# Source code for sympy.logic.boolalg

"""
Boolean algebra module for SymPy
"""
from __future__ import print_function, division

from collections import defaultdict
from itertools import combinations, product

from sympy.core.basic import Basic
from sympy.core.cache import cacheit
from sympy.core.core import C
from sympy.core.numbers import Number
from sympy.core.decorators import deprecated
from sympy.core.operations import LatticeOp
from sympy.core.function import Application
from sympy.core.compatibility import ordered, xrange, with_metaclass
from sympy.core.sympify import converter, _sympify, sympify
from sympy.core.singleton import Singleton, S

class Boolean(Basic):
"""A boolean object is an object for which logic operations make sense."""

__slots__ = []

def __and__(self, other):
return And(self, other)

__rand__ = __and__

def __or__(self, other):
return Or(self, other)

__ror__ = __or__

def __invert__(self):
return Not(self)

def __rshift__(self, other):
return Implies(self, other)

def __lshift__(self, other):
return Implies(other, self)

__rrshift__ = __lshift__
__rlshift__ = __rshift__

def __xor__(self, other):
return Xor(self, other)

__rxor__ = __xor__

def equals(self, other):
"""
Returns if the given formulas have the same truth table.
For two formulas to be equal they must have the same literals.

Examples
========

>>> from sympy.abc import A, B, C
>>> from sympy.logic.boolalg import And, Or, Not
>>> (A >> B).equals(~B >> ~A)
True
>>> Not(And(A, B, C)).equals(And(Not(A), Not(B), Not(C)))
False
>>> Not(And(A, Not(A))).equals(Or(B, Not(B)))
False
"""

from sympy.logic.inference import satisfiable
return self.atoms() == other.atoms() and \
not satisfiable(Not(Equivalent(self, other)))

# Developer note: There is liable to be some confusion as to when True should
# be used and when S.true should be used in various contexts throughout SymPy.
# An important thing to remember is that sympify(True) returns S.true.  This
# means that for the most part, you can just use True and it will
# automatically be converted to S.true when necessary, similar to how you can
# generally use 1 instead of S.One.

# The rule of thumb is:

#   "If the boolean in question can be replaced by an arbitrary symbolic
#   Boolean, like Or(x, y) or x > 1, use S.true. Otherwise, use True"

# In other words, use S.true only on those contexts where the boolean is being
# used as a symbolic representation of truth.  For example, if the object ends
# up in the .args of any expression, then it must necessarily be S.true
# instead of True, as elements of .args must be Basic.  On the other hand, ==
# is not a symbolic operation in SymPy, since it always returns True or False,
# and does so in terms of structural equality rather than mathematical, so it
# should return True. The assumptions system should use True and False. Aside
# from not satisfying the above rule of thumb, the assumptions system uses a
# three-valued logic (True, False, None), whereas S.true and S.false represent
# a two-valued logic.  When it doubt, use True.

# 2. "S.true == True" is True.

# While "S.true is True" is False, "S.true == True" is True, so if there is
# any doubt over whether a function or expression will return S.true or True,
# just use "==" instead of "is" to do the comparison, and it will work in
# either case.  Finally, for boolean flags, it's better to just use "if x"
# instead of "if x is True". To quote PEP 8:

#     Don't compare boolean values to True or False using ==.

#       Yes:   if greeting:
#       No:    if greeting == True:
#       Worse: if greeting is True:

class BooleanAtom(Boolean):
"""
Base class of BooleanTrue and BooleanFalse.
"""

[docs]class BooleanTrue(with_metaclass(Singleton, BooleanAtom)):
"""
SymPy version of True.

The instances of this class are singletonized and can be accessed via
S.true.

This is the SymPy version of True, for use in the logic module. The
primary advantage of using true instead of True is that shorthand boolean
operations like ~ and >> will work as expected on this class, whereas with
True they act bitwise on 1. Functions in the logic module will return this
class when they evaluate to true.

Examples
========

>>> from sympy import sympify, true, Or
>>> sympify(True)
True
>>> ~true
False
>>> ~True
-2
>>> Or(True, False)
True

========
sympy.logic.boolalg.BooleanFalse

"""
def __nonzero__(self):
return True

__bool__ = __nonzero__

def __hash__(self):
return hash(True)

def as_set(self):
"""
Rewrite logic operators and relationals in terms of real sets.

Examples
========

>>> from sympy import true
>>> true.as_set()
UniversalSet()
"""
return S.UniversalSet

[docs]class BooleanFalse(with_metaclass(Singleton, BooleanAtom)):
"""
SymPy version of False.

The instances of this class are singletonized and can be accessed via
S.false.

This is the SymPy version of False, for use in the logic module. The
primary advantage of using false instead of False is that shorthand boolean
operations like ~ and >> will work as expected on this class, whereas with
False they act bitwise on 0. Functions in the logic module will return this
class when they evaluate to false.

Examples
========

>>> from sympy import sympify, false, Or, true
>>> sympify(False)
False
>>> false >> false
True
>>> False >> False
0
>>> Or(True, False)
True

========
sympy.logic.boolalg.BooleanTrue

"""
def __nonzero__(self):
return False

__bool__ = __nonzero__

def __hash__(self):
return hash(False)

def as_set(self):
"""
Rewrite logic operators and relationals in terms of real sets.

