A categorical characterization of general automata
David Rine1971, Information and Control
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Abstract
Let d" be a category of input objects and a/a category of output objects. Assume that d" has limits and d has colimits; then, the behavior of every diagram in the category of abstract machines (d', ~¢) over a connected scheme can be recorded in its limit. Thus, the theory of limit preserving functors describes the interconnections between the various traits that the total system exhibits. Hence, "the systems theorist is told" what jobs an arbitrary system can do for the given system.
Key takeaways
- As we shall see in Theorem 1.6 this coordination is assured only if the diagram of relations is connected in the categorical sense (Mitchell, 1965).
- The proof of the next theorem falls within the scope of Theorem 2.1.
- We can now go to the general argument in order to reach the conclusion of Theorem 2.4.
- The power of Theorem 2.1 lies within the availability of limit preserving functorsfrom S (Mitchell, 1965); for, as we shall see from Theorem 3.1, if P is some other system, every function b of P = ~3 b has a solution set in S which acts like a set of replacements, i.e., what jobs can b do in S. In figure 10, Ci is a component (maximal connected diagram), S3(b ) is a member of b's solution set in S, and ak is the limit of component C k .
- Another way of representing replacement of input objects is illustrated in Fig. 11.
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