Technology-Independent Design of Neurocomputers: The Universal Field Computer 1

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1 Tecnology-Independent Design of Neurocomputers: Te Universal Field Computer 1 Abstract Bruce J. MacLennan Computer Science Department Naval Postgraduate Scool Monterey, CA 9393 We argue tat AI is moving into a new pase caracterized by biological rater tan psycological metapors. Full exploitation of tis new paradigm will require a new class of computers caracterized by massive parallelism: parallelism in wic te number of computational units is so large it can be treated as a continuous quantity. We suggest tat tis leads to a new model of computation based on te transformation of continuous scalar and vector fields. We describe a class of computers, called field computers tat conform to tis model, and claim tat tey can be implemented in a variety of tecnologies (e.g., optical, artificial neural network, molecular). We also describe a universal field computer and sow tat it can be programmed for te parallel computation of a wide variety of field transformations. 1. Te New AI Traditional Artificial Intelligence tecnology is based on psycological metapors, tat is, idealized models of uman cognitive beavior. In particular, models of conscious, goal-directed problem solving ave provided te basis for many of AI s accomplisments to date. As valuable as tese metapors ave been, we believe tat tey are not appropriate for many of te tasks for wic we wis to use computers. In particular, symbolic information processing does not seem to be a good model of te way people (or animals) beave skillfully in subcognitive tasks, suc as pattern recognition and sensorimotor coordination. Tus, te needs of tese applications are driving Artificial Intelligence into a new pase caracterized by biological metapors. We call tis pase, caracterized by a combination of symbolic and nonsymbolic processing, te new AI (MacLennan, in press). Te tecnology of te new AI already includes neural information processing, genetic algoritms, and simulated annealing. Te new AI will allow us to make use of massively parallel computers (including neurocomputers), optical computers, molecular computation, and, we expect, a new generation of analog computers. Current AI tecnology as been quite successful in a number of tasks, for example, cess, diagnosis of blood diseases and teorem proving. Many oter tasks remain beyond its capabilities, including face recognition, autonomous movement and continuous speec recognition. Te interesting ting is tat te tasks tat AI as been most successful wit are tose tat we commonly consider iger cognitive activities, specifically, tose activities tat can be performed by umans, but by few oter animals. On te oter and, te tasks tat currently stretc te capabilities of AI tecnology are tose tat are lower on te scale of cognitive accomplisment. Specifically, tey are activities tat almost any animal can perform wit skill. A rodent may not be able to prove teorems, but it can effectively navigate its way troug a complicated terrain, avoid predators, and find food and accomplis tis wit a comparatively small brain constructed of comparatively slow devices. It as been truly said tat computers will replace matematicians long before tey will replace carpenters. Unfortunately, many important applications of artificial intelligence require just te sort of activities tat stretc te current tecnology. Terefore it is important to seek te reason for te limitations of te current tecnology, and to see if tere is a way around tem. Current AI tecnology is based on psycological metapors; its algoritms mimic conscious, rational tougt. Tus, tis tecnology deals best wit verbalizable knowledge (knowledge tat), deductive reasoning and discrete categories. However, as we ve seen, tere are oter kind of intelligent beavior. 1. Te researc reported erein was supported by te Office of Naval Researc under contract N WR Autor s address after October, 1987: Computer Science Department, Ayres Hall, University of Tennessee, Knoxville, Tenn Paper to appear in proceedings of IEEE First Annual International Conference on Neural Networks, San Diego, June 21-2,

