368 Murray Eden 



16-configurations with tt ~ 34 would be 250,000. These estimations are plotted 

 on the same figure as the computations for configurations of up to eight cells 

 (Fig. 4). It can be seen that none of the estimates obtainable up to a cumulative 

 probability of 0.907 can be distinguished from the ordinate axis. 



The probability of the most probable A-configuration and of the least 

 probable A-configuration are presented in Table III, for several values of Ar. It 

 will be noted that the probability of the most probable configuration decreases 

 slowly with increasing k. While there is no practical way to make exact com- 

 putations of probability for large values of k, it may be conjectured that the 

 probabiUty of the first ranked configuration is proportional to 1/2^'. On the 

 other hand, the probability of the configurations of the lowest ranks falls very 

 rapidly*. As with the estimates of the number of configurations, exact solutions 

 for probability are readily obtained only for the two-ended configurations. 

 As was noted earlier the number of such forms approximates (1 + V^)'' but the 

 probability associated with each such form is 2^//c ! 



Information theory (15) suggests methods of defining appropriate measures 

 for the distribution of probabilities as a function of A'. If 7V[A'] is the number of 

 distinct configurations containing k cells each, the maximal uncertainty for the 

 A'-array can be defined as i/^." = — Ig A^[A']t. In a similar manner, an uncer- 

 tainty can be defined for an arbitrary generating function, Gj, considered as an 



N[k] 



information source. H{G^^ = ~^Pi Ig/'o ^^ which /?j is the probability that 



i = l 



the generating function G^ will terminate after k — 1 adjunctions in configura- 

 tion oj,. Further, a measure of relatedness (16) may be defined as I{Gj^ — 

 [//,o - H{Gj,,)]. 



What does this mean in teiTns of information theory? Supposing we had 

 a generating function or some procedure that produced every one of these 

 unusual configurations with equal probability. Then the two numbers H^9 and 

 H(Gjj,) would be identical. The uncertainty of such a generating function 

 would be maximal. On the other hand, if the generating process were such as 

 to specify, with probability l,only one out of the total number of configurations, 

 then the uncertainty of the generating process H{Gj,.) would be 0. As can be 

 seen, I(Gj^) for a given generating process carried out through k steps has been 

 defined above simply as the difi'erence of these two quantities. Very briefly 

 then, this measure would suggest that if a knowledge of the generating process 

 does not enable us to predict which of the possible configurations to expect 

 after the process has gone along for k steps, then knowledge of the generating 

 process provides no information. On the other hand, if one can devise a 

 mathematical mechanism, that is, a generating process, that can specify the 

 ultimate form of an organism exactly, then the generating process contains all 

 the information it possibly can. 



Applying this measure to the presently available data and the particular 

 generating function introduced earlier, it is observed that liGj k) increases with 



* It will be noted that the probabihty of the most probable configuration exhibits a maxi- 

 mum at A; = 5. This is an accident attributable to the fact that this particular 5-configuration is 

 the only one containing a cluster and it is asymmetric. Such an accident is extremely unlikely to 

 be found when k is large. 



t The symbol 'Ig' is used here to denote 'logarithm to the base 2'. 



