Contains reports on one research project.National Science Foundation (Grant GP-2495)National Aeronautics and Space Administration (Grant NsG-334)National Aeronautics and Space Administration (Grant NsG-496

Spann, R. N.

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XIII. PROCESSING AND TRANSMISSION OF INFORMATIONProf. P. EliasProf. F. C. Hennie IIIProf. R. S. KennedyProf. C. E. ShannonProf. H. L. Van TreesProf. J. M. WozencraftDr. R. E. KahnD. S. ArnsteinR. A. CarpenterD. ChaseR. ColtonM. EbertD. FalconerM. GoodmanJ. JohnsonMaxF. McCannQ. McDowellH. MeynC. MoldenC. W. NiessenG. C. O'LearyR. PilcJ. T. Pinkston IIIE. W. Portner, Jr.J. S. RichtersD. L. SnyderR. N. SpannW. R. SutherlandM. G. TaylorA. BOUNDS ON MULTIPLE-THRESHOLD FUNCTIONSThis report presents some preliminary results regarding multiple-threshold func-tions. 1, Z A lower bound on the number of thresholds required to realize all functionsof n variables will be derived.Multiple-threshold functions will be defined as follows.DEFINITION 1: A Boolian function f(x 1, .. . , xn) is k-threshold threshold realizableiff there exists a set of real numbers w. . . . .wn, T1' . . . . .Tk such thatj 1 wix i -kHj=j=lTj) > 0 f(x, ... x ) = qT. < 0 f (x I ..... xn ) = ,i wixi -where q = 0 or 1. Thus a given set of wi and T. define one function with q = 1 and thecomplement of that function with q = 0. It is also clear that if a function is realizablewith k thresholds it is realizable with m thresholds for m greater than k.It is of substantial theoretical and practical interest to determine the minimum num-ber of thresholds required to realize any function of n variables. We shall give a lowerbound for this minimum number. To the author's knowledge no one has exhibited ann-variable function that requires more than n thresholds. We will show that for suf-ficiently large n such functions must exist.We see that*This work was supported in part by the National Science Foundation (GrantGP-2495), and the National Aeronautics and Space Administration (Grants NsG-334 andNsG-496).QPR No. 78 207(XIII. PROCESSING AND TRANSMISSION OF INFORMATION)r w x. - T = 0 (2)j= 1will be satisfied iff Eq. 3 holds.nw.x. - T. 0 for some j, 1 < j < k. (3)i= 1Consider an (n+k)-dimensional space (called the realization space) with axes labeledw, ... , wn  1' ' . . . . . Tk. Each point in this space corresponds to multiple-thresholdrealizations of a function and its complement. Both these realizations require k orfewer thresholds. Using the vectors W = (w I , ... ,w n ) and X = (x .. . ,x n), we canwrite Eq. 3 asW-X-T. 0. (4)For any particular X, Eq. 4 is the equation of a hyperplane passing through the originof the realization space.For a givenX, the k hyperplanes defined by Eq. 5 below divide the realization spaceinto a finite number of regions, the exact number depending on the relative orientationsof the hyperplanes.W-X-T.= 0 1 j <k (5)The coordinates of any point on any hyperplane are such thatk(W X-T) = 0. (6)j=1The coordinates of a point that is not on any hyperplane (internal to a region) are suchthat eitherkI (W.X-Tj) > 0j=lor (7)kS(W . X-T.) < 0.j= 1Furthermore, the coordinates of all points internal to a given region will yield the samesign for the product in Eq. 7.QPR No. 78 208(XIII. PROCESSING AND TRANSMISSION OF INFORMATION)Now let Xbe a vector in n-dimensional switching space. Each of the Zn possible X'sgenerates k hyperplanes. Thus all 2 X vectors generate k2n hyperplanes, which dividethe realization space into a finite number of regions. The coordinates of a point internalto a given region specify a Boolian function and its complement, both of which requirek or fewer thresholds for their realizations. The coordinates associated with all pointsin a given region correspond to realizations of the same two functions. It is possible,however, that different regions of the realization space may correspond to the same twofunctions.Let S(k, n) be the maximum number of regions into which the realization space canbe divided by k2 n hyperplanes, all passing through the origin. Then 2S(k, n) is an upperbound to T(k, n), the number of n-variable Boolian functions that are realizable with k orfewer thresholds. Using a result of Cameron,3 we haven+k-lS(k, n) = 2 k (kh -1) (8)=O0This givesTHEOREM 1:n+k-lT(k, n) < 4 (k2n) . (9)P= 0Employing a bound of Winder 4 and then using Stirling's approximation, we have4 (k2n) n+k-l ek n+k-lT (k, n) < < ( + k - 1 (10)(n+k-1)! 1Let K(n) be the smallest number of thresholds required to realize all 2 n functionsof n variables. K(n) must be such that)n+K(n)-l1eK(n) n  2 n2 K> T(K(n),n) > 2 (11)qr- (n + K(n) - 1Using the fact2, 5 that K(n) > n and K(n) < 2 n and a series of manipulations on the left-most term of Eq. 11, we can establishTHEOREM 2:n-2K(n) > 2 for n > 2. (12)nThus for values of n > 8, K(n) > n, and hence there must exist functions of 8 variablesQPR No. 78 209(XIII. PROCESSING AND TRANSMISSION OF INFORMATION)that require more than 8 thresholds.Also, with reference to Spann 6 we have shown the following.THEOREM 3: For n > 10 the class of Modular Threshold functions does not containall functions.I would like to thank Dr. D. Haring of the Electronic Systems Laboratory of M. I. T.for bringing Winder's correspondence to my attention.R. N. SpannReferences1. P. Ercoli and L. Mercurio, "Threshold Logic with One or More than One Threshold,"Proceedings of the IFIPS Congress, 1962 (North-Holland Publishing Company,Amsterdam, 1962), pp. 741-745.2. D. R. Haring, "Multi-threshold Elements I and II," Memorandum 4631-M-2 and4631-M-3, Electronic Systems Laboratory, M. I. T., March 1965.3. S. H. Cameron, "An Estimate of the Complexity Requisite in a Universal DecisionNetwork," Bionics Symposium, Dayton, Ohio, December 1960, pp. 197-212; WADDRept. 60-600.4. R. O. Winder, "Bounds on Threshold Gate Realizability,"' IEEE Trans. PGECVol. EC-12, No. 5, pp. 561-564, October 1963.5. B. Elspas and R. A. Short, "A Bound on the Run Measure of Switching Functions,"IEEE Trans. on Electronic Computers, Vol. EC-12, February 1964, pp. 1-4.6. R. N. Spann, "Modular Threshold Functions," QPR No. 76, Research Laboratory ofElectronics, M. I. T., January 15, 1965, pp. 241-247.QPR No. 78 210

