A recursive paradigm to solve Boolean relations

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A recursive paradigm to solve Boolean relations

• David Bañeres Univ. Politècnica de
  Catalunya
• Jordi Cortadella Univ. Politècnica de
  Catalunya
• Mike Kishinevsky Strategic CAD Lab.,
  Intel Corp.

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Boolean relation example
3
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111

a
4
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111

x
0
?
y
z
5
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 0?0 0?0 0?0
x
0
?
y
z
6
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 0?0 0?0 0?0
?
x
0
y
z
7
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 ?01 0?0 ?01 0?0 ?01 0?0 ?01
?
x
0
y
z
8
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 ?01 0?0 ?01 0?0 ?01 0?0 ?01
x
1
?
y
z
9
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 ?01 1?0 0?0 ?01 1?0 0?0 ?01 1?0 0?0
?01
x
1
?
y
z
10
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 ?01 1?0 0?0 ?01 1?0 0?0 ?01 1?0 0?0
?01
?
x
1
y
z
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Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 ?01 1?0 ?11 0?0 ?01 1?0 ?11 0?0
?01 1?0 ?11 0?0 ?01
?
x
1
y
z
12
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 ?01 1?0 ?11 0?0 ?01 1?0 ?11 0?0
?01 1?0 ?11 0?0 ?01
x
?
y
z
13
Boolean relation example
x y z
b
c
a
f
01010?10
00110011
01010101
00001111
0?0 ?01 1?0 ?11 0?0 ?01 1?0 ?11 0?0
?01 ? ? ? 1?0 ?11 0?0 ?01
x
?
y
z
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Boolean relation example
x y z
b
c
a
00110011
01010101
00001111
0?0 ?01 1?0 ?11 0?0 ?01 1?0 ?11 0?0
?01 ? ? ? 1?0 ?11 0?0 ?01
a
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Boolean relation example
x y z
b
c
a
00110011
01010101
00001111
0?0 ?01 1?0 ?11 0?0 ?01 1?0 ?11 0?0
?01 ? ? ? 1?0 ?11 0?0 ?01
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Applications of Boolean Relations

• Boolean matching techniques for library binding
• L. Benini and G.D. Micheli, 1997
• FSM encoding
• B. Lin and F.Somenzi, 1990
• Boolean decomposition
• M. Damiani, J. Yang, and G.D. Micheli, 1995
• and others

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Outline

• Boolean Relations
• The semi-lattice of solutions
• The recursive paradigm
• Experimental results

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Boolean Relations
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Boolean Relations
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Boolean Relations
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Solving Boolean Relations
Finding f and g suchthat ?R is a tautology
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Solving Boolean Relations
Finding f and g suchthat ?R is a tautology
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Solving Boolean Relations
Finding f and g suchthat ?R is a tautology
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Solving Boolean Relations
Finding f and g suchthat ?R is a tautology
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Solving Boolean Relations
Finding f and g suchthat ?R is a tautology
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Solving Boolean Relations
Finding f and g suchthat ?R is a tautology
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Related work

• Exact solvers
• F. Somenzi and R.K. Brayton (1989)
• Method similar to Quine-McCluskey
• Extension of primes to c-primes
• S. Jeong and F.Somenzi (1992)
• Formulate the problem as a BCP

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Related work

• Heuristic approaches (based on ESPRESSO)
• A. Ghosh, S. Devadas and A. Newton (1992)
• Y. Watanabe and R. K. Brayton (1993)

GYOCRO xInitial Solution do REDUCE(x) E
XPAND(x) IRREDUNDANT(x) while (x is
improved)
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f g
a b
Boolean relation
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f g
a b
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(No Transcript)
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00 01 10 11
00 01 10 11
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R?R
?
Semi-lattice
Functions
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(Bn ? Bm)
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Optimum-cost function
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(Bn ? Bm)
Number of variables
Number of functions
? ( )
Size of the semi-lattice
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Shortcut use 2-level minimizers ! (ESPRESSO,
Minato-Morreale, Scherzo)
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Subclasses of Boolean relations
General Boolean Relations
Incompletely Specified Functions
Completely Specified Functions
ESPRESSO Minato-Morreale (BDDs) Scherzo (ZDDs) ...
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10 00 ?1 01 10
10 11 ?1 01 10
10 00 11 01 10
10 00 01 01 10
10 11 01 01 10
10 11 11 01 10
10 00 01 01
10 00 01 10
10 00 11 01
10 00 11 10
10 11 01 01
10 11 01 10
10 11 11 01
10 11 11 10
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Quick solver for Boolean Relations(initial
solution in Gyocro)
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Quick solver for Boolean Relations(initial
solution in Gyocro)
f
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Gyocro
10 00 ?1 01 10
10 11 ?1 01 10
10 00 11 01 10
10 00 01 01 10
10 11 01 01 10
10 11 11 01 10
10 00 01 01
10 00 01 10
10 00 11 01
10 00 11 10
10 11 01 01
10 11 01 10
10 11 11 01
10 11 11 10
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GYOCRO xInitial Solution do REDUCE(x) E
XPAND(x) IRREDUNDANT(x) while (x is
improved)
10 00 11 01
10 11 11 10
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Another example
x y
Quick solver (also Gyocro)
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Quick solver for Boolean Relations

