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Lecture 2-3-4 ASSOCIATIONS, RULES, AND MACHINES CONCEPT OF AN E-MACHINE: simulating symbolic read/write memory by changing dynamical attributes of data in a long-term memory Victor Eliashberg Consulting professor, Stanford University, Department of Electrical Engineering. Slide 1.

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  1. Lecture 2-3-4 ASSOCIATIONS, RULES, AND MACHINES CONCEPT OF AN E-MACHINE: simulating symbolic read/write memory by changing dynamical attributes of data in a long-term memory Victor Eliashberg Consulting professor, Stanford University, Department of Electrical Engineering Slide 1

  2. SCIENTIFIC / EGINEERING APPROACH External system (W,D) Computing system, B, simulating the work of human nervous system Sensorimotor devices, D B D W Human-like robot (D,B) External world, W “When you have eliminated the impossible, whatever remains, however improbable, must be the truth.” (Sherlock Holmes) Slide 2

  3. ZERO-APPROXIMATION MODEL s(ν) s(ν+1) Slide 3

  4. BIOLOGICAL INTERPRETATION Working memory, episodic memory, and mental imagery Motor control AM AS Slide 4

  5. y X X11 X12 AM sel y NM.y 0 1 NM symbol read move type symbol Teacher current state of mind next state of mind PROBLEM 1:LEARNING TO SIMULATE the Teacher This problem issimple: system AM needs to learn a manageable number of fixed rules. Slide 5

  6. PROBLEM 2: LEARNING TO SIMULATE EXTERNAL SYSTEMThis problem ishard: the number of fixed rules needed to represent a RAM with n locations explodes exponentially with n. y 1 2 NS NOTE. System (W,D) shown in slide 3 has the properties of a random access memory (RAM). Slide 6

  7. Programmable logic array (PLA): a logic implementation of a local associative memory (solves problem 1 from slide 5) Slide 7

  8. BASIC CONCEPTS FROM THE AREA OF ARTIFICIAL NEURAL NETWORKS Slide 8

  9. Typical neuron Neuron is a very specialized cell. There are several types of neurons with different shapes and different types of membrane proteins. Biological neuron is a complex functional unit. However, it is helpful to start with a simple artificial neuron (next slide). Slide 9

  10. Neuron as the first-order linear threshold element: Inputs:xk R’ Parameters: g1,… gm R’ R’is the set of real non-negativenumbers Output: yR’ xk xm Equations: x1 gk m du g1 Σgkxk gm τ + u = (1) dt k=1 u y=L( u ) (2) where, { u if u > 0 (3) y L( u) = 0 otherwise A more convenient notation xk is the k-th component of input vector g1 x1 gk is the gain (weight) of the k-th synapse gk xk m gm y=L( u ) xm Σgkxk is the total postsynaptic current s = s k=1 τ u is the postsynaptic potential u y is the neuron output u 0 τis the time constant of the neuron y Slide 10

  11. Input synaptic matrix, input long-term memory (ILTM) and DECODING ILTM gx1k gxnk gxik x1 xk DECODING (computing similarity) x xm s1 si sn si sn s1 An abstract representation of (1): m Σgxikxk (2) fdec: X × Gx S si = (1) i=1,…n k=1 Notation: x=(x1, .. xm)are thesignals from input neurons (not shown) gx = (gxik) i=1,…n, k=1,…m is the matrix of synaptic gains -- we postulate that this matrix represents input long-term memory (ILTM) s=(s1, .. sn)is thesimilarity function Slide 11

  12. Layer with inhibitory connections as the mechanism of the winner-take-all (WTA) choice s1 si sn xinh q α α α ui un u1 Equations: τ τ τ (1) β β β dn di d1 (2) Note. Small white and black circles represent excitatory andinhibitorysynapses, respectively. (3) s1 sn si Procedural representation: RANDOM CHOICE iwin : { i / si=max sj > 0 } (4) ( j ) if (i == iwin) di=1; else di=0; (5) iwin “: “ denotes random equally probable choice Slide 12

  13. Output synaptic matrix, output long-term memory (OLTM) and ENCODING di dn d1 dn d1 di y1 y ENCODING (data retrieval) yk gyki gykn gyk1 yp OLTM An abstract representation of (1): n Σgykidi yk = (2) fenc: D × Gy Y (1) k=1,…p i=1 NOTATION: d=(d1, .. dm)signals from the WTA layer (see previous slide) gy = (gyki) i=1,…n, k=1,…m is the matrix of synaptic gains -- we postulate that this matrix represents output long-term memory (OLTM) y=(y1, .. yp) output vector Slide 13

