1. Interpolating Functions from Large Boolean Relations
Jie-Hong Roland Jiang, Hsuan-Po Lin, and Wei-Lun Hung
Grad. Inst. of Electronics Eng. / Dept of Electrical Eng.
National Taiwan University

2. Outline ◆ Introduction ◆ Our Approach ◆ Experimental Results
◆ Conclusions

3. Relations vs. Functions
Boolean relation R(X, Y)
Boolean function F(X)
Allow one-to-many mappings
Can describe non- deterministic behavior
More generic than functions
Disallow one-to-many mappings
Can only describe deterministic behavior
A special case of relation 11 10 01 00
x1x2
y1y2 11 10 01 00
x1x2
y1y2
f1 = x1 x2
f2 =  x1 x2

4. Total Relations vs. Partial Relations
Every input element is mapped to at least one output element
Some input element is not mapped to any output element 11 10 01 00 1
x1x2 y 11 10 01 00 1
x1x2 y

5. T(X, y) = R(X, y)  y.  R(X, y)
Totalization
A partial relation can be totalized
Assume that the input element not mapped to any output element is a don’t care
Partial relation
Total relation 11 10 01 00 1
x1x2 y 11 10 01 00 1
x1x2 y
Totalize
T(X, y) = R(X, y)  y.  R(X, y)

6. Application of Boolean Relations
In high-level design, Boolean relations can be used to describe (nondeterministic) specifications
In gate-level design, Boolean relations can be used to characterize the flexibility of sub-circuits
Boolean relations are more powerful than traditional don’t-care representations 11 10 01 00
x1x2
y1y2

7. Needs of Relation Determinization
For hardware implement of a system, we need functions rather than relations
Physical realization are deterministic by nature
One input stimulus results in one output response

8. Needs of Relation Determinization
To simplify implementation, we can explore the flexibilities described by a relation for optimization y1 y1 x1 z1 z1 x1 x2 x2 z2 z2 y2 y2
f1 = x1 x2
f2 =  x1 x2
f1 = x2
f2 =  x1
x1x2
y1y2
z1z2 11 10 01 00 11 10 01 00 11 10 01 00

9. Problem Statement
Given a nondeterministic Boolean relation R(X, Y), how to determinize and extract functions from it?
For a deterministic total relation, we can uniquely extract the corresponding functions

10. Related Work Prior Work Ours BDD- or SOP-based representation
Not scalable
Focused on function optimization
Ours
AIG representation
Focus on scalability with reasonable optimization quality

11. Enabling Technique Craig Interpolation Theorem I φB φA
If φA  φB is UNSAT for propositional formulas φA ,φB,
then there exists an interpolant I such that
φA → I
I  φB is UNSAT
I refers only to the common
variables of φA and φB I φB φA

12. Single-Output Relations
For a single-output total relation R(X, y), we derive a function f for variable y using interpolation
 R(X,0)   R(X,1) is UNSAT I φB φA 11 10 01 00 1
x1x2 y 11 10 00
φB :  R(X,1)
Minimal care offset of f
φA :  R(X,0)
Minimal care onset of f

13. Interpolantion vs. Cofactoring
Naïve derivation of f is by cofactoring
R(X,1) (largest onset of f) or
 R(X,0) (smallest onset of f)
 φB= R(X,1)
Largest onset of f I φB φA 11 10 01 00 1
x1x2 y 11
φA= R(X,0)
Smallest onset of f 10 00
The function extracted from a single-output relation using cofactor is usually of similar size as the relation
Interpolation may extract smaller functions and prevent AIG explosion
 R(X,0) ⊆ I ⊆ R(X,1)

14. Multiple-Output Relations
Two-phase computation:
Phase 1 : Backward reduction
Reduce to single-output case
R(X, y1, …, yn) → y2, …, yn. R(x, y1, …, yn)
Existential quantification using expansion
y. φ(x,y) = φ(x,1) + φ(x,0)
Existential quantification using substitution [CAV 2009]
 y. φ(x,y) = φ(x, f(x))
Phase 2 : Forward substitution
Extract functions

15. Multiple-Output Relations
Example (Xp)
R(3) R
R(2)
Phase1: (expansion reduction)
y3.R(X, y1, y2 , y3) → R(3)(X, y1, y2)
y2.R(3)(X, y1, y2) → R(2)(X, y1) X y1 y2 y3
Phase2:
R(2)(X, y1) → y1 = f1 (X)
R(3)(X, y1, y2) → R(3)(X, f1(X), y2) → y2 = f2 (X)
R(X, y1, y2 , y3) → R(X, f1(X), f2(X), y2) → y3 = f3 (X) f1 X f2 X f3 X

