1. To SAT or Not to SAT: Ashenhurst Decomposition in a Large Scale
Hsuan-Po Lin, Jie-Hong Roland Jiang, and Ruei-Rung Lee
Graduate Institute of Electronics Engineering / Department of Electrical Engineering,
National Taiwan University
5/21/2019
ICCAD 2008 1

2. Outline 1 2 3 4 5 Introduction Prior Work Our Approach
Experimental Results
Conclusions 5
5/21/2019
ICCAD 2008

3. Introduction Ashenhurst decomposition f(X) = h(XH,XC,g(XG,XC))
When can f be decomposed as the composition of some g and h?
Comprehensible using decomposition chart
5/21/2019
ICCAD 2008

4. f h Decomposition Chart f(a,b,c,d) = h(c,d,g(a,b,d)) XH : {c}
XG : {a,b}
XC : {d}
f(a,b,c,d) = h(c,d,g(a,b,d)) f
XG XC
a,b,d
000
010
100
110
001
011
101
111 00 10 1 01 11
000
010
100
110
001
011
101
111 00 10 1 01 11
XH XC g a b d h c
c,d h g 1 00 10 01 11 1 00 10 01 11
XH XC
c,d f
5/21/2019
ICCAD 2008

5. Prior Work BDD-based functional decomposition Memory explosion problem
Variable partition cannot be automated
Hard to handle non-disjoint decomposition
Multiple-output decomposition cannot be handled naturally
5/21/2019
ICCAD 2008

6. Our Approach SAT-based computation
Interpolation  get g
Functional dependency  get h
With/without pre-specified variable partition
Single-/multiple-output decomposition
5/21/2019
ICCAD 2008

7. Single-Output Decomposition
Problem formulation
Given a function f(X) with variable partition X={XG|XH|XC}, find g and h such that f(X) = h(XH,XC,g(XG,XC))
5/21/2019
ICCAD 2008

8. Decomposability as SAT
Functions g and h exists iff
(f(XH1, XG1, XC)≠ f(XH1, XG2, XC))Λ
(f(XH2, XG2, XC)≠ f(XH2, XG3, XC))Λ
(f(XH3, XG3, XC)≠ f(XH3, XG1, XC)) is UNSAT f
XH1
XG1 XC
XG2
XH2
XG3
XH3 1
xc[[XC]] f
xg1[[XG1]]
xg2[[XG2]]
xg3[[XG3]]
xh1[[XH1]] 1
xh2[[XH2]]
xh3[[XH3]]
5/21/2019
ICCAD 2008

9. Craig Interpolation P φA φB
For formulas φA and φB with φA Λ φB UNSAT, then there exists an interpolant P of φA such that
1. φA  P
2. P Λ φB is UNSAT
3. P refers only to the
common variables of
φA and φB
We show the connection between P and g P φB φA
5/21/2019
ICCAD 2008

10. Interpolating g-function
P: refer to the common variables XG1, XG2, XC
φA: CP(XG1)≠CP(XG2)
φB: CP(XG2)≠CP(XG3),CP(XG1)≠CP(XG3), i.e., CP(XG1)=CP(XG2)
f(XH1, XG1, XC)≠ f(XH1, XG2, XC)
(f(XH2, XG2, XC)≠ f(XH2, XG3, XC))Λ
(f(XH3, XG3, XC)≠ f(XH3, XG1, XC)) φA φB f
XH1
XG1 XC
XG2
XH2
XG3
XH3 1
5/21/2019
ICCAD 2008

11. Interpolating g-function (cont’d)
XG1
XG2 … p q r s a
offset
onset
Relation characterized by
P(XG1, XG2,c) for some c  [[Xc]]
Relation after cofactoring
P(XG1=a, XG2,c) w.r.t. some a  [[XG1]]
P(XG1=a, XG2, XC) is a feasible implementation of g(XG, XC) !
5/21/2019
ICCAD 2008

12. Computing h-function A naïve approach Non-scalable(w.r.t. XC size)
For XC = , function h can be derived using Shannon expansion
h(XH,xg)=(xg Λ hxg) v (xg Λ hxg)
Non-scalable(w.r.t. XC size)
where
hxg(XH) = h(XH,0) = h(XH,g(XG=a)) = f(XH,XG=a) and
hxg(XH) = h(XH,1) = h(XH,g(XG=b)) = f(XH,XG=b)
for a,b[[XG]] with g(a)=0 and g(b)=1
5/21/2019
ICCAD 2008

