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

104/22/23 ICCAD 2008

2

Outline

Introduction1

Prior Work2

3

4

Conclusions5

Experimental Results

Our Approach

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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

3

Introduction

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4

Decomposition Chart

f(a,b,c,d) = h(c,d,g(a,b,d))

000 010 100 110 001 011 101 111

00 0 0 0 0

10 0 1 1 0

01 0 0 1 1

11 1 1 0 0

a,b,d

c,d

0 1 1 0 0 0 1 1

00 0 0 0 0

10 0 1 1 0

01 0 0 1 1

11 1 1 0 0

g

c,d

XH : {c}XG : {a,b}XC : {d}

XH XC

XH XC

XG XC

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0 1 1 0 0 0 1 1

00 0 0 0 0

10 0 1 1 0

01 0 0 1 1

11 1 1 0 0

h

000 010 100 110 001 011 101 111

00 0 0 0 0

10 0 1 1 0

01 0 0 1 1

11 1 1 0 0

f

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

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6

Our Approach

SAT-based computation Interpolation get g Functional dependency get h

With/without pre-specified variable partition

Single-/multiple-output decomposition

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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))

7

Single-Output Decomposition

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Functions g and h exists iff (f(XH

1, XG1, XC)≠ f(XH

1, XG2, XC))Λ

(f(XH2, XG

2, XC)≠ f(XH2, XG

3, XC))Λ (f(XH

3, XG3, XC)≠ f(XH

3, XG1, XC)) is UNSAT

8

Decomposability as SAT

xg1[[XG

1]] xg2[[XG

2]] xg3[[XG

3]]

xh1[[XH

1]] 1 0

xh2[[XH

2]] 0 1

xh3[[XH

3]] 1 0

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xc[[XC]]

f

f f f f f fXH

1 XG1XC XH

1 XG2XC XH

2 XG2XC XH

2 XG3XC XH

3 XG3XC XH

3 XG1XC

1 1 1

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

φBφA

9

Craig Interpolation

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Interpolating g-function P: refer to the common variables XG

1, XG2, XC

φA: CP(XG1)≠CP(XG

2) φB: CP(XG

2)≠CP(XG3),CP(XG

1)≠CP(XG3), i.e., CP(XG

1)=CP(XG2)

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f f f f f fXH

1XG1XC XH

1XG2XC XH

2XG2XC XH

2XG3XC XH

3XG3XC XH

3XG1XC

1 1 1

φA φB

f(XH1, XG

1, XC)≠ f(XH1, XG

2, XC) (f(XH2, XG

2, XC)≠ f(XH2, XG

3, XC))Λ(f(XH

3, XG3, XC)≠ f(XH

3, XG1, XC))

XG1 XG

2

… …XG

1 XG2

… …

p

q

rs

p

q

rs

Interpolating g-function (cont’d)

onset

offseta

Relation characterized by P(XG

1, XG2,c) for some c [[Xc]]

Relation after cofactoring

P(XG1=a, XG

2,c) w.r.t. some a [[XG1]]

P(XG1=a, XG

2, XC) is a feasible implementation of g(XG, XC) !

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Computing h-function

A naïve approach For XC = , function h can be derived

using Shannon expansion

h(XH,xg)=(xg Λ hxg) v (xg Λ hxg

)

Non-scalable(w.r.t. XC size)

where

hxg(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

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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 XHXC 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]1304/22/23 ICCAD 2008

AshenhurstDecomposition(f, XH, XG, XC):

1. F CircuitInstantiation(f )

2. (φA, φB) CircuitPartition(F)

3. P(XG1,XG

2,XC) Interpolation(φA, φB)

4. g(XG,XC) Cofactor(P, a[[XG1]])

5. h FunctionalDependency(f, g)

return(g, h)

14

Overall Flow

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15

Automating Variable Partition

For each xi X, addconditional clauses for XG: ((xi

2≡xi3)Λ(xi

4≡xi5)Λ(xi

6≡xi1)) V i

conditional clauses for XH: ((xi1≡xi

2)Λ(xi3≡xi

4)Λ(xi5≡xi

6)) V i

(i,i) (0,0) xiXC

(0,1) xiXG

(1,0) xiXH

(1,1) xi can be in either of XG and XH

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f f f f f fXH

1 XG1XC

1 XH2 XG

2XC2 XH

3 XG3XC

3 XH4XG

4XC4 XH

5 XG5XC

5 XH6 XG

6XC6

1 1 1

16

Automating Variable Partition (cont’d)

Make “unit assumptions” on control variables i, i 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

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17

Multi-Output Decomposition

New UNSAT condition:

V(fi(XH1, XG

1, XC)≠ fi(XH1, XG

2, XC))Λ

V(fi(XH2, XG

2, XC)≠ fi(XH2, XG

3, XC))Λ

V(fi(XH3, XG

3, XC)≠ fi(XH3, XG

1, XC))

Computation is same as before

i

i

i

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XG

XH

X

ghf

xg

XC

... ...

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

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Circuit #func #var #fail #SAT_TO #succ #var_succ #VP_avgrate_valid_VP

time_avg(sec)

Mem(Mb)

b15 370 143—306 0 51 319 143—306 1519 0.917 96.62 107.20

b17 1009 76—308 0 148 861 76—308 1645 0.904 87.12 125.84

c7552 36 50—194 0 2 34 50—194 1350 0.455 64.38 36.65

s13207 3 212—212 0 0 3 212—212 569 0.908 70.26 50.62

s38417 256 53—99 6 72 178 53—99 1090 0.523 103.33 136.04

s38584 7 50—147 0 0 7 50—147 1120 0.924 47.13 51.56

19

Experimental Results

Single-output decomposition

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Circuit #pair #var #fail #SAT_TO #succ #var_succ #VP_avgrate_valid_VP

time_avg(sec)

Mem(Mb)

b15 201 145--306 0 31 170 145--269 1176 0.845 113.86 224.07

b17 583 79--310 0 88 495 79--308 676 0.824 103.12 419.35

c7552 21 56--195 0 2 19 56--141 188 0.465 89.57 78.67

s13207 3 212--228 0 0 3 212--228 585 0.7 93.36 118.03

s38417 218 53--116 13 30 175 53--116 689 0.498 109.06 319.48

s38584 9 50--151 0 0 9 50--151 1656 0.713 46.17 207.78

Two-output decomposition

20

Variable partition before / after UNSAT core refinement

Experimental Results (cont’d)

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Experimental Results (cont’d)

Variable partition qualities resulted from different efforts

||XG|-|XH||/|X||XC|/|X|

w/ mini w/o mini

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22

Conclusions

Proposed pure SAT-based Ashenhurst decompositionEasily extendable to non-disjoint and

multiple-output decompositionsScalable to functions with up to 300

input variables

Future workFunctional decomposition beyond

Ashenhurst

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23

Thanks for Your Attention!

04/22/23 ICCAD 2008