Examples
========

>>> from sympy import false
>>> false.as_set()
EmptySet()
"""
from sympy.sets.sets import EmptySet
return EmptySet()

true = BooleanTrue()
false = BooleanFalse()
# We want S.true and S.false to work, rather than S.BooleanTrue and
# S.BooleanFalse, but making the class and instance names the same causes some
# major issues (like the inability to import the class directly from this
# file).
S.true = true
S.false = false

converter[bool] = lambda x: S.true if x else S.false

class BooleanFunction(Application, Boolean):
"""Boolean function is a function that lives in a boolean space
It is used as base class for And, Or, Not, etc.
"""
is_Boolean = True

def __call__(self, *args):
return self.func(*[arg(*args) for arg in self.args])

def _eval_simplify(self, ratio, measure):
return simplify_logic(self)

def to_nnf(self, simplify=True):
return self._to_nnf(*self.args, simplify=simplify)

@classmethod
def _to_nnf(cls, *args, **kwargs):
simplify = kwargs.get('simplify', True)
argset = set([])
for arg in args:
if not is_literal(arg):
arg = arg.to_nnf(simplify)
if simplify:
if isinstance(arg, cls):
arg = arg.args
else:
arg = (arg,)
for a in arg:
if Not(a) in argset:
return cls.zero
else:
return cls(*argset)

[docs]class And(LatticeOp, BooleanFunction):
"""
Logical AND function.

It evaluates its arguments in order, giving False immediately
if any of them are False, and True if they are all True.

Examples
========

>>> from sympy.core import symbols
>>> from sympy.abc import x, y
>>> from sympy.logic.boolalg import And
>>> x & y
And(x, y)

Notes
=====

The & operator is provided as a convenience, but note that its use
here is different from its normal use in Python, which is bitwise
and. Hence, And(a, b) and a & b will return different things if
a and b are integers.

>>> And(x, y).subs(x, 1)
y

"""
zero = false
identity = true

nargs = None

@classmethod
def _new_args_filter(cls, args):
newargs = []
for x in args:
if isinstance(x, Number) or x in (0, 1):
newargs.append(True if x else False)
else:
newargs.append(x)
return LatticeOp._new_args_filter(newargs, And)

def as_set(self):
"""
Rewrite logic operators and relationals in terms of real sets.

Examples
========

>>> from sympy import And, Symbol
>>> x = Symbol('x', real=True)
>>> And(x<2, x>-2).as_set()
(-2, 2)
"""
from sympy.sets.sets import Intersection
if len(self.free_symbols) == 1:
return Intersection(*[arg.as_set() for arg in self.args])
else:
raise NotImplementedError("Sorry, And.as_set has not yet been"
" implemented for multivariate"
" expressions")

[docs]class Or(LatticeOp, BooleanFunction):
"""
Logical OR function

It evaluates its arguments in order, giving True immediately
if any of them are True, and False if they are all False.

Examples
========

>>> from sympy.core import symbols
>>> from sympy.abc import x, y
>>> from sympy.logic.boolalg import Or
>>> x | y
Or(x, y)

Notes
=====

The | operator is provided as a convenience, but note that its use
here is different from its normal use in Python, which is bitwise
or. Hence, Or(a, b) and a | b will return different things if
a and b are integers.

>>> Or(x, y).subs(x, 0)
y

"""
zero = true
identity = false

@classmethod
def _new_args_filter(cls, args):
newargs = []
for x in args:
if isinstance(x, Number) or x in (0, 1):
newargs.append(True if x else False)
else:
newargs.append(x)
return LatticeOp._new_args_filter(newargs, Or)

def as_set(self):
"""
Rewrite logic operators and relationals in terms of real sets.

Examples
========

>>> from sympy import Or, Symbol
>>> x = Symbol('x', real=True)
>>> Or(x>2, x<-2).as_set()
(-oo, -2) U (2, oo)
"""
from sympy.sets.sets import Union
if len(self.free_symbols) == 1:
return Union(*[arg.as_set() for arg in self.args])
else:
raise NotImplementedError("Sorry, Or.as_set has not yet been"
" implemented for multivariate"
" expressions")

[docs]class Not(BooleanFunction):
"""
Logical Not function (negation)

Returns True if the statement is False
Returns False if the statement is True

Examples
========

>>> from sympy.logic.boolalg import Not, And, Or
>>> from sympy.abc import x, A, B
>>> Not(True)
False
>>> Not(False)
True
>>> Not(And(True, False))
True
>>> Not(Or(True, False))
False
>>> Not(And(And(True, x), Or(x, False)))
Not(x)
>>> ~x
Not(x)
>>> Not(And(Or(A, B), Or(~A, ~B)))
Not(And(Or(A, B), Or(Not(A), Not(B))))

Notes
=====

- The ~ operator is provided as a convenience, but note that its use
here is different from its normal use in Python, which is bitwise
not. In particular, ~a and Not(a) will be different if a is
an integer. Furthermore, since bools in Python subclass from int,
~True is the same as ~1 which is -2, which has a boolean
value of True.  To avoid this issue, use the SymPy boolean types
true and false.

