DATE-2002 TED 1

Taylor Expansion Diagrams:A Compact Canonical Representation

for Symbolic Verification

M. Ciesielski, P. Kalla, Z. Zeng B. RouzeyreElectrical & Computer Engineering LIRMMUniversity of Massachusetts, Amherst, USA Montpellier, Franceciesiel@ecs.umass.edu rouzeyre@lirmm.fr

DATE-2002 TED 2

Outline

• Motivation – RTL verification• Background

– BDD (binary), BMD (word-level)• New canonical representation: TED

– Construction and manipulation– Properties – Applications

• Conclusions– Limitations, future work

DATE-2002 TED 3

Motivation – RTL Verification

• Complex RTL designs– Data flow and control– Arithmetic and Boolean

BA

s

10

F

Dak

bk

>

+*-

BA

s

01

F

Dak

bk

<=

+*-

• Equivalence verification– Need representation that can

handle arithmetic/Boolean• Efficient, compact• Canonical

DATE-2002 TED 4

Common Representations

• Boolean functions ( f : B B )– Truth table, Karnaugh map– SoP, PoS, ESoP, Reed-Muller, etc.– Binary Decision diagrams (BDD, ZDD, KFDD, …)

• Arithmetic functions ( f : B Int )– Binary Moment Diagrams (*BMD, K*BMD, …)– Multi-terminal, Algebraic Decision Diagrams (ADD)

• Need more abstract representation for arithmetic functions (f : Int Int )

DATE-2002 TED 5

Binary Decision Diagrams (BDD)

• Based on recursive Shannon expansion: f = x fx + x’ fx’

where: fx = f(x=1), fx’=f(x=0)

• Compact data structure for Boolean logic• Canonical representation

– reduced ordered BDDs (ROBDD)• Essential for verification

– equivalence checking, satisfiability (SAT)

fxfx’

f x

DATE-2002 TED 6

Application to Verification• Canonicity: equivalence checking of logic circuits

10

a

b

c

F = a’bc + abc +ab’c G = (a+b)c

10

a

b

c

• Limitations– Require bit-level expansion of word-level variables– Size explosion for some functions (arithmetic)

DATE-2002 TED 7

Binary Moment Diagrams (BMD)

• Devised for word-level, arithmetic operations• Based on modified Shannon expansion (pos. Davio)

f = x fx + x’ fx’ = x fx + (1-x) fx’ = fx’ + x (fx - fx’ ) = fx’ + x fx

where fx’ = fx=0 is zero moment

f x = (fx - fx’ ) is first moment (derivative)

DATE-2002 TED 8

BMD – Example• Unsigned integer: X = 8x3 + 4x2 + 2x1 + x0

10

x0

x1

x2

1

2

4

x3

8

*BMDx3

8

x2

x1

x0

4210

BMD

DATE-2002 TED 9

*BMD• Efficiently models word-level operators

4

10

x0

x1

x2

12

4

y0

y1

y2

21

X YX + Y

10

x0

x1

x2

y0

y1

y2

12

4

24

1

• Limitation: requires bit-level representation of a word

DATE-2002 TED 10

Symbolic Representation• Why expand words into bits? Can we do better?

– Abstract words into symbolic variables

Word level

4

10

x0

x1

x2

12

4

y0

y1

y2

21

X + Y

10

X

Y

Symbolic

X Y

10

X

Y

Symbolic10

x0

x1

x2

y0

y1

y2

12

4

24

1

Word level

DATE-2002 TED 11

Taylor Expansion Diagram (TED)• F = arithmetic function (F : Int Int )

– Treat F as a continuous function

– Taylor Expansion (around x=0):

F(x) = F(0) + x F’(0) + ½ x2 F’’(0) + …• Notation

– F0(x) = F(x=0) 0-child - - - - - -

– F1(x) = F’(x=0) 1-child ----------– F2(x) = ½ F’’(x=0) 2-child ======– etc.

x

F0(x) F1(x) F2(x) …

F(x)

F(x) = F0(x) + x F1(x) + x2 F2(x) + …

DATE-2002 TED 12

Construction - Your First TEDF = A2B + 2C + 3A F0(A) = F|A=0 = 2C + 3

F1(A) = F’|A=0 = 2AB|A=0 = 0

F2(A) = ½ F’’|A=0 = B

B

1

A

G= 2C + 3

0

B0 = B(0) = 0

B1=B’ = 1

C G0(C) = (2C+3)|C=0 = 3

G1(C) = (2C+3)’ = 2

C

23

BB

(without normalization)

DATE-2002 TED 13

TED – a few Examples

(A+B)C +1

10

B

C

A

1

(A+B)(A+2C)

10

B

C

A

B

1

2

1

x0

x1

x2

x3

2

4

1

0

1x0

x1

x2

1

1

1

44

816

16

64

11

X2 = (8x3 + 4x2 + 2x1 + x0)2

DATE-2002 TED 14

TED - Reduction Rules

a

b 0

f

gb g

1. Eliminate redundant nodes:– Nodes with all edges 0– Nodes with only constant term

A

B

C

10 0 11

BB

CC

6 51

A

B

C

01

6 5 12. Merge isomorphic subgraphs

(A2 + 5A + 6)(B + C)

f = 0 a2 + 0 a + g(b) = g(b)

