Symbolic Generalizations

Use of Sommerhoff's Concept of Directive Correlation:

"This is Sommerhoff's concept of directive correlation. It should be noted that this concept is wholly operational. It specifies a practical test, and judges solely by the result. For this reason it is applicable with equal facility and rigor to the behaviors of living systems and automata. It is one of the fundamental concepts of cybernetics.

"In order to show its precision and objectivity, Sommerhoff specified it in analytic form: that is, in the mathmatics of continuous functions. Continuity, however, is not essential, and it can equally be expressed in the general algebra of sets and relations.

If the actions are:

e: the mapping of disturbances into environment's responses,
r: the mapping of disturbances into regulator's responses,
c: the mapping of the couple (environment's responses and regulator's responses) into the outcomes
G: the subset of outcomes that correspond to "goal achieved",

then the property of the regulator that shows directive correlation is equivalent to

c(r^oe ^-1) ***** G
(in the notation of Bourbaki, and in which, in the composition of r and e^-1, the set eliminated is the set of disturbances).

This expression I believe to be of fundamental importance in the general theory of "brain-like" action, for it shows exactly what is essential. Here I wish only to call attention to the fact that if we convert it to the equivalent form:

r*****[c^-1(G)] ^oc,
r is shown explicitly as a function of e. In other words, a system can be brain-like only against a defined environment. There is no such thing as a system being brain-like in general, or against all environments." ⁷