2 In te past AI as attempted to reduce all intelligent activity to intellectual activity; te computer is in effect a disembodied brain. We claim tat AI is entering a new pase tat recognizes te role of te body in intelligent beavior, and tat empasizes unconscious, tacit knowledge wat we migt call skillful beavior, as opposed to knowledge-based beavior. Tis new pase attempts to come to grips wit suc problems as unverbalized knowledge (knowledge ow), immediate perception, sensorimotor coordination, approximate and context-sensitive categorization, and everyday (as opposed to intellectual) beavior. Te new AI is caracterized by a greater use of biological (as opposed to psycological) metapors. Harbingers of te new AI include te recent researc activity in neurocomputation, genetic algoritms, cellular arcitectures and molecular computation. In tis paper we present tecniques by wic tese nontraditional computers can be designed independently of teir implementation tecnology. 2. Field Transformation Computers 2.1 Massive Parallelism Many of te newer computational paradigms are caracterized by te processing of massive amounts of data in parallel. For example, in neurocomputers and Boltzmann macines (Hinton and Sejnowski, 1983) large numbers of simple processing elements compute in parallel. Similarly, some optical computers process in parallel te elements of an optical wav efront. A key advantage of molecular computers will be te ability of large numbers of molecules to operate in parallel. Consideration of new computing paradigms suc as tese leads us to offer te following definition of massive parallelism: 2 Definition (Massive Parallelism): A computational system is massively parallel if te number of processing elements is so large tat it may conveniently be considered a continuous quantity. Of course, tis definition admits borderline cases. For most purposes, a million processors will qualify, but 16 will not. In some circumstances as few as a tousand may be sufficient. Wy is it relevant tat te number of processors can be taken as a continuous quantity? One reason is tat for some kinds of massively parallel computers te number of processors is in fact continuous, or nearly so. Examples are optical and molecular computers. You don t count processors; you measure teir quantity in terms of some macroscopic unit. Te second reason for seeking continuity is tat te matematics is simpler. Wen te number of processing elements is very large, statistical metods can often be applied. Also, continuous matematics (suc as te infinitesmal calculus) can be applied, wic is muc more tractable tan discrete matematics (e.g. combinatorics). Under our definition of massive parallelism, it doesn t matter weter te implementation tecnology is in fact discrete or continuous (or nearly so, as in molecular computing). In eiter case te design of te computer can be described by continuous matematics. Ten, if te intended implementation tecnology is discrete, we can select out of te continuum of points a sufficiently large finite number. Tis selection may be eiter regular (e.g. in a grid) or random (subject only to statistical constraints). In tis way muc of te design of massively parallel computers can be accomplised independently of te implementation tecnology. 2.2 Field Transformation Given our definition of massive parallelism, it is clear tat te processing elements of a massively parallel computer cannot be individually programmed; tey must be controlled en masse. How can tis be done? We suggest tat te operation of massively parallel computers is best tougt of as field processing. Tat is, we tink of a very large aggregation of data as forming a continuous (scalar or vector) field (analogous to an electrical field). Te individual processing steps operate on entire fields to yield entire fields. Since a continuum of data is transformed in parallel, we acieve massive parallelism. A simple example is an optical convolution, wic operates on an entire optical field in parallel. Conventional digital (and analog) computers perform point processing, tat is, tey operate on one (or a 2. Peraps infinite or continuous parallelism would be a better term. -2-

3 few) points at a time. We suggest tat te full benefit of massive parallelism will be acieved by field processing, te parallel transformation of entire fields of data. (Te distinction between point processing and field processing is analogous to tat between word-at-a-time and vector processing in functional programming; see Backus, 1978.) In te remainder of tis section we discuss field transformation computers: computers designed for field processing. 2.3 Classes of Field Transformations Tere are two classes of field transformations: nonrecursive and recursive (or functional and temporal). In nonrecursive processing, fields are passed troug various transforms and are combined wit one anoter to yield an output field; tere may be feed-forward but no feed-back. Nonrecursive transformation applies a (peraps complex) function to its input fields to yield its output fields. Te input-output dependency is functional: same inputs, same outputs. Recursive processing is like nonrecursive except tat tere is feed-back. Terefore te fields evolve in time according to te differential equations describing te system. Te output of a recursive transform depends on its inputs and on its current state. We expect field