>>> from sympy import true
>>> ~True
-2
>>> ~true
False

"""

is_Not = True

@classmethod
def eval(cls, arg):
if isinstance(arg, Number) or arg in (True, False):
return false if arg else true
if arg.is_Not:
return arg.args[0]
# Simplify Relational objects.
if isinstance(arg, C.Equality):
return C.Unequality(*arg.args)
if isinstance(arg, C.Unequality):
return C.Equality(*arg.args)
if isinstance(arg, C.StrictLessThan):
return C.GreaterThan(*arg.args)
if isinstance(arg, C.StrictGreaterThan):
return C.LessThan(*arg.args)
if isinstance(arg, C.LessThan):
return C.StrictGreaterThan(*arg.args)
if isinstance(arg, C.GreaterThan):
return C.StrictLessThan(*arg.args)

def as_set(self):
"""
Rewrite logic operators and relationals in terms of real sets.

Examples
========

>>> from sympy import Not, Symbol
>>> x = Symbol('x', real=True)
>>> Not(x>0).as_set()
(-oo, 0]
"""
if len(self.free_symbols) == 1:
return self.args[0].as_set().complement
else:
raise NotImplementedError("Sorry, Not.as_set has not yet been"
" implemented for mutivariate"
" expressions")

def to_nnf(self, simplify=True):
if is_literal(self):
return self

expr = self.args[0]

func, args = expr.func, expr.args

if func == And:
return Or._to_nnf(*[~arg for arg in args], simplify=simplify)

if func == Or:
return And._to_nnf(*[~arg for arg in args], simplify=simplify)

if func == Implies:
a, b = args
return And._to_nnf(a, ~b, simplify=simplify)

if func == Equivalent:
return And._to_nnf(Or(*args), Or(*[~arg for arg in args]), simplify=simplify)

if func == Xor:
result = []
for i in xrange(1, len(args)+1, 2):
for neg in combinations(args, i):
clause = [~s if s in neg else s for s in args]
result.append(Or(*clause))
return And._to_nnf(*result, simplify=simplify)

if func == ITE:
a, b, c = args
return And._to_nnf(Or(a, ~c), Or(~a, ~b), simplify=simplify)

raise ValueError("Illegal operator %s in expression" % func)

[docs]class Xor(BooleanFunction):
"""
Logical XOR (exclusive OR) function.

Returns True if an odd number of the arguments are True and the rest are
False.

Returns False if an even number of the arguments are True and the rest are
False.

Examples
========

>>> from sympy.logic.boolalg import Xor
>>> from sympy import symbols
>>> x, y = symbols('x y')
>>> Xor(True, False)
True
>>> Xor(True, True)
False
>>> Xor(True, False, True, True, False)
True
>>> Xor(True, False, True, False)
False
>>> x ^ y
Xor(x, y)

Notes
=====

The ^ operator is provided as a convenience, but note that its use
here is different from its normal use in Python, which is bitwise xor. In
particular, a ^ b and Xor(a, b) will be different if a and
b are integers.

>>> Xor(x, y).subs(y, 0)
x

"""
def __new__(cls, *args, **kwargs):
argset = set([])
obj = super(Xor, cls).__new__(cls, *args, **kwargs)
for arg in obj._args:
if isinstance(arg, Number) or arg in (True, False):
if arg:
arg = true
else:
continue
if isinstance(arg, Xor):
for a in arg.args:
argset.remove(a) if a in argset else argset.add(a)
elif arg in argset:
argset.remove(arg)
else:
if len(argset) == 0:
return false
elif len(argset) == 1:
return argset.pop()
elif True in argset:
argset.remove(True)
return Not(Xor(*argset))
else:
obj._args = tuple(ordered(argset))
obj._argset = frozenset(argset)
return obj

@property
@cacheit
def args(self):
return tuple(ordered(self._argset))

def to_nnf(self, simplify=True):
args = []
for i in xrange(0, len(self.args)+1, 2):
for neg in combinations(self.args, i):
clause = [~s if s in neg else s for s in self.args]
args.append(Or(*clause))
return And._to_nnf(*args, simplify=simplify)

[docs]class Nand(BooleanFunction):
"""
Logical NAND function.

It evaluates its arguments in order, giving True immediately if any
of them are False, and False if they are all True.

Returns True if any of the arguments are False
Returns False if all arguments are True

Examples
========

>>> from sympy.logic.boolalg import Nand
>>> from sympy import symbols
>>> x, y = symbols('x y')
>>> Nand(False, True)
True
>>> Nand(True, True)
False
>>> Nand(x, y)
Not(And(x, y))

"""
@classmethod
def eval(cls, *args):
return Not(And(*args))

[docs]class Nor(BooleanFunction):
"""
Logical NOR function.

It evaluates its arguments in order, giving False immediately if any
of them are True, and True if they are all False.

Returns False if any argument is True
Returns True if all arguments are False

Examples
========

>>> from sympy.logic.boolalg import Nor
>>> from sympy import symbols
>>> x, y = symbols('x y')

>>> Nor(True, False)
False
>>> Nor(True, True)
False
>>> Nor(False, True)
False
>>> Nor(False, False)
True
>>> Nor(x, y)
Not(Or(x, y))

"""
@classmethod
def eval(cls, *args):
return Not(Or(*args))

[docs]class Implies(BooleanFunction):
"""
Logical implication.

