Fast Boolean Minimizer for Completely Specified Functions

1
Fast Boolean Minimizer for Completely Specified
Functions

• Petr Fier1, Premysl Rucký1, Irena Vánová2
• 1Czech Technical University in Prague
• Dept. of Computer Science and Engineering
• 2UTIA, CAS

2
Outline

• Motivation
• Ternary tree
• The minimization algorithm
• Experimental results
• Conclusions

3
Motivation

• Standard two-level minimizers are not able to
  handle huge functions
• Normal functions ? Espresso
• Many input variables ? BOOM
• Many output variables ? FC-Min
• Many defined terms ? ???

4
Motivation

• Functions having many (up to millions) product
  terms often emerge in logic synthesis (e.g.,
  collapsing, Boolean manipulations with functions,
  etc.)
• These functions often contain redundancies,
  easily detectable absorptions,
• Early detection of these would significantly
  reduce memory consumption and the computation
  time
• ? There is a need for a two-level minimizer able
  to handle such functions.
• The priority is speed, not the result quality
• Do something with it, but I want the result
  immediately

5
Ternary Tree

• Recall
• Many terms tenths of thousands, millions
• There could occur redundancies (duplicate terms)
• ? An efficient storage structure is needed

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Ternary Tree
Used as a storage of terms. It is not a function
representation (like BDDs)! Ternary tree node
0
1
-
Depth number of input variables
7
Ternary Tree
a
b
c
d
e
00000
0001-
-0-10
-0-11
101-1
11--0
11111
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Ternary Tree

• Properties
• Insertion of a term in O(n)
• Looking up a term in O(n)
• Size between n and 3n1
• Comparison of look-up speeds

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Ternary Tree Look-up Example
Searching abcd (1001-)
a
b
failure
c
d
e
00000
0001-
-0-10
-0-11
101-1
11--0
11111
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TT Minimization

• Very simple. Only two rules applicable to the
  leaves only
• Absorption of one variable
• a ab a
• (00-) (001) (00-)
• Complement property
• ab ab b
• (000) (001) (00-)

???-
???1
???-
???1
???-
???0
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TT Minimization Example
001010011100101110111
00101-10-11-
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Tree Rotation

• The operations can be performed on leaves only gt
  the tree must be rotated
• Cut off the root node ? TT is split into three
  (at most)
• Append the root variable to all the leaves
• Merge the trees

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Tree Rotation Example
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TT Minimization Example cont.

10-00-01-11-
001-1-10-
15
Experimental Results - Collapsing
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Experimental Results - Collapsing
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Experimental Results Random Functions
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Conclusions

• A new two-level minimizer based on ternary trees
• Good for functions described by many terms
• Ternary tree is a good storage for a large
  number of terms
• The minimization is very fast, the result quality
  is not optimum, though
• Anyway, we are lucky that at least some
  minimization is done, since standard tools are
  completely unusable for these functions
• Practice has shown that such a fast minimization
  is essential for many logic synthesis processes
  (multi-level network collapsing, Boolean
  manipulation with functions)
• If it is used as a preprocessor to Espresso, the
  overall minimization time is reduced and the
  result quality is retained