Contains reports on three research projects.Bell Telephone Laboratories, IncorporatedNational Institutes of HealthTeagle Foundation, IncorporatedUnited States Air Force (WADD Contract AF33(616)-7783

Arbib, M. A.

Cowan, J. D.

Winograd, S.

DSpace@MIT

XIV. NEUROPHYSIOLOGYW. S. McCulloch J. E. Brown R. MelzackJ. A. Aldrich J. D. Cowan N. OnestoF. S. Axelrod R. C. Gesteland W. H. PittsH. J. C. Berendsen K. Kornacker A. TaubM. Blum J. Y. Lettvin P. D. WallDiane MajorA. COMPUTATION IN THE PRESENCE OF NOISE (RELIABLE MACHINESFROM UNRELIABLE COMPONENTS)Information theory may be applied to the problems of computation in the presence ofnoise or reliable processing with unreliable computing elements. It has been demon-strated that redundant computers that achieve arbitrarily low frequencies of error (apartfrom errors in the final outputs) may be constructed, so that they are not completelyredundant but process a finite fraction of information. This depends critically upon theerror behavior of components as a function of complexity. If component errors increasewith complexity, this reliability can be obtained only by decreasing the fraction of infor-mation processed in the computer. However, it is possible to maximize this fraction,for given components and codes. A further result of interest is that such computersneed not be precisely connected and, in fact, a certain bounded fraction of errors inconnection may be tolerated.J. D. Cowan, S. WinogradB. ULTIMATE PERIODICITY OF REAL-TIME COMPUTATION BY A TURINGMACHINE WITHOUT OUTPUTSIn McNaughton's survey, I read of a conjecture by Burks on real-time computa-tion by a "Turing Machine with Output." This led me to consider real-time computationby a Turing machine without output. This report is devoted to the simple but interestingresult that I obtained to the effect that such computation can only produce ultimatelyperiodic sequences - thus whilst Turing has shown 2 that rr and e are computable, wehave now shown that only rational numbers are computable in real time by a Turingmachine without output. Our definition of real-time computation is very general;furthermore, we have found that a number of reasonable alterations in this definitionhas not vitiated the validity of our result.DEFINITION: A Turing machine prints the sequence (ar) in real time if there existsan integer n such that the machine prints a r at time nr, and a r is never printed to theleft of ar-1. (The tape may be printed initially on a finite number of squares.) WeThis work was supported in part by Bell Telephone Laboratories, Inc.; the NationalInstitutes of Health; The Teagle Foundation, Inc.; and in part by the U.S. Air Forceunder WADD Contract AF33(616)-7783.189(XIV. NEUROPHYSIOLOGY)now introduce an auxiliary concept.DEFINITION: A finite printer is a quadruple P = (A, S, s , M), whereA is a finite set (the Alphabet of P);S is a finite set (the set of States of P);s o is a member of S (the initial state of P); andM: S-SXAThe sequence {ar} is printed by P if M(sj_ 1 ) = (s, a ), j = 1, 2, ....LEMMA: Given a Turing machine T and a sequence {ar} that it prints in real time,there exists a finite printer that prints the same sequence.This lemma, whose proof need not be given in this report, leads immediately to ourmain theorem.THEOREM: A sequence printed in real time by a Turing machine is ultimatelyperiodic.PROOF: Immediately follows from the lemma and the finitude of the set of internalstates of a finite printer.M. A. ArbibReferences1. R. McNaughton, The Theory of Automata - A Survey, to appear in Vol. II ofAdvances in Computers, F. L. Alt(ed.) (Academic Press, New York, 1961).2. A. M. Turing, On computable numbers with an application to the entscheidungsproblem, Proc. London Math. Soc., Ser. 2, Vol. 42, pp. 230-265, 1936.C. AN INELEGANT SOLUTION TO THE PROBLEM OF REALIZABILITY OFBOOLIAN FUNCTIONS BY THE FORMAL THRESHOLD TYPE OF NEURONThis report shows that of the 22n possible Boolian functions of n variables, not moren+1 nthan 2 n+l(n) are realizable by the formal threshold type neurons. We give a pro-grammable procedure for listing all such realizable functions, and discuss the numberof steps involved.A formal threshold type of neuron N is determined by n "weights" w l , . . . , wn, andna "threshold" 0. N fires at time t + I iff Z w.X.X 0, where the input to N at time t isi=lgiven by the binary vector (X . . . Xn) (where X = 1 (resp., 0) is to be interpreted as"input line i fires (resp., does not fire)").We say that N realizes the Boolian function f(X1, ... ,X ) if N fires when, and onlywhen, f(X 1, . . .,Xn) = 1.Now let Cn, = {(Xl ... ,Xn) X =0 or 1), be the set of the 2 n vertices of the hypercube190(XIV. NEUROPHYSIOLOGY)nin Euclidean n-space. A hyperplane H: E w.X. = 0 clearly partitions Cn into 2 subsetsi=lnH+, H (i.e., into the set of points for which Z w.X. 80, and the set for whichi=lnE w.X. < ).i=lIt is now clear that the task of listing realizable Boolian functions reduces to that oflisting all partitions of C n by hyperplanes (note, however, that 2 functions are associatedwith each such partition).We observe that any hyperplane H may be moved until it contains n points of Cnand that (H + , H - ) is then determined by these n points, plus the specification of in whichof H+ or H- each of the n points was. Conversely, it is easy to see that any such speci-fication determines a partition of the hypercube by a hyperplane. Hence, our listingprocedure merely consists in looking at all ) sets of n points of Cn ; and then, foreach set, listing the 2n partitions that it determines. This procedure is clearly pro-grammable.By a complicated argument, which will not be given here, we can show that eachhyperplane through n points of C n determines at least one partition of Cn "unique unto2n 2nitself." Hence our procedure produces 2n( ) partitions, of which at least ( n ) aredistinct. For large n, log 2n 2n) ~ log[ (2n) - n2 - n log(n), sothat our procedureis of the optimal "order of order of (sic) magnitude."Since there are 22 possible Boolian functions, and since 2. 2 n2 2  = o 22n asn - oc, we see that, as n increases, only a vanishingly small fraction of Boolian func-tions of n variables can be-realized by the formal threshold type of neurons. This wouldseem to indicate that, at least in the choice of their basic components, computerdesigners should not strive to mimic the human nervous system too closely.In closing, we note that our listing procedure provides an algorithm for testingwhether or not a particular Boolian function is realizable. But it is inelegant, and thenumber of steps is discouraging. Manuel Blum has produced several theorems onrealizability, and is at present extending them in the hope of finding a satisfyingly ele-gant algorithm.M. A. Arbib191

Neurophysiology

http://dspace.mit.edu/bitstream/1721.1/53631/1/RLE_QPR_063_XIV.pdf

Neurophysiology

Abstract

Similar works

Full text

Available Versions

DSpace@MIT