A implies B is equivalent to !A v B

Accepts two Boolean arguments; A and B.
Returns False if A is True and B is False
Returns True otherwise.

Examples
========

>>> from sympy.logic.boolalg import Implies
>>> from sympy import symbols
>>> x, y = symbols('x y')

>>> Implies(True, False)
False
>>> Implies(False, False)
True
>>> Implies(True, True)
True
>>> Implies(False, True)
True
>>> x >> y
Implies(x, y)
>>> y << x
Implies(x, y)

Notes
=====

The >> and << operators are provided as a convenience, but note
that their use here is different from their normal use in Python, which is
bit shifts. Hence, Implies(a, b) and a >> b will return different
things if a and b are integers.  In particular, since Python
considers True and False to be integers, True >> True will be
the same as 1 >> 1, i.e., 0, which has a truth value of False.  To
avoid this issue, use the SymPy objects true and false.

>>> from sympy import true, false
>>> True >> False
1
>>> true >> false
False

"""
@classmethod
def eval(cls, *args):
try:
newargs = []
for x in args:
if isinstance(x, Number) or x in (0, 1):
newargs.append(True if x else False)
else:
newargs.append(x)
A, B = newargs
except ValueError:
raise ValueError(
"%d operand(s) used for an Implies "
"(pairs are required): %s" % (len(args), str(args)))
if A == True or A == False or B == True or B == False:
return Or(Not(A), B)
else:
return Basic.__new__(cls, *args)

def to_nnf(self, simplify=True):
a, b = self.args
return Or._to_nnf(~a, b, simplify=simplify)

[docs]class Equivalent(BooleanFunction):
"""
Equivalence relation.

Equivalent(A, B) is True iff A and B are both True or both False

Returns True if all of the arguments are logically equivalent.
Returns False otherwise.

Examples
========

>>> from sympy.logic.boolalg import Equivalent, And
>>> from sympy.abc import x, y
>>> Equivalent(False, False, False)
True
>>> Equivalent(True, False, False)
False
>>> Equivalent(x, And(x, True))
True
"""
def __new__(cls, *args, **options):
args = [_sympify(arg) for arg in args]

argset = set(args)
for x in args:
if isinstance(x, Number) or x in [True, False]: # Includes 0, 1
if len(argset) <= 1:
return true
if True in argset:
return And(*argset)
if False in argset:
return And(*[~arg for arg in argset])
_args = frozenset(argset)
obj = super(Equivalent, cls).__new__(cls, _args)
obj._argset = _args
return obj

@property
@cacheit
def args(self):
return tuple(ordered(self._argset))

def to_nnf(self, simplify=True):
args = []
for a, b in zip(self.args, self.args[1:]):
args.append(Or(~a, b))
args.append(Or(~self.args[-1], self.args[0]))
return And._to_nnf(*args, simplify=simplify)

[docs]class ITE(BooleanFunction):
"""
If then else clause.

ITE(A, B, C) evaluates and returns the result of B if A is true
else it returns the result of C

Examples
========

>>> from sympy.logic.boolalg import ITE, And, Xor, Or
>>> from sympy.abc import x, y, z
>>> ITE(True, False, True)
False
>>> ITE(Or(True, False), And(True, True), Xor(True, True))
True
>>> ITE(x, y, z)
ITE(x, y, z)
>>> ITE(True, x, y)
x
>>> ITE(False, x, y)
y
>>> ITE(x, y, y)
y
"""
@classmethod
def eval(cls, *args):
try:
a, b, c = args
except ValueError:
raise ValueError("ITE expects exactly 3 arguments")
if a == True:
return b
if a == False:
return c
if b == c:
return b

def to_nnf(self, simplify=True):
a, b, c = self.args
return And._to_nnf(Or(~a, b), Or(a, c), simplify=simplify)

### end class definitions. Some useful methods

def conjuncts(expr):
"""Return a list of the conjuncts in the expr s.

Examples
========

>>> from sympy.logic.boolalg import conjuncts
>>> from sympy.abc import A, B
>>> conjuncts(A & B)
frozenset([A, B])
>>> conjuncts(A | B)
frozenset([Or(A, B)])

"""
return And.make_args(expr)

def disjuncts(expr):
"""Return a list of the disjuncts in the sentence s.

Examples
========

>>> from sympy.logic.boolalg import disjuncts
>>> from sympy.abc import A, B
>>> disjuncts(A | B)
frozenset([A, B])
>>> disjuncts(A & B)
frozenset([And(A, B)])

"""
return Or.make_args(expr)

def distribute_and_over_or(expr):
"""
Given a sentence s consisting of conjunctions and disjunctions
of literals, return an equivalent sentence in CNF.

Examples
========

>>> from sympy.logic.boolalg import distribute_and_over_or, And, Or, Not
>>> from sympy.abc import A, B, C
>>> distribute_and_over_or(Or(A, And(Not(B), Not(C))))
And(Or(A, Not(B)), Or(A, Not(C)))
"""
return _distribute((expr, And, Or))

def distribute_or_over_and(expr):
"""
Given a sentence s consisting of conjunctions and disjunctions
of literals, return an equivalent sentence in DNF.

