python3-deap 1.0.2.post2-2 / usr / lib / python3 / dist-packages / deap / gp.py

#    This file is part of DEAP.
#
#    DEAP is free software: you can redistribute it and/or modify
#    it under the terms of the GNU Lesser General Public License as
#    published by the Free Software Foundation, either version 3 of
#    the License, or (at your option) any later version.
#
#    DEAP is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
#    GNU Lesser General Public License for more details.
#
#    You should have received a copy of the GNU Lesser General Public
#    License along with DEAP. If not, see <http://www.gnu.org/licenses/>.

"""The :mod:`gp` module provides the methods and classes to perform
Genetic Programming with DEAP. It essentially contains the classes to
build a Genetic Program Tree, and the functions to evaluate it.

This module support both strongly and loosely typed GP.
"""
import copy
import random
import re
import sys
import warnings

from collections import defaultdict, deque
from functools import partial, wraps
from inspect import isclass
from operator import eq, lt
import collections

######################################
# GP Data structure                  #
######################################

# Define the name of type for any types.
__type__ = object

class PrimitiveTree(list):
    """Tree spefically formated for optimization of genetic
    programming operations. The tree is represented with a
    list where the nodes are appended in a depth-first order.
    The nodes appended to the tree are required to
    have an attribute *arity* which defines the arity of the
    primitive. An arity of 0 is expected from terminals nodes.
    """
    def __init__(self, content):
        list.__init__(self, content)

    def __deepcopy__(self, memo):
        new = self.__class__(self)
        new.__dict__.update(copy.deepcopy(self.__dict__, memo))
        return new

    def __setitem__(self, key, val):
        # Check for most common errors
        # Does NOT check for STGP constraints
        if isinstance(key, slice):
            if key.start >= len(self):
                raise IndexError("Invalid slice object (try to assign a %s"
                    " in a tree of size %d). Even if this is allowed by the"
                    " list object slice setter, this should not be done in"
                    " the PrimitiveTree context, as this may lead to an"
                    " unpredictable behavior for searchSubtree or evaluate."
                     % (key, len(self)))
            total = val[0].arity
            for node in val[1:]:
                total += node.arity - 1
            if total != 0:
                raise ValueError("Invalid slice assignation : insertion of"
                    " an incomplete subtree is not allowed in PrimitiveTree."
                    " A tree is defined as incomplete when some nodes cannot"
                    " be mapped to any position in the tree, considering the"
                    " primitives' arity. For instance, the tree [sub, 4, 5,"
                    " 6] is incomplete if the arity of sub is 2, because it"
                    " would produce an orphan node (the 6).")
        elif val.arity != self[key].arity:
            raise ValueError("Invalid node replacement with a node of a"
                             " different arity.")
        list.__setitem__(self, key, val)

    def __str__(self):
        """Return the expression in a human readable string.
        """
        string = ""
        stack = []
        for node in self:
            stack.append((node, []))
            while len(stack[-1][1]) == stack[-1][0].arity:
                prim, args = stack.pop()
                string = prim.format(*args)
                if len(stack) == 0:
                    break   # If stack is empty, all nodes should have been seen
                stack[-1][1].append(string)

        return string

    @classmethod
    def from_string(cls, string, pset):
        """Try to convert a string expression into a PrimitiveTree given a
        PrimitiveSet *pset*. The primitive set needs to contain every primitive
        present in the expression.

