def num_lattice_points_fast(n):
# count with inclusion-exclusion
if n == 1:
return 1
if (n + 3) % 2 == 1:
return 0
a = (n + 3) / 2
half_a = (a - 1) / 2
first_lattice_point_x = 3 - (a % 3)
prime_factors_a = [p for p, m in utils.prime_factors(a)]
# A is every 3rd point on the line in the lattice, B1 is all points w/ x
# coord divisible by prime factor i
# #(A and not B) = #A - #(A and (B1 or B2 or ...))
# = #A - #((A and B1) or (A and B2) or ..)
# = #A - (#(A and B1) + #(A and B2) + ... - #(A and B1 and B2) - ...)
num_lattice_points_half = 0
factor_sets = utils.power_set(prime_factors_a)
for factors in factor_sets:
sign = 1 if (len(factors) % 2 == 0) else -1
pr = [(3, first_lattice_point_x)] + [(f, 0) for f in factors]
num_lattice_points_half += sign * count_elts_in_range_with_remainders(first_lattice_point_x, half_a + 1, pr)
return num_lattice_points_half * 2
def all_hypotheses(examples):
"""Build a list of all the possible hypotheses"""
values = values_table(examples)
h_powerset = power_set(values.keys())
hypotheses = []
for s in h_powerset:
hypotheses.extend(build_attr_combinations(s, values))
hypotheses.extend(build_h_combinations(hypotheses))
return hypotheses
def generalizations(examples_so_far, h):
"""Generalize the hypothesis. First delete operations
(including disjunctions) from the hypothesis. Then, add OR operations."""
hypotheses = []
# Delete disjunctions
disj_powerset = power_set(range(len(h)))
for disjs in disj_powerset:
h2 = h.copy()
for d in reversed(list(disjs)):
del h2[d]
if check_all_consistency(examples_so_far, h2):
hypotheses += h2
# Delete AND operations in disjunctions
for i, disj in enumerate(h):
a_powerset = power_set(disj.keys())
for attrs in a_powerset:
h2 = h[i].copy()
for a in attrs:
del h2[a]
if check_all_consistency(examples_so_far, [h2]):
h3 = h.copy()
h3[i] = h2.copy()
hypotheses += h3
# Add OR operations
if hypotheses == [] or hypotheses == [{}]:
hypotheses = add_or(examples_so_far, h)
else:
hypotheses.extend(add_or(examples_so_far, h))
shuffle(hypotheses)
return hypotheses
def build_h_combinations(hypotheses):
"""Given a set of hypotheses, builds and returns all the combinations of the
hypotheses."""
h = []
h_powerset = power_set(range(len(hypotheses)))
for s in h_powerset:
t = []
for i in s:
t.extend(hypotheses[i])
h.append(t)
return h
def add_or(examples_so_far, h):
"""Add an OR operation to the hypothesis. The AND operations in the disjunction
are generated by the last example (which is the problematic one)."""
ors = []
e = examples_so_far[-1]
attrs = {k: v for k, v in e.items() if k != 'GOAL'}
a_powerset = power_set(attrs.keys())
for c in a_powerset:
h2 = {}
for k in c:
h2[k] = attrs[k]
if check_negative_consistency(examples_so_far, h2):
h3 = h.copy()
h3.append(h2)
ors.append(h3)
return ors
def test_power_set(self):
assert list(utils.power_set([1, 2, 3])) == [(), (1, ), (2, ), (3, ),
(1, 2), (1, 3), (2, 3),
(1, 2, 3)]