Even-Parity Problem

Parity is one of the classical GP problems. The goal is to find a program that produces the value of the Boolean even parity given n independent Boolean inputs. Usually, 6 Boolean inputs are used (Parity-6), and the goal is to match the good parity bit value for each of the 2^6 = 64 possible entries. The problem can be made harder by increasing the number of inputs (in the DEAP implementation, this number can easily be tuned, as it is fixed by a constant named PARITY_FANIN_M).

For more information about this problem, see Reference.

Primitives set used

Parity uses standard Boolean operators as primitives, available in the Python operator module :

pset = gp.PrimitiveSet("MAIN", PARITY_FANIN_M, "IN")
pset.addPrimitive(operator.and_, 2)
pset.addPrimitive(operator.or_, 2)
pset.addPrimitive(operator.xor, 2)
pset.addPrimitive(operator.not_, 1)
pset.addTerminal(1)
pset.addTerminal(0)

In addition to the n inputs, we add two constant terminals, one at 0, one at 1.

Note

As Python is a dynamic typed language, you can mix Boolean operators and integers without any issue.

Evaluation function

In this implementation, the fitness of a Parity individual is simply the number of successful cases. Thus, the fitness is maximized, and the maximum value is 64 in the case of a 6 inputs problems.

def evalParity(individual):
    func = toolbox.compile(expr=individual)
    return sum(func(*in_) == out for in_, out in zip(inputs, outputs)),

inputs and outputs are two pre-generated lists, to speedup the evaluation, mapping a given input vector to the good output bit. The Python sum() function works on booleans (false is interpreted as 0 and true as 1), so the evaluation function boils down to sum the number of successful tests : the higher this sum, the better the individual.

Conclusion

The other parts of the program are mostly the same as the Symbolic Regression algorithm.

The complete examples/gp/parity.

Reference

John R. Koza, “Genetic Programming II: Automatic Discovery of Reusable Programs”, MIT Press, 1994, pages 157-199.