Note that the output is NOT simplified.

Examples
========

>>> from sympy.logic.boolalg import distribute_or_over_and, And, Or, Not
>>> from sympy.abc import A, B, C
>>> distribute_or_over_and(And(Or(Not(A), B), C))
Or(And(B, C), And(C, Not(A)))
"""
return _distribute((expr, Or, And))

def _distribute(info):
"""
Distributes info[1] over info[2] with respect to info[0].
"""
if info[0].func is info[2]:
for arg in info[0].args:
if arg.func is info[1]:
conj = arg
break
else:
return info[0]
rest = info[2](*[a for a in info[0].args if a is not conj])
return info[1](*list(map(_distribute,
[(info[2](c, rest), info[1], info[2]) for c in conj.args])))
elif info[0].func is info[1]:
return info[1](*list(map(_distribute,
[(x, info[1], info[2]) for x in info[0].args])))
else:
return info[0]

def to_nnf(expr, simplify=True):
"""
Converts expr to Negation Normal Form.
A logical expression is in Negation Normal Form (NNF) if it
contains only And, Or and Not, and Not is applied only to literals.
If simpify is True, the result contains no redundant clauses.

Examples
========

>>> from sympy.abc import A, B, C, D
>>> from sympy.logic.boolalg import Not, Equivalent, to_nnf
>>> to_nnf(Not((~A & ~B) | (C & D)))
And(Or(A, B), Or(Not(C), Not(D)))
>>> to_nnf(Equivalent(A >> B, B >> A))
And(Or(A, And(A, Not(B)), Not(B)), Or(And(B, Not(A)), B, Not(A)))
"""
if is_nnf(expr, simplify):
return expr
return expr.to_nnf(simplify)

[docs]def to_cnf(expr, simplify=False):
"""
Convert a propositional logical sentence s to conjunctive normal form.
That is, of the form ((A | ~B | ...) & (B | C | ...) & ...)
If simplify is True, the expr is evaluated to its simplest CNF form.

Examples
========

>>> from sympy.logic.boolalg import to_cnf
>>> from sympy.abc import A, B, D
>>> to_cnf(~(A | B) | D)
And(Or(D, Not(A)), Or(D, Not(B)))
>>> to_cnf((A | B) & (A | ~A), True)
Or(A, B)

"""
expr = sympify(expr)
if not isinstance(expr, BooleanFunction):
return expr

if simplify:
return simplify_logic(expr, 'cnf', True)

# Don't convert unless we have to
if is_cnf(expr):
return expr

expr = eliminate_implications(expr)
return distribute_and_over_or(expr)

[docs]def to_dnf(expr, simplify=False):
"""
Convert a propositional logical sentence s to disjunctive normal form.
That is, of the form ((A & ~B & ...) | (B & C & ...) | ...)
If simplify is True, the expr is evaluated to its simplest DNF form.

Examples
========

>>> from sympy.logic.boolalg import to_dnf
>>> from sympy.abc import A, B, C
>>> to_dnf(B & (A | C))
Or(And(A, B), And(B, C))
>>> to_dnf((A & B) | (A & ~B) | (B & C) | (~B & C), True)
Or(A, C)

"""
expr = sympify(expr)
if not isinstance(expr, BooleanFunction):
return expr

if simplify:
return simplify_logic(expr, 'dnf', True)

# Don't convert unless we have to
if is_dnf(expr):
return expr

expr = eliminate_implications(expr)
return distribute_or_over_and(expr)

def is_nnf(expr, simplified=True):
"""
Checks if expr is in Negation Normal Form.
A logical expression is in Negation Normal Form (NNF) if it
contains only And, Or and Not, and Not is applied only to literals.
If simpified is True, checks if result contains no redundant clauses.

Examples
========

>>> from sympy.abc import A, B, C
>>> from sympy.logic.boolalg import Not, is_nnf
>>> is_nnf(A & B | ~C)
True
>>> is_nnf((A | ~A) & (B | C))
False
>>> is_nnf((A | ~A) & (B | C), False)
True
>>> is_nnf(Not(A & B) | C)
False
>>> is_nnf((A >> B) & (B >> A))
False
"""

expr = sympify(expr)
if is_literal(expr):
return True

stack = [expr]

while stack:
expr = stack.pop()
if expr.func in (And, Or):
if simplified:
args = expr.args
for arg in args:
if Not(arg) in args:
return False
stack.extend(expr.args)

elif not is_literal(expr):
return False

return True

[docs]def is_cnf(expr):
"""
Test whether or not an expression is in conjunctive normal form.

Examples
========

>>> from sympy.logic.boolalg import is_cnf
>>> from sympy.abc import A, B, C
>>> is_cnf(A | B | C)
True
>>> is_cnf(A & B & C)
True
>>> is_cnf((A & B) | C)
False

"""
return _is_form(expr, And, Or)

[docs]def is_dnf(expr):
"""
Test whether or not an expression is in disjunctive normal form.