        :param string: String representation of a Python expression.
        :param pset: Primitive set from which primitives are selected.
        :returns: PrimitiveTree populated with the deserialized primitives.
        """
        tokens = re.split("[ \t\n\r\f\v(),]", string)
        expr = []
        ret_types = deque()
        for token in tokens:
            if token == '':
                continue
            if len(ret_types) != 0:
                type_ = ret_types.popleft()
            else:
                type_ = None

            if token in pset.mapping:
                primitive = pset.mapping[token]

                if type_ is not None and not issubclass(primitive.ret, type_):
                    raise TypeError("Primitive {} return type {} does not "
                                    "match the expected one: {}."
                                    .format(primitive, primitive.ret, type_))

                expr.append(primitive)
                if isinstance(primitive, Primitive):
                    ret_types.extendleft(reversed(primitive.args))
            else:
                try:
                    token = eval(token)
                except NameError:
                    raise TypeError("Unable to evaluate terminal: {}.".format(token))

                if type_ is None:
                    type_ = type(token)

                if not issubclass(type(token), type_):
                    raise TypeError("Terminal {} type {} does not "
                                    "match the expected one: {}."
                                    .format(token, type(token), type_))

                expr.append(Terminal(token, False, type_))
        return cls(expr)

    @property
    def height(self):
        """Return the height of the tree, or the depth of the
        deepest node.
        """
        stack = [0]
        max_depth = 0
        for elem in self:
            depth = stack.pop()
            max_depth = max(max_depth, depth)
            stack.extend([depth+1] * elem.arity)
        return max_depth

    @property
    def root(self):
        """Root of the tree, the element 0 of the list.
        """
        return self[0]

    def searchSubtree(self, begin):
        """Return a slice object that corresponds to the
        range of values that defines the subtree which has the
        element with index *begin* as its root.
        """
        end = begin + 1
        total = self[begin].arity
        while total > 0:
            total += self[end].arity - 1
            end += 1
        return slice(begin, end)


class Primitive(object):
    """Class that encapsulates a primitive and when called with arguments it
    returns the Python code to call the primitive with the arguments.

        >>> pr = Primitive("mul", (int, int), int)
        >>> pr.format(1, 2)
        'mul(1, 2)'
    """
    __slots__ = ('name', 'arity', 'args', 'ret', 'seq')
    def __init__(self, name, args, ret):
        self.name = name
        self.arity = len(args)
        self.args = args
        self.ret = ret
        args = ", ".join(map("{{{0}}}".format, list(range(self.arity))))
        self.seq = "{name}({args})".format(name=self.name, args=args)

    def format(self, *args):
        return self.seq.format(*args)

class Terminal(object):
    """Class that encapsulates terminal primitive in expression. Terminals can
    be values or 0-arity functions.
    """
    __slots__ = ('name', 'value', 'ret', 'conv_fct')
    def __init__(self, terminal, symbolic, ret):
        self.ret = ret
        self.value = terminal
        self.name = str(terminal)
        self.conv_fct = str if symbolic else repr

    @property
    def arity(self):
        return 0

    def format(self):
        return self.conv_fct(self.value)

class Ephemeral(Terminal):
    """Class that encapsulates a terminal which value is set when the
    object is created. To mutate the value, a new object has to be
    generated. This is an abstract base class. When subclassing, a
    staticmethod 'func' must be defined.
    """
    def __init__(self):
        Terminal.__init__(self, self.func(), symbolic=False, ret=self.ret)

    @staticmethod
    def func():
        """Return a random value used to define the ephemeral state.
        """
        raise NotImplementedError

class PrimitiveSetTyped(object):
    """Class that contains the primitives that can be used to solve a
    Strongly Typed GP problem. The set also defined the researched
    function return type, and input arguments type and number.
    """
    def __init__(self, name, in_types, ret_type, prefix="ARG"):
        self.terminals = defaultdict(list)
        self.primitives = defaultdict(list)
        self.arguments = []
        # setting "__builtins__" to None avoid the context
        # being polluted by builtins function when evaluating
        # GP expression.
        self.context = {"__builtins__" : None}
        self.mapping = dict()
        self.terms_count = 0
        self.prims_count = 0

        self.name = name
        self.ret = ret_type
        self.ins = in_types
        for i, type_ in enumerate(in_types):
            arg_str = "{prefix}{index}".format(prefix=prefix, index=i)
            self.arguments.append(arg_str)
            term = Terminal(arg_str, True, type_)
            self._add(term)
            self.terms_count += 1

