General Schema Theory for Genetic Programming with Subtree ...
General Schema Theory for Genetic Programming with Subtree ...
General Schema Theory for Genetic Programming with Subtree ...
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<strong>General</strong> <strong>Schema</strong> <strong>Theory</strong> <strong>for</strong> GP: Part II<br />
5.2 Exact <strong>Schema</strong> Theorem <strong>for</strong> GP <strong>with</strong> Standard Crossover Acting on Linear<br />
Structures<br />
As an example let us further specialise the result in the previous section to the case of<br />
linear structures. In the case of function sets including only unary functions, programs<br />
and schemata can be represented as linear sequences of symbols like ������� ����� ��� (see also<br />
Section 6.1), and so:<br />
�<br />
�<br />
�<br />
�<br />
The hyperschema ��¥�� � � � ����� � � §������ � � � � ����� � � � .<br />
The VA hyperschema ��¥�������� �<br />
§���§������ ¥ ��� �<br />
�<br />
� � ����� ��� , where the notation<br />
��� �����<br />
means ��� ¥�� � repeated times. This is equivalent to the non-VA hyperschema<br />
�<br />
�<br />
�<br />
�<br />
� � ����� � � since, in the case of linear structures, � ’s in function nodes can be<br />
¥ ���<br />
� replaced by ’s.<br />
The set ��¥�������� ����� ��� §������ �<br />
The set ��¥�������� ����� ��� §���§������ �<br />
�<br />
���<br />
�<br />
����� <strong>for</strong> ,<br />
otherwise.<br />
�<br />
���<br />
� ����� � � ¥ �<br />
��� � � �<br />
�<br />
�<br />
�<br />
� � ����� ��� if ������������� ,<br />
��� ¥<br />
�<br />
� otherwise.<br />
�<br />
By substituting these quantities into Equation 12 and per<strong>for</strong>ming some simplifications<br />
one obtains the following result<br />
� xo ¥���� ����� ��� §�©���� ���<br />
�<br />
�<br />
��� � � �<br />
�¡ £¢<br />
�<br />
� � � �<br />
� ���<br />
� � ¥ �<br />
��� §�©�� � � ����� ¥���� �<br />
�<br />
��� �<br />
�<br />
� � ����� ��� §�©�� ¥�¥ �<br />
���������<br />
� (13)<br />
Equation 13 can be shown to be equivalent to the schema theorem <strong>for</strong> linear structures<br />
reported in (Poli and McPhee, 2001b).<br />
5.3 A Different Form of Macroscopic <strong>Schema</strong> Theorem <strong>for</strong> One-point Crossover<br />
As an additional example, we will derive a GP schema theorem <strong>for</strong> one-point crossover<br />
equivalent to the one described in Section 2.2.<br />
Again, in order to use Theorem 2 we need to first check whether one-<br />
point crossover is node invariant, i.e. whether it � ¥ ��§���� ��� § ��� ���<br />
is true that<br />
��¥���� ��§ ��¥���� ��� . It is easy to see that � ¥ ��§���� ��� ���<br />
��� ¥���¥���� ��§ ��¥���� ��� , and that<br />
¥���� �<br />
�<br />
§ ���<br />
§ ��� � if and only if � � ¥���¥���� ��§ ��¥���� ��� . There<strong>for</strong>e, the expression of<br />
� � ¥����<br />
¥���� §�� � § ��� §�� � � ��� § ��� � in Part I can be trans<strong>for</strong>med as follows �<br />
� �¥¤ �<br />
� �¦¤ � ¥ ��§���� � � § � � ��� � �<br />
So, we can replace � ¥ ��§���� �<br />
��� § ��� � if ����� and � �<br />
§ ¥����<br />
� otherwise,<br />
§<br />
�<br />
��� ¥���¥�� � ��§ ��¥�� � ��� if ����� and � �<br />
� �<br />
otherwise,<br />
�<br />
� �¥¤ � ¥ ��§���� ��¥���� ��§ ��¥���� �����<br />
�<br />
�<br />
§ �<br />
�<br />
¥���� § ��� � ,<br />
� in Equation 10 <strong>with</strong> the expression<br />
� ¥ � �<br />
obtaining, after some simplification:<br />
�<br />
�<br />
§ �<br />
�<br />
��� � ¥ ������� ¥��<br />
¥��<br />
�<br />
§ �<br />
� §<br />
���<br />
�<br />
�<br />
¥���¥�� � ��§ ��¥�� � ��� ,<br />
Evolutionary Computation Volume ?, Number ? 19
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