  14. A neural implementation of a local associative memory (solves problem 1 from slide 5)(WTA.EXE) addressing by content DECODING S21(I,j) S21(i,j) Input long-term memory (ILTM) N1(j) RANDOM CHOICE Output long-term memory (OLTM) ENCODING retrieval Slide 14

  15. A functional model of the previous network[7],[8],[11] (WTA.EXE) (1) (2) (3) (4) (5) Slide 15

  16. HOW CAN WE SOLVE THE HARD PROBLEM 2 from slide 6? Slide 16

  17. External system as a generalized RAM Slide 17

  18. Concept of a generalized RAM (GRAM) Slide 18

  19. Slide 18 Slide 19

  20. Representation of local associative memory in terms of three “one-step” procedures: DECODING, CHOICE, ENCODING Slide 20

  21. INTERPRETATION PROCEDURE Slide 21

  22. At the stage of training, sel=1; at the stage of examination sel=0. System AS simply “tape-records” its experience,(x1,x2,xy)(0:ν). y 1 2 NS GRAM NOTE. System (W,D) shown in slide 3 has the properties of a random access memory (RAM). Slide 22

  23. EXPERIMENT 1: Fixed rules and variable rules Slide 23

  24. EXPERIMENT 1 (continued 1) Slide 24

  25. EXPERIMENT 1 (continued 2) Slide 25

  26. A COMPLETE MEMORY MACHINE (CMM) SOLVES PROBLEM 2, but this solution can be easily falsified! Slide 26

  27. GRAM as a state machine: combinatorial explosion of the number of fixed rules Slide 27

  28. Concept of a primitive E-machine Slide 28

  29. s(i) > c ; Slide 29 (α< .5)

  30. Effect of a RAM w/o a RAM buffer G-state E-state 1 2 3 4 2 1 2 3 4 1 3 4 c c c c c c c c c c c c b b b b b b b b b b b b a a a a a a a a a a a a 1 2 3 4 1 2 3 4 1 2 3 4 a c b a c b a c b a c b 3 4 2 1 a b c a b c a b c a b c Slide 30

  31. EFFECT OF“MANY MACHINES IN ONE” 2 6 1 3 4 5 7 8 n=8 locations of LTM X(1) 0 0 0 0 1 1 1 1 G-state X(2) 0 0 1 1 0 0 1 1 y(1) 0 1 0 1 0 1 0 1 E-state AND m+1 A table with n=2 OR m 2 N=2 represents XOR different m-input 1-output Boolean functions. Let m=10. Then n=2048 and N=2 NAND 1024 NOR Slide 31

  32. Simulation of GRAM with A={1,2}, and D={a,b,ε} i 2 6 1 3 4 5 7 addr 1 1 2 2 1 2 din a b b a dout a b b a b a s (i) is the number of matches in the first two rows. Input(addr,din) = (1,ε) producess(i)=1fori=1andi=2. ν= 5 s(i) if ( s(i)>e(i) ) e(i)(ν+1) = s(i)(ν); else e(i)(ν+1) = c · e(i)(ν) ; τ=1/(1-c) e(i) se(i) = s(i) · ( 1+a ·e(i) ); (a<.5) se(i) dout=bis read fromi=2that hasse(i)=max(se) Slide 32

  33. iwin : {i : se(i)=max(se)>0}; Assume that the E-machine starts with the state of LTM shown in the table and doesn’t learn more, so this state remains the same. What changes is the E-state, e(1),…e(4). Assume that at ν=1, e(1)=..e(4)=0. Let us send the input sequence(addr,din)(1:5) = (1,a), (1,b),(2,a),(2,b),(1,ε). As can be verified, atν= 5, the state e(i) and functions s(i) and se(i) for i=1,..4 are as shown below. Accordingly, iwin=2 and dout=b. s (i) is the number of matches in the first two rows. Input(addr,din) = (1,ε) producess(i)=1fori=1andi=2. i 1 3 4 2 gx(1,1:4) addr 1 1 2 2 din gx(2,1:4) a b b a if ( s(i)>e(i) ) e(i)(ν+1) = s(i)(ν); else e(i)(ν+1) = c · e(i)(ν) ; τ=1/(1-c) dout gy(1,1:4) a b b a ν= 5 se(i) = s(i) · ( 1+a ·e(i) ); (a<.5) s(i) e(i) y =gy(iwin); (a<.5) se(i) Slide 33

  34. What can be efficiently computed in this “nonclassical” symbolic/dynamical computational paradigm (call it the E-machine paradigm)? What computational resources are available in the brain -- especially in the neocortex -- for the implementation of this paradigm? How can dynamical equations (such as the last equation in slide 29) be efficiently implemented in biologically plausible neural network models? Slide 34

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