    def renameArguments(self, **kargs):
        """Rename function arguments with new names from *kargs*.
        """
        for i, old_name in enumerate(self.arguments):
            if old_name in kargs:
                new_name = kargs[old_name]
                self.arguments[i] = new_name
                self.mapping[new_name] = self.mapping[old_name]
                self.mapping[new_name].value = new_name
                del self.mapping[old_name]

    def _add(self, prim):
        def addType(dict_, ret_type):
            if not ret_type in dict_:
                new_list = []
                for type_, list_ in list(dict_.items()):
                    if issubclass(type_, ret_type):
                        for item in list_:
                            if not item in new_list:
                                new_list.append(item)
                dict_[ret_type] = new_list

        addType(self.primitives, prim.ret)
        addType(self.terminals, prim.ret)

        self.mapping[prim.name] = prim
        if isinstance(prim, Primitive):
            for type_ in prim.args:
                addType(self.primitives, type_)
                addType(self.terminals, type_)
            dict_ = self.primitives
        else:
            dict_ = self.terminals

        for type_ in dict_:
            if issubclass(prim.ret, type_):
                dict_[type_].append(prim)

    def addPrimitive(self, primitive, in_types, ret_type, name=None):
        """Add a primitive to the set.

        :param primitive: callable object or a function.
        :parma in_types: list of primitives arguments' type
        :param ret_type: type returned by the primitive.
        :param name: alternative name for the primitive instead
                     of its __name__ attribute.
        """
        if name is None:
            name = primitive.__name__
        prim = Primitive(name, in_types, ret_type)

        assert name not in self.context or \
               self.context[name] is primitive, \
               "Primitives are required to have a unique name. " \
               "Consider using the argument 'name' to rename your "\
               "second '%s' primitive." % (name,)

        self._add(prim)
        self.context[prim.name] = primitive
        self.prims_count += 1

    def addTerminal(self, terminal, ret_type, name=None):
        """Add a terminal to the set. Terminals can be named
        using the optional *name* argument. This should be
        used : to define named constant (i.e.: pi); to speed the
        evaluation time when the object is long to build; when
        the object does not have a __repr__ functions that returns
        the code to build the object; when the object class is
        not a Python built-in.

        :param terminal: Object, or a function with no arguments.
        :param ret_type: Type of the terminal.
        :param name: defines the name of the terminal in the expression.
        """
        symbolic = False
        if name is None and isinstance(terminal, collections.Callable):
            name = terminal.__name__

        assert name not in self.context, \
               "Terminals are required to have a unique name. " \
               "Consider using the argument 'name' to rename your "\
               "second %s terminal." % (name,)

        if name is not None:
            self.context[name] = terminal
            terminal = name
            symbolic = True
        elif terminal in (True, False):
            # To support True and False terminals with Python 2.
            self.context[str(terminal)] = terminal

        prim = Terminal(terminal, symbolic, ret_type)
        self._add(prim)
        self.terms_count += 1

    def addEphemeralConstant(self, name, ephemeral, ret_type):
        """Add an ephemeral constant to the set. An ephemeral constant
        is a no argument function that returns a random value. The value
        of the constant is constant for a Tree, but may differ from one
        Tree to another.

        :param name: name used to refers to this ephemeral type.
        :param ephemeral: function with no arguments returning a random value.
        :param ret_type: type of the object returned by *ephemeral*.
        """
        module_gp = globals()
        if not name in module_gp:
            class_ = type(name, (Ephemeral,), {'func' : staticmethod(ephemeral),
                                               'ret' : ret_type})
            module_gp[name] = class_
        else:
            class_ = module_gp[name]
            if issubclass(class_, Ephemeral):
                if class_.func is not ephemeral:
                    raise Exception("Ephemerals with different functions should "
                                    "be named differently, even between psets.")
                elif class_.ret is not ret_type:
                    raise Exception("Ephemerals with the same name and function "
                                    "should have the same type, even between psets.")
            else:
                raise Exception("Ephemerals should be named differently "
                                "than classes defined in the gp module.")
        
        self._add(class_)
        self.terms_count += 1

    def addADF(self, adfset):
        """Add an Automatically Defined Function (ADF) to the set.

