Post on 13-Dec-2015
Marc Riedel – EE5393
The Synthesis of Robust Polynomial Arithmetic with Stochastic Logic
The Synthesis of Robust Polynomial Arithmetic with Stochastic Logic
Electrical & Computer EngineeringUniversity of Minnesota
A
B
C
Opportunities & Challenges
• Topological constraints.• Inherent structural randomness.• High defect rates.
Novel materials, devices, technologies:
Challenges for logic synthesis:
• High density of bits/logic/interconnects.
{
{
N Wires
M Wires
Opportunities & Challenges
Strategy:• Cast synthesis in terms of arithmetic
operations on real values.• Synthesize circuits that compute
logical values with probability corresponding to the real-valued inputs and outputs.
{
{
N Wires
M Wires
Probabilistic Signals
Claude E. Shannon1916 –2001
“A Mathematical Theory of Communication” Bell System Technical Journal, 1948.
deterministic
random
deterministic
Stochastic LogicStructure logical computation probabilistically.
p1 = Prob(1)
p2 = Prob(1)
1,1,0,1,0,1,1,0…
1,0,0,0,1,1,0,0,…
combinationalcircuit
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,…
Stochastic Logic
5/8
3/8
Structure logical computation probabilistically.
1,1,0,1,0,1,1,0…
1,0,0,0,1,1,0,0,…
combinationalcircuit
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,…
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.
parallel bit streams
4/8
3/8
4/8
8/8
5/8
3/8
A real value x in [0, 1] is encoded as a stream of bits X.For each bit, the probability that it is one is: P(X=1) = x.
Probabilistic Bundles
01001
xX
A
VDD
A{{
N Wires
M Wires
Nanowire Crossbar (idealized)
Randomized connections,yet nearly one-to-one.
Interfacing with Stochastic LogicInterpret outputs according to fractional weighting.
combinationalcircuit
Arithmetic Operations
AND
A
BC
A
BC
MUX
S
Multiplication (Scaled) Addition
ba
BPAP
CPc
)()(
)(
)
)1(
()](1[)()(
)(
bsas
BPSPAPSP
CPc
Computing with Probabilities
combinationalcircuit
X2
X1
Xn
independentrandomBooleanvariables
YrandomBooleanvariable
ii xX )1Pr(
10 ix
),,()1Pr( 1 nxxfyY
10 y
Computing with Probabilities
combinationalcircuit
independentrandomBooleanvariables
YrandomBooleanvariable
Constrain the problem:
(independently)
constant
)1Pr(
tX i
X2
X1
Xn
Computing with Probabilities
combinationalcircuit
Constrain the problem:
(independently)
constant
)1Pr(
tX i
)(tft
X
Y
X
Y
Z
Z
C
S
Computing with Probabilities
(independently)tZX )1Pr()1Pr(
3.0)1Pr( Y
t
t
t
t
0.3
0.3
24.06.0 tt
3.08.08.0 2 tt
SynthesisAnalysis
“There are known ‘knowns’; and there are unknown ‘unknowns’; but today I’ll speak of the known ‘unknowns’.”
– Donald Rumsfeld, 2004
combinationalcircuit
Probabilistic Inputs
Probabilistic Outputs
KnownKnown
UnknownSpecified /Independent
UnknownGiven
Synthesis
“There are known ‘knowns’; and there are unknown ‘unknowns’; but today I’ll speak of the known ‘unknowns’.”
– Donald Rumsfeld, 2004
Questions:
• What kinds of functions can be implemented in the probabilistic domain?
• How can we synthesize the logic to implement these?
Probabilistic Domain
A polynomial g(t) is either:1. g(t) ≡ 0 or 1, or2. 0 < g(t) < 1, for 0 < t < 1 and 0 ≤ g(0), g(1) ≤ 1
combinationalcircuit )(tgt
A necessary and sufficient condition:
Probabilistic Domain
combinationalcircuit )(tgt
Synthesis steps:
