Contains a report on a research project.M.I.T. Research Laboratory of ElectronicsM.I.T. Computation Cente

Edwards, D. J.

Goldberg, S. H.

Rochester, N.

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X. ARTIFICIAL INTELLIGENCEProf. J. McCarthy D. G. Bobrow D. C. LuckhamProf. M. L. Minsky R. K. Brayton K. MalingProf. N. Rochestert D. J. Edwards I). M. R. ParkProf. C. E. Shannon S. H. Goldberg S. R. RussellP. W. Abrahams L. Hodes J. R. SlagleL. KleinrockA. MACHINE MANIPULATION OF ALGEBRAIC EXPRESSIONSA list processing computer programming language (1, 3), called LISP, is used toprogram the IBM 704 computer to analyze a linear bilateral electric network whose com-ponents have literal or numerical values. LISP was chosen because it is particularlysuited to the expression of mathematical and logical algorithms such as are associatedwith manipulations and simplification of algebraic expressions of the type found in elec-trical circuit analysis. Basically, our problem can be stated: Given a network madeup of ideal resistors, capacitors, and inductors, calculate literal expressions for circuitcharacteristics such as the driving point and transfer impedance between given nodes.A network is described to the machine as a list of circuit elements in which eachelement is described as a resistor, capacitor, or inductor connected between two nodes.Each element is also given a name thatcan be any algebraic expression. Pro-grams (which take the form of functionsin LISP) have been written to accept suchS- N2  a network; these programs will allow theR < calculation of a literal expression for thedriving-point and transfer characteristics.In the following example (see Fig. X-1),Fig. X-1. Simple resistor network usedin a transfer impedance cal- the transfer impedance between N 1 andculation. N2 with respect to ground was calculated.This simple network was chosen becausethe computer result was easy to check and because it was a good test for our programs.First, the machine constructed a 3 X 3 admittance matrix for the network and reducedthis to a 2 X 2 matrix by the method of matrix postmultiplication. Next, the determinantof the 2 X 2 admittance matrix was found and used to find the transfer impedance:e 2  -1. 33/R-1 (1)1 (2. 67/R)(2. 67/R) - (-1. 33/R) 2This work is supported jointly by Research Laboratory of Electronics and theComputation Center, M.I. T.tVisiting Professor of Electrical Engineering, M. I. T.132(X. ARTIFICIAL INTELLIGENCE)An important by-product of our electric -circuit-analysis problem was the simplifica-tion functions that were written in such a way that they could operate on an algebraicexpression such as Eq. 1 and reduce it to its simplest form. The simplification func-tions were used to simplify the expression for the transfer impedance (Eq. 1) to 0. 25 R,which is the correct transfer impedance. The computer took less than 18 seconds forthis entire transfer impedance calculation. In a similar manner the driving-point imped-ance between N 1 and ground was found to be 0. 5 R.For the network shown in Fig. X-2, the machine required 2.2 minutes to computethe driving-point admittance,e 2 3 3 4 2 2e I 2.0 CRL s + LRs + CLs 4 + Lsi 2 23 2 24 2 351 2.0 CLRs 2 + 2.0 CL s + 2.0 RC L s + Ls + CLsand approximately 3. 1 minutes to calculate the transfer impedance,34 2e 2  CRL s + RL si 223 2 23 23 2241 2.0RC Ls +CRLs + CLs  + CRL s + CL s + LRs + Ls + C2 L -RIn this instance, a complicated program was required for clearing fractions. Becausethis program was run in the interpretive mode, it was slow. When the LISP compiler isavailable and the clear fractions program is compiled, the time will be greatly reduced.The machine also calculated the driving-point impedance between nodes N1 and No ofthe circuit of Fig. X-3 in 2.9 minutes asel - 0.872485 R1The success of this method of electric circuit analysis depends to a great extent uponthe ability of the simplification functions to find the simplest form of any algebraicexpression. Simplification is accomplished by the systematic application of an algorithmthat embodies the usual simplification rules. Some examples of algebraic simplificationNI R N60.5RNI  L N 2  N2  R 3RR N 7  025RN3  LN4 R R72R R N5R 2RN 3  N0N0 0Fig. X-2. An example of an RLC network that Fig. X-3. A complex nonplanaris more complicated than that of resistor network.Fig. X-l.133(X. ARTIFICIAL INTELLIGENCE)performed by the machine are:(a+b)(c+d) - b(c+d) - ca = da (2)an + 1/a 2 + b + ca + d = d + b + a(c+l/a3+an- l )  (3)Each of these examples required less than 6 seconds of machine time.During the evaluation of characteristics for complicated electric networks, themachine can produce algebraic expressions that are far too complicated for any humanbeing to understand. Because simplifying assumptions such as eliminating negligiblequantities are needed in most engineering problems, a LISP function, makaprox, thatwould make approximations by eliminating negligible addends in a given algebraic expres-sion was written. Makaprox is given an expression, a list of approximate values, anda limiting ratio; addends should be ignored after the limiting number is reached. Theexpressionab(bc+b+e) [(a+b+e)2(b+e)d(a+c) ] (a+b+c) (4)was simplified with the approximate valuesa z 3.2 b z 5.7 c z 0. 04 d = 2.0 e z 0. 02 (5)and a limiting ratio of 0.01. The simplified expression isba(b+bc)(b+a) 1+2db(c+a) (6)This investigation has demonstrated that a digital computer is a practical means ofperforming algebraic processes, as well as numerical calculations. A more detaileddescription of the project will be found in the theses of Goldberg, Edwards, andRubenstein (3, 4, 5).N. Rochester, S. H. Goldberg, D. J. EdwardsReferences1. J. McCarthy, LISP: A programming system for manipulating symbolic expres-sions, a paper presented at the Annual Meeting of the Association for Computing Machin-ery, Massachusetts Institute of Technology, Cambridge, Massachusetts, September 2-4,1959.2. J. McCarthy, Recursive functions of symbolic expressions and their computationby machine, Quarterly Progress Report No. 53, Research Laboratory of Electronics,M.I.T., April 15, 1959, pp. 124-152.3. S. H. Goldberg, Solution of an electrical network using a digital computer,S. M. Thesis, Department of Electrical Engineering, M.I. T., September 1959.4. D. J. Edwards, Symbolic circuit analysis with the 704 electronic calculator,S. B Thesis, Department of Electrical Engineering, M. I. T., June 1959.5. S. Rubenstein, The construction of the admittance matrix with a digital computer,S. B. Thesis, Department of Electrical Engineering, M. I. T., June 1959.134

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