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.numbers import Number
from sympy.core.operations import LatticeOp
from sympy.core.function import Application, Derivative
from sympy.core.compatibility import ordered, range, with_metaclass, as_int
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):
        """Overloading for & operator"""
        return And(self, other)

    __rand__ = __and__

    def __or__(self, other):
        """Overloading for |"""
        return Or(self, other)

    __ror__ = __or__

    def __invert__(self):
        """Overloading for ~"""
        return Not(self)

    def __rshift__(self, other):
        """Overloading for >>"""
        return Implies(self, other)

    def __lshift__(self, other):
        """Overloading for <<"""
        return Implies(other, self)

    __rrshift__ = __lshift__
    __rlshift__ = __rshift__

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

    __rxor__ = __xor__

    def equals(self, other):
        """
        Returns True 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
        from sympy.core.relational import Relational

        if self.has(Relational) or other.has(Relational):
            raise NotImplementedError('handling of relationals')
        return self.atoms() == other.atoms() and \
                not satisfiable(Not(Equivalent(self, other)))


class BooleanAtom(Boolean):
    """
    Base class of BooleanTrue and BooleanFalse.
    """
    is_Boolean = True
    is_Atom = True
    _op_priority = 11  # higher than Expr

    def simplify(self, *a, **kw):
        return self

    def expand(self, *a, **kw):
        return self

    @property
    def canonical(self):
        return self

    def _noop(self, other=None):
        raise TypeError('BooleanAtom not allowed in this context.')

    __add__ = _noop
    __radd__ = _noop
    __sub__ = _noop
    __rsub__ = _noop
    __mul__ = _noop
    __rmul__ = _noop
    __pow__ = _noop
    __rpow__ = _noop
    __rdiv__ = _noop
    __truediv__ = _noop
    __div__ = _noop
    __rtruediv__ = _noop
    __mod__ = _noop
    __rmod__ = _noop
    _eval_power = _noop


