Toward a theory of evolutionary computation.

We outline a theory of evolutionary computation using a formal model of evolutionary computation--the Evolutionary Turing Machine--which is introduced as the extension of the Turing Machine model. Evolutionary Turing Machines provide a better and a more complete model for evolutionary computing than conventional Turing Machines, algorithms, and Markov chains. The convergence and convergence rate are defined and investigated in terms of this new model. The sufficient conditions needed for the completeness and optimality of evolutionary search are investigated. In particular, the notion of the total optimality as an instance of the multiobjective optimization of the Universal Evolutionary Turing Machine is introduced. This provides an automatic way to deal with the intractability of evolutionary search by optimizing the quality of solutions and search costs simultaneously. Based on a new model a very flexible classification of optimization problem hardness for the evolutionary techniques is proposed. The expressiveness of evolutionary computation is investigated. We show that the problem of the best evolutionary algorithm is undecidable independently of whether the fitness function is time dependent or fixed. It is demonstrated that the evolutionary computation paradigm is more expressive than Turing Machines, and thus the conventional computer science based on them. We show that an Evolutionary Turing Machine is able to solve nonalgorithmically the halting problem of the Universal Turing Machine and, asymptotically, the best evolutionary algorithm problem. In other words, the best evolutionary algorithm does not exist, but it can be potentially indefinitely approximated using evolutionary techniques.