QUANTITY OF VARIETY 7/21 



7/20. Within the set of determinate machines further constraints 

 may be appHed. Thus the set can be restricted to those that have 

 a certain set of states as operands, or to those that have only one 

 basin, or to those that are not reducible. 



A common and very powerful constraint is that of continuity. 

 It is a constraint because whereas the function that changes arbi- 

 trarily can undergo any change, the continuous function can change, 

 at each step, only to a neighbouring value. Exercise 4 gives but a 

 feeble impression of the severity of this constraint. 



Ex. 1 : The set of closed single-valued transformations (absolute systems) on 

 three states «, b, c has 27 members (compare Ex. 7/7/7). How many 

 members remain if we add the restriction that the absolute system is to 

 have no state of equihbrium? 



Ex. 2: (Continued.) Similarly, but the restriction is that there must be only 

 one basin. 



Ex. 3 : (Continued.) Similarly, but the restriction is that the transitions a ^ b 

 and 6 -> c may not occur. 



Ex. 4: A vector has ten components, each of which can take one of the values: 

 1, 2, 3, 4. How much variety has the set of vectors if (i) the components 

 vary independently (S.7/I2); (ii) under the rule that no two adjacent compo- 

 nents may differ in value by more than one unit? 



7/21. Learning and constraint. For the psychologist, an important 

 example of constraint occurs in learning. Pavlov, for instance, in 

 one experiment gave both thermal and tactile stimuli, as well as 

 reinforcement by meat powder, in the following combinations: 



(The fourth combination occurred, of course, in the intervals.) 

 Now the total combinations possible are eight; Pavlov presented 

 only four. It was an essential part of the experiment that the full 

 set should not be given, for otherwise there would be nothing par- 

 ticular for the animal to learn. Constraint was an essential feature 

 of the experiment. 



The same principle can be seen more simply in learning by associa- 

 tion. Suppose one wanted the subject, given a letter, to reply with 

 a number according to the rule 



A given 

 B „ 

 C „ 



reply with 2 



3 

 133 



