Source code for sympy.logic.boolalg

"""Boolean algebra module for SymPy"""
from sympy.core.basic import Basic
from sympy.core.operations import LatticeOp
from sympy.core.function import Application, sympify

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

__slots__ = []

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

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

def __invert__(self):
return Not(self)

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

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

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

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])

[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
>>> x & y
And(x, y)
"""
zero = False
identity = True

[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.
"""
zero = True
identity = False

[docs]class Xor(BooleanFunction):
"""
Logical XOR (exclusive OR) function.
"""
@classmethod
def eval(cls, *args):
"""
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
>>> Xor(True, False)
True
>>> Xor(True, True)
False

>>> Xor(True, False, True, True, False)
True
>>> Xor(True, False, True, False)
False
"""
if not args: return False
args = list(args)
A = args.pop()
while args:
B = args.pop()
A = Or(And(A, Not(B)), And(Not(A), B))
return A

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

Note: De Morgan rules applied automatically
"""

is_Not = True

@classmethod
def eval(cls, *args):
"""
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
>>> Not(True)
False
>>> Not(False)
True
>>> Not(And(True, False))
True
>>> Not(Or(True, False))
False

If multiple statements are given, returns an array of each result

>>> Not(True, False)
[False, True]
>>> Not(True and False, True or False, True)
[True, False, False]

>>> Not(And(And(True, x), Or(x, False)))
Not(x)
"""
if len(args) > 1:
return list(map(cls, args))
arg = args[0]
if type(arg) is bool:
return not arg
# apply De Morgan Rules
if arg.func is And:
return Or(*[Not(a) for a in arg.args])
if arg.func is Or:
return And(*[Not(a) for a in arg.args])
if arg.func is Not:
return arg.args[0]

[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.
"""
@classmethod
def eval(cls, *args):
"""
Logical NAND function.

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

Examples
========

>>> from sympy.logic.boolalg import Nand
>>> Nand(False, True)
True
>>> Nand(True, True)
False
"""
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.
"""
@classmethod
def eval(cls, *args):
"""
Logical NOR function.

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

Examples
========

>>> from sympy.logic.boolalg import Nor
>>> Nor(True, False)
False
>>> Nor(True, True)
False
>>> Nor(False, True)
False
>>> Nor(False, False)
True
"""
return Not(Or(*args))

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

A implies B is equivalent to !A v B
"""
@classmethod
def eval(cls, *args):
"""
Logical implication.

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
>>> Implies(True, False)
False
>>> Implies(False, False)
True
>>> Implies(True, True)
True
>>> Implies(False, True)
True
"""
try:
A, B = args
except ValueError:
raise ValueError("%d operand(s) used for an Implies (pairs are required): %s" % (len(args), str(args)))
if A is True or A is False or B is True or B is False:
return Or(Not(A), B)
else:
return Basic.__new__(cls, *args)

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

Equivalent(A, B) is True if and only if A and B are both True or both False
"""
@classmethod
def eval(cls, *args):
"""
Equivalence relation.

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
>>> Equivalent(False, False, False)
True
>>> Equivalent(True, False, False)
False
>>> Equivalent(x, And(x, True))
True

"""

argset = set(args)
if len(argset) <= 1:
return True
if True in argset:
return And(*argset)
if False in argset:
return Nor(*argset)
return Basic.__new__(cls, *set(args))

[docs]class ITE(BooleanFunction):
"""
If then else clause.
"""
@classmethod
def eval(cls, *args):
"""
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
>>> x = True
>>> y = False
>>> z = True
>>> ITE(x,y,z)
False
>>> ITE(Or(x, y), And(x, z), Xor(z, x))
True
"""
args = list(args)
if len(args) == 3:
return Or(And(args[0], args[1]), And(Not(args[0]), args[2]))
raise ValueError("ITE expects 3 arguments, but got %d: %s" % (len(args), str(args)))

### end class definitions. Some useful methods

def fuzzy_not(arg):
"""
Not in fuzzy logic

Will return Not if arg is a boolean value, and None if argument
is None.

Examples:

>>> from sympy.logic.boolalg import fuzzy_not
>>> fuzzy_not(True)
False
>>> fuzzy_not(None)
>>> fuzzy_not(False)
True

"""
if arg is None:
return
return not arg

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)))
"""
if expr.func is Or:
for arg in expr.args:
if arg.func is And:
conj = arg
break
else:
return expr
rest = Or(*[a for a in expr.args if a is not conj])
return And(*list(map(distribute_and_over_or,
[Or(c, rest) for c in conj.args])))
elif expr.func is And:
return And(*list(map(distribute_and_over_or, expr.args)))
else:
return expr

[docs]def to_cnf(expr):
"""
Convert a propositional logical sentence s to conjunctive normal form.
That is, of the form ((A | ~B | ...) & (B | C | ...) & ...)

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)))

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

expr = sympify(expr)
expr = eliminate_implications(expr)
return distribute_and_over_or(expr)

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

"""
expr = sympify(expr)

# Special case of a single disjunction
if expr.func is Or:
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 And:
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 Or:
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)))
"""
expr = sympify(expr)
if expr.is_Atom:
return expr     ## (Atoms are unchanged.)
args = list(map(eliminate_implications, expr.args))
if expr.func is Implies:
a, b = args[0], args[-1]
return (~a) | b
elif expr.func is Equivalent:
a, b = args[0], args[-1]
return (a | Not(b)) & (b | Not(a))
else:
return expr.func(*args)

def compile_rule(s):
"""
Transforms a rule into a sympy expression
A rule is a string of the form "symbol1 & symbol2 | ..."
See sympy.assumptions.known_facts for examples of rules

TODO: can this be replaced by sympify ?

Examples
========

>>> from sympy.logic.boolalg import compile_rule
>>> compile_rule('A & B')
And(A, B)
>>> compile_rule('(~B & ~C)|A')
Or(A, And(Not(B), Not(C)))
"""
import re
from sympy.core import Symbol
return eval(re.sub(r'([a-zA-Z0-9_.]+)', r'Symbol("\1")', s), {'Symbol' : Symbol})

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(range(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]