The Problem Space

As proof of concept, we have applied our system to a trigonometric identity problem described by Koza [Koza 92]. The particular function examined was $Cos \; 2x$ , and the desired trigonometric identity was $1 - 2Sin^2\;x$. The system was given a set of input and output pairs, and must determine the function that maps one onto the other, with the input values in the range $[0, 2 \pi]$. Table  2 contains a tableau which summarizes Koza's experiments.

 

Table 2: A Koza-style tableau

Objective :	Find a new mathematical expression, in symbolic form
	that equals a given mathematical expression, for all
	values of its independent variables.
GPTerminal Set:	X , the constant 1.0.
GPFunction Set	$+, -, *, \%, \sin$
Fitness cases	The given sample of 20 data points in the
	interval [$0, 2 \pi$]
Raw Fitness	The sum, taken over the 20 fitness cases,
	of the error
Standardised Fitness	Same as raw fitness
Hits	The number of fitness cases for which the error
	is less than 0.01
Wrapper	None
Parameters	M = 500, G = 51

Other identities exist for $Cos \; 2x$, such as 2 Cos²x - 1. If we were to include Cos as one of the <pre-op> rules, GE would naturally produce simple Cos identities for $Cos \; 2x$, which was verified in early experiments which included Cos. These runs consistently found the target expression $Cos \; 2x$ and, therefore, we decided to exclude Cos from the terminal operator set. Koza included the constant 1.0 in his terminal operator set, although Genetic Programming has shown an ability to generate the constant 1.0, as has GE with expressions such as x/x. However, we included 1.0 for consistency reasons.
We adopt a similar style to Koza of summarizing information, using a modified version of his tableau in Table  3. Notice how our terminal operands and terminal operators are analogous to GPTerminals and GPfunctions respectively.

 

Table 3: Grammatical Evolution Tableau

Objective :	Find a new mathematical expression, in symbolic form
	that equals a given mathematical expression, for all
	values of its independent variables.
Terminal Operands:	X , the constant 1.0
Terminal Operators	The binary operators +, *, /, and -
	The unary operator Sin
Fitness cases	The given sample of 20 data points in the
	interval [$0, 2 \pi$]
Raw Fitness	The sum, taken over the 20 fitness cases,
	of the error
Standardised Fitness	Same as raw fitness
Hits	The number of fitness cases for which the error
	is less than 0.01
Wrapper	Standard productions to generate
	C functions
Parameters	M = 500, G = 51

The production rules for <expr> are as given earlier. As this and subsequent rules are the only ones that require a choice, they are the ones that will be evolved.