logic.py

                    *[standardize_variables(a, dic) for a in sentence.args])

standardize_variables.counter = itertools.count()

# ______________________________________________________________________________


class FolKB(KB):

    """A knowledge base consisting of first-order definite clauses.
    >>> kb0 = FolKB([expr('Farmer(Mac)'), expr('Rabbit(Pete)'),
    ...              expr('(Rabbit(r) & Farmer(f)) ==> Hates(f, r)')])
    >>> kb0.tell(expr('Rabbit(Flopsie)'))
    >>> kb0.retract(expr('Rabbit(Pete)'))
    >>> kb0.ask(expr('Hates(Mac, x)'))[x]
    Flopsie
    >>> kb0.ask(expr('Wife(Pete, x)'))
    False
    """

    def __init__(self, initial_clauses=[]):
        self.clauses = []  # inefficient: no indexing
        for clause in initial_clauses:
            self.tell(clause)

    def tell(self, sentence):
        if is_definite_clause(sentence):
            self.clauses.append(sentence)
        else:
            raise Exception("Not a definite clause: {}".format(sentence))

    def ask_generator(self, query):
        return fol_bc_ask(self, query)

    def retract(self, sentence):
        self.clauses.remove(sentence)

    def fetch_rules_for_goal(self, goal):
        return self.clauses


test_kb = FolKB(
    list(map(expr, ['Farmer(Mac)',
                    'Rabbit(Pete)',
                    'Mother(MrsMac, Mac)',
                    'Mother(MrsRabbit, Pete)',
                    '(Rabbit(r) & Farmer(f)) ==> Hates(f, r)',
                    '(Mother(m, c)) ==> Loves(m, c)',
                    '(Mother(m, r) & Rabbit(r)) ==> Rabbit(m)',
                    '(Farmer(f)) ==> Human(f)',
                    # Note that this order of conjuncts
                    # would result in infinite recursion:
                    # '(Human(h) & Mother(m, h)) ==> Human(m)'
                    '(Mother(m, h) & Human(h)) ==> Human(m)'
                    ]))
)

crime_kb = FolKB(
    list(map(expr,
             ['(American(x) & Weapon(y) & Sells(x, y, z) & Hostile(z)) ==> Criminal(x)',  # noqa
              'Owns(Nono, M1)',
              'Missile(M1)',
              '(Missile(x) & Owns(Nono, x)) ==> Sells(West, x, Nono)',
              'Missile(x) ==> Weapon(x)',
              'Enemy(x, America) ==> Hostile(x)',
              'American(West)',
              'Enemy(Nono, America)'
              ]))
)


def fol_bc_ask(KB, query):
    """A simple backward-chaining algorithm for first-order logic. [Fig. 9.6]
    KB should be an instance of FolKB, and goals a list of literals.
    >>> test_ask('Farmer(x)')
    ['{x: Mac}']
    >>> test_ask('Human(x)')
    ['{x: Mac}', '{x: MrsMac}']
    >>> test_ask('Hates(x, y)')
    ['{x: Mac, y: MrsRabbit}', '{x: Mac, y: Pete}']
    >>> test_ask('Loves(x, y)')
    ['{x: MrsMac, y: Mac}', '{x: MrsRabbit, y: Pete}']
    >>> test_ask('Rabbit(x)')
    ['{x: MrsRabbit}', '{x: Pete}']
    >>> test_ask('Criminal(x)', crime_kb)
    ['{x: West}']
    """
    return fol_bc_or(KB, query, {})


def fol_bc_or(KB, goal, theta):
    for rule in KB.fetch_rules_for_goal(goal):
        lhs, rhs = parse_definite_clause(standardize_variables(rule))
        for theta1 in fol_bc_and(KB, lhs, unify(rhs, goal, theta)):
            yield theta1


def fol_bc_and(KB, goals, theta):
    if theta is None:
        pass
    elif not goals:
        yield theta
    else:
        first, rest = goals[0], goals[1:]
        for theta1 in fol_bc_or(KB, subst(theta, first), theta):
            for theta2 in fol_bc_and(KB, rest, theta1):
                yield theta2

# ______________________________________________________________________________

# Example application (not in the book).
# You can use the Expr class to do symbolic differentiation.  This used to be
# a part of AI; now it is considered a separate field, Symbolic Algebra.


def diff(y, x):
    """Return the symbolic derivative, dy/dx, as an Expr.
    However, you probably want to simplify the results with simp.
    >>> diff(x * x, x)
    ((x * 1) + (x * 1))
    >>> simp(diff(x * x, x))
    (2 * x)
    """
    if y == x:
        return ONE
    elif not y.args:
        return ZERO
    else:
        u, op, v = y.args[0], y.op, y.args[-1]
        if op == '+':
            return diff(u, x) + diff(v, x)
        elif op == '-' and len(args) == 1:
            return -diff(u, x)
        elif op == '-':
            return diff(u, x) - diff(v, x)
        elif op == '*':
            return u * diff(v, x) + v * diff(u, x)
        elif op == '/':
            return (v*diff(u, x) - u*diff(v, x)) / (v * v)
        elif op == '**' and isnumber(x.op):
            return (v * u ** (v - 1) * diff(u, x))
        elif op == '**':
            return (v * u ** (v - 1) * diff(u, x) +
                    u ** v * Expr('log')(u) * diff(v, x))
        elif op == 'log':
            return diff(u, x) / u
        else:
            raise ValueError("Unknown op: {} in diff({}, {})".format(op, y, x))


def simp(x):
    if not x.args:
        return x
    args = list(map(simp, x.args))
    u, op, v = args[0], x.op, args[-1]
    if op == '+':
        if v == ZERO:
            return u
        if u == ZERO:
            return v
        if u == v:
            return TWO * u
        if u == -v or v == -u:
            return ZERO
    elif op == '-' and len(args) == 1:
        if u.op == '-' and len(u.args) == 1:
            return u.args[0]  # --y ==> y
    elif op == '-':
        if v == ZERO:
            return u
        if u == ZERO:
            return -v
        if u == v:
            return ZERO
        if u == -v or v == -u:
            return ZERO
    elif op == '*':
        if u == ZERO or v == ZERO:
            return ZERO
        if u == ONE:
            return v
        if v == ONE:
            return u
        if u == v:
            return u ** 2
    elif op == '/':
        if u == ZERO:
            return ZERO
        if v == ZERO:
            return Expr('Undefined')
        if u == v:
            return ONE
        if u == -v or v == -u:
            return ZERO
    elif op == '**':
        if u == ZERO:
            return ZERO
        if v == ZERO:
            return ONE
        if u == ONE:
            return ONE
        if v == ONE:
            return u
    elif op == 'log':
        if u == ONE:
            return ZERO
    else:
        raise ValueError("Unknown op: " + op)
    # If we fall through to here, we can not simplify further
    return Expr(op, *args)


def d(y, x):
    "Differentiate and then simplify."
    return simp(diff(y, x))

# _________________________________________________________________________

# Utilities for doctest cases
# These functions print their arguments in a standard order
# to compensate for the random order in the standard representation






# ________________________________________________________________________