Karva notation was developed specifically for gene expression programming and consists of a universal way of representing any mathematical or logical expression that can be represented as a tree. Here we are going to present it briefly, but you can learn more about it in the chapter The Architectures of APS Learning Algorithms.

The chromosomes of gene expression programming code for one or more K-expressions (from Karva notation), and each K-expression is encoded in a gene. Genes and K-expressions are very easy to interpret. For example, the gene:

0123456
+/*abcd

can be represented as a diagram or expression tree (ET):

The translation of a gene into its corresponding ET is straightforward: The first element in the gene (position 0) corresponds to the root of the ET; then, below that node, are attached as many nodes as there are arguments to that function (two, in this case); then these nodes are filled consecutively with the elements in the gene (in this case, positions 1 and 2), and so forth. The process is repeated until a line composed of only terminals is formed (in this case, the third line).

More formally, the gene and ET above are expressed by the expression:

Usually the genes evolved by GEP are more complex than the gene presented above, with noncoding regions at the end. Consider the following gene (the head is shown in blue):

01234567890123456
Q/a*+b-cbabaccbac

where “Q” represents the square root function. This gene has a head of length 8 and a tail of length 9. Note that the head contains both functions and terminals, whereas the tail contains exclusively terminals. The translation of such genes is done exactly as in the previous example, obtaining:

Note that, in this case, not all the elements in the gene were used to construct the ET, as the translation ends whenever a line containing only terminals is formed. In this case, the gene ends at position 16 whereas the K-expression ends at position 9.

Furthermore, GEP chromosomes are usually multigenic, and each gene codes for a sub-ET or sub-program. After translation, the sub-ETs are linked by a particular linking function: addition, subtraction, multiplication, or division in APS 3.0.

For example, the following chromosome composed of three genes (position 0 indicates the beginning of each gene):

012345678012345678012345678
*aQ+abbaa/Q*/aababa*+Qaabba

encodes the following sub-ETs:

Then the sub-ETs are afterwards linked by one of the available linking functions. For instance, if the linking function were addition, then the following program would be obtained:

–

Linking Function

Note that the ET above can be easily linearized into a single K-expression:

01234567890123456
++a*/aQQ*+/aaabba

These manipulations are important in order to fully explore all the APS features, especially the Change Seed method, which allows you to change or create a particular program and then use this program to breed even better programs.

Obviously you’ll need to know the symbols used to represent all the 70 built-in mathematical functions of APS 3.0 in Karva if you wish to create seed programs with them. But they are quite easy to learn as most of the times their names obey common rules of naming mathematical functions in programming languages.

So, Automatic Problem Solver 3.0 evolves models in its native Karva code, but the models evolved by APS in its native language can be automatically parsed into visually appealing expression trees, allowing a quicker and more complete understanding of their mathematical intricacies.

Additionally, Automatic Problem Solver 3.0 offers a total of eight built-in grammars so that the models evolved by APS in its native language can be automatically translated into the most commonly used programming languages (C, C++, C#, Visual Basic, VB.Net, Java, Java Script, and Fortran). This code can then be used in other applications.

And finally, Automatic Problem Solver 3.0 allows also the design of User Defined Grammars so that the models evolved by APS in its native language can be automatically translated into the programming language of your choice if you happen to prefer a programming language not already covered by the eight built-in grammars of APS 3.0 (C, C++, C#, Visual Basic, VB.Net, Java, Java Script, and Fortran).