Introduction It has been long noticed that there are emergent phenomena in genetic programming (GP) runs unintended by the human designer of the algorithm. Early on it was observed that code which does not change the output of the program (i.e. non-e#ective code) appears in many GP runs [34, 38, 2]. It was also noted that bloat a#ects many GP systems. Reasons for bloat and non-e#ective code have been examined in years past [25, 4, 7] and remedies have been developed more or less e#ective under particular circumstances (e.g. [29, 15, 22, 17]). Here we would like to argue that non-e#ective code and bloat are only the tip of an iceberg and that there is more to be discovered about "emergent phenomena" in GP runs. Particularly, we would like to study repetition of patterns in GP-evolved programs. These are instructions, or more interestingly, groups of instructions, that occur several times in a program. In fact long sequences of instructions which are repeated can sometimes be decompose

We model the distribution of functions implemented by non-recursive programs, similar to linear genetic programming (GP). Most functions are constants, the remainder are mostly parsimonious. The effect of ad-hoc rules on GP are described and new heuristics are proposed. Bounds on how long programs need to be before the distribution of their functionality is close to its limiting distribution are provided in general and for average computers. Results for average computers and a model like genetic programming are experimentally tested.

Abstract. Theoretical models of Turing complete linear genetic programming (GP) programs suggest the fraction of halting programs is vanishingly small. Convergence results proved for an idealised machine, are tested on a small T7 computer with (finite) memory, conditional branches and jumps. Simulations confirm Turing complete fitness landscapes of this type hold at most a vanishingly small fraction of usable solutions. 1

The distribution of reversible programs tends to a limit as their size increases. For problems with a Hamming distance fitness function the limiting distribution is binomial with an exponentially small chance (but non zero) chance of perfect solution. Sufficiently good reversible circuits are more common. Expected RMS error is also calculated. Random unitary matrices may suggest possible extension to quantum computing. Using the genetic programming (GP) benchmark, the six multiplexor, circuits of Toffoli gates are shown to give a fitness landscape amenable to evolutionary search. Minimal CCNOT solutions to the six multiplexer are found but larger circuits are more evolvable.

Point mutation has no effect on almost all linear programs. In two genetic

Abstract. Biological chromosomes are replete with repetitive sequences, microsatellites, SSR tracts, ALU, etc. in their DNA base sequences. We discover hierarchical repeating sequences (building blocks?) are evolved by genetic programming in linear time series prediction programs. “DNA whose sequence is not maintained by selection will develop periodicities as a result of random crossover ” George P Smith, Science, 1976. 1

We use the minimal instruction set F-4 computer to define a minimal Turing complete T7 computer suitable for genetic programming (GP) and amenable to theoretical analysis. Experimental runs and mathematical analysis of the T7, show the fraction of halting programs is drops to zero as bigger programs are run. 1

This paper focuses on the halting probability and the number of instructions executed by programs that halt for Turing-complete assembly-like languages and register based machines.

Abstract. Conventional genetic programming research excludes memory and iteration. We have begun an extensive analysis of the space through which GP or other unconventional AI approaches search and extend it to consider explicit program stop instructions (T8) and any time models (T7). We report halting probability, run time and functionality (including entropy of binary functions) of both halting and anytime programs. Turing complete program fitness landscapes, even with halt, scale poorly. 1

We have analysed runs of 12 different genetic programming problems. Some of the problems are the ‘toy ’ problems used in generic programming research and some are significant real world applications. We have generated log files of the runs and looked for recurring and unusual patterns and whether there are any differences between the toy problems and the real world problems. The major finding is that some programs are being evaluated many times. In the real-world problems 30-78 % of the time was spent on reevaluating programs that had already been evaluated. For problems where the evaluation function is expensive significant savings are possible if evaluated programs are cached. A surprising finding was that, for two of the real world problems, a very large number of the evaluations were of 1-node programs. 1.