Genetic Programming

Genetic programming is a special field of evolutionary computation that aims at building programs automatically to solve problems independently of their domain. Although there exist diverse representations used to evolve programs, the most common is the syntax tree.

../../_images/gptree.png

For example, the above figure presents the program \max(x + 3 * y, x +
x). For this tree and further examples, the leaves of the tree, in green, are called terminals, while the internal nodes, in red, are called primitives. The terminals are divided in two subtypes: the constants and the arguments. The constants remain the same for the entire evolution while the arguments are the program inputs. For the last presented tree, the arguments are the variables x and y, and the constant is the number 3.

In DEAP, user defined primitives and terminals are contained in a primitive set. For now, two kinds of primitive set exists: the loosely and the strongly typed.

Loosely Typed GP

Loosely typed GP does not enforce a specific type between the nodes. More specifically, primitives’ arguments can be any primitives or terminals present in the primitive set.

The following code define a loosely typed PrimitiveSet for the previous tree

pset = PrimitiveSet("main", 2)
pset.addPrimitive(max, 2)
pset.addPrimitive(operator.add, 2)
pset.addPrimitive(operator.mul, 2)
pset.addTerminal(3)

The first line creates a primitive set. Its arguments are the name of the procedure it will generate ("main") and its number of inputs, 2. The next three lines add functions as primitives. The first argument is the function to add and the second argument the function arity. The last line adds a constant terminal. Currently, the default names for the arguments are "ARG0" and "ARG1". To change it to "x" and "y", simply call

pset.renameArguments(ARG0="x")
pset.renameArguments(ARG1="y")

In this case, all functions take two arguments. Having a 1 argument negation function, for example, could be done with

pset.addPrimitive(operator.neg, 1)

Our primitive set is now ready to generate some trees. The gp module contains three prefix expression generation functions genFull(), genGrow(), and genHalfAndHalf(). Their first argument is a primitive set. They return a valid prefix expression in the form of a list of primitives. The content of this list can be read by the PrimitiveTree class to create a prefix tree.

expr = genFull(pset, min_=1, max_=3)
tree = PrimitiveTree(expr)

The last code produces a valid full tree with height randomly chosen between 1 and 3.

Strongly Typed GP

In strongly typed GP, every primitive and terminal is assigned a specific type. The output type of a primitive must match the input type of another one for them to be connected. For example, if a primitive returns a boolean, it is guaranteed that this value will not be multiplied with a float if the multiplication operator operates only on floats.

def if_then_else(input, output1, output2):
    return output1 if input else output2

pset = PrimitiveSetTyped("main", [bool, float], float)
pset.addPrimitive(operator.xor, [bool, bool], bool)
pset.addPrimitive(operator.mul, [float, float], float)
pset.addPrimitive(if_then_else, [bool, float, float], float)
pset.addTerminal(3.0, float)
pset.addTerminal(1, bool)

pset.renameArguments(ARG0="x")
pset.renameArguments(ARG1="y")

In the last code sample, we first define an if then else function that returns the second argument if the first argument is true and the third one otherwise. Then, we define our PrimitiveSetTyped. Again, the procedure is named "main". The second argument defines the input types of the program. Here, "x" is a bool and "y" is a float. The third argument defines the output type of the program as a float. Adding primitives to this primitive now requires to set the input and output types of the primitives and terminal. For example, we define our "if_then_else" function first argument as a boolean, the second and third argument have to be floats. The function is defined as returning a float. We now understand that the multiplication primitive can only have the terminal 3.0, the if_then_else function or the "y" as input, which are the only floats defined.

The previous code can produce the tree on the left but not the one on the right because the type restrictions.

../../_images/gptypedtrees.png

Note

The generation of trees is done randomly while making sure type constraints are respected. If any primitive has an input type that no primitive and terminal can provide, chances are that this primitive will be picked and placed in the tree, resulting in the impossibility to complete the tree within the limit fixed by the generator. For example, when generating a full tree of height 2, suppose "op" takes a boolean and a float, "and" takes 2 boolean and "neg" takes a float, no terminal is defined and the arguments are booleans. The following situation will occur where no terminal can be placed to complete the tree.


../../_images/gptypederrtree.png

In this case, DEAP raises an IndexError with the message "The gp.generate function tried to add a terminal of type float, but there is none available."

