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_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_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 _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 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))))))
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