computers to permit elementary field transforms to be connected in a variety of ways to yield more complex recursive and nonrecursive field transforms. We also expect field computers to permit limited point processing. Scalar values are often useful as global parameters for controlling field processing operations. For example, te average ligt intensity of a scene (a scalar) migt be used to control a field transformation for contrast enancement. Point processing can also be used for controlling te tresolds of large numbers of neural units (e.g., in simulated annealing; see Kirkpatrick et al.), or for determining global reaction parameters for molecular processes. Many field processing tasks will depend on a number of fixed or constant fields tat must be properly initialized. Tere are a number of sources for tese fixed fields. For example, tey may be computed by anoter field transformation process and loaded into read-only memories. Fixed fields can also be generated by training processes, wic build tem up by recursive field processing. Finally, fixed fields can be modified adaptively as te system runs, in wic case tey are only relatively fixed (i.e., tey cange at a muc slower rate tan te variable fields). 2. General Purpose Field Computers We can imagine implementing various recursive and nonrecursive field processing systems by assembling te appropriate elementary field transforms. We expect tat many special purpose field computers will be implemented in just tis way (indeed, some already are). On te oter and, te flexibility of general purpose digital computers as sown us te value of programmability. In tese te connection of te processing elements is transitory and under te control of an easily alterable program. Is it possible to design a general purpose field computer, tat is, a field computer tat can be programmed to emulate any oter field computer? We argue tat it is, and muc of te rest of tis paper is in pursuit of tis goal. Wat would a general purpose field computer be like? We expect tat it would ave a number of field storage units, of various dimensionalities, for olding (bounded) scalar and vector fields. Some of tese would old fixed fields for controlling te processing. Oters would old variable fields (1) captured from input devices, or (2) to be presented to output devices, or (3) as intermediate fields in recursive processes. Tere would also be some scalar registers. Field transformation processes would be implemented by programmed connections between elementary field transforms. Tese elementary operations sould permit programming any useful field transformation in a modest number of steps. Note tat we are not too concerned about te number of steps, since eac processes in parallel a massive amount of data. Some of te elementary transforms may be sensitive to scalar parameters, tus permitting global control. Is it possible to find a set of elementary transforms tat can be assembled to yield any useful field transformation? Tis is exactly wat we establis in te next section. We sow ow a limited variety of -3-

4 processing units can be assembled to compute almost any field transformation to any desired accuracy. Of course, te more accuracy we want, te more units it will take, but tat is acceptable. Wat is not acceptable is to replace field processing by point processing. To do so would be completely impractical: you can t do operations serially. Tus we must identify a universal set of field transforms in terms of wic all oters can be implemented. 3. A Universal Field Computer 3.1 Introduction Te value of te Turing macine as a model of digital computation is tat it allows establising te limitations and capabilities of discrete symbol processing. In particular, te universal Turing macine establises te possibility of general purpose digital computers. On te oter and, te universal Turing macine is an idealization; it as te minimum capabilities required to compute all computable functions, so it is muc less efficient tan real computers. Real computers extend te facilities of te universal Turing macine for te sake of practical (efficient) computation. In tis section we outline an analogous idealized model of computation for massively parallel and analog computation, tat is, for field computers. Tis universal field computer is capable of implementing any function defined on fields. Of course, tere are some limitations on te functions tat can be so computed, just as tere are limitations on te functions tat can be computed by Turing macines. We claim tat te class of implementable functions is sufficiently broad to include all tose required for practical applications. Also, we expect tat real (practical) general purpose field computers will provide more tan tis minimum of facilities. Tere are a number of ways we migt design a universal field computer, just as tere are many alternatives to te universal Turing macine tat compute te same class of functions. Fourier analysis and interpolation teory bot suggest ways of implementing arbitrary functions in terms of a limited class of primitives. In te rest of tis section we explore a particular approac, based on an extension of Taylor s Teorem to field transformations. 