def test_miniscope(self):
""" miniscoped normal form """
f = expr("forall z: exists x: P(z) ==> Q(z) | ~Q(x)")
e = expr("forall z: ~P(z) | Q(z) | ~forall x: Q(x)")
self.assertEqual(e, miniscope(f))
self.assertTrue(prover.test_equivalent(e, f, []))
f = expr("exists x: forall z: P(z) ==> Q(z) | ~Q(x)")
# e = expr("forall z: ~P(z) | Q(z) | ~forall x: Q(x)")
# do not swap the quantifiers
e = expr("exists x: forall z: ~P(z) | Q(z) | ~Q(x)")
self.assertTrue(prover.test_equivalent(e, f, []))
self.assertEqual(e, miniscope(f))
def bfs(f, h, operators, max=0):
"""Return the best formula given a heuristic `h`. """
get_log().debug('running search for shortest formula using ops {}'
.format(operators))
if max == 0:
max = h.formula_cost(f)
closed = set()
atoms = sorted(collect_propositions(f), key=lambda x: str(x))
queue = atoms[:]
best = None
best_cost = max if max > 0 else 1000
while queue:
current = queue.pop(0)
get_log().debug('...current: {}'.format(str(current)))
if h.formula_cost(current) < best_cost:
if prover.test_equivalent(f, current, []):
best = current
best_cost = h.formula_cost(best)
if (max == -1):
return best
new = [n for n in create_combinations(current, atoms, operators)
if h.formula_cost(n) < best_cost and n not in closed]
queue.extend(new)
closed.update(new)
return best or f
def test_pushquants(self):
# easy forall cases
f = expr('forall x: P(x) & Q')
e = expr('(forall x: P(x)) & Q')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
f = expr('forall x: P & Q(x)')
e = expr('P & (forall x: Q(x))')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
f = expr('forall x: P(x) | Q')
e = expr('(forall x: P(x)) | Q')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
f = expr('forall x: P | Q(x)')
e = expr('P | forall x: Q(x)')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
# easy exists cases
f = expr('exists x: P(x) & Q')
e = expr('(exists x: P(x)) & Q')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
f = expr('exists x: P & Q(x)')
e = expr('P & (exists x: Q(x))')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
f = expr('exists x: P(x) | Q')
e = expr('(exists x: P(x)) | Q')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
f = expr('exists x: P | Q(x)')
e = expr('P | (exists x: Q(x))')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
# variable reduction
f = expr('(forall x: P(x) & Q(x))')
e = expr('(forall x: P(x)) & (forall x: Q(x))')
self.assertEqual(e, push_quants(f))
self.assertTrue(prover.test_equivalent(e, f, []))
# catch
f = expr("(forall x: forall x': P(x) | Q(x'))")
e = expr("(forall x: P(x)) | (forall x': Q(x'))")
self.assertEqual(e, push_quants(f))
# variable reduction
f = expr('(exists x: P(x) | Q(x))')
e = expr('(exists x: P(x)) | (exists x: Q(x))')
self.assertEqual(e, push_quants(f))
# catch
f = expr("(exists x: exists x': P(x) & Q(x'))")
e = expr("(exists x: P(x)) & (exists x': Q(x'))")
self.assertEqual(e, push_quants(f))
# no rename
f = expr('(forall y: P(x) & Q(y))')
e = expr('P(x) & (forall y: Q(y))')
self.assertEqual(e, push_quants(f))
# rename required
f = expr("(forall x': P(x) & Q(x'))")
e = expr("P(x) & (forall x': Q(x'))")
self.assertEqual(e, push_quants(f))
# rename required
f = expr("(exists x': P(x) & Q(x'))")
e = expr("P(x) & (exists x': Q(x'))")
self.assertEqual(e, push_quants(f))
f = expr("forall x: forall z: ( ( Q or P(x)) ==> R(z) )")
e = expr("(Q | exists x: P(x)) ==> forall z: R(z)")
self.assertEqual(e, push_quants(f))
f = expr('(forall x: exists y: P(x) & Q(y))')
e = expr('(forall x: P(x)) & (exists y: Q(y))')
self.assertEqual(e, push_quants(f))
f = expr('(forall x: exists y: P(x) | Q(y))')
e = expr('(forall x: P(x)) | (exists y: Q(y))')
self.assertEqual(e, push_quants(f))
f = expr('(forall x: exists y: P(x) ==> Q(y))')
e = expr('(exists x: P(x)) ==> (exists y: Q(y))')
self.assertEqual(e, push_quants(f))
f = expr('(forall x: exists y: P(x) ==> Q(y, x))')
e = expr('(forall x: P(x) ==> (exists y: Q(y, x)))')
self.assertEqual(e, push_quants(f))
f = expr('(forall x: exists y: P(x) <== Q(y))')
e = expr('(forall x: P(x)) <== (forall y: Q(y))')
self.assertEqual(e, push_quants(f))
def test_simplification_complex(self):
f = expr('forall x: P(x)')
expected = expr('forall x: P(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x: P(x) & true')
