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WALTER R. STAHL

can reproduce all living phenomena, including even such subtle matters as creative activity or emotions; he chooses to exclude entirely the sub-jective aspect of such behavior.

B. Modeling of Self-Reproduction

Self-reproduction of elementary automata or biological units may be included under concept of cybemetic self-organization modeling. It was evidently first recognized as a legitimate problem of mathematics by von Neumann (1949, 1956), who also madę important contributions to the problem of brain models, described in “The Computer and the Brain” (von Neumann, 1958). He proposed two types of elementary models of self-reproduction, which have been discussed and extended in other reports of the writer (Stahl, 1964, 1966) and also in reports by Moore (1962), Lofgren (1962), Kroi’ (1962), and KeyPman (1961). Details of self-reproducing models are beyond the scope of this review. The perti-nent similarity invariants or tests are primarily of the “black box,” relational, and performance types, though numerical criteria, e.g., time or number of steps to reproduce, may also apply.

Physical analogs of the self-reproduction process have also been ad-vanced, as in the work of Morowitz (1959), and it is interesting to notę that mechanical and symbolic-computer reproduction can be compared to each other if the mechanical system is represented by state transition notation. Many conceptual schemes of the origins of life are possible, as reviewed, for example, by Pattee (1966), and should be represented as some sort of state machinę for purposes of comparison with other forms of self-reproduction. Letichevskiy and Dorodnitsyna (1963) have ad-vanced a model of natural selection based on elementary automata operating in a defined medium.

I. Abstract and Axiomatic Models

This fina of models may be included under the broad cybemetic modeling category or considered as an independent concept. Abstract and axiomatic models are paper-and-pencil symbolic constructs based on one or another well-defined form of mathematics, such as group and set theory, relational algebra, the predicate całculus, metamathematic proof theory, solvability and unsolyability theory, Boolean and threshold logie, topology, signal coding theory, etc. The similarity criteria are of an abstract sort and generally in the spirit of the very formal definition of a mathematical model, based on “sentences in a defined language,” as discussed in Section II of this chapter. If two mathematical constructs