# Logic Module¶

## Introduction¶

The logic module for SymPy allows to form and manipulate logic expressions using symbolic and Boolean values.

## Forming logical expressions¶

You can build Boolean expressions with the standard python operators & (And), | (Or), ~ (Not):

>>> from sympy import *
>>> x, y = symbols('x,y')
>>> y | (x & y)
Or(And(x, y), y)
>>> x | y
Or(x, y)
>>> ~x
Not(x)


You can also form implications with >> and <<:

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


Like most types in SymPy, Boolean expressions inherit from Basic:

>>> (y & x).subs({x: True, y: True})
True
>>> (x | y).atoms()
set([x, y])


The logic module also includes the following functions to derive boolean expressions from their truth tables-

sympy.logic.boolalg.SOPform(variables, minterms, dontcares=None)[source]

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.

References

 [R216] en.wikipedia.org/wiki/Quine-McCluskey_algorithm

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))

sympy.logic.boolalg.POSform(variables, minterms, dontcares=None)[source]

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.

References

 [R217] en.wikipedia.org/wiki/Quine-McCluskey_algorithm

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)


## Boolean functions¶

class sympy.logic.boolalg.BooleanTrue[source]

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

class sympy.logic.boolalg.BooleanFalse[source]

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

class sympy.logic.boolalg.And[source]

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.

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


Examples

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


Attributes

 nargs
class sympy.logic.boolalg.Or[source]

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.

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


Examples

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


Attributes

 nargs
class sympy.logic.boolalg.Not[source]

Logical Not function (negation)

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

Notes

• De Morgan rules are applied automatically.
• 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


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
>>> Not(And(And(True, x), Or(x, False)))
Not(x)
>>> ~x
Not(x)


Attributes

 nargs
class sympy.logic.boolalg.Xor[source]

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.

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


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
Or(And(Not(x), y), And(Not(y), x))


Attributes

 nargs
class sympy.logic.boolalg.Nand[source]

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)
Or(Not(x), Not(y))


Attributes

 nargs
class sympy.logic.boolalg.Nor[source]

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)
And(Not(x), Not(y))


Attributes

 nargs
class sympy.logic.boolalg.Implies[source]

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.

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


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)


Attributes

 nargs
class sympy.logic.boolalg.Equivalent[source]

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


Attributes

 nargs
class sympy.logic.boolalg.ITE[source]

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)
Or(And(Not(x), z), And(x, y))


Attributes

 nargs

The following functions can be used to handle Conjunctive and Disjunctive Normal forms-

sympy.logic.boolalg.to_cnf(expr, simplify=False)[source]

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)

sympy.logic.boolalg.to_dnf(expr, simplify=False)[source]

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)

sympy.logic.boolalg.is_cnf(expr)[source]

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

sympy.logic.boolalg.is_dnf(expr)[source]

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


## Simplification and equivalence-testing¶

sympy.logic.boolalg.simplify_logic(expr, form=None, deep=True)[source]

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. The input can be a string or a boolean expression. form can be ‘cnf’ or ‘dnf’ or None. If its ‘cnf’ or ‘dnf’ the simplest expression in the corresponding normal form is returned. If form is None, the answer is returned according to the form with lesser number of args (CNF by default) The optional parameter deep 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))


SymPy’s simplify() function can also be used to simplify logic expressions to their simplest forms.

sympy.logic.boolalg.bool_map(bool1, bool2)[source]

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})


## Inference¶

This module implements some inference routines in propositional logic.

The function satisfiable will test that a given Boolean expression is satisfiable, that is, you can assign values to the variables to make the sentence $$True$$.

For example, the expression x & ~x is not satisfiable, since there are no values for x that make this sentence True. On the other hand, (x | y) & (x | ~y) & (~x | y) is satisfiable with both x and y being True.

>>> from sympy.logic.inference import satisfiable
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> y = Symbol('y')
>>> satisfiable(x & ~x)
False
>>> satisfiable((x | y) & (x | ~y) & (~x | y))
{x: True, y: True}


As you see, when a sentence is satisfiable, it returns a model that makes that sentence True. If it is not satisfiable it will return False.

sympy.logic.inference.satisfiable(expr, algorithm='dpll2')[source]

Check satisfiability of a propositional sentence. Returns a model when it succeeds

Examples:

>>> from sympy.abc import A, B
>>> from sympy.logic.inference import satisfiable
>>> satisfiable(A & ~B)
{A: True, B: False}
>>> satisfiable(A & ~A)
False