3.2 Taylor Series Approximation of Field Transforms In tis section we develop te basic teory of functions on scalar and vector fields and of teir approximation by Taylor series. Once it is understood tat fields are treated as continuous-dimensional vectors, it will seen tat te matematics is essentially tat of finite-dimensional vectors. Tus te treatment ere is euristic rater tan rigorous. First we consider scalar fields; later we turn to vector fields. As usual we take a scalar field to be a function! from an underlying set " to an algebraic field K, tus!: " # K. For our purposes K will be te field of real numbers, R. We use te notation $(") for te set of all scalar fields over te underlying set " (K = R being understood). Tus, $(") is a function space, and in fact a linear space under te following definitions of field sum and scalar product: (! + % ) t =! t + % t (&!) t = &(! t ) Note tat we often write! t for!(t), te value of te field at te point t. As a basis for tis linear space we take te unit functions ' t for eac t ( ". Tey are defined ' t (t) = 1 ' t (s) = 0, if s ) t Te preceding definitions sow tat we can tink of scalar fields as vectors over te set ". Since we want to be quite general, we assume only tat " is a measurable space. In practice, it will usually be a closed and bounded subspace of E n, n-dimensional Euclidean space. Tus we typically ave one, two and tree dimensional closed and bounded scalar fields. Since " is a measure space, we can define an inner product between scalar fields: (1) (2) --

5 ! * % +! t % t dt. (3) We also define te norm:! =! t dt. () Tus $(") is te function space L 1 ("). Note tat te ' t are not an ortogonal set under tis norm, since ' t = 0. We first consider scalar valued functions of scalar fields, tat is functions f : $(") # R. We prove some basic properties of tese functions, culminating in Taylor s teorem. Definition (Differentiability): Suppose f is a scalar valued function of scalar fields, f : $(") # R, and tat f is defined on a neigborood of! ( $("). Ten we say tat f is differentiable at! if tere is a field D ( $(") suc tat for all - ( $(") were / # 0 as - # 0. Teorem: If f is differentiable at! ten f is continuous at!. Proof: Since f is differentiable at! we know Terefore, f (% ). f (!) = (%.!) * D + / %.! 0 (%.!) * D + / %.! 0 D %.! + / %.! = ( D + / ) %.!. Tus f is continuous at!. f (! + - ). f (!) = - * D + / - (5) f (% ). f (!) = (%.!) * D + / %.!. Since our vectors are continuous dimensional, partial derivatives are wit respect to a coordinate t ( " rater tan wit respect to a coordinate variable. To accomplis tis it s convenient to make use of te Dirac delta functions: 1 t (t) = 2 (6) 1 t (s) = 0, for s ) t Of course, by te first equation above we mean 1 t (s) = lim #0 for s. t < 3 /2. Note te following properties of te delta functions (fields): 1 t = 1 1 t *! =! t (7) Definition (Partial Derivative): Te partial derivative, at coordinate t ( ", of f : $(") # R, evaluated at!, is defined: f (!) = lim f (! + 1 t). f (!). 1 t (8) Teorem: If f is differentiable at! ten te first order partial derivatives exist at!. Proof: First observe tat by differentiability -5-

6 f (! + 1 t ). f (!) Recalling tat / # 0 as # 0, observe = f (!) + 1 t * D + / 1 t. f (!) = 1 t * D + / 1 t / = 1 t * D + / / = D t + / / lim 5 f (! + 1 t ). f (!) 5 5. D 5 t55 = lim D t + / /. D t = lim / = 0 Hence, f (!) = D t, were D is te field wose existence is guaranteed by differentiability. Tus te 1 t partial derivative exists. Wat is te field D wose points are te partial derivatives? It is just te gradient of te function. Definition (Gradient): te gradient of f (!) is a field wose value at a point t is te partial derivative at tat point, f (!): 1 t [6 f (!)] t = 1 t f (!). Te gradient can also be expressed in terms of te basis functions: 6 f (!) = ' t 1 t f (!) dt. Wen no confusion will result, we use te following operator notations: (9) (10) 6 f = ' t f /1 t dt 6 = ' t /1 t dt (11) /1 t = 1 t * 6 Note tat by te definitions of differentiability (Eq. 5) and te gradient (Eq. 9) we ave tat were / # 0 as - f (! + - ). f (!) = - * 6 f (!) + / -, (12) # 0. Tis leads to te concept of a directional derivative. Definition (Directional Derivative): Te directional derivative in te direction - is given by te following formulas (sown in bot explicit and operator forms): 6 - f (!) = - * 6 f (!) = -, t f (!) dt " 1 t. 6 - = - * 6 = -, t /1 t dt " Note tat te notation is accurate in tat (- * 6) f (!) = - * [6 f (!)]. Also note tat /1 t = 6 1 t. Lemma: If f is differentiable in a neigborood of!, ten Proof: By te definition of te derivative: (13) d dx f (! + x- ) = - * 6 f (! + x- ). (1) -6-

