Finding General Solutions to the Parity Problem by Evolving Machine-Language Representations

Abstract

Evolutionary search on a machine-language program representation is used to nd exact and general solutions to the bit-counting problem|abstractly, given a set B of boolean variables, compute the numb e r o f v ariables in B with value true. Solutions to the bit-counting problem directly solve the parity problem that decides whether the number of true variables in B is even or odd. Our virtual machine language contains generic instructions for manipulating a register set and includes unrestricted branches and bit-wise logical operations. It does not contain instructions specic only to bit-counting or parity. A two-level lex-icographic tness function rst selects for programs that closely approximate (or solve) the problem and then selects programs that properly terminate. We compare two population-based search o p-erators: crossover and macro-mutation. For this problem, macro-mutation sig-nicantly outperforms crossover. Introducing single-point mutation significantly improves the crossover search but only slightly improves the macro-mutation search. Comparison with random search veries the eecacy of the directed search methods on the bit-counting problem.