1. Fast Boolean Minimizer for Completely Specified Functions
Petr Fišer1, Přemysl Rucký1, Irena Váňová2
1Czech Technical University in Prague
Dept. of Computer Science and Engineering
2UTIA, CAS

2. Outline Motivation Ternary tree The minimization algorithm
Experimental results
Conclusions
DDECS’2008, Bratislava

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 → ???
DDECS’2008, Bratislava

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”
DDECS’2008, Bratislava

5. Ternary Tree Recall: Many terms = tenths of thousands, millions
There could occur redundancies (duplicate terms)
→ An efficient storage structure is needed
DDECS’2008, Bratislava

6. Ternary Tree Used as a storage of terms.
It is not a function representation (like BDDs)!
Ternary tree node: 1 -
Depth = number of input variables
DDECS’2008, Bratislava

7. Ternary Tree a b c d e 00000 0001- -0-10 -0-11 101-1 11--0 11111
DDECS’2008, Bratislava

8. Ternary Tree Properties: Insertion of a term in O(n)
Looking up a term in O(n)
Size: between n and 3n+1
Comparison of look-up speeds:
On success
On failure
Ternary tree n
1 … (n-1)
Truth table
n … n.p
n.p
DDECS’2008, Bratislava

9. Ternary Tree Look-up Example
Searching: ab‘c‘d (1001-) a b
failure c d e
00000
0001-
-0-10
-0-11
101-1
11--0
11111
DDECS’2008, Bratislava

10. TT Minimization Absorption of one variable Complement property
Very simple. Only two rules applicable to the leaves only:
Absorption of one variable
a + ab = a
(00-) + (001) = (00-)
Complement property
ab + a’b = b
(000) + (001) = (00-)
???-
???1
???-
???0
???1
???-
DDECS’2008, Bratislava

11. TT Minimization Example
DDECS’2008, Bratislava

12. Tree Rotation
The operations can be performed on leaves only => 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
DDECS’2008, Bratislava

13. Tree Rotation Example
DDECS’2008, Bratislava

14. TT Minimization Example cont.…
DDECS’2008, Bratislava

15. Experimental Results - Collapsing
Benchmark
Size (n / p)
Espresso
TT-Min
TT-min + Espresso
Terms
Time [s]
c880_3
60 /
15673
470
57853
23.11
305
c880_4
60 / 67136
440 37
12046
6.44
439
8.5
c1908_4
33 / 32768
7040 20
20990
1.1 17
c1908_5
33 / 6464
2144 1
4640
0.19 2
c1908_6
33 / 13700
3200 4
8704
0.41
4.5
c3540_1
50 /
32455
4740
101512
32.9
32396
1569
c3540_2
50 / 42568
10202
254
19620
4.2
10065
193
c3540_3
50 / 4464
1022
2112
0.34
2.5
c3540_4
50 / 1912
184
0.1
459
0.07
c3540_5
50 / 6657
1516 5
3654
0.62
1522
c3540_6
50 /
14397
900
43442
9.09
382
c3540_7
50 / 6933
611
2360
0.37
DDECS’2008, Bratislava

16. Experimental Results - Collapsing
Benchmark
Size (n / p)
Espresso
TT-Min
TT-min + Espresso
Terms
Time [s]
c880_3
60 /
15673
470
57853
23.11
305
c880_4
60 / 67136
440 37
12046
6.44
439
8.5
c1908_4
33 / 32768
7040 20
20990
1.1 17
c1908_5
33 / 6464
2144 1
4640
0.19 2
c1908_6
33 / 13700
3200 4
8704
0.41
4.5
c3540_1
50 /
32455
4740
101512
32.9
32396
1569
c3540_2
50 / 42568
10202
254
19620
4.2
10065
193
c3540_3
50 / 4464
1022
2112
0.34
2.5
c3540_4
50 / 1912
184
0.1
459
0.07
c3540_5
50 / 6657
1516 5
3654
0.62
1522
c3540_6
50 /
14397
900
43442
9.09
382
c3540_7
50 / 6933
611
2360
0.37
DDECS’2008, Bratislava

17. Experimental Results – Random Functions
Benchmark
Espresso
TT-Min
Terms
Time [s]
30 / 5000 35
17.82
1004
0.63
35 / 5000
1169
1719
1.01
35 / 8000
200
400.98
3703
2.41
40 / 5000 -
> 24 h
3033
5.5
40 / 10000
5054
7.0
40 / 20000
6606
13.72
50 / 30000
13232
33.6
60 / 50000
38203
94.9
DDECS’2008, Bratislava

18. 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…)
DDECS’2008, Bratislava