7 d dx f (! + x- ) = lim f [! + (x + )- ]. f (! + x- ) = lim f (! + x- + - ). f (! + x- ) - * 6 f (! + x- ) + / - = lim = lim - * 6 f (! + x- ) + / - / since / # 0 as # 0. = - * 6 f (! + x- ) Teorem (Mean Value): Suppose f : $(") # R is continuous on a neigborood containing! and %. Ten, tere is a 7, , suc tat f (% ). f (!) = (%.!) * 6 f (8 ). were 8 =! + 7 (%.!) (15) Proof: Let - = %.! and consider te function F(x) = f (! + x- ). f (!). x[ f (% ). f (!)]. Since f is continuous, so is F. Now, since F(0) = F(1) = 0, we ave by Rolle s Teorem tat tere is a 7, , suc tat F9(7 ) = 0. Note tat F9(x) = d { f (! + x- ). f (!). x[ f (% ). f (!)]} dx = d dx f (! + x- ). [ f (% ). f (!)]. By te preceding lemma Hence, substituting 7 for x, Terefore, transposing we ave and te teorem is proved. F9(x) = - * 6 f (! + x- ). [ f (% ). f (!)] 0 = F9(7 ) = - * 6 f (! + 7- ). [ f (% ). f (!)]. f (% ). f (!) = - * 6 f (! + 7- ) Teorem (Taylor): Suppose tat f and all its partial derivatives troug order n + 1 are continuous in a neigborood of!. Ten for all - suc tat! + - is in tat neigborood tere is a 7, , suc tat f (! + - ) = f (!) f (!) f (!) n! 6ṉ f (!) + 1 (n + 1)! 6n+1 - f (! + 7- ). (16) Proof: By te Taylor teorem on real variables, f (! + t- ) = f (!) + d/dt f (!)t d2 /dt 2 f (!)t n! d n /dt n f (!)t n + 1 (n + 1)! d n+1 /dt n+1 f (! + 7- )t n+1. Observe tat by te preceding lemma -7-

8 Terefore, d n dt n f (! + t- ) = 6ṉ f (! + t- ). f (! + t- ) = f (!) f (!)t f (!)t n! 6ṉ f (!)t n + Setting t = 1 giv es te desired result. Te extension to a function of several scalar fields is routine. 3.3 A Universal Field Computer Based on Taylor Series Approximation 1 (n + 1)! 6n+1 - f (! + 7- )t n+1. We can use Taylor s Teorem to derive approximations of quite a general class of scalar valued functions of scalar fields. Tus, if we equip our universal field computer wit te ardware necessary to compute Taylor series approximations, ten we will be able to compute any of a wide class of functions (namely, tose functions wose first n partial derivatives exist and are continuous). Terefore, consider te general form of an n-term Taylor series: f (!) = n : 1 k! 6ḵ f (! 0 ), were - =!.! 0 (17) k=1 Wat ardware is required? Clearly we will need a field subtractor for computing te difference field - =!.! 0. We will also need a scalar multiplier for scaling eac term by 1/k!; we will also need a scalar adder for adding te terms togeter. Te arder problem is to find a way to compute 6 ḵ f (! 0 ) for a vector - tat depends on te (unknown) input!. Te trouble is tat te - s and te 6s are interleaved, as can be seen ere: 6 ḵ f (! 0 ) = (- * 6) k f (! 0 ) = (- * 6) k.1 [- * 6 f (! 0 )] = (- * 6) k.1 -, t1 f (! 0 ) dt 1 " 1 t1. =... - t1 - t tk k 1 t1 1 t tk f (! 0 ) dt 1 dt 2... dt k We want to separate everyting tat depends on -, and is tus variable, from everyting tat depends on f (! 0 ), and is tus fixed. Tis can be accomplised (albeit, wit extravagant use of our dimensional resources) by means of an outer product operation. Terefore we define te outer product of two scalar fields: Note tat if!, % ( $(") ten! % ( $(" 2 ). (! % ) s,t =! s % t (18) To see ow te outer product allows te variable and fixed parts to be separated, consider first te case : f (! 0 ) = - s - t 1 s 1 t f (! 0 ) dt ds = (- - ) s,t (6) s (6) t f (! 0 ) dt ds = (- - ) s,t (6 6) s,t f (! 0 ) dt ds = 2(- - ) x (6 6) x dx f (! 0 ) = (- - ) * (6 6) f (! 0 ) Now we can see ow te general case goes. First we define te k-fold outer product: Ten,! [1] =! (19)! [k+1] =!! [k] -8-

9 Te n-term Taylor series ten becomes f (!) = 6 ḵ f (!) = - [k] * 6 [k] f (!) (20) n : 1 k! (!.! 0) [k] * 6 [k] f (! 0 ) k=1 Since! 0 is fixed, we can compute eac 6 [k] f (! 0 ) once, wen te field computer is programmed. Ten, for any giv en input! we can compute (!.! 0 ) [k] and take te inner product of tis wit 6 [k] f (! 0 ). Tus, in addition to te components mentioned above, computing te Taylor series approximation also requires outer and inner product units tat will accommodate spaces up to tose in $(" n ). We consider a very simple example of Taylor series approximation. Suppose we want to approximate defint!, wic computes te definite integral of!, defint! =! s ds. First we determine its partial derivative at t by observing: Tus, lim defint (! + 1 t). defint! 1 t defint! = 1, and we can see tat,! s + 1 t (s) ds.! ", s ds = lim ",! s ds + 1 ", t (s) ds.! = lim ", s ds " = lim 1 t / = 1 (21) 6 defint! = 1, (22) were 1 is te constant 1 function, 1 t = 1. Tis leads to a one term Taylor series, wic is exact: Note tat 1 is a fixed field tat must be loaded into te computer. 