Bez nadpisu

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Boolean Minimizer FC-Min Coverage Finding Process
Petr Fier, Hana Kubátová Czech Technical
University Department of Computer Science and
Engineering
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Outline

• Main Features of FC-Min
• Basic Principles
• An Example
• Details on the Find Cover Phase
• Experimental Results
• Conclusions

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Main Features

• Extremely fast two-level Boolean minimizer
• Capable to handle functions with a large number
  of input output variables
• Advantageous for highly unspecified functions
• Low memory demands

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Principles

• Implicant generation is different from standard
  methods (Q-M based)
• No prime implicants are being generated
• Only necessary group implicants are produced
• The cover of the on-set is found first
• Then the implicants with the properties of the
  cover are computed

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Preliminaries
Given Input matrix In, p Output matrix Om,
p ? A Boolean function of n input variables m
output variables p defined terms, the rest are
dont cares
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Three Main Phases

• Find Cover (thus FC-Min)
• Find Implicants
• Expand Input

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Three Main Phases

• Find Cover (thus FC-Min)
• Find Implicants
• Expand Input

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Find Cover Phase

• We try to find a rectangle cover of the on-set
• The elements of the cover will determine the
  implicants of the final solution
• The tentative implicants are being derived from
  the Output matrix only!

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Example
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The Algorithm

• Find Cover is NP-hard
• ?
• Some heuristic has to be used
• We use a greedy heuristic based on a gradual
  search for coverage elements consisting of the
  maximum number of 1s

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The Algorithm

• Select a row containing the most of 1s
• Continue the search for a next row to add in
  order to increase the number of the covered 1s
• Repeat 2. until the number of 1s increases (or
  stop see next slide)
• Repeat all until the whole on-set is covered

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The Depth Factor

• Finding a cover consisting of many 1s in the
  output matrix is advantageous but the
  implicants are hard to find and the IG phase
  fails
• Solution the Depth Factor DF
• With a given probability we decide whether to
  prolong C(ti) or not

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The Depth Factor
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The Depth Factor
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The Depth Factor
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Find Implicants Phase

• Main Idea
• When a term (cube) should cover a particular
  output vector, the corresponding input vector
  must be contained in this cube
• ? Thus the minimum term satisfying the particular
  cover can be constructed as a minimum supercube
  of all the input vectors corresponding to C(ti)

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Find Implicants Phase
Both Matrices 0 11010 10000 1 10000 11100 2
01001 01100 3 01111 01010 4 00110 00111 5 01110
00000 6 10110 00011 7 00001 01101 8 10101 10111 9
11100 10100
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Find Implicants Phase
All the implicants
SOP Forms y0 t3 t5 x0 x2 x3' x0 x2'
x4 y1 t2 t4 x2'x3' x0' x1 x2 x3 x4 y2
t2 t3 t6 x2'x3' x0 x2 x3' x0' x1' y3
t1 t4 x1'x2 x0' x1 x2 x3 x4 y4 t1 t6
x1'x2 x0' x1'
t1 -01-- 00011 t2 --00- 01100 t3 1-10-
10100 t4 01111 01010 t5 1-0-0 10000 t6 00---
00101
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Experimental Results

• Significantly faster than ESPRESSO
• Result quality comparable to ESPRESSO BOOM
• Produces results having a low number of terms
• Details in the Proceedings

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MCNC Benchmarks

• 120 Benchmarks were solved
• 72 were solved in a shorter time than ESPRESSO
• In 86 FC-Min reached the same or better result
• In 67 both

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Time Complexity
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Time Complexity
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Time Complexity
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Conclusions

• A new two-level Boolean minimizer has been
  proposed
• Novel method of implicant generation
• Usable for extremely large problems
• Extremely good for problems with a large number
  of output variables