        :param adfset: PrimitiveSetTyped containing the primitives with which
                       the ADF can be built.
        """
        prim = Primitive(adfset.name, adfset.ins, adfset.ret)
        self._add(prim)
        self.prims_count += 1

    @property
    def terminalRatio(self):
        """Return the ratio of the number of terminals on the number of all
        kind of primitives.
        """
        return self.terms_count / float(self.terms_count + self.prims_count)

class PrimitiveSet(PrimitiveSetTyped):
    """Class same as :class:`~deap.gp.PrimitiveSetTyped`, except there is no
    definition of type.
    """
    def __init__(self, name, arity, prefix="ARG"):
        args = [__type__]*arity
        PrimitiveSetTyped.__init__(self, name, args, __type__, prefix)

    def addPrimitive(self, primitive, arity, name=None):
        """Add primitive *primitive* with arity *arity* to the set.
        If a name *name* is provided, it will replace the attribute __name__
        attribute to represent/identify the primitive.
        """
        assert arity > 0, "arity should be >= 1"
        args = [__type__] * arity
        PrimitiveSetTyped.addPrimitive(self, primitive, args, __type__, name)

    def addTerminal(self, terminal, name=None):
        """Add a terminal to the set."""
        PrimitiveSetTyped.addTerminal(self, terminal, __type__, name)

    def addEphemeralConstant(self, name, ephemeral):
        """Add an ephemeral constant to the set."""
        PrimitiveSetTyped.addEphemeralConstant(self, name, ephemeral, __type__)


######################################
# GP Tree compilation functions      #
######################################
def compile(expr, pset):
    """Compile the expression *expr*.

    :param expr: Expression to compile. It can either be a PrimitiveTree,
                 a string of Python code or any object that when
                 converted into string produced a valid Python code
                 expression.
    :param pset: Primitive set against which the expression is compile.
    :returns: a function if the primitive set has 1 or more arguments,
              or return the results produced by evaluating the tree.
    """
    code = str(expr)
    if len(pset.arguments) > 0:
        # This section is a stripped version of the lambdify
        # function of SymPy 0.6.6.
        args = ",".join(arg for arg in pset.arguments)
        code = "lambda {args}: {code}".format(args=args, code=code)
    try:
        return eval(code, pset.context, {})
    except MemoryError:
        _, _, traceback = sys.exc_info()
        raise MemoryError("DEAP : Error in tree evaluation :"
        " Python cannot evaluate a tree higher than 90. "
        "To avoid this problem, you should use bloat control on your "
        "operators. See the DEAP documentation for more information. "
        "DEAP will now abort.").with_traceback(traceback)

def compileADF(expr, psets):
    """Compile the expression represented by a list of trees. The first
    element of the list is the main tree, and the following elements are
    automatically defined functions (ADF) that can be called by the first
    tree.


    :param expr: Expression to compile. It can either be a PrimitiveTree,
                 a string of Python code or any object that when
                 converted into string produced a valid Python code
                 expression.
    :param psets: List of primitive sets. Each set corresponds to an ADF
                  while the last set is associated with the expression
                  and should contain reference to the preceding ADFs.
    :returns: a function if the main primitive set has 1 or more arguments,
              or return the results produced by evaluating the tree.
    """
    adfdict = {}
    func = None
    for pset, subexpr in reversed(list(zip(psets, expr))):
        pset.context.update(adfdict)
        func = compile(subexpr, pset)
        adfdict.update({pset.name : func})
    return func

######################################
# GP Program generation functions    #
######################################
def genFull(pset, min_, max_, type_=None):
    """Generate an expression where each leaf has a the same depth
    between *min* and *max*.

