THE BOOLEAN ALGEBRA OF NEURAL NETS 107 



then c 3 fires if either c x or c 2 has fired: 



N 3 (t) . = .SNAt)vSN 2 (t), 



the symbol v denoting disjunction. Finally, if c x has two endfeet on 

 c 3 , but c 2 forms an inhibitory connection with c 3 (Figure Id), then 

 c 3 fires provided d has fired and c 2 has not: 



N 3 (t) . = .™SN 2 (t) .SNAt). 



It may be that c ; is itself a member of a class cu . For the mo- 

 ment, however, we consider the case when the net contains no cycles 

 so that if we pass successively from any neuron to any of its effer- 

 ents in the net we shall never pass twice over the same neuron. Then 

 associated with each neuron Oj except the peripheral afferents of the 

 net there is a single equivalence of the form (2), and neither the 

 assertion nor the negation of Ni(t) occurs anywhere on the right in 

 this equivalence. If there occurs on the right of (2) any N (t) , 



where c. r is not a peripheral afferent, then N. t (t) can be replaced by 



the right member of the equivalence associated with c . . By continu- 



ing sequentially we shall find ultimately that Ni(t) is equivalent to a 

 certain disjunction of conjunctions of propositions of the form 

 S n N k (t) and of the negations of such, n being everywhere at least 1, 

 and every c^ being a peripheral afferent. Moreover, since every term 

 in the disjunction is a sufficient condition for the firing of Ci at the 

 time t , no term in the disjunction can consist exclusively of nega- 

 tions. The set of all equivalences of the type just described consti- 

 tutes a solution of the net, since this set contains the necessary and 

 sufficient condition for the firing at time t of every neuron in the net, 

 in terms of the firing and non-firing of the peripheral afferents at 

 earlier times. 



The right member of an equivalence of the type just described 

 McCulloch and Pitts call a temporal propositional expression (abbre- 

 viated TPE) and it denotes a temporal propositional function. A 

 TPE is any sentence formed out of primitive sentences of the type 

 Ni(t) operated upon any number of times by the operator S , the 

 sentences being combined by disjunctions and conjunctions, as well 

 as any sentence formed by conjoining a sentence of the foregoing 

 type with the negation of another sentence of this type. Otherwise 

 put, a TPE is any disjunction of conjunctions of sentences S n Nj(t) 

 and M S n N k (t), except that no conjunction can consist exclusively of 

 negations. The importance of this notion of a TPE lies in the fact 

 that any TPE is "realizable," which is to say that given any TPE, it 

 is possible to describe a non-cyclic net of such a sort that this TPE 



