def to_NNF(expr):
"""
Generates the Negation Normal Form of any boolean expression in terms
of AND, OR, and Literal objects.
"""
if isinstance(expr, Not):
arg = expr.args[0]
tmp = to_NNF(arg) # Strategy: negate the NNF of expr
return ~tmp
if isinstance(expr, Or):
return OR(*[to_NNF(x) for x in Or.make_args(expr)])
if isinstance(expr, And):
return AND(*[to_NNF(x) for x in And.make_args(expr)])
if isinstance(expr, Nand):
tmp = AND(*[to_NNF(x) for x in expr.args])
return ~tmp
if isinstance(expr, Nor):
tmp = OR(*[to_NNF(x) for x in expr.args])
return ~tmp
if isinstance(expr, Xor):
cnfs = []
for i in range(0, len(expr.args) + 1, 2):
for neg in combinations(expr.args, i):
clause = [~to_NNF(s) if s in neg else to_NNF(s)
for s in expr.args]
cnfs.append(OR(*clause))
return AND(*cnfs)
if isinstance(expr, Xnor):
cnfs = []
for i in range(0, len(expr.args) + 1, 2):
for neg in combinations(expr.args, i):
clause = [~to_NNF(s) if s in neg else to_NNF(s)
for s in expr.args]
cnfs.append(OR(*clause))
return ~AND(*cnfs)
if isinstance(expr, Implies):
L, R = to_NNF(expr.args[0]), to_NNF(expr.args[1])
return OR(~L, R)
if isinstance(expr, Equivalent):
cnfs = []
for a, b in zip_longest(expr.args, expr.args[1:], fillvalue=expr.args[0]):
a = to_NNF(a)
b = to_NNF(b)
cnfs.append(OR(~a, b))
return AND(*cnfs)
if isinstance(expr, ITE):
L = to_NNF(expr.args[0])
M = to_NNF(expr.args[1])
R = to_NNF(expr.args[2])
return AND(OR(~L, M), OR(L, R))
else:
return Literal(expr)
def to_NNF(expr, composite_map=None):
"""
Generates the Negation Normal Form of any boolean expression in terms
of AND, OR, and Literal objects.
Examples
========
>>> from sympy import Q, Eq
>>> from sympy.assumptions.cnf import to_NNF
>>> from sympy.abc import x, y
>>> expr = Q.even(x) & ~Q.positive(x)
>>> to_NNF(expr)
(Literal(Q.even(x), False) & Literal(Q.positive(x), True))
Supported boolean objects are converted to corresponding predicates.
>>> to_NNF(Eq(x, y))
Literal(Q.eq(x, y), False)
If ``composite_map`` argument is given, ``to_NNF`` decomposes the
specified predicate into a combination of primitive predicates.
>>> cmap = {Q.nonpositive: Q.negative | Q.zero}
>>> to_NNF(Q.nonpositive, cmap)
(Literal(Q.negative, False) | Literal(Q.zero, False))
>>> to_NNF(Q.nonpositive(x), cmap)
(Literal(Q.negative(x), False) | Literal(Q.zero(x), False))
"""
from sympy.assumptions.ask import Q
from sympy.assumptions.assume import AppliedPredicate, Predicate
if composite_map is None:
composite_map = dict()
binrelpreds = {Eq: Q.eq, Ne: Q.ne, Gt: Q.gt, Lt: Q.lt, Ge: Q.ge, Le: Q.le}
if type(expr) in binrelpreds:
pred = binrelpreds[type(expr)]
expr = pred(*expr.args)
if isinstance(expr, Not):
arg = expr.args[0]
tmp = to_NNF(arg, composite_map) # Strategy: negate the NNF of expr
return ~tmp
if isinstance(expr, Or):
return OR(*[to_NNF(x, composite_map) for x in Or.make_args(expr)])
if isinstance(expr, And):
return AND(*[to_NNF(x, composite_map) for x in And.make_args(expr)])
if isinstance(expr, Nand):
tmp = AND(*[to_NNF(x, composite_map) for x in expr.args])
return ~tmp
if isinstance(expr, Nor):
tmp = OR(*[to_NNF(x, composite_map) for x in expr.args])
return ~tmp
if isinstance(expr, Xor):
cnfs = []
for i in range(0, len(expr.args) + 1, 2):
for neg in combinations(expr.args, i):
clause = [
~to_NNF(s, composite_map) if s in neg else to_NNF(
s, composite_map) for s in expr.args
]
cnfs.append(OR(*clause))
return AND(*cnfs)
if isinstance(expr, Xnor):
cnfs = []
for i in range(0, len(expr.args) + 1, 2):
for neg in combinations(expr.args, i):
clause = [
~to_NNF(s, composite_map) if s in neg else to_NNF(
s, composite_map) for s in expr.args
]
cnfs.append(OR(*clause))
return ~AND(*cnfs)
if isinstance(expr, Implies):
L, R = to_NNF(expr.args[0],
composite_map), to_NNF(expr.args[1], composite_map)
return OR(~L, R)
if isinstance(expr, Equivalent):
cnfs = []
for a, b in zip_longest(expr.args,
expr.args[1:],
fillvalue=expr.args[0]):
a = to_NNF(a, composite_map)
b = to_NNF(b, composite_map)
cnfs.append(OR(~a, b))
return AND(*cnfs)
if isinstance(expr, ITE):
L = to_NNF(expr.args[0], composite_map)
M = to_NNF(expr.args[1], composite_map)
R = to_NNF(expr.args[2], composite_map)
return AND(OR(~L, M), OR(L, R))
if isinstance(expr, AppliedPredicate):
pred, args = expr.function, expr.arguments
newpred = composite_map.get(pred, None)
if newpred is not None:
return to_NNF(newpred.rcall(*args), composite_map)
if isinstance(expr, Predicate):
newpred = composite_map.get(expr, None)
if newpred is not None:
return to_NNF(newpred, composite_map)
return Literal(expr)
