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_error(self):
a_sym = Symbol("a")
alphabet = Alphabet({a_sym})
a = Next(AtomicFormula(a_sym))
pl = PL(alphabet)
with self.assertRaises(ValueError) as ve:
pl.expand_formula(a)
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 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 test_expand_formula_allowed_formulas(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.expand_formula(a), a)
self.assertEqual(pl.expand_formula(Not(b)), Not(b))
self.assertEqual(pl.expand_formula(And(a, b)), And(a, b))
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 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 _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 _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
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 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 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))
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 __init__(self, alphabet: Alphabet):
super().__init__(alphabet)
self.pl = PL(self.alphabet)
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)
}
