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© C. FERREIRA, 2002 (Terms of Use) ISBN: 9729589054

Gene Expression Programming: Mathematical Modeling by an Artificial Intelligence

Finding solutions to odd-parity functions with UDFs
 
Before immersing ourselves in the evolution of odd-n-parity functions using user defined functions, letís first analyze how UDFs work and how they are implemented in GEP.

In gene expression programming, UDFs may represent functions of several variables although their arity in terms of expression rules is equal to zero. That is, nodes with UDFs behave as leaf nodes. So, the implementation of UDFs in GEP can be done using at least two different methods: either they are treated as terminals and are used both in the heads and tails or they are treated as functions and are used exclusively in the heads. Either way, the algorithm works very well. Thus, it is a matter of taste which one to choose. I chose the latter as it seems more consistent and less confusing. Consequently, UDFs are only used in the heads, they can occupy the first positions in the genes of the individuals of the initial population, and they can also occupy the first position of an RIS element.

Letís see now how genes containing UDFs are expressed. Consider, for instance, the chromosome below:

0123456789012345678

AON1cAA11ccacbccaba

(4.37)

where ď1Ē represents a UDF. Its expression results in the following ET:

This program encodes a solution to the odd-3-parity function using XOR as a UDF. The difference between this UDF and a normal XOR is that the arguments to the UDF are fixed during the definition of the function whereas the arguments to the XOR are flexible and depend on the particular configuration of the expression tree. For instance, in the chromosome (4.37) above, the arguments to the UDF1 are by definition a and b. Only now are we able to check the solution encoded in chromosome (4.37) and confirm that, indeed, it encodes a perfect solution to the odd-3-parity function. Compare this solution with the following solution discovered using a normal XOR instead of a rigid, user defined XOR:

01234

XXabc

(4.38)

Nevertheless, UDFs are extremely useful, especially if they are carefully crafted to the problem at hand. As shown in Table 4.20, the use of an inflexible XOR encapsulated in a UDF is not much help in the evolution of solutions to the odd-5-parity and odd-6-parity functions (Table 4.20, columns 3 and 4). However, the use of other, higher order parity functions as UDFs proves to be extremely efficient (see Table 4.21, especially columns 3 and 4).


Table 4.20
Parameters for the odd-n-parity problem using XOR as UDF.

  Odd-3 Odd-4 Odd-5 Odd-6
Number of runs 100 100 100 100
Number of generations 50 50 100 300
Population size 10 30 30 30
Number of fitness cases 8 16 32 64
Function set A O N A O N A O N (A O N)2
User defined functions X X X (X)2
Terminal set a b c a b c d a b c d e a b c d e f
Head length 7 7 7 7
Number of genes 3 3 3 3
Linking function X X X X
Chromosome length 45 45 45 45
Mutation rate 0.044 0.044 0.044 0.044
One-point recombination rate 0.3 0.3 0.3 0.3
Two-point recombination rate 0.3 0.3 0.3 0.3
Gene recombination rate 0.1 0.1 0.1 0.1
Gene transposition rate 0.1 0.1 0.1 0.1
IS transposition rate 0.1 0.1 0.1 0.1
IS elements length 1,2,3 1,2,3 1,2,3 1,2,3
RIS transposition rate 0.1 0.1 0.1 0.1
RIS elements length 1,2,3 1,2,3 1,2,3 1,2,3
Success rate 95% 100% 2% 1%



Table 4.21
Parameters for the odd-n-parity problem using odd-n-parity functions as UDFs.

  Odd-3 Odd-4 Odd-5 Odd-6
Number of runs 100 100 100 100
Number of generations 50 50 100 200
Population size 10 30 30 30
Number of fitness cases 8 16 32 64
Function set A O N A O N A O N (A O N)2
User defined functions Odd-2 Odd-3 Odd-4 Odd-5
Terminal set a b c a b c d a b c d e a b c d e f
Head length 7 7 7 7
Number of genes 3 3 3 3
Linking function X X X X
Chromosome length 45 45 45 45
Mutation rate 0.044 0.044 0.044 0.044
One-point recombination rate 0.3 0.3 0.3 0.3
Two-point recombination rate 0.3 0.3 0.3 0.3
Gene recombination rate 0.1 0.1 0.1 0.1
Gene transposition rate 0.1 0.1 0.1 0.1
IS transposition rate 0.1 0.1 0.1 0.1
IS elements length 1,2,3 1,2,3 1,2,3 1,2,3
RIS transposition rate 0.1 0.1 0.1 0.1
RIS elements length 1,2,3 1,2,3 1,2,3 1,2,3
Success rate 95% 100% 100% 100%


In the next section we will analyze yet another method for dealing with building blocks manipulation, namely, the use of automatically defined functions and compare it with the methods discussed before.

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