• The solution is determined by theminimization
  order.
• Solutions are typically unbalanced.
• Difficult to escape from the local minimum.

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The recursive approach(BREL)
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Projections
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Projections
f
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Projections
f
f g
a b
00 01 10 11
10 ?? ?1 ??
merge
g
R
R
R is an incompletely specified function, but
has more flexibility than R (R ? R)
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ISF
projection
BR
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ISF
ISF minimizer
done !
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ISF
ISF minimizer
incompatible !
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The recursive paradigm
ISF with more flexibility
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The recursive paradigm
ISF with more flexibility
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ISF
ISF minimizer
incompatible !
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The recursive paradigm
ISF with more flexibility
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The recursive paradigm
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ISF
ISF minimizer
incompatible !
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BR1 BR2
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The recursive paradigm
?
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The recursive paradigm
f g
10 00 11 01
?
ISF
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The recursive paradigm
f g
10 00 11 01
?
ISF
ISF
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The recursive paradigm
f g
10 00 11 01
?
ISF
ISF
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10 00 ?1 01 10
10 11 ?1 01 10
10 00 11 01 10
10 00 01 01 10
10 11 01 01 10
10 11 11 01 10
10 00 01 01
10 00 01 10
10 00 11 01
10 00 11 10
10 11 01 01
10 11 01 10
10 11 11 01
10 11 11 10
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10 00 ?1 01 10
10 11 ?1 01 10
10 11 01 01 10
10 11 11 01 10
10 00 11 01
10 11 01 01
10 11 01 10
10 11 11 01
10 11 11 10
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10 00 ?1 01 10
10 11 ?1 01 10
10 00 11 01
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(No Transcript)
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BREL (R,bestF ) R project (R) F
ISF-minimize (R) if cost(F) ? cost(bestF)
return if F ? R then bestF F else
x pick-one-conflict (F,R) (R1,R2) split
(R,x) BREL(R1,bestF) BREL(R2,bestF)
endif
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Algorithmic engineering details

• BDDs for the exploration of solutions
• Efficient manipulation of characteristic
  functions
• ISF minimizers (Minato-Morreale, Restrict, )
• Intensive use of the cache (many identical
  sub-problems)
• Store the best k solutions
• ESPRESSO factorization (accurate cost
  estimation)
• Hybrid exploration of the tree (BFS DFS quick
  solver)
• Clever conflict selection (with many neighboring
  conflicts)
• Highly customizable cost function, traversal
  strategy, best
  solution after timeout

