Cyclic Combinational Circuits and Other Novel Constructs

1
Cyclic Combinational Circuitsand Other Novel
Constructs
Marc D. Riedel
California Institute of Technology
Marrella splendens
Cyclic circuit
(500 million year old Trilobite)
(novel construct)
2
Combinational Circuits
The current outputs depend only on the current
inputs.
3
Combinational Circuits
Acyclic (i.e., feed-forward) circuits are always
combinational.
1
1
0
1
0
0
0
1
0
1
1
1
4
Combinational Circuits
Acyclic (i.e., feed-forward) circuits are always
combinational.
Are combinational circuits always acyclic?
Combinational networks can never have feedback
loops.
A combinational circuit is a directed acyclic
graph (DAG)...
5
Combinational Circuits
Acyclic (i.e., feed-forward) circuits are always
combinational.
Are combinational circuits always acyclic?
Combinational networks can never have feedback
loops.
A combinational circuit is a directed acyclic
graph (DAG)...
Designers and EDA tools follow this practice.
6
Circuits with Cycles
x
a




)))
(
(
(
f
x
c
d
x
a
b
f
1
1
b
x
c
d
7
Circuits with Cycles
0
0
x
AND
a
OR




x
0
)))
(
(
(
f
c
d
x
a
b
f
0
1
1
b
AND
0
x
OR
c
AND
d
OR
8
Circuits with Cycles
0
0
x
AND
a
OR




x
)))
(
(
(
f
x
c
d
a
b
f
1
1
b
AND
0
x
OR
c
AND
d
OR
9
Circuits with Cycles
1
x
AND
a
OR




x
1
x
1
)))
(
(
(
f
c
d
a
b
f
1
1
b
AND
1
1
x
OR
c
AND
d
OR
10
Circuits with Cycles
Circuit is cyclic yet combinational computes
functions f1 and f2 with 6 gates.
1
x
AND
An acyclic circuit computing these functions
requires 8 gates.
a
OR



x
))
(
(
c
d
a
b
f
1
b
AND
1
1
x
OR
c
AND



)
(
a
b
x
c
d
f
2
d
OR
11
Circuits with Cycles
Circuit is cyclic yet combinational computes
functions f1 and f2 with 6 gates.
There is no feedback in a functional sense.
x
A cyclic topology permits greater overlap in the
computation of the two functions
AND
An acyclic circuit computing these functions
requires 8 gates.
a
OR



x
))
(
(
c
d
a
b
f
1
b
AND
x
OR
c
AND



)
(
a
b
x
c
d
f
2
d
OR
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Prior Work (early era)

• Kautz and Huffman discussed the concept of
  feedback in logic circuits (in 1970 and 1971,
  respectively).
• McCaw and Rivest presented simple examples (in
  1963 and 1977, respectively).

13
Prior Work (later era)

• Stok observed that designers sometimes introduce
  cycles among functional units (in 1992).
• Malik, Shiple and Du et al. proposed techniques
  for analyzing such circuits (in 1994,1996, and
  1998 respectively).

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Cyclic Circuits Key Contributions
Theory

• Formulated a precise model for analysis.

• Provided constructions and lower bounds proving
  thatcyclic designs can be more compact.

Practice

• Devised efficient techniques for analysis and
  synthesis.

• Implemented the ideas and demonstrated they are
  applicable for a wide range of circuits.

15
Outline of Talk
Cyclic Circuits

• Analysis circuit model, symbolic techniques.
• Synthesis framework, implementation, and
  results.
• Theory circuit complexity (limited).

Current Future Research Directions

• Application of circuit design techniques to
  biological systems.

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Circuit Model
A circuit must produces definite output values
for each input combination (in the care set)

• Regardless of the prior values.
• Independently of all timing assumptions.

Formally

• Fixed-point analysis over a ternary-valued (0, 1,
  ?) domain.

Informally

• A sequence of controlling values always
  determines the output.

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Controlling Values
0
0
?
1
1
a controlling input
1
1
?
full set ofnon-controlling inputs
?
unknown/undefinedoutput
18
Timing Model
The arrival time at a gate output is determined

• either by the earliest controlling input

02
Each gate has delay in 0, td
13
03
06
(Assume td 1)
19
Timing Model
The arrival time at a gate output is determined

• either by the earliest controlling input

• or by the latest non-controlling input.

12
02
Each gate has delay in 0, td
13
17
13
03
16
06
(Assume td 1)
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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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Synthesis
Design a circuit to meet a specification.

• General methodology optimize by introducing
  feedback in the substitution/minimization phase.
• Developed a tool called CYCLIFY within Berkeley
  SIS Environment.
• Optimizations are significant and applicable to a
  wide range of circuits.

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Example 7 Segment Display
Inputs
27
Example 7 Segment Display
Output
28
Substitution
Basic minimization/restructuring operation
express a function in terms of other functions.
(cost 9)
Substitute b into a
(cost 8)
Substitute c into a
(cost 5)
Substitute c, d into a
(cost 4)
29
Substitution/Minimization
substitutional set
Þ
target function
Þ
Þ
low-cost expression
30
Acyclic Substitution
Select an acyclic topological ordering
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Acyclic Substitution
Select an acyclic topological ordering

a

b

c

d

e

f

g
Area (literal count) 37
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Acyclic Substitution
Select an acyclic topological ordering

a

b

c

d

e

f

g
Nodes at the top benefit little from substitution.
33
Cyclic Substitution
How can we find a cyclic solution that is
combinational?

a

b

c
?

