Statement

This month William got interested in genetic algorithms, that he is now using for solving even the simplest everyday tasks. However, he is unsatisfied with the performance of the crossover phase. In this phase, two strings s and t formed by features of selected individuals are combined together into a new string s & t, in order to mix them up exactly a bit. After days of excruciating research, William finally came up with few requirements that (in his opinion) would characterize a perfect crossover (where s, t stand for strings and a, b, c stand for single characters): (1) s & "" = "" & s = s, i.e., a crossover with an empty string does not change the other string; (2) (cs) & (tc) = t & s, i.e., a crossover among strings such that the first character of the first string equals the last character of the last string results in that equal characters being removed and the remaining strings swapped; (3) (as) & (tb) = b (s & t) a, i.e...