Examples
========

>>> from sympy.logic.boolalg import is_dnf
>>> from sympy.abc import A, B, C
>>> is_dnf(A | B | C)
True
>>> is_dnf(A & B & C)
True
>>> is_dnf((A & B) | C)
True
>>> is_dnf(A & (B | C))
False

"""
return _is_form(expr, Or, And)

def _is_form(expr, function1, function2):
"""
Test whether or not an expression is of the required form.

"""
expr = sympify(expr)

# Special case of an Atom
if expr.is_Atom:
return True

# Special case of a single expression of function2
if expr.func is function2:
for lit in expr.args:
if lit.func is Not:
if not lit.args[0].is_Atom:
return False
else:
if not lit.is_Atom:
return False
return True

# Special case of a single negation
if expr.func is Not:
if not expr.args[0].is_Atom:
return False

if expr.func is not function1:
return False

for cls in expr.args:
if cls.is_Atom:
continue
if cls.func is Not:
if not cls.args[0].is_Atom:
return False
elif cls.func is not function2:
return False
for lit in cls.args:
if lit.func is Not:
if not lit.args[0].is_Atom:
return False
else:
if not lit.is_Atom:
return False

return True

def eliminate_implications(expr):
"""
Change >>, <<, and Equivalent into &, |, and ~. That is, return an
expression that is equivalent to s, but has only &, |, and ~ as logical
operators.

Examples
========

>>> from sympy.logic.boolalg import Implies, Equivalent, \
eliminate_implications
>>> from sympy.abc import A, B, C
>>> eliminate_implications(Implies(A, B))
Or(B, Not(A))
>>> eliminate_implications(Equivalent(A, B))
And(Or(A, Not(B)), Or(B, Not(A)))
>>> eliminate_implications(Equivalent(A, B, C))
And(Or(A, Not(C)), Or(B, Not(A)), Or(C, Not(B)))
"""

def is_literal(expr):
"""
Returns True if expr is a literal, else False.

Examples
========

>>> from sympy import Or, Q
>>> from sympy.abc import A, B
>>> from sympy.logic.boolalg import is_literal
>>> is_literal(A)
True
>>> is_literal(~A)
True
>>> is_literal(Q.zero(A))
True
>>> is_literal(A + B)
True
>>> is_literal(Or(A, B))
False
"""
if isinstance(expr, Not):
return not isinstance(expr.args[0], BooleanFunction)
else:
return not isinstance(expr, BooleanFunction)

@deprecated(
def compile_rule(s):
"""
Transforms a rule into a SymPy expression
A rule is a string of the form "symbol1 & symbol2 | ..."

Note: This function is deprecated.  Use sympify() instead.

"""
import re
return sympify(re.sub(r'([a-zA-Z_][a-zA-Z0-9_]*)', r'Symbol("\1")', s))

def to_int_repr(clauses, symbols):
"""
Takes clauses in CNF format and puts them into an integer representation.

Examples
========

>>> from sympy.logic.boolalg import to_int_repr
>>> from sympy.abc import x, y
>>> to_int_repr([x | y, y], [x, y]) == [set([1, 2]), set([2])]
True

"""

# Convert the symbol list into a dict
symbols = dict(list(zip(symbols, list(xrange(1, len(symbols) + 1)))))

def append_symbol(arg, symbols):
if arg.func is Not:
return -symbols[arg.args[0]]
else:
return symbols[arg]

return [set(append_symbol(arg, symbols) for arg in Or.make_args(c))
for c in clauses]

def _check_pair(minterm1, minterm2):
"""
Checks if a pair of minterms differs by only one bit. If yes, returns
index, else returns -1.
"""
index = -1
for x, (i, j) in enumerate(zip(minterm1, minterm2)):
if i != j:
if index == -1:
index = x
else:
return -1
return index

"""
Converts a term in the expansion of a function from binary to it's
variable form (for SOP).
"""
temp = []
for i, m in enumerate(minterm):
if m == 0:
temp.append(Not(variables[i]))
elif m == 1:
temp.append(variables[i])
else:
pass  # ignore the 3s
return And(*temp)

def _convert_to_varsPOS(maxterm, variables):
"""
Converts a term in the expansion of a function from binary to it's
variable form (for POS).
"""
temp = []
for i, m in enumerate(maxterm):
if m == 1:
temp.append(Not(variables[i]))
elif m == 0:
temp.append(variables[i])
else:
pass  # ignore the 3s
return Or(*temp)

def _simplified_pairs(terms):
"""
Reduces a set of minterms, if possible, to a simplified set of minterms
with one less variable in the terms using QM method.
"""
simplified_terms = []
todo = list(range(len(terms)))
for i, ti in enumerate(terms[:-1]):
for j_i, tj in enumerate(terms[(i + 1):]):
index = _check_pair(ti, tj)
if index != -1:
todo[i] = todo[j_i + i + 1] = None
newterm = ti[:]
newterm[index] = 3
if newterm not in simplified_terms:
simplified_terms.append(newterm)
simplified_terms.extend(
[terms[i] for i in [_ for _ in todo if _ is not None]])
return simplified_terms

def _compare_term(minterm, term):
"""
Return True if a binary term is satisfied by the given term. Used
for recognizing prime implicants.
"""
for i, x in enumerate(term):
if x != 3 and x != minterm[i]:
return False
return True

