STOCHASTIC LOGIC Architectures for post-cmos switches David S. Ricketts Electrical & Computer Carnegie Mellon University www.ece.cmu.edu/~ricketts Jehoshua (Shuki) Bruck Engineering Electrical Engineering California Institute of Technology bruck@caltech.com
Random at the Nanoscale Statistical Variation Probabilistic Stochastic ti.
Statistical Variation Random Dopant Fluctuations SiO 2 Gate wl M p1 M p2 Source Drain A. Brown et al., IEEE Trans. Nanotechnology, p. 195, 2002 1 2 M s1 M n1 M n2 M s2 Line Edge Roughness we data M mux Column mux M wr we data 1µm K. Shepard, U. Columbia Gate Oxide Variation At nanoscale, nothing is deterministic Year L eff 3σ T ox 3σ V T 3σ W 3σ H 3σ ρ 3σ 90 nm 47% 16% 13% 33% 36% 33% Momose et al, IEEE Trans. Electron Devices, 45(3), 1998 BOX Devices Wires [Sani Nassif, Proc. IEEE CICC, May 2001]
Probabilistic Random events during operation, dice are rolled continuously Example: Probabilistic bili CMOS (PCMOS), K. V. Palem, Rice Univ, et. al. VLSI-SoC: Research Trends in VLSI and Systems on Chip, Springer Boston,
Probabilistic (2) Markov Random Field (MRF) Logic, A. Zaslavsky, Brown Univ. Nepal, et. al. Designing Logic Circuits for Probabilistic Computation in the Presence of Noise RAZOR, T. Austin, T. Mudge, Univ Michigan D. Ernst, et. al., Razor: A Low-Power Pipeline Based on Circuit-Level Timing Speculation
Stochastic Logic Goal: Leverage random fluctuations of nanoscale switches to build a new paradigm in logic/computation. Random events during operation, dice are rolled continuously We don t look to fix randomness, but rather exploit it. We are investigating architectures/logic families that are based on stochastic logic using stochastic switches. We leverage the inherent random state of certain nanoscale devices.
Switch: Bistable Elements Basic idea of a switch * is a bistable element Transition between states t can be external, i.e. input, or internal, e.g. thermal energy. In nature, many physical systems have two stable, or meta-stable states with an energy barrier between them. Transitions occur randomly due to the thermal energy of the system Energy Barrier 0 1
Switch: Bistable Elements (2) Transition between states can be calculated from the barrier height and the thermal energy of the system ~ kt. Energy Barrier P exp E k T B E R 1/ exp flip f0 kt B E 0 1
Switch: Bistable Elements (3) Probability of state determined by symmetry of potential well Energy Barrier P exp E E 1 E 2 kbt 0 1
Example #1 Rotaxane Molecule Used for dense memory cross bar arrays Switching/relaxation R CHEMPHYSCHEM 2002, 3, 519 525 1 kt B G exp h RT : s days Eyring equation Phil. Trans. R. Soc. A (2007) 365, 1607 1625
Example #2 Data storage (disk drive) bit E KuV V 1 FePt Nanoparticle arrays E K V Low Moderate V 2 High V 3 < V 2 < V 1 V 3 20 nm d d 6.3 0.3 nm 0.0505 nm S. Sun, C. Murray, D. Weller, L. Folks, A. Moser, Science, 287, pp. 1989 2000 %) Magnetization ( 1 Stored Data Magnetization Decay V 3 V 2 Data Lost V 1 0 1 minute 1 year 10 years Log Time (s)
Stochastic Logic (J. Bruck) Shannon s work focused on deterministic switching circuits, circuits where each switch is associated with a Boolean variable defining whether the switch is closed. We instead focus on stochastic switching circuits, circuits where each pswitch is associated with a Bernoulli random variable defining the (independent) probability that the pswitch is closed. D. Wilhelm and J. Bruck, Stochastic switching circuit synthesis, IEEE Int.Sym. on Inf. Theory, Toronto, 2008. Deterministic switch Deterministic switch Switches are ON or OFF with a known probability (not necessarily 50%) We construct global probabilities based upon a logical connection of probabilistic switches (Pswitch) and deterministic i ti switches.
Stochastic Logic (2) Series/Parallel Pswitches
Stochastic Logic (3) Examples: Series/Parallel Pswitches 75% 37.5% 43.8% 1) Let circuit C1 be the single-pswitch circuit. 2) For bit Fi, i = 2 to n, let circuit Ci be: a) If Fi = 0, C1 in series with Ci 1, or b) If Fi = 1, C1 in parallel with Ci 1 1101.011 2 2 2 2.2 2 2 3 2 1 0 1 2 3
Stochastic Logic: Universal Probability Generator Generating a deterministic input to desired probabilistic states Could generate 2 n individual id probability generators and select one Inefficient exponential increase in switch count. Need an algorithm that creates the desired probabilities with the minimum hardware Need universal Pswitch network synthesizer
Universal Probability Generator Recursive architecture generates a minimal sized probability generator using 4n-2 switches
Universal Probability Generator (2) Example
Stochastic Logic Summary Goal: Leverage random fluctuations of nanoscale switches to build a new paradigm in logic/computation. Utilize thermal energy of nano-scale switches to generate switches that open/close randomly Probability of open/close is determined by energy states, i.e. double well geometry. Build stochastic ti switches Pswitches from these random devices. Are able to synthesize arbitrary probabilities given a known, but not designed, probability. Universal Probability Generator allows for minimal designs of arbitrary deterministic to probability logic networks.
An Application Example 1 Million RF temperature sensors distributed uniformly over Indiana, powered by solar energy scavenging circuits that operate sensors for 15 min per 24 hour period
An Application Example (2) How to ensure temperature readings throughout the state 24 hours a day? Need 1/100 of sensors on at all times. Need sensors that are on to be even distributed, so that no area is missed. What is the overhead of organization? Of communication? Build each sensor with a 1/100 probability bilit of being turned ON. Sensor network will inherently provide even coverage and time/energy organization. What if 15 min varied by weather, by adjusting random wake-up, energy adaptive networks could be implemented.
Outlook Applications General probability generators Energy/distribution management Stochastic computation in eoltionar evolutionary systems, sstemseg e.g. Stochastic switching as a survival strategy in fluctuating environments, Nature Genetics Efficient implementation of Pswithces at the nanoscale Development of more complex synthesis and computation theory/ algorithms.
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