1. Faster Logic Manipulation for Large Designs
Alan Mishchenko Robert Brayton
UC Berkeley

2. Logic Representations
Truth tables
Sum of products (SOP, CNF)
Factored forms
BDDs
AIGs and TTs
DSDs
(since 1854?)
(since 1959)
(since 1983)
(since 1986)
(since 2000)
(since 2012)

3. Pros and Cons of Logic Representations
Efficiency for
small / large
functions
Canonicity
Speed of computing
Robustness
Size of represen- tation
Reveals Boolean properties** TT
+++/--
yes ++ -- -
SOP/CNF
+/+ no +
no/yes
+/++++ FF
+++
BDD
++/++
AIG
++/+++
++++
DSD
++++/-
** symmetry, unateness, NPN canonicity, decomposability, etc

4. Motivation for DSD Use Boolean network in SIS Equivalent AIG in ABC ax y f z
Equivalent AIG in ABC a b c d f e x y z
AIG is a Boolean network of 2-input AND nodes and invertors (dotted lines)

5. One AIG Node – Many Cuts Combinational AIG
Each AIG cut represents a different logic node
AIG manipulation with cuts is equivalent to working on many Boolean networks simultaneously f
4-input logic function a b c d e
Different cuts for the same node

6. Computing Logic Function of a CutS P E D
SOP
BDD TT
TTs are better than BDDs in terms of memory and runtime
However, both TTs and BDDs obscure Boolean properties
DSD
DSDs take less memory/runtime than BDDs/TTs for practical functions of K inputs (8 < K < 16)
DSDs explicitly represent Boolean properties
Symmetry, unateness, NPN canonicity, decomposition, etc
Very important for practical applications!

7. Logic Operations on Cut Functions
AIG Rewriting
Technology mapping with Boolean matching
Rewriting node A  a b c A
Subgraph 1
Subgraph 2
Rewriting node B a b c  B
Subgraph 2
Subgraph 1
6-LUT
16-input structure
composed of three 6-LUTs

8. DSD and 1-Step DSD DSD 1-step DSD
Represented as a tree whose leaves have disjoint support
DSD is full if functions at nodes are only {AND, XOR, MUX}.
Invertors are edge attributes.
A node without DSD is prime.
1-step DSD
There exists variable x such that both cofactors w.r.t. x are full DSDs ()
<> [] a b d e f g c
F(a,b,c,d,e,f,g) = AND(a, MUX(b,!XOR(c,d,e),!AND(f,g)))
()=and, <>=mux, []=xor, !=not

9. DSD Computation
DSDs and 1-DSDs represent functions of the cuts in the AIG during mapping or rewriting
At each node, find all K-cuts
Each cut defines a function with at most K inputs
Compute cut’s DSD by composing DSDs of the fanin cuts
Compute NPN canonical form and find it in (or add it to) the unique table in DSD manager
Use computed table to reuse the results of intermediate Boolean operations (such as computing AND of two DSDs)
How many cut functions are DSD or 1-DSD?

10. DSD: Prevalence in Benchmarks
Table 1A: Percentages of unique functions, NPN classes, and DSDs for different cut sizes (industrial benchmarks).
% of functions and classes
% of DSDs relative to NPN classes
(#DSDs / total n-input NPNs) * 100
% of DSDs relative to the number of cuts
(#DSDs / #cuts) * 100
Cut size
Cuts
Func
NPN
Full
1-step
Total 6
30.0 M
0.88
0.04
6.15
64.32
70.47
93.16
6.24
99.40 8
33.6 M
2.25
0.24
10.91
58.42
69.33
86.28
11.07
97.35 10
35.0 M
3.09
0.59
17.40
49.96
67.36
80.78
14.22
95.00 12
35.6 M
3.73
0.97
21.03
42.08
63.11
75.06
16.87
91.93 14
35.9 M
4.20
1.26
22.02
36.86
58.88
71.06
17.71
88.77
Table 1B: Percentages of unique functions, NPN classes, and DSDs for different cut sizes (public benchmarks).
% of functions and classes
% of DSDs relative to NPN classes
(#DSDs / total n-input NPNs) * 100
% of DSDs relative to the number of cuts
(#DSDs / #cuts) * 100
Cut size
Cuts
Func
NPN
Full
1-step
Total 6
19.8 M
1.68
0.07
4.74
58.38
63.12
89.72
9.32
99.04 8
21.1 M
4.66
0.56
8.27
47.22
55.49
78.48
16.36
94.84 10
21.3 M
6.71
1.46
11.57
37.79
49.37
69.96
19.60
89.56 12
21.7 M
7.92
2.23
12.55
32.69
45.24
63.51
20.93
84.44 14
22.0 M
8.89
2.92
11.99
28.40
40.39
60.36
19.25
79.61

11. DSD Package Similar to BDD package Different from BDD package
Maintains canonical forms
Performs Boolean operations
Employs computed table
Different from BDD package
Uses different data structure
Used different normalization rules
Limited to small practical functions (6-16 inputs)
The computed table is more reusable

