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3.3 Step 3: Fitness Function

The first two preparatory steps define the primitive set for GP, and therefore indirectly define the search space GP will explore. This includes all the programs that can be constructed by composing the primitives in all possible ways. However, at this stage, we still do not know which elements or regions of this search space are good. I.e., which regions of the search space include programs that solve, or approximately solve, the problem. This is the task of the fitness measure, which is our primary (and often sole) mechanism for giving a high-level statement of the problem's requirements to the GP system. For example, suppose the goal is to get GP to synthesise an amplifier automatically. Then the fitness function is the mechanism which tells GP to synthesise a circuit that amplifies an incoming signal. (As opposed to evolving a circuit that suppresses the low frequencies of an incoming signal, or computes its square root, etc. etc.)

Fitness can be measured in many ways. For example, in terms of: the amount of error between its output and the desired output; the amount of time (fuel, money, etc.) required to bring a system to a desired target state; the accuracy of the program in recognising patterns or classifying objects; the payoff that a game-playing program produces; the compliance of a structure with user-specified design criteria.

There is something unusual about the fitness functions used in GP that differentiates them from those used in most other evolutionary algorithms. Because the structures being evolved in GP are computer programs, fitness evaluation normally requires executing all the programs in the population, typically multiple times. While one can compile the GP programs that make up the population, the overhead of building a compiler is usually substantial, so it is much more common to use an interpreter to evaluate the evolved programs.

Interpreting a program tree means executing the nodes in the tree in an order that guarantees that nodes are not executed before the value of their arguments (if any) is known. This is usually done by traversing the tree recursively starting from the root node, and postponing the evaluation of each node until the values of its children (arguments) are known. Other orders, such as going from the leaves to the root, are possible. If none of the primitives have side effects, the two orders are equivalent.3 This depth-first recursive process is illustrated in Figure  3.1 . Algorithm  3.1 gives a pseudocode implementation of the interpretation procedure. The code assumes that programs are represented as prefix-notation expressions and that such expressions can be treated as lists of components.


PIC

Figure 3.1: Example interpretation of a syntax tree (the terminal x is a variable and has a value of -1). The number to the right of each internal node represents the result of evaluating the subtree root at that node.



procedure: eval( expr )

1: if expr is a list then

2: proc = expr(1) {Non-terminal: extract root}

 3: if proc is a function then

4: value = proc( eval(expr(2)), eval(expr(3)), ... ) {Function: evaluate arguments}

5: else

6: value = proc( expr(2), expr(3), ...) {Macro: don't evaluate arguments}

7: end if

8: else

9: if expr is a variable or expr is a constant then

10: value = expr {Terminal variable or constant: just read the value}

11: else

12: value = expr() {Terminal 0-arity function: execute}

13: end if

14: end if

 15: return  value

Notes: expr is an expression in prefix notation, expr(1) represents the primitive at the root of the expression, expr(2) represents the first argument of that primitive, expr(3) represents the second argument, etc.

Algorithm 3.1: Interpreter for genetic programming


In some problems we are interested in the output produced by a program, namely the value returned when we evaluate the tree starting at the root node. In other problems we are interested in the actions performed by a program composed of functions with side effects. In either case the fitness of a program typically depends on the results produced by its execution on many different inputs or under a variety of different conditions. For example the program might be tested on all possible combinations of inputs x1, x2, ..., xN. Alternatively, a robot control program might be tested with the robot in a number of starting locations. These different test cases typically contribute to the fitness value of a program incrementally, and for this reason are called fitness cases.

Another common feature of GP fitness measures is that, for many practical problems, they are multi-objective, i.e., they combine two or more different elements that are often in competition with one another. The area of multi-objective optimisation is a complex and active area of research in GP and machine learning in general. See Chapter  9 and also (Deb2001).


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