[docs]class BooleanTrue(with_metaclass(Singleton, BooleanAtom)): """ SymPy version of True, a singleton that 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. Notes ===== 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 in doubt, use ``True``. "``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:`` Examples ======== >>> from sympy import sympify, true, Or >>> sympify(True) True >>> ~true False >>> ~True -2 >>> Or(True, False) True See Also ======== 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, a singleton that 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. Notes ====== See note in :py:class`sympy.logic.boolalg.BooleanTrue` Examples ======== >>> from sympy import sympify, false, Or, true >>> sympify(False) False >>> false >> false True >>> False >> False 0 >>> Or(True, False) True See Also ======== 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 _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 argset.add(a) else: argset.add(arg) 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 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 = [] rel = [] for x in reversed(list(args)): if isinstance(x, Number) or x in (0, 1): newargs.append(True if x else False) continue if x.is_Relational: c = x.canonical if c in rel: continue nc = (~c).canonical if any(r == nc for r in rel): return [S.false] rel.append(c) 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() Interval.open(-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 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 = [] rel = [] for x in args: if isinstance(x, Number) or x in (0, 1): newargs.append(True if x else False) continue if x.is_Relational: c = x.canonical if c in rel: continue nc = (~c).canonical if any(r == nc for r in rel): return [S.true] rel.append(c) 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() Union(Interval.open(-oo, -2), Interval.open(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))) ~x >>> ~x ~x >>> Not(And(Or(A, B), Or(~A, ~B))) ~((A | B) & (~A | ~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): from sympy import ( Equality, GreaterThan, LessThan, StrictGreaterThan, StrictLessThan, Unequality) 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, Equality): return Unequality(*arg.args) if isinstance(arg, Unequality): return Equality(*arg.args) if isinstance(arg, StrictLessThan): return GreaterThan(*arg.args) if isinstance(arg, StrictGreaterThan): return LessThan(*arg.args) if isinstance(arg, LessThan): return StrictGreaterThan(*arg.args) if isinstance(arg, GreaterThan): return 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() Interval(-oo, 0) """ if len(self.free_symbols) == 1: return self.args[0].as_set().complement(S.Reals) 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 range(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: argset.add(arg) rel = [(r, r.canonical, (~r).canonical) for r in argset if r.is_Relational] odd = False # is number of complimentary pairs odd? start 0 -> False remove = [] for i, (r, c, nc) in enumerate(rel): for j in range(i + 1, len(rel)): rj, cj = rel[j][:2] if cj == nc: odd = ~odd break elif cj == c: break else: continue remove.append((r, rj)) if odd: argset.remove(true) if true in argset else argset.add(true) for a, b in remove: argset.remove(a) argset.remove(b) 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 range(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) ~(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) ~(x | y) """ @classmethod def eval(cls, *args): return Not(Or(*args))
class Xnor(BooleanFunction): """ Logical XNOR function. Returns False if an odd number of the arguments are True and the rest are False. Returns True if an even number of the arguments are True and the rest are False. Examples ======== >>> from sympy.logic.boolalg import Xnor >>> from sympy import symbols >>> x, y = symbols('x y') >>> Xnor(True, False) False >>> Xnor(True, True) True >>> Xnor(True, False, True, True, False) False >>> Xnor(True, False, True, False) True """ @classmethod def eval(cls, *args): return Not(Xor(*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) elif A == B: return S.true elif A.is_Relational and B.is_Relational: if A.canonical == B.canonical: return S.true if (~A).canonical == B.canonical: return 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): from sympy.core.relational import Relational 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 argset.discard(x) argset.add(True if x else False) rel = [] for r in argset: if isinstance(r, Relational): rel.append((r, r.canonical, (~r).canonical)) remove = [] for i, (r, c, nc) in enumerate(rel): for j in range(i + 1, len(rel)): rj, cj = rel[j][:2] if cj == nc: return false elif cj == c: remove.append((r, rj)) break for a, b in remove: argset.remove(a) argset.remove(b) argset.add(True) if len(argset) <= 1: return true if True in argset: argset.discard(True) return And(*argset) if False in argset: argset.discard(False) 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 else: if b == True and c == False: return a if b == False and c == True: return Not(a) def to_nnf(self, simplify=True): a, b, c = self.args return And._to_nnf(Or(~a, b), Or(a, c), simplify=simplify) def _eval_derivative(self, x): return self.func(self.args[0], *[a.diff(x) for a in self.args[1:]]) # the diff method below is copied from Expr class def diff(self, *symbols, **assumptions): new_symbols = list(map(sympify, symbols)) # e.g. x, 2, y, z assumptions.setdefault("evaluate", True) return Derivative(self, *new_symbols, **assumptions)
### 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({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({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)))) (A | ~B) & (A | ~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)) (B & C) | (C & ~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 simplify 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))) (A | B) & (~C | ~D) >>> to_nnf(Equivalent(A >> B, B >> A)) (A | ~B | (A & ~B)) & (B | ~A | (B & ~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) (D | ~A) & (D | ~B) >>> to_cnf((A | B) & (A | ~A), True) 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)) (A & B) | (B & C) >>> to_dnf((A & B) | (A & ~B) | (B & C) | (~B & C), True) 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)) B | ~A >>> eliminate_implications(Equivalent(A, B)) (A | ~B) & (B | ~A) >>> eliminate_implications(Equivalent(A, B, C)) (A | ~C) & (B | ~A) & (C | ~B) """ return to_nnf(expr) 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) 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]) == [{1, 2}, {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] def term_to_integer(term): """ Return an integer corresponding to the base-2 digits given by ``term``. Parameters ========== term : a string or list of ones and zeros Examples ======== >>> from sympy.logic.boolalg import term_to_integer >>> term_to_integer([1, 0, 0]) 4 >>> term_to_integer('100') 4 """ return int(''.join(list(map(str, list(term)))), 2) def integer_to_term(k, n_bits=None): """ Return a list of the base-2 digits in the integer, ``k``. Parameters ========== k : int n_bits : int If ``n_bits`` is given and the number of digits in the binary representation of ``k`` is smaller than ``n_bits`` then left-pad the list with 0s. Examples ======== >>> from sympy.logic.boolalg import integer_to_term >>> integer_to_term(4) [1, 0, 0] >>> integer_to_term(4, 6) [0, 0, 0, 1, 0, 0] """ s = '{0:0{1}b}'.format(abs(as_int(k)), as_int(abs(n_bits or 0))) return list(map(int, s)) def truth_table(expr, variables, input=True): """ Return a generator of all possible configurations of the input variables, and the result of the boolean expression for those values. Parameters ========== expr : string or boolean expression variables : list of variables input : boolean (default True) indicates whether to return the input combinations. Examples ======== >>> from sympy.logic.boolalg import truth_table >>> from sympy.abc import x,y >>> table = truth_table(x >> y, [x, y]) >>> for t in table: ... print('{0} -> {1}'.format(*t)) [0, 0] -> True [0, 1] -> True [1, 0] -> False [1, 1] -> True >>> table = truth_table(x | y, [x, y]) >>> list(table) [([0, 0], False), ([0, 1], True), ([1, 0], True), ([1, 1], True)] If input is false, truth_table returns only a list of truth values. In this case, the corresponding input values of variables can be deduced from the index of a given output. >>> from sympy.logic.boolalg import integer_to_term >>> vars = [y, x] >>> values = truth_table(x >> y, vars, input=False) >>> values = list(values) >>> values [True, False, True, True] >>> for i, value in enumerate(values): ... print('{0} -> {1}'.format(list(zip( ... vars, integer_to_term(i, len(vars)))), value)) [(y, 0), (x, 0)] -> True [(y, 0), (x, 1)] -> False [(y, 1), (x, 0)] -> True [(y, 1), (x, 1)] -> True """ variables = [sympify(v) for v in variables] expr = sympify(expr) if not isinstance(expr, BooleanFunction) and not is_literal(expr): return table = product([0, 1], repeat=len(variables)) for term in table: term = list(term) value = expr.xreplace(dict(zip(variables, term))) if input: yield term, value else: yield value 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 def _convert_to_varsSOP(minterm, variables): """ 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 >>> from sympy import symbols >>> w, x, y, z = symbols('w x y z') >>> 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) (y & z) | (z & ~w) References ========== .. [1] en.wikipedia.org/wiki/Quine-McCluskey_algorithm """ 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 >>> from sympy import symbols >>> w, x, y, z = symbols('w x y z') >>> 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) z & (y | ~w) References ========== .. [1] en.wikipedia.org/wiki/Quine-McCluskey_algorithm """ 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 {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) ~x & ~y >>> S(b) (z & ~x & ~y) | (~x & ~y & ~z) >>> simplify_logic(_) ~x & ~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) (y & ~z, {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) ((x & y) | (w & ~y) | (z & ~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)) (c & d & (a | b) & (~a | ~b), {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