1. Analog Logic and Differential Belief Propagation Benjamin Vigoda, Ph.D. NSF CCR-0122419 Physics and Media Group MIT Media Laboratory Benjamin Vigoda, November, 2003

2. 2 Optimization/Inference (Belief Propagation) Benjamin Vigoda, November, 2003 Using Physical Systems (Analog Circuits)

3. 3 Conjecture 1: Optimization is probably not how we want to perform all computation. Avogadro-Scale Computing as Optimization Benjamin Vigoda, November, 2003

4. 4 Conjecture 2: But as the number of gates becomes large, the majority of gates will be used to solve inference/optimization problems. (That’s how we use super-computers now.) Avogadro-Scale Computing as Optimization Benjamin Vigoda, November, 2003

5. 5 Implementing Optimization/Inference Natively with Physical Dynamics No clock necessary Parallel programming of 10 23 transistors by saying, “optimize this.” Robustness to noise and fault tolerance arise naturally Enables smoother design trade-offs between: –Speed, Power, Accuracy, and Complexity Implement natively with physical dynamics, “dL{x}/dt=0” Benjamin Vigoda, November, 2003

6. 6 Linear Feedback Shift Register (LFSR) (000111101011001)* or 0* Benjamin Vigoda, November, 2003

7. 7 Transmitter: LFSR Transmitter LFSR: Benjamin Vigoda, November, 2003

8. 8 Receiver: Noise Lock Loop Transmitter: LFSR Low-Complexity LFSR Synchronization by Forward-Only Message Passing, Vigoda, Dauwels, Gershenfeld and Loeliger, submitted IEEE Transactions on Information Theory, 2003 Benjamin Vigoda, November, 2003

9. 9 Noise Lock Loop Performs LFSR Synchronization Benjamin Vigoda, PhD, August, 2003

10. 10 “Soft-XOR gate” Circuit Translinear circuits: A proposed classification. Barrie Gilbert. Electronics Letters, 1975. Benjamin Vigoda, November, 2003

11. 11 “soft-EQUALS Gate” Circuit Benjamin Vigoda, PhD, August, 2003

12. 12 “Soft-DELAY” Continuous-Time Analog Memory Elements Benjamin Vigoda, November, 2003

13. 13 Noise Lock Loop Circuit Synchronization of a Chaotic Dynamical System Using a Soft-Decision Receiver. Justin Dauwels, Matthias Frey, Tobias Koch, Hans-Andrea Loeliger, Patrick Merkli, Benjamin Vigoda*.

14. 14 Design Flow: Compile MATLAB to Spice to Circuits * Waveform probing commands.probe.options probefilename="D:+ probesdbfile="D:+ probetopmodule="3 softxors".SUBCKT softxor p_x0 p_x1 p_y0 p_y1 p_z0 p_z1 GND Vdd M4 N5 p_y1 N7 GND NMOS_49 W='48*l' L='4*l' AS='40*l*l' AD='40*l*l' PS='24*l' PD='24*l' M=1 M8 N1 p_y0 N7 GND NMOS_49 W='48*l' L='4*l' AS='40*l*l' AD='40*l*l' PS='24*l' PD='24*l' M=1 M7 N8 p_y1 N1 GND NMOS_49 W='48*l' L='4*l' AS='40*l*l' AD='40*l*l' PS='24*l' PD='24*l' M=1 M3 N8 p_y0 N5 GND NMOS_49 W='48*l' L='4*l' AS='40*l*l' AD='40*l*l' PS='24*l' PD='24*l' M=1 M9 GND p_x1 N1 GND NMOS_49 W='48*l' L='4*l' AS='40*l*l' AD='40*l*l' PS='24*l' PD='24*l' M=1 M6 N5 p_x0 GND GND NMOS_49 W='48*l' L='4*l' AS='40*l*l' AD='40*l*l' PS='24*l' PD='24*l' M=1 … M12 p_z1 N7 Vdd N6 PMOS_49 W='48*l' L='10*l' AS='66*l*l' AD='66*l*l' PS='60*l' PD='60*l' M=1 M11 Vdd N7 N7 N2 PMOS_49 W='48*l' L='10*l' AS='66*l*l' AD='66*l*l' PS='60*l' PD='60*l' M=1.ENDS * Main circuit: 3 softxors.tran 1n 1600n.include Mami_15.md.options abstol=1e-15.param l=0.8u R1 N1 GND 50 TC=0.0, 0.0 R2 N7 GND 50 TC=0.0, 0.0 Xsoftxor_1 N5 N3 N4 N19 N11 N10 GND Vdd softxor Xsoftxor_2 N11 N10 N9 N8 N7 N1 GND Vdd softxor Xsoftxor_3 N2 N16 N15 N14 N9 N8 GND Vdd softxor * SPICE netlist written by Bayes2Gates (c) 2002 Ben Vigoda, MIT Media Lab * Written on 28-Jun-2002 MATLAB Bayes Net Toolbox Spice Netlist Benjamin Vigoda, November, 2003

15. 15 Fault Tolerant Soft-Logic Benjamin Vigoda, November, 2003 Xu Sun, Mehdi Gazor, Bernard Yen, Ben Vigoda

16. 16 Differential Belief Propagation (advertisement) How do algorithmic dynamics of belief propagation relate to the physical dynamics of the computing substrate? Belief Propagation + high-frequency electronics = ? –Parasitic capacitances in transistors –Imperfect delay elements Belief Propagation + molecular dynamics = ? Use belief propagation to estimate solutions of partial differential equations Benjamin Vigoda, November, 2003