d

e

f

g
34
Cyclic Substitution
Simpler Example
Candidates
Target
35
Cyclic Substitution
Simpler Example
Candidates
Target
36
Cyclic Substitution
Simpler Example
Candidates
Target
37
Branch and Bound
Break-Down approach

• Search performed outside space of combinational
  solutions.
• Terminates on optimal solution

38
Branch (without Bounding)
Build-Up approach
Search performed inside space of combinational
solutions
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Example 7 Segment Display
Combinational solution

a

b

c

d

e

f

g
Area (literal count) 34
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Branch and Bound
Large search space
Heuristics

• Limit the density of edges a priori
• Limit breadth
• Tunnel depth-wise (with backtracking)

(See The Synthesis of Cyclic Circuits, DAC, 03)
41
Optimization for Area
Number of NAND2/NOR2 gates for Berkeley SIS vs.
CYCLIFYsolutions
Based on script.rugged sequence and technology
mapping.
42
Optimization for Area and Delay
Number of NAND2/NOR2 gates and the Delay
of Berkeley SIS vs. CYCLIFY solutions
Based on script.delay sequence and technology
mapping.
43
Practice

• Improvements in area (and consequently power) and
  delay are significant.
• Similar improvements were obtained for larger
  scale circuits e.g., the ALU of an 8051
  microprocessor.
• E.D.A. companies (Altera and Synopsys) have
  expressed strong interest.

44
Theory
Prove that cyclic implementations can have fewer
gates than equivalent acyclic ones.
(optimal)
45
6/7 Construction
Cyclic Circuit 6 functions, 3 variables, 6
fan-in 2 gates.
AND
OR
AND
OR
AND
OR
Acyclic Circuit at least 7 fan-in 2 gates.
46
f1
Acyclic Circuit at least 7 fan-in 2 gates.
47
Theory
Strategy

• Exhibit a cyclic circuit that is optimal in terms
  of the number of gates, say with C(n) gates, for
  n variables.
• Prove a lower bound on the size of an acyclic
  circuit implementing the same functions, say A(n)
  gates.

Main Result
48
Current Future Research Interests

• Logic Synthesis and Verification functional
  decomposition, symbolic data structures, cyclic
  decision diagrams.
• Novel Platforms asynchronous models,
  nanotechnology, noisy/probabilistic gates.
• Computational Biology analysis of intracellular
  biochemical networks.

49
Computational Biology
Information encoded in biological systems

• One-dimensional digital (quaternary) code of DNA.

• Three-dimensional structure of proteins.

• Multi-dimensional intra-cellular biochemical
  networks.

• Vast complexity of multi-cellular biological
  organisms.

50
Example of a Biological Circuit
Lambda Phage model of Arkin et al., 1998
51
Intracellular Biochemical Networks
Formulation varies from qualitative and imprecise

52
Intracellular Biochemical Networks
... to quantitative and highly precise
53
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.)
54
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)

55
Biochemical Reactions
Lingua Franca of computational biology.
Coupled Set Reactions
Goal given initial conditions, analyze
(predict) the evolution of such a system.
56
Biochemical Reactions
Convential Approach numerical calculations based
on coupled ordinary differential equations.

• Computationally challenging (sometimes
  intractable).

• Assumes that molecular quantities are
  continuous values that vary deterministically
  over time.

57
Biochemical Reactions
Convential Approach numerical calculations based
on coupled ordinary differential equations.

• In intracellular networks, the number of
  molecules of each complex type is generally
  small (10s, 100s, at most 1000s).

• Individual reactions matter.

58
Biochemical Reactions
Discrete Quantities of Molecular Species
Reactions
States
A
B
C
S1
4
7
5
S2
3
5
7
S3
0
0
997
A reaction transforms one state into another
59
Analysis of Biochemical Reactions
Large domain, small range?
For m species, each with max. quantity N
inputs
Yes/No
Nm possibilities
Yes/No possibilities
60
Analysis of Biochemical Reactions
Discrete Quantities of Molecular Species
Types of Questions

• Can a certain state, S1, be transformed into
  another state, 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?

61
Decision Diagrams
e.g., set of possible initial states
62
System of Biochemical Equations
State Evolution
R1 occurs
or
R2 occurs
states before
states after
or
R3 occurs
63
Decision Diagrams
reaction 1 occurs
64
Decision Diagrams
reaction 2 occurs
65
Decision Diagrams
reaction 3 occurs
66
Decision Diagrams
Reachable States After The Next Reaction
S1
or
S2
or
S3
67
Decision Diagrams
Evolution of Reachable States
S
Track evolution of a large number of states in
parallel.
68
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
69
Future Directions
Novel data structures that may allow us to ask
(and answer) quantitative and probabilistic
questions
70
Papers Related to Cyclic Circuits
Computational Biology

• Collaboration with Molecular Sciences Institute,
  Berkeley(NIH Grant for Centers of Excellence in
  Genomic Sciences).

• The Synthesis of Cyclic Combinational Circuits,
  M. Riedel and J. Bruck, DAC 03. Received Best
  Paper Award.
• Cyclic Combinational Circuits Analysis for
  Synthesis, M. Riedel and J. Bruck, IWLS 03.
• Timing Analysis of Cyclic Combinational
  Circuits, M. Riedel and J. Bruck, IWLS, 04.

Patents

• A Method for the Synthesis of Cyclic
  Combinational Circuits, M. Riedel and J. Bruck
  (pending).

More Information

• www.paradise.caltech.edu/riedel,
  riedel_at_caltech.edu