    :param pset: Primitive set from which primitives are selected.
    :param min_: Minimum height of the produced trees.
    :param max_: Maximum Height of the produced trees.
    :param type_: The type that should return the tree when called, when
                  :obj:`None` (default) no return type is enforced.
    :returns: A full tree with all leaves at the same depth.
    """
    def condition(height, depth):
        """Expression generation stops when the depth is equal to height."""
        return depth == height
    return generate(pset, min_, max_, condition, type_)

def genGrow(pset, min_, max_, type_=None):
    """Generate an expression where each leaf might have a different depth
    between *min* and *max*.

    :param pset: Primitive set from which primitives are selected.
    :param min_: Minimum height of the produced trees.
    :param max_: Maximum Height of the produced trees.
    :param type_: The type that should return the tree when called, when
                  :obj:`None` (default) no return type is enforced.
    :returns: A grown tree with leaves at possibly different depths.
    """
    def condition(height, depth):
        """Expression generation stops when the depth is equal to height
        or when it is randomly determined that a a node should be a terminal.
        """
        return depth == height or \
               (depth >= min_ and random.random() < pset.terminalRatio)
    return generate(pset, min_, max_, condition, type_)

def genHalfAndHalf(pset, min_, max_, type_=None):
    """Generate an expression with a PrimitiveSet *pset*.
    Half the time, the expression is generated with :func:`~deap.gp.genGrow`,
    the other half, the expression is generated with :func:`~deap.gp.genFull`.

    :param pset: Primitive set from which primitives are selected.
    :param min_: Minimum height of the produced trees.
    :param max_: Maximum Height of the produced trees.
    :param type_: The type that should return the tree when called, when
                  :obj:`None` (default) no return type is enforced.
    :returns: Either, a full or a grown tree.
    """
    method = random.choice((genGrow, genFull))
    return method(pset, min_, max_, type_)

def genRamped(pset, min_, max_, type_=None):
    """
    .. deprecated:: 1.0
        The function has been renamed. Use :func:`~deap.gp.genHalfAndHalf` instead.
    """
    warnings.warn("gp.genRamped has been renamed. Use genHalfAndHalf instead.",
                  FutureWarning)
    return genHalfAndHalf(pset, min_, max_, type_)

def generate(pset, min_, max_, condition, type_=None):
    """Generate a Tree as a list of list. The tree is build
    from the root to the leaves, and it stop growing when the
    condition is fulfilled.

    :param pset: Primitive set from which primitives are selected.
    :param min_: Minimum height of the produced trees.
    :param max_: Maximum Height of the produced trees.
    :param condition: The condition is a function that takes two arguments,
                      the height of the tree to build and the current
                      depth in the tree.
    :param type_: The type that should return the tree when called, when
                  :obj:`None` (default) no return type is enforced.
    :returns: A grown tree with leaves at possibly different depths
              dependending on the condition function.
    """
    if type_ is None:
        type_ = pset.ret
    expr = []
    height = random.randint(min_, max_)
    stack = [(0, type_)]
    while len(stack) != 0:
        depth, type_ = stack.pop()
        if condition(height, depth):
            try:
                term = random.choice(pset.terminals[type_])
            except IndexError:
                _, _, traceback = sys.exc_info()
                raise IndexError("The gp.generate function tried to add "\
                                  "a terminal of type '%s', but there is "\
                                  "none available." % (type_,)).with_traceback(traceback)
            if isclass(term):
                term = term()
            expr.append(term)
        else:
            try:
                prim = random.choice(pset.primitives[type_])
            except IndexError:
                _, _, traceback = sys.exc_info()
                raise IndexError("The gp.generate function tried to add "\
                                  "a primitive of type '%s', but there is "\
                                  "none available." % (type_,)).with_traceback(traceback)
            expr.append(prim)
            for arg in reversed(prim.args):
                stack.append((depth+1, arg))
    return expr


######################################
# GP Crossovers                      #
######################################

def cxOnePoint(ind1, ind2):
    """Randomly select in each individual and exchange each subtree with the
    point as root between each individual.