3. Transformations on Scalar and Vector Fields defint! =! * 1 (23) Te previous results apply to scalar valued functions of scalar fields. Tese kinds of functions are useful (e.g., to compute te average value of a scalar field), but tey do not exploit te full parallelism of a field computer. Acieving tis requires te use of functions tat accept a (scalar or vector) field as input, and return a field as output. We briefly sketc te teory for scalar field valued functions of scalar fields; transformations on vector fields are an easy extension of tis. By a scalar field valued function of scalar fields we mean a function F: $(") # $("). Suc a function is considered a family of scalar valued functions f t : $(") # R for eac t ( "; tese are te component functions of F. Note tat F can be expressed in terms of its components: F(!) = f t (!) ' t dt (2) More briefly, F = f t ' t dt. F is decomposed into its components by 1 t * F(!) = f t (!). Next we consider te directional derivative of a field transformation. For a scalar function f, 6 - f (!) is a scalar tat describes ow muc f (!) canges wen its argument is perturbed by a small amount in te direction -. For a field transformation F, 6 - F(!) sould be a field, eac component of wic reflects ow muc te corresponding component of F(!) canges wen! moves in te direction -. Tat is, [6 - F(!)] t = 6 - f t (!). Hence, 6 - F(!) = ' t 6 - f t (!) dt$, (25) or, more briefly, 6 - F = ' t 6 - F dt. It s easy to sow tat 6 - = - * 6. Te corresponding Taylor series -9-

10 approximation is: F(!) = n : 1 k! [(!.! 0) * 6] k F(! 0 )$ As before, outer products can be used to separate te variable and fixed components. k=1 We consider vector fields briefly. Recall tat any tree-dimensional vector field $ can be considered tree scalar fields!, %, 8 were (26) $ t =! t i + % t j + 8 t k (27) Similarly, a function tat returns a tree-dimensional vector field can be broken down into tree functions tat return scalar fields. Tus, we see tat a transformation on finite dimensional vector fields can be implemented by a finite number of transformations on scalar fields. To ensure te continuity of field valued functions, certain restrictions must be placed on te fields permitted as arguments. Altoug tese restrictions are still under investigation, we believe tat it is sufficient tat te input field s gradient be bounded at eac stage. Tis will be te case for all pysically realizable fields. Tis restriction on allowable inputs finds its analogy in digital computers: legal input numbers are restricted to some range; numbers outside tat range may cause underflow or overflow in te subsequent computation. In te same way ere, fields wose gradients are too large may lead to incorrect results.. Conclusions We av e argued tat AI is moving into a new pase caracterized by biological rater tan psycological metapors. Full exploitation of tis new paradigm will require a new class of computers caracterized by massive parallelism: parallelism in wic te number of computational units is so large it can be treated as a continuous quantity. We suggest tat tis leads to a new model of computation based on te transformation of continuous scalar and vector fields. We av e described a class of computers, called field computers tat conform to tis model, and ave indicated tat tey may be implemented in a variety of tecnologies (e.g., optical, artificial neural network, molecular). To illustrate te capabilities and limitations of tis model we ave described a universal field computer and sown tat it can be programmed for te parallel computation of a wide variety of field transformations. Te universal field computer is not practical as it stands; it s an idealized computing engine. Neverteless, just as te universal Turing macine suggests ways of designing practical von Neumann computers, so te universal field computer suggests ways of designing practical general-purpose field computers. 5. References 1. Backus, Jon, Can Programming be Liberated from te von Neumann Style? A Functional Style and its Algebra of Programs, Comm. ACM, Vol. 21, No. 8 (August 1978), pp Hinton, G. E., and Sejnowski, T. J., Optimal Perceptual Inference, in Proc. IEEE Computer Society Conf. on Computer Vision and Pattern Recognition (Wasington, D.C., 1983), pp Kirkpatrick, S., Gelatt, C. D., Jr., and Vecci, M. P., Optimization by Simulated Annealing, Science, Vol. 220 (1983), pp MacLennan, B. J., Logic for te New AI, in Aspects of Artificial Intelligence, J. H. Fetzer (ed.), D. Reidel, in press. Note: Tis paper was presented June 22 at te IEEE First International Conference on Neural Networks, San Diego, CA, June 21-2, It is included in te proceedings. -10-