12. Primitives of DSD Manager
One constant 0 node
One primary input node n
Multi-input AND nodes with ordered fanins
Multi-input XOR nodes with ordered fanins
Three-input MUX nodes
Multi-input PRIME nodes (w/o DSD; with 1-step DSD)
Multi-input PRIME nodes (w/o DSD; w/o 1-step DSD)

13. Usefulness of DSDs Constructed on-the-fly during cut enumeration
No BDDs
No TTs
Start from elementary variables
Derive resulting DSDs by ANDing fanin DSDs
Canonicized
Unique table (only stores NPN classes of Boolean functions)
Cached
Computed table
If available, the result is returned (e.g. LUT structures matching)
If not available, the result is computed and stored

14. Rules for Canonizing a DSD
Collapsing similar functions
AND((a,b), AND(c,AND(d,e))) = AND(a,b,c,d,e)
Propagating complements to inputs or output
AND(!a, !AND(!b, c))  AND(a, !AND(b, c))
AND(!a, !XOR(b, c))  AND(a, XOR(b, c))
AND(a, MUX(b,!AND(c,d),!AND(e,f)))  AND(a,!MUX(b,AND(c,d),AND(e,f)))
Using a single variable
AND(MUX(d, e, f), a, XOR(b, c))  AND(n, XOR(n, n), MUX(n, n, n))
F = (ab)c and G = d(ef)  AND(n, XOR(n, n)) - same NPN class
(Arbitrary but fixed) ordering rules, applied in the increasing order
The node fanins are ordered by their support size
If tie, AND precedes XOR precedes MUX precedes PRIME
If tie, a non-complemented fanin precedes a complemented fanin
If tie, fanins’ fanins are ordered and compared in their selected order
If recursive comparison of fanin subtrees fails to produce a unique order, the fanins’ DSDs are isomorphic and their order is immaterial

15. Example of Canonicizing a DSD
F(a,b,c,d,e,f,g) = OR( a, !MUX( b, !XOR(c,!e,d), !AND(f,g) ) ) OR ()
<> [] a b d e f g c ()
<> [] !a b d e f g c () !a
<> b () f g [] d e c
()=and, <>=mux, []=xor, !=not
PN = !abfgced

16. Examples of Full DSDs 2 inputs 3 inputs 4 inputs 5 inputs 6 inputs
(ab)
(abc)
(abcd)
(abcde)
(abcdef)
[ab]
(a!(bc))
(a!(b!(cd)))
(a[(bc)(de)])
(ab[(cd)(ef)])
[abc]
[abcd]
(a!(!(bc)!(de))
(!(abc)!(def))
[a(bc)]
(a[b(cd)])
(a<b(cd)e>)
[a<b[cd][ef]>]
(a[bc])
(a<bcd>)
<a(bc)(de)>
<a(bc)<def>>
<abc>
<ab[cd]>
<a<bc!(de)>>
<(ab!(cd))ef>
(ab) denotes AND(a,b)
[ab] denotes XOR(a,b)
<abc> denotes MUX(a, b, c) = ab + !ac
First two columns contain all canonical forms using this basis

17. Case Study: LUT structure mapping
16-input structure
composed of three 6-LUTs
Comparing run times
with truth tables (TTs)
with DSDs
Application selected because:
LUT structure mapping mitigates structural bias of AIG and improves the quality of regular LUT-mapping
useful for new generations of programmable devices
mapping requires heavy Boolean manipulation
implementation in ABC using TTs is prohibitively slow for large designs
Preliminary Results (DSDs vs. TTs)
Boolean matching
on average 30X faster
Total runtime of mapping
on average 7X faster

18. References Canonical form Computation from cofactors
R. L. Ashenhurst, “The decomposition of switching functions”. Computation Lab, Harvard University, 1959, Vol. 29, pp
Computation from cofactors
V. Bertacco and M. Damiani, "Disjunctive decomposition of logic functions," Proc. ICCAD ‘97, pp
Computation from cofactors (corrections)
Y. Matsunaga, "An exact and efficient algorithm for disjunctive decomposition", Proc. SASIMI '98, pp
Alternative computations
T. Sasao and M. Matsuura, "DECOMPOS: An integrated system for functional decomposition," Proc. IWLS ’98, pp
S.-I. Minato and G. De Micheli, “Finding all simple disjunctive decompositions using irredundant sum-of-products forms”. Proc. ICCAD’98, pp
Boolean operations
S. Plaza and V. Bertacco, "Boolean operations on decomposed functions", Proc. IWLS ’05.
Applications in synthesis and mapping
A. Mishchenko, R. K. Brayton, and S. Chatterjee, "Boolean factoring and decomposition of logic networks", Proc. ICCAD'08, pp

19. Conclusion DSDs are an interesting representation
Effective only for small, practical functions!
DSDs reveal Boolean properties and facilitate
Rewriting
Technology mapping
Factoring
DSDs lead to larger scope (cut depth) than TTs
Most operations are local but repeated often
Memory usage and runtime are improved due to
Canonicity (unique table in the DSD manager)
Result caching (computed table in the DSD manager)

20. The End