinstance(f, PathExpressionUnion):
return truth(f.p1, trace, start, end) or truth(
f.p2, trace, start, end)
if isinstance(f, PathExpressionSequence):
return any(
truth(f.p1, trace, start, k) and truth(f.p2, trace, k, end)
for k in range(start, end + 1))
if isinstance(f, PathExpressionStar):
return end == start or any(
truth(f.p, trace, start, k) and truth(f, trace, k, end)
for k in range(start, end + 1))
else:
pl, I = PL._from_set_of_propositionals(trace.get(start),
self.alphabet)
assert pl.is_formula(f)
return end == start + 1 and end <= trace.length() and pl.truth(
f, I)
def to_nnf(self, f: Formula):
if isinstance(f, PathExpressionUnion):
return PathExpressionUnion(self.to_nnf(f.p1), self.to_nnf(f.p2))
elif isinstance(f, PathExpressionSequence):
return PathExpressionSequence(self.to_nnf(f.p1), self.to_nnf(f.p2))
elif isinstance(f, PathExpressionStar):
return PathExpressionStar(self.to_nnf(f.p))
elif isinstance(f, Formula):
pl = PL(self.alphabet)
assert pl.is_formula(f)
return f
else:
raise ValueError
def to_equivalent_formula(self, derived_formula: Formula):
return self.pl.to_equivalent_formula(derived_formula)
def _expand_formula(self, f: Formula):
if self.pl.is_formula(f):
return self.pl.expand_formula(f)
else:
return f