    :param ind1: First tree participating in the crossover.
    :param ind2: Second tree participating in the crossover.
    :returns: A tuple of two trees.
    """
    if len(ind1) < 2 or len(ind2) < 2:
        # No crossover on single node tree
        return ind1, ind2

    # List all available primitive types in each individual
    types1 = defaultdict(list)
    types2 = defaultdict(list)
    if ind1.root.ret == __type__:
        # Not STGP optimization
        types1[__type__] = range(1, len(ind1))
        types2[__type__] = range(1, len(ind2))
        common_types = [__type__]
    else:
        for idx, node in enumerate(ind1[1:], 1):
            types1[node.ret].append(idx)
        for idx, node in enumerate(ind2[1:], 1):
            types2[node.ret].append(idx)
        common_types = set(types1.keys()).intersection(set(types2.keys()))

    if len(common_types) > 0:
        type_ = random.choice(list(common_types))

        index1 = random.choice(types1[type_])
        index2 = random.choice(types2[type_])

        slice1 = ind1.searchSubtree(index1)
        slice2 = ind2.searchSubtree(index2)
        ind1[slice1], ind2[slice2] = ind2[slice2], ind1[slice1]

    return ind1, ind2


def cxOnePointLeafBiased(ind1, ind2, termpb):
    """Randomly select crossover point in each individual and exchange each
    subtree with the point as root between each individual.

    :param ind1: First typed tree participating in the crossover.
    :param ind2: Second typed tree participating in the crossover.
    :param termpb: The probability of chosing a terminal node (leaf).
    :returns: A tuple of two typed trees.

    When the nodes are strongly typed, the operator makes sure the
    second node type corresponds to the first node type.

    The parameter *termpb* sets the probability to choose between a terminal
    or non-terminal crossover point. For instance, as defined by Koza, non-
    terminal primitives are selected for 90% of the crossover points, and
    terminals for 10%, so *termpb* should be set to 0.1.
    """

    if len(ind1) < 2 or len(ind2) < 2:
        # No crossover on single node tree
        return ind1, ind2

    # Determine wether we keep terminals or primitives for each individual
    terminal_op = partial(eq, 0)
    primitive_op = partial(lt, 0)
    arity_op1 = terminal_op if random.random() < termpb else primitive_op
    arity_op2 = terminal_op if random.random() < termpb else primitive_op

    # List all available primitive or terminal types in each individual
    types1 = defaultdict(list)
    types2 = defaultdict(list)

    for idx, node in enumerate(ind1[1:], 1):
        if arity_op1(node.arity):
            types1[node.ret].append(idx)

    for idx, node in enumerate(ind2[1:], 1):
        if arity_op2(node.arity):
            types2[node.ret].append(idx)

    common_types = set(types1.keys()).intersection(set(types2.keys()))

    if len(common_types) > 0:
        # Set does not support indexing
        type_ = random.sample(common_types, 1)[0]
        index1 = random.choice(types1[type_])
        index2 = random.choice(types2[type_])

        slice1 = ind1.searchSubtree(index1)
        slice2 = ind2.searchSubtree(index2)
        ind1[slice1], ind2[slice2] = ind2[slice2], ind1[slice1]

    return ind1, ind2


######################################
# GP Mutations                       #
######################################
def mutUniform(individual, expr, pset):
    """Randomly select a point in the tree *individual*, then replace the
    subtree at that point as a root by the expression generated using method
    :func:`expr`.

    :param individual: The tree to be mutated.
    :param expr: A function object that can generate an expression when
                 called.
    :returns: A tuple of one tree.
    """
    index = random.randrange(len(individual))
    slice_ = individual.searchSubtree(index)
    type_ = individual[index].ret
    individual[slice_] = expr(pset=pset, type_=type_)
    return individual,


def mutNodeReplacement(individual, pset):
    """Replaces a randomly chosen primitive from *individual* by a randomly
    chosen primitive with the same number of arguments from the :attr:`pset`
    attribute of the individual.

