Output GroupingBased Decomposition of Logic Functions

1
Output Grouping-Based Decomposition of Logic
Functions

• Petr Fier, Hana Kubátová
• Department of Computer Science and Engineering
• Czech Technical University

2
Outline

• Motivation
• Single-Level Partitioning
• Proposed Method
• Basic FC-Min Principles
• Output Grouping
• Experimental Results
• Conclusions

3
Motivation

• Typical logic synthesis process
• Perform Boolean minimization
• Multi-level synthesis, decomposition
• Technology mapping
• These phases are often independent on each other
  - ineffective

4
Motivation

• Boolean minimization should be driven towards the
  target technology

5
Single-Level Partitioning

• Two-level AND-OR network
• Limited number of inputs/outputs in real devices
• Solution divide the circuit into stand-alone
  blocks, while reducing the number of their inputs
  and complexity of the blocks

6
Single-Level Partitioning

• The way how to minimize the number of inputs of
  the blocks proposed before
• Fier, P. - Kubátová, H. Single-Level
  Partitioning Support in BOOM-II, Proc. 2nd
  Descrete-Event System Design 2004 (DESDes'04),
  Dychów, Poland, 15.-17.9.04, pp. 149-154
• Reducing the input set means an increase of the
  size of the circuits (some of the group terms
  cannot be shared)
• The issue how to group the outputs to minimize
  the complexity of the blocks

7
Proposed Method

• Method to determine the output grouping
• Based on FC-Min, even when no minimization is
  involved
• Significant reduction in area overhead, for both
  the two-level and multi-level implementations
• Input set is reduced as well

8
FC-Min

• Two-level Boolean minimizer
• Primarily designed for group minimization
• Produces only the necessary group implicants no
  excessive implicants are generated
• First, the on-set cover is found (Find Cover)
• Then implicants are computed

9
Find Cover

• Generates rectangle cover of the on-set
• Determines the number of product terms in the
  solution, not their structure
• Independent on literals
• Directly determines group implicants

10
Find Cover
11
Output Grouping

• Main idea outputs that share many group
  implicants should be grouped together
• The effects are obvious for two-level
  minimization, however the same can be observed
  for multi-level implementation

12
Output Grouping

• Grouping matrix (G-Matrix)
• Combines influences of various group implicants
• Symmetric matrix of dimensions m, m
• The value of Gi, j defines the strength binding
  the two output variables i and j together

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Output Grouping

• G-Matrix Example

Initial State
12345 1 00000 2 00000 3 00000 4 00000 5 00000
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Output Grouping

• G-Matrix Example

T1 added
01234 0 00000 1 00000 2 00000 3 00001 4 00010
15
Output Grouping

• G-Matrix Example

T2 added
01234 0 00000 1 00100 2 01000 3 00001 4 00010
16
Output Grouping

• G-Matrix Example

T3 added
01234 0 00100 1 00100 2 11000 3 00001 4 00010
17
Output Grouping

• G-Matrix Example

T4 added
01234 0 00100 1 00110 2 11000 3 01001 4 00010
18
Output Grouping

• G-Matrix Example

T6 added
01234 0 00100 1 00110 2 11001 3 01001 4 00110
19
Output Grouping

• G-Matrix Example

T1 added
Another cover, 2nd G-Matrix
01234 0 00000 1 00000 2 00011 3 00101 4 00110
20
Output Grouping

• G-Matrix Example

T2 added
Another cover, 2nd G-Matrix
01234 0 00000 1 00101 2 01012 3 00101 4 01210
21
Output Grouping

• G-Matrix Example

T3 added
Another cover, 2nd G-Matrix
01234 0 00000 1 00111 2 01012 3 01101 4 01210
22
Output Grouping

• G-Matrix Example

T4 added
Another cover, 2nd G-Matrix
01234 0 00100 1 00111 2 11012 3 01101 4 01210
23
Output Grouping

• G-Matrix Example

T5 added
Another cover, 2nd G-Matrix
01234 0 00100 1 00211 2 12012 3 01101 4 01210
24
Output Grouping

• G-Matrix Example

T6 added
Another cover, 2nd G-Matrix
01234 0 00100 1 00211 2 12012 3 01102 4 01220
25
Output Grouping

• G-Matrix Example

Normalization transform into lt0, 1gt
01234 0 00100 1 00211 2 12012 3 01102 4 01220
0 1 2 3 4 0 0 0 0.5 0 0 1 0 0
1 0.5 0.5 2 0.5 1 0 0.5 1 3 0 0.5 0.5 0
1 4 0 0.5 1 1 0
2
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Output Grouping

• G-Matrix Example

Summing the two matrices
0 1 2 3 4 0 0 0 1.5 0 0 1 0 0
2 1.5 0.5 2 1.5 2 0 0.5 2 3 0 1.5 0.5 0
2 4 0 0.5 2 2 0
0 1 2 3 4 0 0 0 0.5 0 0 1 0 0
1 0.5 0.5 2 0.5 1 0 0.5 1 3 0 0.5 0.5 0
1 4 0 0.5 1 1 0
01234 0 00100 1 00110 2 11001 3 01001 4 00110


27
Output Grouping

• G-Matrix Example

• Final output grouping
• Task
• Divide the 5-output circuit into max. 3-output
  blocks

28
Output Grouping

• G-Matrix Example

Final output grouping
Find maximum
0 1 2 3 4 0 0 0 1.5 0 0 1 0 0
2 1.5 0.5 2 1.5 2 0 0.5 2 3 0 1.5 0.5 0
2 4 0 0.5 2 2 0
B1 1, 2
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Output Grouping

• G-Matrix Example

Final output grouping
Find maximum in row 1 and column 2
0 1 2 3 4 0 0 0 1.5 0 0 1 0 0
2 1.5 0.5 2 1.5 2 0 0.5 2 3 0 1.5 0.5 0
2 4 0 0.5 2 2 0
Select 4
B1 1, 2, 4
30
Output Grouping

• G-Matrix Example

Final output grouping
B1 1, 2, 4 B2 0, 3
31
Experimental Results

• Hard MCNC benchmarks
• 3 experiments for each
• Minimize by Boom and decompose into2-input gate
  network using SIS
• Randomly divide the circuit into several blocks,
  then 1.
• Divide the circuit using the proposed method, the
  1.

32
Experimental Results
33
Conclusions

• A new output-grouping method
• Based on FC-Min
• Significant area reduction observed, with respect
  to the random technique
• Input support reduced too
• Very fast