Kinematic and tessellation models of self-repair

One of the causes of the great survival capability of some biological systems is the fact that they are systems of individuals such that the system behavior does not critically depend on anyone individual. Such systems can be regarded as redundant systems with different kinds and levels of redundancy. For example, although each specimen of a biological society has a limited life span, the whole society can exist for a much longer time. This is primarly due to self-reproduction, one form of redundancy. On a lower level, the lifespan of a specimen can be larger than the lifespan of parts of the specimen. Again this is due to redundancy, but of another form if the specimen is required to have a definite internal structure. The concept of self-repair is studied in tenus of automata theory. Different classes of automata (systems), like well-localized and non-well-localized automata, are considered. The parts (components) of the automata are uniformly exposed to errors. It is shown that if an automaton of a certain class has a lifespan not exceeded by any other automaton of the class, then it must contain a "repairing" mechanism. Such automata can be said to be self-repairing with respect to the class. A definition of self-repair is suggested. It is found that a self-repairing system which is well-localized with respect to its inputs and outputs has a finite lifespan. This corresponds to the finite life span we observe in nature for any animal or for any well-localized machine. On the other hand, if we relax the condition that the automaton be well-localized, then infinite lifespans can be obtained. Such automata also have self-reproducing properties and we obtain here a connection between the concepts of self-repair and self-reproduction. These self-repairing automata are, in a way, similar to growing biological societies with loosely specified internal structures. Automata have been studied by von Neumann and others with different kinds of models, namely kinematic models and tessellation (this name was suggested by Moore) models. We shall see that these two models can be traced back to the particle and wave aspects of matter. We will develop the concept of self-repair from both aspects