    :param individual: The normal or typed tree to be mutated.
    :returns: A tuple of one tree.
    """
    if len(individual) < 2:
        return individual,

    index = random.randrange(1, len(individual))
    node = individual[index]

    if node.arity == 0: # Terminal
        term = random.choice(pset.terminals[node.ret])
        if isclass(term):
            term = term()
        individual[index] = term
    else:   # Primitive
        prims = [p for p in pset.primitives[node.ret] if p.args == node.args]
        individual[index] = random.choice(prims)

    return individual,

def mutEphemeral(individual, mode):
    """This operator works on the constants of the tree *individual*. In
    *mode* ``"one"``, it will change the value of one of the individual
    ephemeral constants by calling its generator function. In *mode*
    ``"all"``, it will change the value of **all** the ephemeral constants.

    :param individual: The normal or typed tree to be mutated.
    :param mode: A string to indicate to change ``"one"`` or ``"all"``
                 ephemeral constants.
    :returns: A tuple of one tree.
    """
    if mode not in ["one", "all"]:
        raise ValueError("Mode must be one of \"one\" or \"all\"")

    ephemerals_idx = [index
                      for index, node in enumerate(individual)
                      if isinstance(node, Ephemeral)]

    if len(ephemerals_idx) > 0:
        if mode == "one":
            ephemerals_idx = (random.choice(ephemerals_idx),)

        for i in ephemerals_idx:
            individual[i] = type(individual[i])()

    return individual,

def mutInsert(individual, pset):
    """Inserts a new branch at a random position in *individual*. The subtree
    at the chosen position is used as child node of the created subtree, in
    that way, it is really an insertion rather than a replacement. Note that
    the original subtree will become one of the children of the new primitive
    inserted, but not perforce the first (its position is randomly selected if
    the new primitive has more than one child).

    :param individual: The normal or typed tree to be mutated.
    :returns: A tuple of one tree.
    """
    index = random.randrange(len(individual))
    node = individual[index]
    slice_ = individual.searchSubtree(index)
    choice = random.choice

    # As we want to keep the current node as children of the new one,
    # it must accept the return value of the current node
    primitives = [p for p in pset.primitives[node.ret] if node.ret in p.args]

    if len(primitives) == 0:
        return individual,

    new_node = choice(primitives)
    new_subtree = [None] * len(new_node.args)
    position = choice([i for i, a in enumerate(new_node.args) if a == node.ret])

    for i, arg_type in enumerate(new_node.args):
        if i != position:
            term = choice(pset.terminals[arg_type])
            if isclass(term):
                term = term()
            new_subtree[i] = term

    new_subtree[position:position+1] = individual[slice_]
    new_subtree.insert(0, new_node)
    individual[slice_] = new_subtree
    return individual,

def mutShrink(individual):
    """This operator shrinks the *individual* by chosing randomly a branch and
    replacing it with one of the branch's arguments (also randomly chosen).

    :param individual: The tree to be shrinked.
    :returns: A tuple of one tree.
    """
    # We don't want to "shrink" the root
    if len(individual) < 3 or individual.height <= 1:
        return individual,

    iprims = []
    for i, node in enumerate(individual[1:], 1):
        if isinstance(node, Primitive) and node.ret in node.args:
            iprims.append((i, node))

    if len(iprims) != 0:
        index, prim = random.choice(iprims)
        arg_idx = random.choice([i for i, type_ in enumerate(prim.args) if type_ == prim.ret])
        rindex = index+1
        for _ in range(arg_idx+1):
            rslice = individual.searchSubtree(rindex)
            subtree = individual[rslice]
            rindex += len(subtree)

        slice_ = individual.searchSubtree(index)
        individual[slice_] = subtree

    return individual,

######################################
# GP bloat control decorators        #
######################################

def staticLimit(key, max_value):
    """Implement a static limit on some measurement on a GP tree, as defined
    by Koza in [Koza1989]. It may be used to decorate both crossover and
    mutation operators. When an invalid (over the limit) child is generated,
    it is simply replaced by one of its parents, randomly selected.

