EE 5940: Circuits, Computation and Biology

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EE 5940 Circuits, Computation and Biology
Prof. Marc Riedel, ECE, Univ. Minnesota
A Decision Diagram
The Goddess Durga
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Analysis of Digital Circuits
Large domain, small range.
inputs
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Analysis of Digital Circuits
Large domain, small range.
inputs
2m possibilities
2 possibilities
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Analysis of Systems of Chemical Equations
Large domain, small range?
For m species, each with max. quantity N
inputs
?
Yes/No
Nm possibilities
Yes/No possibilities
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Data Structures
Truth Tables
Example
4 rows
2 variables
8 rows
3 variables
264 rows
64 variables
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Data Structures
Decision Diagrams
Example
x1
x2
x3
f
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
0
1
1
1
1
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Data Structures
Decision Diagrams
Optimize by merging nodes
x3
x3
x3
x3
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Data Structures
Decision Diagrams
S
Optimize by merging nodes
x1
1
0
x2
1
x3
x3
x3
0
1
0
1
1
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Data Structures
Decision Diagrams
Optimize by merging nodes
x2
x2
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Data Structures
Decision Diagrams
S
Optimize by merging nodes
x1
0
x2
1
0
x3
1
0
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Data Structures
Logic Operations

AND
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Decision Diagrams
Properties

• Canonical unique up to variable ordering
• Compact represent functions of up to 1000
  variables
• Efficient perform logic operations in
  linear-time

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Ordered Binary Decision Diagrams
(a.k.a. Branching Programs)
Example
Directed Acyclic Graph variables are inspected
in order.
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Reducing OBDDs
Terminal Rule eliminate duplicate terminals.
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Reducing OBDDs
Elimination Rule eliminate a node if its 0 and
1 edges lead to the same node.
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Reducing OBDDs
Merging Rule merge two nodes that reference
the same variable and point to the same
successors.
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Reduced OBDDs
Question
What is the optimal strategy for eliminate and
merge operations?
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Reduced OBDDs
Strategy eliminate and merge nodes repeatedly,
in any order, until no further simplifications
are possible.
Is the result unique?
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Uniqueness (proof)
By induction on the number of variables.
Induction Hypothesis Assume that any two ROBDDs
for a function with k 1 variables, k gt 0, are
isomorphic.
Inductive Step Show that any two ROBDDs for a
function with k variables are isomorphic.
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Inductive Step
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Inductive Step
Show that this mapping is well-defined and
one-to-one.
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Inductive Step
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Mapping Well-Defined
low(v)
high(v)
Counter Example (Unreduced BDDs)
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Mapping One-To-One
Counter Example (Unreduced BDDs)
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Logic Operations
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Logic Operations
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Logic Operations
Compute
0
1
0
1
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Logic Operations
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Logic Operations
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Logic Operations
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Logic Operations
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Logic Operations
0
1
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Logic Operations
0
1
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Logic Operations
0
1
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Logic Operations
0
1
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Logic Operations
Simplify
0
1
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Logic Operations
Simplify
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Variable Ordering
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Analysis

• Functional Analysis determine what is computed.
• Timing Analysis determine how long it takes
  to compute it.

l1 1
l4 3
10
a
11
12
l3 2
10
b
10
c
02
l2 1
a
10
l5 2
c
10
01
10
b
12
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Analysis

• Functional Analysis determine what is computed.
• Timing Analysis determine how long it takes
  to compute it.

Explicit analysis
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Analysis

• Functional Analysis determine what is computed.
• Timing Analysis determine how long it takes
  to compute it.

Explicit analysis
00
00
00
OR
AND
AND
01
02
01
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Analysis

• Functional Analysis determine what is computed.
• Timing Analysis determine how long it takes
  to compute it.

Explicit analysis
00
00
00
00
10
00
OR
AND
AND
02
01
02
01
01
03
m inputs
explict evaluation intractable
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Analysis

• Functional Analysis determine what is computed.
• Timing Analysis determine how long it takes
  to compute it.

Symbolic analysis
binary, multi-terminal decision diagrams.
00
10
00
OR
AND
AND
02
01
03
(See Timing Analysis of Cyclic Circuits, IWLS,
04)
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Analysis of Systems of Chemical Equations
Large domain, small range?
For m species, each with max. quantity N
inputs
?
Yes/No
Nm possibilities
Yes/No possibilities
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Biochemical Reactions
Lingua Franca of computational biology.
Reaction
1 molecule of type A combines with 2
molecules of type B to produce 2 molecules of
type C.
Reaction is annotated with a rate constant and
physical constraints (localization, gradients,
etc.)
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Biochemical Reactions
Lingua Franca of computational biology.
Reaction
Species

• Elementary molecules (e.g., hydrogen,
  phosphorous, ...)

• Complex molecules (e.g., proteins,
  enzymes, RNA ...)

Reaction

• Conglomeration of steps (e.g., transcription
  of gene product)

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Biochemical Reactions
Lingua Franca of computational biology.
Coupled Set Reactions
Goal given initial conditions, analyze
(predict) the evolution of such a system.
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System of Chemical Equations
Discrete Quantities of Molecular Species
Types of Questions

• Can a certain state, S1, be transformed into
  another state, S2? If so, in how many different
  ways? S2

• Can S1 be transformed into S2 without passing
  through a third state S3?

• Can S1 be reached from at least one state in a
  set of states T? From all the states in a set of
  states U?

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Decision Diagrams
e.g., set of possible initial states
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System of Chemical Equations
State Evolution
R1 occurs
or
R2 occurs
state before
state after
or
R3 occurs
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Decision Diagrams
reaction 1 occurs
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Decision Diagrams
reaction 2 occurs
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Decision Diagrams
reaction 3 occurs
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Decision Diagrams
Reachable States After The Next Reaction
S1
or
S2
or
S3
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Decision Diagrams
Evolution of Reachable States
S
Track evolution of a large number of states in
parallel.
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Yes/No Questions
Can ask (and answer) arbitrarily complicated
yes/no questions pertaining to reachability
C1 state S is reachable after 100 reactions
C2 state T is reachable from state U or
from state V but not from both
C3 state X is never reachable