BDS : A BDDBased Logic Optimization System

1
BDS A BDD-BasedLogic Optimization System
Congguang Yang
Synopsys, Inc.
Maciej Ciesielski
University of Massachusetts, Amherst Department
of Electrical and Computer Engineering
Vigyan Singhal
Tempus Fugit, Inc.
2
Outline

• Introduction Background, Motivation, Goals.
• Logic Optimization Through BDD Decomposition
  -BDDlopt, ICCD99
• BDD-Based Logic Synthesis System - BDS
• Conclusion

3
Binary Decision Diagrams

• Proposed by Lee 1959
• A switching function can be efficiently
  implementedby a series of binary switch.
• Akers 1978
• Binary Decision Diagram (BDDs)
• A set of reduction rules
• Bryant 1986
• Ordered BDDs (OBDDs)
• A set of efficient operators

4
Binary Decision Diagrams
Example F ac bc abc
5
Drawback of Traditional Logic Optimization

• A Boolean formula is treated as a polynomial
  formula.
• Weak Boolean factorization capability.
• Two-level techniques are used to perform Boolean
  simplification.
• Separate platforms for Boolean simplification and
  factorization.
• Our goal to use a common platform to carry out
  both Boolean simplification and factorization.

6
Mini-history of Logic Representation
Minterm
Sum-of-product
Cube
Factored Form
BDD
7
Advantages of BDDs

• BDD is canonical.

8
Advantages of BDDs

• BDD is canonical.
• Implicit factored representation.

• BDD variable reordering performs implicit logic
  simplification.
• Some redundancy removed during BDD construction.
• Problem How to translate a BDD into a
  multi-level representation?

9
Previous work on BDD Decomposition Karplus
1988
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BDD Decompositions in BDS

• Boolean AND/OR Decomposition
• Boolean XOR Decomposition (skip)
• MUX Decomposition (skip)

11
Logic Optimization Flow of BDDlopt
Boolean Network
12
Experimental Results
XOR-intensive
AND/OR-intensive
13
Trade-off Between Local and Global Representation
14
BDD-Based Logic Optimization System (BDS)
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Sweep Boolean Network
Goal Network preprocessing.

• Constant propagation.
• Remove single-input Boolean nodes.
• Remove functionally equivalent Boolean nodes.

16
Sweep Boolean Network (continued)
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Functionally Duplicated Nodes
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Boolean Node Elimination
19
Results of Iterative Eliminate
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Sharing Extraction
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BDD-Based Logic Optimization System (BDS)
Boolean Network
Eliminate
Sweep
Sharing Extraction
Boolean Factorization
Final Factoring Tree Processing
Technology Mapping
22
Experimental Results
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Conclusions Future Work

• First unified approach to both AND/OR- and
  XOR-intensive Boolean functions.
• Implementation of first working BDD-based
  multi-level logic optimization system (BDS).
• Network partitioning. Group collapse.
• More general BDD decompositions.
• http//www-unix.ecs.umass.edu/cyang/bds.html

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Boolean XOR Decomposition Generalized
x-Dominators
XOR Decomposition (generalized x-dominator)
Given F and g,
h F g h h F g.
h
F
a
a
g
Generalized x-dominator
b
b
c
c
c
c
d
d
d
1
1
1
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Special Case of Generalized x-Dominator
F
a
cd
ab
x-dominator
b
c
a
c
b
d
d
1
1
1
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Algorithm for Iterative Eliminate
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Algorithm for Iterative Eliminate
Naive
BDS
28
Experimental Results
29
Trade-off Between Local and Global Representation
30
Factoring Tree Processing
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Lessons Learnt from BDDlopt

• BDD decomposition is a good alternative to
  traditional logic optimization.
• BDD decomposition-based logic optimization is
  fast.
• Stand-alone BDD decomposition scheme is not
  amenable to large circuits.

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Boolean AND/OR Decomposition (Boolean Division)
Goal F Q D.
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Boolean AND/OR Decomposition

• Boolean DivisionIf D gt F, F F D Q D
  Q F/D.

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Boolean AND/OR Decomposition

• Boolean DivisionIf D gt F, F F D Q D
  Q F/D.

Boolean Space
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Boolean AND/OR Decomposition

• Boolean DivisionIf D gt F, F F D Q D
  Q F/D.

• Boolean SubtractionIf D lt F, F F D R
  D R F/D.

• D Generalized Dominator ?

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Generation of Generalized Dominator
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Generation of Generalized Dominator
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Example of Boolean Division

• F abg acg adf aef afg bd ce be
  cd.
• Algebraic F (b c)(d e ag) (d e
  g)af. (11 lit)
• Boolean F (af b c)(ag d e). (8 lit)

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Boolean Division Based on Generalized Dominator