16. Multiple-Output Relations
Example (St)
R<2> R
R<3>
Phase1: (substitution reduction)
R(X, y1, y2 , y3) → y3 = f3’(X , y1, y2 ) → R<3>(X, y1, y2)
R<3>(X, y1, y2) → y2 = f2’(X , y1 ) → R<2>(X, y1) X y1 y2 y3
Phase2:
R<2>(X, y1) → y1 = f1 (X)
R<3>(X, y1, y2) → R<3>(X, f1(X), y2) → y2 = f2 (X)
R(X, y1, y2 , y3) → R(X, f1(X), f2(X), y2) → y3 = f3 (X) y1 X
f3' y2 y1 X
f2' f1 X f2 X f3 X

17. Multiple-Output Relations
Example (SD)
R<3>
R<2> R
Phase1: (substitution reduction)
R(X, y1, y2 , y3) → y3 = f3’(X , y1, y2 ) → R<3>(X, y1, y2)
R<3>(X, y1, y2) → y2 = f2’(X , y1 ) → R<2>(X, y1)
R<2>(X, y1, y2) → y1 = f1’(X) X y1 y2 y3
Phase2: (interpolation for deterministic rel.)
D = R Λ ∏i (yi ≡ fi’)
fi (X) = Interpolant(Dyi, D yi) y1 X
f3' y2 y1 X
f2' X
f1' f1 X f2 X f3 X

18. Experimental Setup Our methods were implemented in ABC using MiniSAT
We prepare Boolean relations by computing the transition relations of ISCAS and ITC circuits
Insert don’t cares to introduce nondeterminism
Represent transition relations using AIGs

19. Experimental Results – w/o DC
Original
BDD Xp
Circuit
(#pi, #po) #n #l #v
time
s5378
(214, 179)
624 12
1570
783 10
1561
286.4
1412 25
49.6
s9234.1
(247, 211)
1337
3065
---
7837 59
2764
158.6
s13207
(700, 669)
1979 23
3836
5772
140
3554
769.3
s15850
(611, 597)
2648 36
15788
42622
188
13348
2700
s35932
(1763, 1728)
8820
7099
7280
6843
4178.5
s38584
(1464, 1452)
9664 26
19239
22589
277
17678
5772.8
b10
(28, 17)
167 11
159
200
152
0.1
197 8
0.9
b11
(38, 31)
482 21
416
1301 18
394
1504 57
5.1
b12
(126, 121)
953 16
1639
1663 14
1574
56.7
2166 24
b13
(63, 53)
231
383
240
349
3.1
224
2.2
Ratio 1 1
3.40
4.16
0.91
Ratio 2
1.70
0.89
0.97
2.24
1.79
Ratio 3
1.31
2.02
--- : BDD nodes are over 500K during computation

20. Experimental Results – w/ DC
Original
BDD Xp
Circuit
(#pi, #po) #n #l #v
time
s5378
(214, 179)
624 12
1570
769 11
1561
200.2
1332 25
49.1
s9234.1
(247, 211)
1337
3065
---
7696 55
2765
166.7
s13207
(700, 669)
1979 23
3836
5818
202
3554
897.9
s15850
(611, 597)
2648 36
15788
40078
136
13309
2596.9
s35932
(1763, 1728)
8820
7099
7360
6843
4811.1
s38584
(1464, 1452)
9664 26
19239
23726
331
17676
5476.7
b10
(28, 17)
167
159
199 9
152
0.1
193 8
1.0
b11
(38, 31)
482 21
416
1221 20
394
0.9
1562 52
5.5
b12
(126, 121)
953 16
1639
1619 15
1574
452.5
2261
27.0
b13
(63, 53)
231 10
383
243
349
1.6
229
2.5
Ratio 1 1
3.35
4.53
0.91
Ratio 2
1.65
0.94
0.97
2.27
1.71
Ratio 3
1.38
1.82
--- : BDD nodes are over 500K during computation

21. Experimental Results (cont’d)
Further synthesis by ABC commands “collapse” and “dc2”
Original Xp
Circuit #n #l #v
#n_c
#l_C
#v_C
time_c
s5378
624 12
1570
772 11
1561
1412 25
760
0.4
s9234.1
1337
3065
2791
2764
7837 59
2751 22
6.4
s13207
1979 23
3836
2700 20
3554
5772
140
2709
5.2
S15850*
2648 36
15788
---
42622
188
13348
s35932
8820
7099
7825 9
6843
7280 10
7857
39.9
s38584
9664 26
19239
12071
17676
22589
277
17678
12132 21
36.5
b10
167
159
195
152
197 8
b11
482
416
1187
394
1504 57
1226
0.2
b12
953 16
1639
1556 17
1574
2166
1645
b13
231
383
237
349
224
0.1
Ratio 1 1
1.21
0.93
2.02
3.92
1.22
0.89
Ratio 2
1.01
0.96

22. Experimental Results (cont’d)
Interpolant vs. cofactor

23. Conclusions Scalable function extraction from Boolean relations
Interpolation plays an essential role
Relations with up to thousands of variables can be determinized inexpensively
Extracted functions are of reasonable sizes
Future work
Determinization scheduling
Simple function extraction