13. Computing h-function (cont’d)
A scalable approach
Function h can be obtained with functional dependency formulation
f(X) = h(φ1(X), φ2(X),…, φm(X)), where h is the unknown to be computed
Let the above gi be such that
φi = xi with xi  XHXC for i = 1,…,m-1
φm = g(XG,XC)
So-derived h is what we want
Computation can be done with pure SAT solving [Lee et al.07]
5/21/2019
ICCAD 2008

14. Overall Flow AshenhurstDecomposition(f, XH, XG, XC):
F CircuitInstantiation(f )
(φA, φB) CircuitPartition(F)
P(XG1,XG2,XC) Interpolation(φA, φB)
g(XG,XC) Cofactor(P, a[[XG1]])
h FunctionalDependency(f, g)
return(g, h)
5/21/2019
ICCAD 2008

15. Automating Variable Partition
For each xi  X, add
conditional clauses for XG: ((xi2≡xi3)Λ(xi4≡xi5)Λ(xi6≡xi1)) V ai
conditional clauses for XH: ((xi1≡xi2)Λ(xi3≡xi4)Λ(xi5≡xi6)) V bi
(ai,bi)
(0,0) xi  XC
(0,1) xi  XG
(1,0) xi  XH
(1,1) xi can
be in either of
XG and XH f
XH1
XG1
XC1
XH2
XG2
XC2
XH3
XG3
XC3
XH4
XG4
XC4
XH5
XG5
XC5
XH6
XG6
XC6 1
5/21/2019
ICCAD 2008

16. Automating Variable Partition (cont’d)
Make “unit assumptions” on control variables ai, bi for each xi
Avoid trivial partition
Impose seed partition
Enforce one variable in XH
Enforce two variables in XG
Enumerate seed partitions
Incremental SAT solving
Apply minimal UNSAT core refinement to reduce XC
5/21/2019
ICCAD 2008

17. Multi-Output Decomposition
New UNSAT condition:
V(fi(XH1, XG1, XC)≠ fi(XH1, XG2, XC))Λ
V(fi(XH2, XG2, XC)≠ fi(XH2, XG3, XC))Λ
V(fi(XH3, XG3, XC)≠ fi(XH3, XG1, XC))
Computation is same as before i i i XG XH X g h f xg XC
...
5/21/2019
ICCAD 2008

18. Experimental Setup Implemented in ABC using MiniSAT
Linux, Xeon 3.4GHz CPU, and 6Gb RAM
Decompose primary-output and transition functions
Only functions with no less than 50 inputs are considered
ISCAS, MCNC, and ITC
5/21/2019
ICCAD 2008

19. Experimental Results Single-output decomposition
Circuit
#func
#var
#fail
#SAT_TO
#succ
#var_succ
#VP_avg
rate_
valid_VP
time_avg
(sec)
Mem
(Mb)
b15
370
143—306 51
319
1519
0.917
96.62
107.20
b17
1009
76—308
148
861
1645
0.904
87.12
125.84
c7552 36
50—194 2 34
1350
0.455
64.38
36.65
s13207 3
212—212
569
0.908
70.26
50.62
s38417
256
53—99 6 72
178
1090
0.523
103.33
136.04
s38584 7
50—147
1120
0.924
47.13
51.56
Two-output decomposition
Circuit
#pair
#var
#fail
#SAT_TO
#succ
#var_succ
#VP_avg
rate_
valid_VP
time_avg
(sec)
Mem
(Mb)
b15
201 31
170
1176
0.845
113.86
224.07
b17
583 88
495
676
0.824
103.12
419.35
c7552 21 2 19
188
0.465
89.57
78.67
s13207 3
585
0.7
93.36
118.03
s38417
218 13 30
175
689
0.498
109.06
319.48
s38584 9
1656
0.713
46.17
207.78
5/21/2019
ICCAD 2008

20. Experimental Results (cont’d)
Variable partition before / after UNSAT core refinement
5/21/2019
ICCAD 2008

21. Experimental Results (cont’d)
Variable partition qualities resulted from different efforts
||XG|-|XH||/|X|
|XC|/|X|
w/ mini
w/o mini
5/21/2019
ICCAD 2008

22. Conclusions Proposed pure SAT-based Ashenhurst decomposition
Easily extendable to non-disjoint and multiple-output decompositions
Scalable to functions with up to 300 input variables
Future work
Functional decomposition beyond Ashenhurst
5/21/2019
ICCAD 2008

23. Thanks for Your Attention!
5/21/2019
ICCAD 2008