Evolving Turing Complete representations

Standard GP, chiefly concerned with evolving functions, which are mappings from inputs to output, is not Turing Complete. We raise issues resulting from attempts at extending standard GP to Turing Complete representations. Firstly, there is a problem when a contiguous piece of code is moved to a new location (in a different program) by crossover. In general its functionality will be altered if global memory is used, as other parts of the program may access the same piece of memory. Secondly, traditional crossover does not respect modules. Crossover can disrupt a group of instructions that were working together (e.g. in the body of a loop) in one parent, but end up separated in two different offspring after reproduction. A crossover operator is proposed that only operates at the boundaries of modules. The identification of module boundaries is made easy by using a representation in which explicit modules are denned, in contrast with other representations where the module boundaries would have to be identified by some other means. The halting problem is a central issue, however as a consequence of this crossover operator we are more likely to produce self terminating programs, thus saving time when testing.