APPENDIX A 



INTRODUCTION 



Michael H. Arbib 



"A 



LOGICAL Calculus of the Ideas Immanent in Nervous 

 Activity" by Warren S. McCulloch and Walter Pitts is the classic 

 paper on neurophysiological automata theory and still merits 

 reading today, almost twenty years after its publication. Section I. 

 which gives the neurophysiological basis for the model, is still valid 

 in all its essentials and remains the most readable discussion of this 

 basis. Section II, on the theory of nets without circles, and the dis- 

 cussion of Section IV are equally excellent. 



However, Section III, the theory of nets with circles, was only 

 intended as a sketchy account. It was presented in Carnap's 

 notation, which was not apt for the task at hand, and is incomplete, 

 hard to read, and contains many errors. Hence, for this part of 

 the theory, we advise the reader to turn to more recent publications. 

 The theory of nets with circles was first fully worked out by 

 Kleene (1) and has since been given an elegant re-presentation 

 by Copi, Elgot, and Wright (2). The assertions of McCulloch and 

 Pitts concerning the connection between the neural nets and Turing 

 machines [Turing (3)] have been fully worked out by Arbib (4). 



REFERENCES 



\. Kleene, S. C: Representation of Events in Nerve Nets and Finite 

 Automata. In Automata Studies, ed. by C. E. Shannon and J. Mc- 

 Carthy, Princeton, Princeton University Press, 1956, p. 3. 



2. Copi, I. M., Elgot, C. C, and Wright, J. B.: Reahzation of events by 



logical nets. J. Assn. Computing Mchy., 5.- 181-1 96, 1958. 



3. Turing, A. M.: On computable numbers, with an application to the 



Entscheidungs-problem. Proc. London Math. Soc. (2) ^i.-230-265, 

 1936; with a correction, ibid., 43:544-546, 1947. 



4. Arbib, M.: Turing machines, finite automata and neural nets. J. 



Assn. Computing Mchy., 8:461-415, 1961. 



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