Ephemeral Constants

An ephemeral constant is a terminal encapsulating a value that is generated from a given function at run time. Ephemeral constants allow to have terminals that don’t have all the same values. For example, to create an ephemeral constant that takes its value in [-1, 1) we use

pset.addEphemeralConstant(lambda: random.uniform(-1, 1))

The ephemeral constant value is determined when it is inserted in the tree and never changes unless it is replaced by another ephemeral constant. Since it is a terminal, ephemeral constant can also be typed.

pset.addEphemeralConstant(lambda: random.randint(-10, 10), int)

Generation of Tree Individuals

The code presented in the last two sections produces valid trees. However, as in the Operators and Algorithms tutorial, these trees are not yet valid individuals for evolution. One must combine the creator and the toolbox to produce valid individuals. We need to create the Fitness and the Individual classes. We add a reference to the primitive set to the Individual in addition to the fitness. This is used by some of the gp operators to modify the individuals.

creator.create("FitnessMin", base.Fitness, weights=(-1.0,))
creator.create("Individual", gp.PrimitiveTree, fitness=creator.FitnessMin,
               pset=pset)

We then register the generation functions into a Toolbox.

toolbox = base.Toolbox()
toolbox.register("expr", gp.genFull, pset=pset, min_=1, max_=3)
toolbox.register("individual", tools.initIterate, creator.Individual,
                 toolbox.expr)

Calling toolbox.individual() readily returns an individual of type PrimitiveTree.

Evaluation of Trees

In DEAP, trees can be translated to readable Python code and compiled to Python code objects using functions provided by the gp module. The first function, str() takes an expression or a PrimitiveTree and translates it into readable Python code. For example, the following lines generate a tree and output the code from the first example primitive set.

>>> expr = genFull(pset, min_=1, max_=3)
>>> tree = PrimitiveTree(expr)
>>> str(tree)
'mul(add(x, x), max(y, x))'

Now, this string represents the program we just generated, but it cannot yet be executed. To make it executable, we have to compile the expression to a the Python code object. Since this function has two inputs, we wish to compile the code into a callable object. This is possible with compile(). The function takes two arguments: the expression to compile and the associated primitive set. The following example compiles the previous tree and evaluates the resulting function for x=1 and y=2.

>>> function = compile(tree, pset)
>>> function(1, 2)
4

When the generated program has no input argument, the expression can be compiled to byte code using the same compile() function. An example of this sort of problem is the Artificial Ant Problem.

Tree Size Limit and Bloat Control

Since DEAP uses the Python parser to compile the code represented by the trees, it inherits from its limitations. The most commonly encountered restriction is the parsing stack limit. The Python interpreter parser stack limit is usually fixed between 92 and 99. This means that an expression can at most be composed of 91 succeeding primitives. In other words, a tree can have a maximum depth of 91. When the limit is exceeded, Python raises the following error

s_push: parser stack overflow
Traceback (most recent call last):
[...]
MemoryError

Since this limit is hard-coded in the interpreter, there exists no easy way to increase it. Furthermore, this error commonly stems from a phenomena known in GP as bloat. That is, the produced individuals have reached a point where they contain too much primitives to effectively solve the problem. This problem leads to evolution stagnation. To counteract this, DEAP provides different functions that can effectively restrain the size and height of the trees under an acceptable limit. These operators are listed in the GP section of Operators.

Plotting Trees

The function deap.gp.graph() returns the necessary elements to plot tree graphs using NetworX or pygraphviz. The graph function takes a valid PrimitiveTree object and returns a node list, an edge list and a dictionary associating a label to each node. It can be used like following with pygraphviz.

from deap import base, creator, gp

pset = gp.PrimitiveSet("MAIN", 1)
pset.addPrimitive(operator.add, 2)
pset.addPrimitive(operator.sub, 2)
pset.addPrimitive(operator.mul, 2)
pset.renameArguments(ARG0='x')

creator.create("Individual", gp.PrimitiveTree)

toolbox = base.Toolbox()
toolbox.register("expr", gp.genHalfAndHalf, pset=pset, min_=1, max_=2)
toolbox.register("individual", tools.initIterate, creator.Individual, toolbox.expr)

expr = toolbox.individual()
nodes, edges, labels = gp.graph(expr)

### Graphviz Section ###
import pygraphviz as pgv

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")

Using NetworkX, the last section becomes:

import matplotlib.pyplot as plt
import networkx as nx

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()

Depending on the version of graphviz, the nodes may appear in an unpredictable order. Two plots of the same tree may have sibling nodes swapped. This does not affect the primitive tree representation nor the numerical results.

How to Evolve Programs

The different ways to evolve program trees are presented through the Genetic Programming (GP) examples.