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Contains report on one research project.Joint Services Electronics Program (Contract DA36-039-AMC-03200(E))National Science Foundation (Grant GP-2495)National Aeronautics and Space Administration (Grant NsG-334)National Aeronautics and Space Administration (Grant NsG-496

Kennedy, Robert S.

XIV. PROCESSING AND TRANSMISSION OF INFORMATION*Prof. P. Elias D. Chase C. W. NiessenProf. R. G. Gallager J. R. Colton J. H. NymanProf. F. C. Hennie III P. M. Ebert G. C. O'LearyProf. I. M. Jacobs D. D. Falconer R. PilcProf. R. E. Kahn L. M. Goodman J. T. Pinkston IIIProf. R. S. Kennedy C. J. Johnson E. W. Portner, Jr.Prof. C. E. Shannon J. Max J. S. RichtersProf. H. L. Van Trees R. F. McCann D. L. SnyderProf. J. M. Wozencraft G. Q. McDowell R. N. SpannD. S. Arnstein J. H. Meyn W. R. SutherlandR. A. Carpenter J. C. Molden M. G. TaylorA. OPTICAL COMMUNICATION SYSTEMSAn interferometric demodulator for use with frequency-position modulation has beeninvestigated by R. E. Olsen in a thesis entitled "Interferometric Demodulation of FM atOptical Frequencies." His work suggests that interferometric techniques can be used toimplement practical and nearly optimal processors for such alphabets. The alphabetsize is limited by the Finesse if the best performance is to be realized.Methods of generating desirable communication signals have been studied byC. J. Johnson in a Master's thesis entitled "Electro-optic Properties and CommunicationApplications of BaTiO3." He investigated those properties of large single crystalsamples of BaTiO3 which are relevant to optical modulators. These crystals weregrown by Dr. A. Linz and V. Belruss of the Massachusetts Institute of Technology.In a third investigation, J. R. Colton has established a mathematical model fortransmission channels employing optical waveguides that distort the signals because ofsurface imperfections. He envisaged the imperfections as random in character andapplied a Central Limit Theorem argument to Helmholtz's Equation; he concluded thatthe resultant channel output may be approximated by a Gaussian random process. Hiswork will culminate in a thesis to be entitled "Analysis of an Optical Fiber as a DiscreteCommunications Channel," which will be submitted to the Department of ElectricalEngineering, M. I. T., in partial fulfillment of the requirements for the degree of Masterof Science.R. S. KennedyThis work was supported in part by the Joint Services Electronics Program (Con-tract DA36-039-AMC-03200(E)), the National Science Foundation (Grant GP-2495), andthe National Aeronautics and Space Administration (Grants NsG-334 and NsG-496).QPR No. 79 205

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