    This operator can be used to avoid memory errors occuring when the tree
    gets higher than 90 levels (as Python puts a limit on the call stack
    depth), because it can ensure that no tree higher than this limit will ever
    be accepted in the population, except if it was generated at initialization
    time.
    
    :param key: The function to use in order the get the wanted value. For 
                instance, on a GP tree, ``operator.attrgetter('height')`` may
                be used to set a depth limit, and ``len`` to set a size limit.
    :param max_value: The maximum value allowed for the given measurement.
    :returns: A decorator that can be applied to a GP operator using \
    :func:`~deap.base.Toolbox.decorate`

    .. note::
       If you want to reproduce the exact behavior intended by Koza, set
       *key* to ``operator.attrgetter('height')`` and *max_value* to 17.

    .. [Koza1989] J.R. Koza, Genetic Programming - On the Programming of
        Computers by Means of Natural Selection (MIT Press,
        Cambridge, MA, 1992)

    """
    def decorator(func):
        @wraps(func)
        def wrapper(*args, **kwargs):
            keep_inds = [copy.deepcopy(ind) for ind in args]
            new_inds = list(func(*args, **kwargs))
            for i, ind in enumerate(new_inds):
                if key(ind) > max_value:
                    new_inds[i] = random.choice(keep_inds)
            return new_inds
        return wrapper
    return decorator

def graph(expr):
    """Construct the graph of a tree expression. The tree expression must be
    valid. It returns in order a node list, an edge list, and a dictionary of
    the per node labels. The node are represented by numbers, the edges are
    tuples connecting two nodes (number), and the labels are values of a
    dictionary for which keys are the node numbers.

    :param expr: A tree expression to convert into a graph.
    :returns: A node list, an edge list, and a dictionary of labels.

    The returned objects can be used directly to populate a
    `pygraphviz <http://networkx.lanl.gov/pygraphviz/>`_ graph::

        import pygraphviz as pgv

        # [...] Execution of code that produce a tree expression

        nodes, edges, labels = graph(expr)

        g = pgv.AGraph()
        g.add_nodes_from(nodes)
        g.add_edges_from(edges)
        g.layout(prog="dot")

        for i in nodes:
            n = g.get_node(i)
            n.attr["label"] = labels[i]

        g.draw("tree.pdf")

    or a `NetworX <http://networkx.github.com/>`_ graph::

        import matplotlib.pyplot as plt
        import networkx as nx

        # [...] Execution of code that produce a tree expression

        nodes, edges, labels = graph(expr)

        g = nx.Graph()
        g.add_nodes_from(nodes)
        g.add_edges_from(edges)
        pos = nx.graphviz_layout(g, prog="dot")

        nx.draw_networkx_nodes(g, pos)
        nx.draw_networkx_edges(g, pos)
        nx.draw_networkx_labels(g, pos, labels)
        plt.show()


    .. note::

       We encourage you to use `pygraphviz
       <http://networkx.lanl.gov/pygraphviz/>`_ as the nodes might be plotted
       out of order when using `NetworX <http://networkx.github.com/>`_.
    """
    nodes = list(range(len(expr)))
    edges = list()
    labels = dict()

    stack = []
    for i, node in enumerate(expr):
        if stack:
            edges.append((stack[-1][0], i))
            stack[-1][1] -= 1
        labels[i] = node.name if isinstance(node, Primitive) else node.value
        stack.append([i, node.arity])
        while stack and stack[-1][1] == 0:
            stack.pop()

    return nodes, edges, labels

if __name__ == "__main__":
    import doctest
    doctest.testmod()