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 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 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_is_formula_not(self):
fol = self.fol
john = ConstantTerm.fromString("john")
paul = ConstantTerm.fromString("paul")
not_a_term = ConstantTerm.fromString("NotATerm")
right_equal = Equal(john, paul)
wrong_equal = Equal(john, not_a_term)
self.assertTrue(fol.is_formula(Not(right_equal)))
self.assertFalse(fol.is_formula(Not(wrong_equal)))
self.assertTrue(
fol.is_formula(
Not(PredicateFormula(PredicateSymbol("Person", 2), john,
paul))))
self.assertFalse(
fol.is_formula(
Not(
PredicateFormula(PredicateSymbol("Person", 3), john, paul,
john))))
self.assertFalse(
fol.is_formula(
Not(
PredicateFormula(PredicateSymbol("Person_fake", 2), john,
paul))))
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_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_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 test_truth_union(self):
ref = self.ref
a = self.a
b = self.b
c = self.c
self.assertTrue(
ref.truth(PathExpressionUnion(a, b), self.trace_1, 0, 1))
self.assertTrue(
ref.truth(PathExpressionUnion(a, c), self.trace_1, 4, 5))
self.assertFalse(
ref.truth(PathExpressionUnion(Not(b), Not(c)), self.trace_1, 4, 5))
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 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 _expand_formula(self, f: Formula):
if isinstance(f, AtomicFormula):
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))
else:
raise ValueError("Formula to expand not recognized")
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 _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 _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 _expand_formula(self, f: Formula):
if isinstance(f, AtomicFormula):
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(f.p, self.expand_formula(f.f))
else:
raise ValueError("Not valid Formula to expand")
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 _build_automata(self):
rows = self.row_symbols
atoms = [AtomicFormula(r) for r in rows]
alphabet = Alphabet(set(rows))
ldlf = LDLf_EmptyTraces(alphabet)
f = PathExpressionEventually(
PathExpressionSequence.chain([
PathExpressionStar(
And.chain([Not(atoms[0]),
Not(atoms[1]),
Not(atoms[2])])),
PathExpressionStar(
And.chain([atoms[0],
Not(atoms[1]),
Not(atoms[2])])),
# Not(atoms[3]), Not(atoms[4]), Not(atoms[5])]),
PathExpressionStar(
And.chain([atoms[0], atoms[1],
Not(atoms[2])])),
# Not(atoms[3]), Not(atoms[4]), Not(atoms[5])]),
# And.chain([atoms[0], atoms[1], atoms[2]]), # Not(atoms[3]), Not(atoms[4]), Not(atoms[5])]),
# And.chain([atoms[0], atoms[1], atoms[2], atoms[3], Not(atoms[4]), Not(atoms[5])]),
# And.chain([atoms[0], atoms[1], atoms[2], atoms[3], atoms[4], Not(atoms[5])]),
# And.chain([atoms[0], atoms[1], atoms[2], atoms[3], atoms[4], atoms[5] ])
]),
And.chain([atoms[0], atoms[1], atoms[2]]))
nfa = ldlf.to_nfa(f)
dfa = _to_pythomata_dfa(nfa)
return dfa
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 _expand_formula(self, f: Formula):
if isinstance(f, AtomicFormula):
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, Until):
return Until(self.expand_formula(f.f1), self.expand_formula(f.f2))
elif isinstance(f, Next):
return Next(self.expand_formula(f.f))
else:
raise ValueError("Not valid Formula to expand")
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 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 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 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_truth(self):
self.assertFalse(self.ldlf.truth(self.not_a, self.trace_1, 0))
self.assertTrue(self.ldlf.truth(self.not_a, self.trace_1, 4))
self.assertTrue(self.ldlf.truth(self.a_and_b, self.trace_1, 0))
self.assertFalse(self.ldlf.truth(self.a_and_b, self.trace_1, 1))
self.assertTrue(self.ldlf.truth(self.a_or_b, self.trace_1, 1))
self.assertTrue(
self.ldlf.truth(Not(And(self.b, self.c)), self.trace_1, 0))
self.assertTrue(
self.ldlf.truth(self.eventually_seq_a_and_b__a_and_c__not_c,
self.trace_1, 0))
self.assertFalse(
self.ldlf.truth(self.eventually_seq_a_and_b__a_and_c__not_c,
self.trace_1, 1))
self.assertTrue(
self.ldlf.truth(self.eventually_propositional_a_and_b__a_and_c,
self.trace_1, 0))
self.assertFalse(
self.ldlf.truth(self.eventually_test_a__c, self.trace_1, 0))
self.assertTrue(
self.ldlf.truth(self.eventually_test_a__b, self.trace_1, 0))
self.assertTrue(
self.ldlf.truth(self.eventually_seq_a_and_b__a_and_c__not_c,
self.trace_1, 0))
self.assertFalse(
self.ldlf.truth(self.eventually_seq_a_and_b__a_and_c__c,
self.trace_1, 0))
self.assertTrue(self.ldlf.truth(self.next_a_and_c, self.trace_1, 0))
self.assertTrue(self.ldlf.truth(self.liveness_b_and_c, self.trace_1,
0))
self.assertFalse(self.ldlf.truth(self.liveness_abc, self.trace_1, 0))
self.assertFalse(self.ldlf.truth(self.always_true__a, self.trace_1, 0))
self.assertTrue(
self.ldlf.truth(self.always_true__a,
self.trace_1.segment(0,
self.trace_1.length() - 1),
0))
self.assertTrue(
self.ldlf.truth(self.always_true__b_or_c, self.trace_1, 0))
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 _falseFormula_to_equivalent_formula(*args) -> Formula:
return And(DUMMY_ATOMIC, Not(DUMMY_ATOMIC))
def _or_to_and(f: Or) -> Formula:
return Not(And(Not(f.f1), Not(f.f2)))
def _implies_to_or(f: Implies) -> Formula:
equivalent_formula = Or(Not(f.f1), f.f2)
return _or_to_and(equivalent_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))))))
