Sufficient conditions for deceptive and easy binary functions

Abstract

This paper finds sufficient conditions for fully or partially deceptive binary functions by calculating schema average fitness values. Deception conditions are first derived for functions of unitation (functions that depend only on the number of 1s in the string) and then extended for any binary function. The analysis is also extended to find a set of sufficient conditions for fully easy binary functions. It is found that the computational effort required to investigate full or partial deception in a problem of sizel using these sufficient conditions isO(2 ^l ) and using all necessary conditions of deception isO(4 ^l ). This calculation suggests that these sufficient conditions can be used to quickly test deception in a function. Furthermore, it is found that these conditions may also be systematically used to design a fully deceptive function by performing onlyO(l ²) comparisons and to design a partially deceptive function to orderk by performing onlyO(k^l) comparisons. The analysis shows that in the class of functions of unitation satisfying these conditions of deception, an order-k partially deceptive function is also partially deceptive to any lower order. Finally, these sufficient conditions are used to investigate deception in a number of currently-used deceptive problems.