The paper is in three parts. First, we use simple adversary arguments to redevelop and explore some of the no-free-lunch (NFL) theorems and perhaps extend them a little. Second, we clarify the relationship of NFL theorems to algorithm theory and complexity classes such as NP. We claim that NFL is weaker in the sense that the constraints implied by the conjectures of traditional algorithm theory on what an evolutionary algorithm may be expected to accomplish are far more severe than those implied by NFL. Third, we take a brief look at how natural evolution relates to computation and optimization. We suggest that the evolution of complex systems exhibiting high degrees of orderliness is not equivalent in difficulty to optimizing hard (in the complexity sense) problems, and that the optimism in genetic algorithms (GAs) as universal optimizers is not justified by natural evolution. This is an informal tutorial paper—most of the information presented is not formally proven, and is either “common knowledge” or formally proven elsewhere. Some of the claims are intuitions based on experience with algorithms, and in a more formal setting should be classified as conjectures.

We compare the search power of crossover and mutation in genetic algorithms. Our discussion is framed within a model of computation using search space structures induced by these operators. Isomorphisms between the search spaces generated by these operators on small populations are identified and explored. These are closely related to the binary reflected Gray code. Using these we generate discriminating functions that are hard for one operator but easy for the other and show how to transform from one case to the other. We use these functions to provide theoretical evidence that traditional GAs use mutation more effectively than crossover, but dispute claims that mutation is a better search mechanism than crossover. To the contrary, we show that methods that exploit crossover more effectively can be designed and give evidence that these are powerful search mechanisms. Experimental results using GIGA, the Gene Invariant Genetic Algorithm, and the well-known GENESIS program support these theoretical claims. Finally, this paper provides the initial approach to a different method of analysis of GAs that does not depend on schema analysis or the notions of increased allocations of trials to hyperplanes of above-average fitness. Instead it focuses on the search space structure induced by the operators and the effect of a population search using them.