propcalc.py

r"""
Module that creates formulas of propositional calculus

Formulas consist of the following operators:

* ``&`` -- and
* ``|`` -- or
* ``~`` -- not
* ``^`` -- xor
* ``->`` -- if-then
* ``<->`` -- if and only if

Operators can be applied to variables that consist of a leading letter and
trailing underscores and alphanumerics.  Parentheses may be used to explicitly
show order of operation.

AUTHORS:

- Chris Gorecki (2006): initial version, propcalc, boolformula,
  logictable, logicparser, booleval

- Michael Greenberg -- boolopt

- Paul Scurek (2013-08-05): updated docstring formatting

EXAMPLES:

We can create boolean formulas in different ways::

    sage: import sage.logic.propcalc as propcalc
    sage: f = propcalc.formula("a&((b|c)^a->c)<->b")
    sage: g = propcalc.formula("boolean<->algebra")
    sage: (f&~g).ifthen(f)
    ((a&((b|c)^a->c)<->b)&(~(boolean<->algebra)))->(a&((b|c)^a->c)<->b)

We can create a truth table from a formula::

    sage: f.truthtable()
    a      b      c      value
    False  False  False  True
    False  False  True   True
    False  True   False  False
    False  True   True   False
    True   False  False  True
    True   False  True   False
    True   True   False  True
    True   True   True   True
    sage: f.truthtable(end=3)
    a      b      c      value
    False  False  False  True
    False  False  True   True
    False  True   False  False
    sage: f.truthtable(start=4)
    a      b      c      value
    True   False  False  True
    True   False  True   False
    True   True   False  True
    True   True   True   True
    sage: propcalc.formula("a").truthtable()
    a      value
    False  False
    True   True

Now we can evaluate the formula for a given set of input::

    sage: f.evaluate({'a':True, 'b':False, 'c':True})
    False
    sage: f.evaluate({'a':False, 'b':False, 'c':True})
    True

And we can convert a boolean formula to conjunctive normal form::

    sage: f.convert_cnf_table()
    sage: f
    (a|~b|c)&(a|~b|~c)&(~a|b|~c)
    sage: f.convert_cnf_recur()
    sage: f
    (a|~b|c)&(a|~b|~c)&(~a|b|~c)

Or determine if an expression is satisfiable, a contradiction, or a tautology::

    sage: f = propcalc.formula("a|b")
    sage: f.is_satisfiable()
    True
    sage: f = f & ~f
    sage: f.is_satisfiable()
    False
    sage: f.is_contradiction()
    True
    sage: f = f | ~f
    sage: f.is_tautology()
    True

The equality operator compares semantic equivalence::

    sage: f = propcalc.formula("(a|b)&c")
    sage: g = propcalc.formula("c&(b|a)")
    sage: f == g
    True
    sage: g = propcalc.formula("a|b&c")
    sage: f == g
    False

It is an error to create a formula with bad syntax::

    sage: propcalc.formula("")
    Traceback (most recent call last):
    ...
    SyntaxError: malformed statement
    sage: propcalc.formula("a&b~(c|(d)")
    Traceback (most recent call last):
    ...
    SyntaxError: malformed statement
    sage: propcalc.formula("a&&b")
    Traceback (most recent call last):
    ...
    SyntaxError: malformed statement
    sage: propcalc.formula("a&b a")
    Traceback (most recent call last):
    ...
    SyntaxError: malformed statement

    It is also an error to not abide by the naming conventions.
    sage: propcalc.formula("~a&9b")
    Traceback (most recent call last):
    ...
    NameError: invalid variable name 9b: identifiers must begin with a letter and contain only alphanumerics and underscores
"""
#*****************************************************************************
#       Copyright (C) 2006 William Stein <[email protected]>
#       Copyright (C) 2006 Chris Gorecki <[email protected]>
#       Copyright (C) 2013 Paul Scurek <[email protected]>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#  as published by the Free Software Foundation; either version 2 of
#  the License, or (at your option) any later version.
#                  http://www.gnu.org/licenses/
#*****************************************************************************

### TODO:
### converts (cnf) returns w/o change

import boolformula
import logicparser


def formula(s):
    r"""
    Return an instance of :class:`BooleanFormula`

    INPUT:

    - ``s`` -- a string that contains a logical expression.

    OUTPUT:

    An instance of :class:`BooleanFormula`

    EXAMPLES:

    This example illustrates ways to create a boolean formula.

    ::

        sage: import sage.logic.propcalc as propcalc
        sage: f = propcalc.formula("a&~b|c")
        sage: g = propcalc.formula("a^c<->b")
        sage: f&g|f
        ((a&~b|c)&(a^c<->b))|(a&~b|c)

    We now demonstrate some possible errors.

    ::

        sage: propcalc.formula("((a&b)")
        Traceback (most recent call last):
        ...
        SyntaxError: malformed statement
        sage: propcalc.formula("_a&b")
        Traceback (most recent call last):
        ...
        NameError: invalid variable name _a: identifiers must begin with a letter and contain only alphanumerics and underscores
    """
    try:
        parse_tree, vars_order = logicparser.parse(s)
        f = boolformula.BooleanFormula(s, parse_tree, vars_order)
        f.truthtable(0, 1)
    except (KeyError, RuntimeError, IndexError, SyntaxError):
        msg = "malformed statement"
        raise SyntaxError(msg)
    return f