Finding Trigonometric Identities

The second problem involves applying our system to a trigonometric identity problem summarised in Table 2 [Ryan 98b]. 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 $[0, 2 \pi]$. Other identities exist for $Cos \; 2x$, such as 2 Cos²x - 1. If we were to include Cos as one of the terminals in our BNF grammar definition, 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 reasons of consistency.

 

Table 2: Grammatical Evolution Tableau for finding Trigonometric Identities

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