DATE-2002 TED 15

TED Normalization• TED can be normalized

– weights of edges of a given node must be relatively prime (to allow sharing isomorphic graphs)

26

B

A

2

2A + 2B + 6

3

0

B

A

1

2

2

1 3

B

A

1

2

11

2(A + B + 3)

DATE-2002 TED 16

TED: Composition

• Operation depends on relative order of variables x, y – if x = y, then z = x, and

h(x) = f(x) OP g(x) = f0(x) OP g0(y) + x [f1(x) OP g1(y)] + x2 [f2(x) OP g2], …

– if x > y, then z = x, and h(x) = f0(x) OP g(y) + x [f1(x) OP g(y)] + x2 [f2(x) OP g], …

– else ….

u

f

x v

g

yOP =

• Recursive composition of nodes, starting at the top

h = f OP gqz

OP = (+, - , •)

DATE-2002 TED 17

COMPOSE Operator – ADD/SUB• Nodes indexed by same variable

• Nodes indexed by different variable (x > y)

u

u0 u1

x v

v0 v1

x+ =

u

u0 u1

x v

v0 v1

y+ =

u + v

u0+v0 u1+v1

x

u + v

u0+ v

x

u1

DATE-2002 TED 18

COMPOSE Operator – MULT• Nodes indexed by same variable

• Nodes indexed by different variable (x > y)

u

u0 u1

x v

v0 v1

x• =

u

u0 u1

x v

v0 v1

y• = u • v

u0 • v u1 • v

x

u • v

u0•v0

x

u0•v1+u1•v0

u1•v1

DATE-2002 TED 19

COMPOSE Operation - Example

*

A+B

0 1

4

3

A

B = 3•5

4•6A

0•5

C

A+2C

0 1

6

5

A

21•5

B

0•0 0•2 1•0 1•2

C C1•5

1•0 1•2

+

1•1

B3•1

0•1 1•1

C

0

7

2

B

0

8

1

+=0+7

8+7B

0+0 0+2 1

C

(A+B)(A+2C)

20 1

C

A

B B

2

DATE-2002 TED 20

Properties of TED

• Canonical• Compact• Linear for polynomials of

arbitrary degree– TED for Xk, k = const, with

n bits, has k(n-1)+1 nodes.

• Can contain symbolic, word-level, and Boolean variables

• It is not a Decision Diagram 1

x0

x1

x2

x3

2

4

1

0

1x0

x1

x2

1

1

1

44

816

16

64

11

X2=(8x3+4x2+2x1+x0)2

n = 4, k = 2

DATE-2002 TED 21

TED for Boolean logic

AND

10

x

y

x y = x y

10

x1 -1

x’ = (1-x)

NOTOR

x y = (x + y – x y)

10

x

yy-1

1

XOR

x

10

yy-2

1

x y = (x + y – 2 x y)

• Needed to model arithmetic-Boolean interface

DATE-2002 TED 22

TED for Arithmetic Circuits• Arithmetic circuits contain related word-level (A, B) and

Boolean (ak, bk) variables

A = [ an-1, …, ak , …,a0 ] = 2(k+1)Ahi + 2k ak + Alo

BA

s1

10

F1

Dak

bk

>

+*-

s1 = ak (1-bk)

Ahi Alo

0 1

2k

2(k+1)

Ahi

ak

Alo

DATE-2002 TED 23

Applications to RTL Verification

• Equivalence checking with TEDs– interacting word-level and Boolean variables

A

B

s2

01

F2

bk

ak

*

*-

D

BA

s1

10

F1

Dak

bk

>

+*-

F1 = s1(A+B)(A-B) + (1-s1)Ds1 = (ak > bk) = ak (1-bk)

F2 = (1-s2) (A2-B2) + s2 Ds2 = ak’ bk = 1 - ak + ak bk

A = [Ahi,ak,Alo], B = [Bhi,bk,Blo]

DATE-2002 TED 24

RTL Verification – Result

• Related word-level and Boolean variables F1 = s1(A+B)(A-B) + (1-s1)D s1 = (ak > bk) = ak (1-bk)

F1 = (ak-akbk){ (2k+1Ahi + 2kak +Alo)2

- (2k+1Bhi + 2k bk + Blo)2 } + (1–ak + akbk) D

1

ak

1

Ahi

D

ak

bk bk

Bhi

Alo

Blo

1

122k+2

2k+2

-2k+

2

2k

-2 2k+2

-1-1

F1 = F2

Alo

2k

2 k+10

DATE-2002 TED 25

Conclusions• TED representation, features

– Canonical– Compact (linear for polynomials)– Represents arithmetic (word-level) operators + Boolean logic

• Applications– Equivalence checking, symbolic simulation

• Limitations– TED still needs to be implemented: efficiency, robustness?– How to represent relations (X < Y) without bit expansion?– Normalization: more complicated than in *BMD– Infinite series for some functions (ax)

• Open problems– Application to satisfiability (SAT), functional test generation