def _rem_redundancy(l1, terms):
"""
After the truth table has been sufficiently simplified, use the prime
implicant table method to recognize and eliminate redundant pairs,
and return the essential arguments.
"""
essential = []
for x in terms:
temporary = []
for y in l1:
if _compare_term(x, y):
temporary.append(y)
if len(temporary) == 1:
if temporary[0] not in essential:
essential.append(temporary[0])
for x in terms:
for y in essential:
if _compare_term(x, y):
break
else:
for z in l1:
if _compare_term(x, z):
if z not in essential:
essential.append(z)
break

return essential

[docs]def SOPform(variables, minterms, dontcares=None):
"""
The SOPform function uses simplified_pairs and a redundant group-
eliminating algorithm to convert the list of all input combos that
generate '1' (the minterms) into the smallest Sum of Products form.

The variables must be given as the first argument.

Return a logical Or function (i.e., the "sum of products" or "SOP"
form) that gives the desired outcome. If there are inputs that can
be ignored, pass them as a list, too.

The result will be one of the (perhaps many) functions that satisfy
the conditions.

Examples
========

>>> from sympy.logic import SOPform
>>> minterms = [[0, 0, 0, 1], [0, 0, 1, 1],
...             [0, 1, 1, 1], [1, 0, 1, 1], [1, 1, 1, 1]]
>>> dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
>>> SOPform(['w','x','y','z'], minterms, dontcares)
Or(And(Not(w), z), And(y, z))

References
==========

.. [1] en.wikipedia.org/wiki/Quine-McCluskey_algorithm

"""
from sympy.core.symbol import Symbol

variables = [sympify(v) for v in variables]
if minterms == []:
return false

minterms = [list(i) for i in minterms]
dontcares = [list(i) for i in (dontcares or [])]
for d in dontcares:
if d in minterms:
raise ValueError('%s in minterms is also in dontcares' % d)

old = None
new = minterms + dontcares
while new != old:
old = new
new = _simplified_pairs(old)
essential = _rem_redundancy(new, minterms)
return Or(*[_convert_to_varsSOP(x, variables) for x in essential])

[docs]def POSform(variables, minterms, dontcares=None):
"""
The POSform function uses simplified_pairs and a redundant-group
eliminating algorithm to convert the list of all input combinations
that generate '1' (the minterms) into the smallest Product of Sums form.

The variables must be given as the first argument.

Return a logical And function (i.e., the "product of sums" or "POS"
form) that gives the desired outcome. If there are inputs that can
be ignored, pass them as a list, too.

The result will be one of the (perhaps many) functions that satisfy
the conditions.

Examples
========

>>> from sympy.logic import POSform
>>> minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1],
...             [1, 0, 1, 1], [1, 1, 1, 1]]
>>> dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
>>> POSform(['w','x','y','z'], minterms, dontcares)
And(Or(Not(w), y), z)

References
==========

.. [1] en.wikipedia.org/wiki/Quine-McCluskey_algorithm

"""
from sympy.core.symbol import Symbol

variables = [sympify(v) for v in variables]
if minterms == []:
return false

minterms = [list(i) for i in minterms]
dontcares = [list(i) for i in (dontcares or [])]
for d in dontcares:
if d in minterms:
raise ValueError('%s in minterms is also in dontcares' % d)

maxterms = []
for t in product([0, 1], repeat=len(variables)):
t = list(t)
if (t not in minterms) and (t not in dontcares):
maxterms.append(t)
old = None
new = maxterms + dontcares
while new != old:
old = new
new = _simplified_pairs(old)
essential = _rem_redundancy(new, maxterms)
return And(*[_convert_to_varsPOS(x, variables) for x in essential])

def _find_predicates(expr):
"""Helper to find logical predicates in BooleanFunctions.

A logical predicate is defined here as anything within a BooleanFunction
that is not a BooleanFunction itself.

"""
if not isinstance(expr, BooleanFunction):
return set([expr])
return set.union(*(_find_predicates(i) for i in expr.args))

[docs]def simplify_logic(expr, form=None, deep=True):
"""
This function simplifies a boolean function to its simplified version
in SOP or POS form. The return type is an Or or And object in SymPy.

Parameters
==========

expr : string or boolean expression
form : string ('cnf' or 'dnf') or None (default).
If 'cnf' or 'dnf', the simplest expression in the corresponding
normal form is returned; if None, the answer is returned
according to the form with fewest args (in CNF by default).
deep : boolean (default True)
indicates whether to recursively simplify any
non-boolean functions contained within the input.

