386 Information Storage and Neural Control 



realizes p\{z\) i.n.s., and Figure 1-a sliows a net tliat realizes 

 Spi{zi) and hence SS-i, i.n.s., if So can be realized i.n.s. Now if 

 So and S3 are realizable then S"'S-2. and S"Sz are realizable i.n.s., 

 for suitable m and n. Hence so are S^'^'^So and ^''""'""Sa. Now the 

 nets of Figures lb, c and d respectively realize S{pi{zi)\ p2{z\)), 

 S{pi{zi) . p2{zx)), and S\pi{z,) . ~ poiz,)) i.n.s. Hence S'-+"+' (SiV 

 S2), ^"'+"+1 (Si . So), and ^''«+"+i (Si . ~ So) are realizable i.n.s. 

 Therefore Si v SoSi . SoSi . ~ So are realizable if Si and So are. 

 By complete induction, all TPE are realizable. In this way all 

 nets may be regarded as built out of the fundamental elements 

 of Figures la, b, c, d, precisely as the temporal propositional ex- 

 pressions are generated out of the operations of precession, dis- 

 junction, conjunction, and conjoined negation. In particular, 

 corresponding" to any description of state, or distribution of the 

 values true and false for the actions of all the neurons of a net save 

 that which makes them all false, a single neuron is constructible 

 whose firing is a necessary and sufficient condition for the validity 

 of that description. Moreover, there is always an indefinite number 

 of topologically different nets realizing any TPE. 



Theorem III 



Let there be given a complex sentence Si built up in any manner out 

 of elementary sentences of the form p(zi — zz) where zz is any numeral, 

 by ary of the propositional connections: negation, disfunction, conjunction, 

 implication, and equivalence. Then Si is a TPE and only if it is false 

 when its constituent p(zi — zz) are all assumed false — i.e., replaced 

 by false sentences — or that the last line in its truth-table contains an 

 'F', — or there is no term in its Hilbert disjunctive normal form com- 

 posed exclusively of negated terms. 



These latter three conditions are of course equivalent (Hilbert 

 and Ackermann, 1938). We see by induction that the first of them 

 is necessary, since p{zi — zz) becomes false when it is replaced 

 by a false sentence, and Si v So, Si . S2 and Si . ~ S2 are all 

 false if both their constituents are. We see that the last condition 

 is sufficient by remarking that a disjunction is a TPE when its 

 constituents are, and that any term 



Si . So . . . . Sm . -^ S,„+i . '^ . . . . -^ s„ 

 can be written as 



