Polysynchronous Clocking for Molecular Computing

Marc RiedelAssociate Professor, Electrical and Computer Engineering

Graduate Faculty, Biomedical Informatics and Computational BiologyUniversity of Minnesota

University Southern California – MBMC Workshop, Dec. 4, 2015

Digital Circuits

Computation Elements State Holding Elements

Synchronous Sequential Computation

Computation

Memory

Clock

Input Output

Bistable Bit Representation(dual-rail encoding)

AND Gate

Outputting 0

Outputting 1

OR Gate

Outputting1

Outputting 0

XOR Gate

Outputting 1

Outputting 0

Logic Gates

Clock

Implementing D Flip Flop

Master-slave configuration

Synchronous Sequential Computation

Computation

Memory

Clock

Input Output

Molecular Clocking

Clock Distribution Network in Silicon Circuits

• Significant Fraction of Design Effort

• 15% of Area, 20% of Power Dissipation

Positional Encodings

75710 = 7·102 + 5·101 + 7·100

• A positional representation scheme is compact: 2n distinct numbers can be represented with n bits.

• Operating on this representation is complex.

Human

Computer 10101112 = 26 + 24 + 22 + 21 + 20

Stochastic (Fractional) Encoding

A real value x in [0, 1] is represented by a sequence of random bits, each of which has probability x of being one and probability of 1 − x of being zero.

0,1,0,1,1,0,0

x = 3/7

Positional Multiplication

AB

C1,1,0,0,0,0,1,01,1,0,1,0,1,1,1

1,1,0,0,1,0,1,0

a = 6/8

b = 4/8

c = 3/8

Assume two input bit streams are independent

6/8 · 4/8 = 3/8

Stochastic Multiplication

Scaled Addition

))1(

()](1[)()()(

bsasBPSPAPSP

CPc

0

1

Stochastic Logic

5/8

3/8

4/8

3/8

4/8

8/8

Probability values are the input and output signals.

combinationalcircuit

Stochastic LogicProbability values are the input and output signals.

1,1,0,1,0,1,1,0…

1,0,0,0,1,1,0,0,…

0,1,1,0,1,0,1,0,…

0,1,1,0,1,0,0,0,…

1,0,1,0,1,0,1,0,…

1,1,1,1,1,1,1,1,…

serial bit streams

combinationalcircuit

combinationalcircuit

Stochastic LogicProbability values are the input and output signals.

t

Functions of a probability value t.

3.08.08.0 2+- tt

Virtual 1: Fewer Gates

Across many types of benchmark circuits and applications, consistently find 50 to 100 times fewer logic gates required.

Stochastic vs. Conventional Circuits

Compare conventional implementation to stochasticimplementation of polynomial functions.

Virtual 1: Fewer GatesStochastic vs. Conventional Circuits

Virtue 2: Fault Tolerance

• Stochastic Encoding– A bit flip does not substantially change the probability:

1010111001 → 1010011001

• Binary Radix Encoding– A bit flip in the most significant bit causes a huge

change in the value:(1010)2 → (0010)2

0.6 0.5

10 2

Stochastic vs. Conventional Circuits

Deterministic v.s. Stochastic Implementation of Gamma correction function with 10% noise injection.

Conventional Implementation

Stochastic Implementation

1% 2% 10%

Stochastic Implementation: no pixels with errors > 25%!

Deterministic implementation:37% pixels with errors > 25%

Virtue 3: Clock Skew ToleranceStochastic vs. Conventional Circuits

Virtue 3: Clock Skew ToleranceStochastic vs. Conventional Circuits

Replace Global Clock with(Crappy) Local Clocks

Domain 1

Domain 1

Domain 1

Replace Global Clock with(Crappy) Local Clocks

Domain 1

Domain 1

Domain 1

Virtue 3: Clock Skew ToleranceStochastic vs. Conventional Circuits

Fractional Representation

Represent a value in the range [0..1] with a pair of molecular types X0 and X1:

Multiplication

a x b = c

a 

b 

c

(scaled) Addition

0 0 → 0

0 1 → 0

1 0 → 1

1 1 → 1

10 1

1. 0 1. 10 1 0 0 1 1

1. 0 1. 10 1 0 1 1

Synchronous Sequential Computation

Computation

Memory

Clock

Input Output

Discussion

• Synthesize a design for a precise, robust, programmable probability distribution on outcomes – for arbitrary types and reactions.

Computational Synthetic Biology vis-a-vis

Technology-Independent Logic Synthesis

• Implement design by selecting specific types and reactions – say from “toolkit”.

Experimental Design vis-a-vis

Technology Mapping in Circuit Design

• Methods and CAD tools for implementing logical and arithmetic with molecular reactions.

Where are we?

• Methods and CAD tools for generating sequential and iterative computation (e.g., DSP functions) with molecular reactions.

Where are we headed?

• A technology-independent chemical CPU.

Discussion

• Mapping designs onto chemical substrates (e.g., DNA)

Acknowledgements

Kia BazarganDavid Lillja

Weikang Qian Peng Li

Keshab Parhi