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Gyocro Gyocro Gyocro BREL BREL BREL
PI PO LIT AREA CPU LIT AREA CPU
int1 4 3 8 9280 0.03 9 8352 0.01
int10 6 4 32 44544 0.08 34 41296 0.02
c17i 5 3 34 34336 0.04 30 32016 0.01
she1 5 3 16 16240 0.04 15 17632 0.00
she2 5 5 31 30624 0.09 24 26488 0.01
she3 6 4 23 24592 0.08 21 21344 0.01
she4 5 6 56 62176 0.14 40 46864 0.03
gr 14 11 318 346608 3.43 313 322016 6.79
b9 16 5 321 382336 4.68 256 306240 0.19
int15 22 14 506 525248 21.94 459 472352 19.14
vtx 22 6 117 151728 30.10 101 94656 0.58
Normalized sum Normalized sum Normalized sum 1.00 1.00 1.00 0.89 0.86 0.44
BREL Configuration Cost function ? Sum of BDDs
sizes ISF Minimizer ? Minato-Morreale
explored relations 10
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Gyocro Gyocro Gyocro BREL BREL BREL
PI PO LIT AREA CPU LIT AREA CPU
int1 4 3 8 9280 0.03 9 8352 0.01
int10 6 4 32 44544 0.08 34 41296 0.02
c17i 5 3 34 34336 0.04 30 32016 0.01
she1 5 3 16 16240 0.04 15 17632 0.00
she2 5 5 31 30624 0.09 24 26488 0.01
she3 6 4 23 24592 0.08 21 21344 0.01
she4 5 6 56 62176 0.14 40 46864 0.03
gr 14 11 318 346608 3.43 313 322016 6.79
b9 16 5 321 382336 4.68 256 306240 0.19
int15 22 14 506 525248 21.94 459 472352 19.14
vtx 22 6 117 151728 30.10 101 94656 0.58
Normalized sum Normalized sum Normalized sum 1.00 1.00 1.00 0.89 0.86 0.44
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Gyocro Gyocro Gyocro BREL BREL BREL
PI PO LIT AREA CPU LIT AREA CPU
int1 4 3 8 9280 0.03 9 8352 0.01
int10 6 4 32 44544 0.08 34 41296 0.02
c17i 5 3 34 34336 0.04 30 32016 0.01
she1 5 3 16 16240 0.04 15 17632 0.00
she2 5 5 31 30624 0.09 24 26488 0.01
she3 6 4 23 24592 0.08 21 21344 0.01
she4 5 6 56 62176 0.14 40 46864 0.03
gr 14 11 318 346608 3.43 313 322016 6.79
b9 16 5 321 382336 4.68 256 306240 0.19
int15 22 14 506 525248 21.94 459 472352 19.14
vtx 22 6 117 151728 30.10 101 94656 0.58
Normalized sum Normalized sum Normalized sum 1.00 1.00 1.00 0.89 0.86 0.44
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Gyocro Gyocro Gyocro BREL BREL BREL
PI PO LIT AREA CPU LIT AREA CPU
int1 4 3 8 9280 0.03 9 8352 0.01
int10 6 4 32 44544 0.08 34 41296 0.02
c17i 5 3 34 34336 0.04 30 32016 0.01
she1 5 3 16 16240 0.04 15 17632 0.00
she2 5 5 31 30624 0.09 24 26488 0.01
she3 6 4 23 24592 0.08 21 21344 0.01
she4 5 6 56 62176 0.14 40 46864 0.03
gr 14 11 318 346608 3.43 313 322016 6.79
b9 16 5 321 382336 4.68 256 306240 0.19
int15 22 14 506 525248 21.94 459 472352 19.14
vtx 22 6 117 151728 30.10 101 94656 0.58
Normalized sum Normalized sum Normalized sum 1.00 1.00 1.00 0.89 0.86 0.44
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Recursive n-way decomposition
subsumes bi-decomposition
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Recursive n-way decomposition
0
subsumes bi-decomposition
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Recursive n-way decomposition
subsumes bi-decomposition
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ALGEBR SPEED ALGEBR SPEED BREL-DECOMPOSITION BREL-DECOMPOSITION BREL-DECOMPOSITION
PI PO PO AREA DELAY AREA DELAY CPU
9symml 9 1 1 293712 9.79 149872 7.85 192.34
majority 5 1 1 8352 3.06 7424 2.86 3.32
b1 3 4 4 12528 2.86 10672 2.81 0.97
cm150a 21 1 1 63568 7.25 63568 5.96 46.02
cm85a 11 3 3 68208 5.57 57072 4.82 30.42
cm162a 14 5 5 83520 5.93 75632 5.37 42.26
cm163a 16 5 5 61248 6.19 63104 5.44 29.26
frg1 28 3 3 139664 9.07 68672 5.67 679.48
c8 28 18 18 150800 6.99 198128 6.01 84.92
cc 21 20 20 87696 5.98 89088 5.16 29.20
ttt2 24 21 21 232464 8.66 271904 7.29 162.19
cht 47 36 36 174000 6.08 270976 5.16 40.67
i5 133 66 66 447296 8.20 600880 6.99 503.50
i7 199 67 67 724304 8.71 817568 7.29 283.52
x3 135 99 99 832416 11.10 1215680 7.50 1082.86
Normalized sum Normalized sum Normalized sum Normalized sum 1.00 1.00 1.17 0.81
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Conclusions

• Boolean relations carry a huge space of
  solutions.
• Easy to be trapped on sub-optimal local minima.
• The semi-lattice is infested by ISFs !
• BREL finds a path to the best ISF by
  recursivelysolving conflicts
• Future work
• Exploitation of symmetries
• Boolean decomposition targeting at delay
  minimization
• BREL publicly available (send e-mail)