Synchronous Sequential Computation
Sustained Oscillations
Transfer concentrations in
alternating phases:
red, green and blue.
First Attempt
R, G, and B converge!
Clock
Decrement x
g
x
X0=
5
Decrement x
X0=
5
x
x
x
x x
Decrement x
xr
x
x
X0=
5
x
x
x x
D
Communicating Ideas
Computing With Limited Memory
Small RAM
e.g., chimpanzee
Large (but not infinite) ROM
e.g., stack of instruction cards
Computing With Limited Memory
m states S1 , , S m
(log2 m bits memory)
n Boolean inputs x1 , , xn
Assume n much grea
Example: Verification
e.g., input/output specification of multiplier
A
x1 , , xn
f ( x1 , , xn )
B
e.g., multi-level logic representation
Binary Decision Diagrams
Graph-based Representation of Boolean Functions
a
Introduced by Lee (1959).
Popularized by
EE i5393
Circuits, Computation, and Biology
UMN
Winter 2013
Homework # 1
Due March 1, 2013
1. Two-Terminal Switches
This problem is based on C. E. Shannons Masters Thesis from 1938, A Symbolic Analysis of Relay and Switching Circuits. This classical paper
EE 5393
Circuits, Computation, and Biology
UMN
Winter 2013
Homework # 2
Due March 29, 2013 [in class]
1. Analyzing Stochasticity in Molecular Reactions
Randomness is inherent to all forms of biochemical computation: at any given
instant, the choice of whi
EE 5393
Circuits, Computation, and Biology
UMN
Winter 2013
Homework # 3
Due May 1, 2013 [in class]
1. Percolation
Percolation theory is a rich mathematical topic that forms the basis of explanations of physical phenomena such as diusion and phase changes
EE 5393
Circuits, Computation and Biology
UMN
(Yay! Its) Spring13
Homeworks # 4
due May 19, 2013 (Hawaii Time)
1. Cyclic Combinational Circuits
In class we discussed combinational versus sequential circuits. Combinational
circuits are memoryless, i.e., th
EE i5393
Circuits, Computation, and Biology
UMN
Winter 2014
Homework # 1
Due March 14, 2013
The theory of reaction kinetics underpins our understanding of biological and chemical systems. It is a simple and elegant formalism: chemical reactions dene rules
EE5393
Circuits, Computation, and Biology
UMN
Winter 2014
Homework # 2
Due April 2, 2014
1. Mapping Molecular Reactions to DNA
In this question, you will map generic molecular reactions into DNA reactions.
I suggest that you use the software DSD from Micr
Gillespies Framework
Track precise (integer) quantities of molecular species.
States
A
Reactions
B
C
S1 4
7
5
R1
S2 2
6
8
R2
S3 22
0
997
R3
k1
2A B
k2
B 2C
AC
A reaction transforms one state into another:
e.g., S1 S2
R
1
k3
3C
3A
2B
Stochastic
Simulati
Design Scenario
Bacteria are engineered to produce an anti-cancer drug:
triggering
compound
E. Coli
drug
Design Scenario
Bacteria invade the cancerous tissue:
cancerous
tissue
Design Scenario
The trigger elicits the bacteria to produce the drug:
Bacteria
Q3:
The following program can be run by a given value. In the main loop, the program
can stop whenever c1, c2, or c3 conditions can be reached. Otherwise the loop is
running forever. After jumping out the while loop, the code goes to for loop so that it
c
Solutions or answers to Final exam in Error Control Coding, October 24, 2010
Solution to Problem 1
a) G(D) = 1 + D 3 , 1 + D 2 + D 4
b) The rate R = 1/2 and i = = m = 4.
c) Yes, since gcd 1 + D 3 , 1 + D 2 + D 4 = 1 + D + D 2 = D j .
d) An equivalent enco
Switch-based Boolean computation
Shannons work: A Symbolic Analysis
of Relay and Switching Circuits(1938)
x1
x1
x2
Series: x1 . x2
Parallel: x1 + x2
x1
x3
x2
x1
x2
x2
x3
1D and 2D switches
ON
1D
switch
2D
switch
OFF
A lattice of 2D switches
TOP
LEFT
RIGHT
EE5393, Circuits, Computation, and
Biology
Computing with Probabilities
a = 6/8
A
B
1,1,0,1,0,1,1,1
1,1,0,0,1,0,1,0 AND
b = 4/8
c = 3/8
1,1,0,0,0,0,1,0
C
Positional Encodings
Human
75710 = 7102 + 5101 + 7100
Computer
10101112 = 26 + 24 + 22 + 21 + 20
A po
EE5393, Circuits, Computation, and
Biology
Computing with Probabilities
a = 6/8
A
B
1,1,0,1,0,1,1,1
1,1,0,0,1,0,1,0 AND
b = 4/8
c = 3/8
1,1,0,0,0,0,1,0
C
Bernstein Polynomial
Bernstein basis polynomial of degree n
Bernstein polynomial of degree n
is a Ber
Playing by the Rules
Biochemical Reactions: how types of molecules combine.
2a +
+
b
c
Biochemical Reactions
+
species count
cell
9
8
6
5
7
9
Discrete chemical kinetics; spatial homogeneity.
Biochemical Reactions
Relative rates or (reaction propensities):
EE5393, Circuits, Computation, and
Biology
Computing with Probabilities
a = 6/8
A
B
1,1,0,1,0,1,1,1
1,1,0,0,1,0,1,0 AND
b = 4/8
c = 3/8
1,1,0,0,0,0,1,0
C
Sequential Constructs
What about complex functions such as tanh, exp, and
abs?
Sequential Constructs
Computing with Defects
TOP
LEFT
Real
case
BOTTOM
BOTTOM
DEFECT
VApplied
LEFT
BOTTOM
Real
case
RIGHT
RIGHT
Ideal
case
TOP
LEFT
TOP
o Ideally,
RIGHT
RIGHT
Ideal
case
LEFT
TOP
BOTTOM
if the applied voltage is 0, then all the crosspoints are OFF
and so there
EE 5393
Circuits, Computation, and Biology
UMN
Winter 2014
Homework # 3
Due April 30, 2014, 2:30pm
1. Two-Terminal Switches
This problem is based on C. E. Shannons Masters Thesis from 1938, A Symbolic Analysis of Relay and Switching Circuits. This classic