384 Information Storage and Neural Control 



and a series of A^, in it such that M (zi) . = . Pr^ (M, A^o, •• • , 

 Zi, sai) is true of it, where sax has the form A^(0). We shall call it 

 realizable in the extended sense, or simply realizable, if for some n S"{Pri) 

 ipi, • ... pp. Zu s) is realizable in the above sense. Cp, is here the 

 realizing neuron. We shall say of two laws of nervous excitation 

 which are such that every S which is realizable in either sense 

 upon one supposition is also realizable, perhaps by a different 

 net, upon the other, that they aie equivalent assumptions, in 



that sense. 



The following theorems about realizability all refer to the ex- 

 tended sense. In some cases, sharper theorems about narrow 

 realizability can be obtained; but in addition to greater com- 

 plication in statement this were of little practical value, since our 

 present neurophysiological knowledge determines the law of ex- 

 citation only to extended equivalence, and the more precise 

 theorems differ according to which possible assumption we make. 

 Our less precise theorems, however, are invariant under equiva- 

 lence, and are still sufficient for all purposes in which the exact 

 time for impulses to pass through the whole net is not crucial. 

 Our central problems may now be stated exactly: first, to find 

 an effective method of obtaining a set of computable S constituting 

 a solution of any given net; and second, to characterize the class 

 of realizable S in an effective fashion. Materially stated, the 

 problems are to calculate the behavior of any net, and to find a 

 net which will behave in a specified way, when such a net exists. 

 A net will be called cyrlic if it contains a circle: i.e., if there 

 exists a chain c„ C/+i , ... of neurons on it, each member of the 

 chain synapsing upon the next, with the same beginning and end. 

 If a set of its neurons Ci , c-i , . . . , Cp is such that its removal from 

 VX leaves it without circles, and no smaller class of neurons has this 

 property, the set is called a cj>clic set, and its cardinality is the 

 order o/vX. In an important sense, as we shall see, the order of a 

 net is an index of the complexity of its behavior. In particular, 

 nets of zero order have especially simple properties; we shall 

 discuss them first. 



Let us define a temporal propositional expression (a TPE), desig- 

 nating a temporal propositional function {TPF), by the following 

 recursion: 



