No free lunch, Kolmogorov complexity and the information landscape

The permutation closure of a single function is the finest level of granularity at which a no-free-lunch result can hold (Schumacher et al., 2001). Using the information landscape framework which was introduced in "Information Landscapes" by Borenstein and Poli (2005), we are able to identify the unique properties of each closure. In particular, we associate each closure with the amount of information its members contain. This allows us to compute bounds on the expected performance of an algorithm on members of that closure. Moreover, we suggest a new way to measure the Kolmogorov complexity of a landscape. This allows us to associate each permutation closure with a particular Kolmogorov complexity.