1. Convert the polynomial into a Bernstein form.
2. Elevate it until all coefficients are in the unit interval.
3. Implement this with “generalized multiplexing”.
A little math…
( ) (1 ) ,n i n ii
nB t t t
i
0,1, ,i n
Bernstein basis polynomial of degree n
A little math…
( ) (1 ) ,n i n ii
nB t t t
i
0,1, ,i n
0
( ) ( )n
n n ni i
i
B t b B t
Bernstein basis polynomial of degree n
Bernstein polynomial of degree n
nib is a Bernstein coefficient
A little math…
0
( ),
( )
iijn n
i jnj j
b a
0,1, ,i n
Obtain Bernstein coefficients from power-form coefficients:
Given0 0
( ) ( )n n
n i n ni i i
i i
g t a t b B t
, we have
A little math…
0
11
0
(1 ) 11 1
1
m
m m mi i i
mm
b i
i ib b b i m
m mb m
Elevate the degree of the Bernstein polynomial:
Given1
1 1
0 0
( ) ( ) ( )m m
m m m mi i i i
i i
g t b B t b B t
, we have
A Mathematical Contribution
Given
• A Bernstein polynomial (satisfying the necessary and sufficient condition)
Iteratively elevating the degree yields:
• A Bernstein polynomial with coefficients in the unit interval (in very few steps).
Example: Converting a Polynomial
32 683)( ttttg Power-Form Polynomial
Example: Converting a Polynomial
32 683)( ttttg
)()(3
2)()( 3
332
31 tBtBtBtg
)()(5
2)(
5
3
)()(4
1)(
6
1)(
4
3
55
52
51
44
43
42
41
tBtBtB
tBtBtBtB
Power-Form Polynomial
Bernstein Polynomial
coefficients in unit interval
Probabilistic Multiplexing
A
BC
MUX
T
)
)1(
()](1[)()(
)(
btat
BPTPAPTP
CPc
Bernstein polynomial
Generalized Multiplexing
DecodingBlock Multiplexing
Blocky
s0...nx1...n
z0...n
n n+1
n+1
Generalized Multiplexing
DecodingBlock Multiplexing
Blocky
s0...nx1...n
z0...n
n n+1
n+1
X1, …, Xn are independent Boolean random variables with , for 1 ≤ i ≤ niXp t
Generalized Multiplexing
DecodingBlock Multiplexing
Blocky
s0...nx1...n
z0...n
n n+1
n+1
X1, …, Xn are independent Boolean random variables with , for 1 ≤ i ≤ niXp t
Pr( 1) ( )ni iS B t
Generalized Multiplexing
DecodingBlock Multiplexing
Blocky
s0...nx1...n
z0...n
n n+1
n+1
Z0, …, Zn are independent Boolean random variables with , for 0 ≤ i ≤ ni
nZ ip b
Generalized Multiplexing
DecodingBlock Multiplexing
Blocky
s0...nx1...n
z0...n
n n+1
n+1
Z0, …, Zn are independent Boolean random variables with , for 0 ≤ i ≤ ni
nZ ip b
0
Pr( 1) ( )n
n nY i i
i
p Y b B t
Experimental Results
Compare [conventional] deterministic to [new] stochastic implementation of polynomial computation:
Deterministic implementation: based on benchmark circuit (ISCAS’85 multiplier circuit C6288).
• Area-delay product.• Error tolerance with noise injection.
Experimental ResultsSixth-order Maclaurin polynomial approx., 10 bits:sin(x), cos(x), tan(x), arcsin(x), arctan(x), sinh(x),
cosh(x), tanh(x), arcsinh(x), exp(x), ln(x+1)
0
10
20
30
40
50
60
0 0.001 0.002 0.005 0.01 0.02 0.05 0.1
error ratio of input data
rela
tiv
e e
rro
r
Stochastic Deterministic
Experimental Results100 randomly chosen Bernstein polynomials with
coefficients in the unit interval, 10 bits:
0
100
200
300
400
500
600
700
800
900
1000
0 0.001 0.002 0.005 0.01 0.02 0.05 0.1
error ratio of input data
rela
tiv
e e
rro
r
Stochastic
Deterministic
Conclusions
• The area-delay product of the stochastic computation is comparable to the deterministic.
• The stochastic implementation is much more input-error tolerant.
Future Directions• Extend to the synthesis of multivariate polynomials.• Implement sequential computation.
Engineering novel functionality in biological systems.
BiochemicalReactions
View engineered biochemistry as a form of computation.
Synthetic Biology
E. Coli
computationinputs outputs
Molecular Triggers
Molecular Products
View engineered biochemistry as a form of computation.
BiochemicalReactions
Synthetic Biology
computationinputs outputs
Quantities of Different
Types
Quantities of Different
Types
Probability Distribution on Different Types
Bacteria are engineered to produce an anti-cancer drug:
Design Scenario
drugtriggering compound E. Coli
produce drug
triggering compound E. Coli
Approach: engineer a probabilistic response in each bacterium.
with Prob. 0.3
don’t produce drugwith Prob. 0.7
Synthesizing Stochasticity
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