Logic Module ============ .. module:: sympy.logic 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 ``&`` (:class:`And`), ``|`` (:class:`Or`), ``~`` (:class:`Not`):: >>> from sympy import * >>> x, y = symbols('x,y') >>> y | (x & y) y | (x & y) >>> x | y x | y >>> ~x ~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 :class:`Basic`:: >>> (y & x).subs({x: True, y: True}) True >>> (x | y).atoms() {x, y} The logic module also includes the following functions to derive boolean expressions from their truth tables- .. autofunction:: sympy.logic.boolalg.SOPform .. autofunction:: sympy.logic.boolalg.POSform Boolean functions ----------------- .. autoclass:: sympy.logic.boolalg.BooleanTrue .. autoclass:: sympy.logic.boolalg.BooleanFalse .. autoclass:: sympy.logic.boolalg.And .. autoclass:: sympy.logic.boolalg.Or .. autoclass:: sympy.logic.boolalg.Not .. autoclass:: sympy.logic.boolalg.Xor .. autoclass:: sympy.logic.boolalg.Nand .. autoclass:: sympy.logic.boolalg.Nor .. autoclass:: sympy.logic.boolalg.Implies .. autoclass:: sympy.logic.boolalg.Equivalent .. autoclass:: sympy.logic.boolalg.ITE The following functions can be used to handle Conjunctive and Disjunctive Normal forms- .. autofunction:: sympy.logic.boolalg.to_cnf .. autofunction:: sympy.logic.boolalg.to_dnf .. autofunction:: sympy.logic.boolalg.is_cnf .. autofunction:: sympy.logic.boolalg.is_dnf Simplification and equivalence-testing -------------------------------------- .. autofunction:: sympy.logic.boolalg.simplify_logic SymPy's simplify() function can also be used to simplify logic expressions to their simplest forms. .. autofunction:: sympy.logic.boolalg.bool_map Inference --------- .. module:: sympy.logic.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``. .. autofunction:: sympy.logic.inference.satisfiable .. TODO: write about CNF file format