expected = expr('forall x: P(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x: P(x) & Q(x) ==> false')
expected = expr('forall x: ~(P(x) & Q(x))')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x, y: P(x) & true')
expected = expr('forall x: P(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x: forall y: P(x) & Q(x) ==> false')
expected = expr('forall x: ~(P(x) & Q(x))')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x: exists y: P(x) & Q(x) ==> false')
expected = expr('forall x: ~(P(x) & Q(x))')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x, y: exists z: x =/= y ==> x < z & z < y')
expected = expr('forall x, y: exists z: x =/= y ==> x < z & z < y')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x, y: (P(x1) & true) | (Q(y1) & false)')
expected = expr('P(x1)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x, y: exists z: (P(x1) & true) ==> (Q(y1) & false)')
expected = expr('~P(x1)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x, y: exists z: ' \
'(P(x1) & true) | (Q(y1) & false) ==> Z(xy) ')
expected = expr('P(x1) ==> Z(xy)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x, y: exists z: ' \
'(P(x1) & true) | (Q(y1) & false) <=> true ')
expected = expr('P(x1)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('forall x, y: exists z: ' \
'(P(x1) & true) | (Q(y1) & false) <== Z(xy) | true ')
expected = expr('true')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('true ==> (P <=> (P <=> false))')
expected = expr('P <=> ~P')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('exists x, y, z: P(x) ==> Q(z) ==> false')
expected = expr('exists x, z: P(x) ==> ~Q(z)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('(forall x, y: P(x) | (P(y) & false)) ==> exists z: Q')
expected = expr('(forall x: P(x)) ==> Q')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
def test_simplification_basic(self):
f = expr('~true')
expected = expr('false')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('~false')
expected = expr('true')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('~~true')
expected = expr('true')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('~~~true')
expected = expr('false')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P | true')
expected = expr(OP_TRUE)
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P | false')
expected = expr('P')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P & false')
expected = expr(OP_FALSE)
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P & true')
expected = expr('P')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P ==> false')
expected = expr('~P')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('true ==> P')
expected = expr('P')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P <=> true')
expected = expr('P')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P <=> false')
expected = expr('~P')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
def test_simplification_none(self):
f = expr('1')
expected = expr('1')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P')
expected = expr('P')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('true')
expected = expr('true')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('false')
expected = expr('false')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P(x)')
expected = expr('P(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P(x) | Q(x)')
expected = expr('P(x) | Q(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P(x) & Q(x)')
expected = expr('P(x) & Q(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P(x) ==> Q(x)')
expected = expr('P(x) ==> Q(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P(x) <== Q(x)')
expected = expr('P(x) <== Q(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P(x) <=> Q(x)')
expected = expr('P(x) <=> Q(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('P(x, y) ==> Q(x)')
expected = expr('P(x, y) ==> Q(x)')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('x > 0')
expected = expr('x > 0')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('x = y')
expected = expr('x = y')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
f = expr('x =/= 0')
expected = expr('x =/= 0')
self.assertTrue(prover.test_equivalent(expected, kleene(f), []))