Examples
========

>>> from sympy.logic import simplify_logic
>>> from sympy.abc import x, y, z
>>> from sympy import S
>>> b = '(~x & ~y & ~z) | ( ~x & ~y & z)'
>>> simplify_logic(b)
And(Not(x), Not(y))

>>> S(b)
Or(And(Not(x), Not(y), Not(z)), And(Not(x), Not(y), z))
>>> simplify_logic(_)
And(Not(x), Not(y))

"""

if form == 'cnf' or form == 'dnf' or form is None:
expr = sympify(expr)
if not isinstance(expr, BooleanFunction):
return expr
variables = _find_predicates(expr)
truthtable = []
for t in product([0, 1], repeat=len(variables)):
t = list(t)
if expr.xreplace(dict(zip(variables, t))) == True:
truthtable.append(t)
if deep:
from sympy.simplify.simplify import simplify
variables = [simplify(v) for v in variables]
if form == 'dnf' or \
(form is None and len(truthtable) >= (2 ** (len(variables) - 1))):
return SOPform(variables, truthtable)
elif form == 'cnf' or form is None:
return POSform(variables, truthtable)
else:
raise ValueError("form can be cnf or dnf only")

def _finger(eq):
"""
Assign a 5-item fingerprint to each symbol in the equation:
[
# of times it appeared as a Symbol,
# of times it appeared as a Not(symbol),
# of times it appeared as a Symbol in an And or Or,
# of times it appeared as a Not(Symbol) in an And or Or,
sum of the number of arguments with which it appeared,
counting Symbol as 1 and Not(Symbol) as 2
]

>>> from sympy.logic.boolalg import _finger as finger
>>> from sympy import And, Or, Not
>>> from sympy.abc import a, b, x, y
>>> eq = Or(And(Not(y), a), And(Not(y), b), And(x, y))
>>> dict(finger(eq))
{(0, 0, 1, 0, 2): [x], (0, 0, 1, 0, 3): [a, b], (0, 0, 1, 2, 8): [y]}

So y and x have unique fingerprints, but a and b do not.
"""
f = eq.free_symbols
d = dict(list(zip(f, [[0] * 5 for fi in f])))
for a in eq.args:
if a.is_Symbol:
d[a][0] += 1
elif a.is_Not:
d[a.args[0]][1] += 1
else:
o = len(a.args) + sum(ai.func is Not for ai in a.args)
for ai in a.args:
if ai.is_Symbol:
d[ai][2] += 1
d[ai][-1] += o
else:
d[ai.args[0]][3] += 1
d[ai.args[0]][-1] += o
inv = defaultdict(list)
for k, v in ordered(iter(d.items())):
inv[tuple(v)].append(k)
return inv

[docs]def bool_map(bool1, bool2):
"""
Return the simplified version of bool1, and the mapping of variables
that makes the two expressions bool1 and bool2 represent the same
logical behaviour for some correspondence between the variables
of each.
If more than one mappings of this sort exist, one of them
is returned.
For example, And(x, y) is logically equivalent to And(a, b) for
the mapping {x: a, y:b} or {x: b, y:a}.
If no such mapping exists, return False.

Examples
========

>>> from sympy import SOPform, bool_map, Or, And, Not, Xor
>>> from sympy.abc import w, x, y, z, a, b, c, d
>>> function1 = SOPform(['x','z','y'],[[1, 0, 1], [0, 0, 1]])
>>> function2 = SOPform(['a','b','c'],[[1, 0, 1], [1, 0, 0]])
>>> bool_map(function1, function2)
(And(Not(z), y), {y: a, z: b})

The results are not necessarily unique, but they are canonical. Here,
(w, z) could be (a, d) or (d, a):

>>> eq =  Or(And(Not(y), w), And(Not(y), z), And(x, y))
>>> eq2 = Or(And(Not(c), a), And(Not(c), d), And(b, c))
>>> bool_map(eq, eq2)
(Or(And(Not(y), w), And(Not(y), z), And(x, y)), {w: a, x: b, y: c, z: d})
>>> eq = And(Xor(a, b), c, And(c,d))
>>> bool_map(eq, eq.subs(c, x))
(And(Or(Not(a), Not(b)), Or(a, b), c, d), {a: a, b: b, c: d, d: x})

"""

def match(function1, function2):
"""Return the mapping that equates variables between two
simplified boolean expressions if possible.

By "simplified" we mean that a function has been denested
and is either an And (or an Or) whose arguments are either
symbols (x), negated symbols (Not(x)), or Or (or an And) whose
arguments are only symbols or negated symbols. For example,
And(x, Not(y), Or(w, Not(z))).

Basic.match is not robust enough (see issue 4835) so this is
a workaround that is valid for simplified boolean expressions
"""

# do some quick checks
if function1.__class__ != function2.__class__:
return None
if len(function1.args) != len(function2.args):
return None
if function1.is_Symbol:
return {function1: function2}

# get the fingerprint dictionaries
f1 = _finger(function1)
f2 = _finger(function2)

# more quick checks
if len(f1) != len(f2):
return False

# assemble the match dictionary if possible
matchdict = {}
for k in f1.keys():
if k not in f2:
return False
if len(f1[k]) != len(f2[k]):
return False
for i, x in enumerate(f1[k]):
matchdict[x] = f2[k][i]
return matchdict

a = simplify_logic(bool1)
b = simplify_logic(bool2)
m = match(a, b)
if m:
return a, m
return m is not None

@deprecated(
def bool_equal(bool1, bool2, info=False):
"""Return True if the two expressions represent the same logical
behaviour for some correspondence between the variables of each
(which may be different). For example, And(x, y) is logically
equivalent to And(a, b) for {x: a, y: b} (or vice versa). If the
mapping is desired, then set info to True and the simplified
form of the functions and mapping of variables will be returned.
"""

mapping = bool_map(bool1, bool2)
if not mapping:
return False
if info:
return mapping
return True