By Tariq Bashir Ahmad

By Tariq Bashir Ahmad

Taylor Expansion Diagrams (TED)

Adapted from the paperM. Ciesielski, P. Kalla, Z. Zeng, B. Rouzeyre,”Taylor Expansion Diagrams:A Compact Canonical Representation for Symbolic Verification”, in DATE, 2002.

ECE 697B Spring 2006 2

Presentation Structure

Motivation – RTL verification Background

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

Construction and manipulation Properties Applications


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


Levels of Abstraction

A[1:0], B[1:0],C[3:0]

C = A1•B1*22 +A1•B0*2 +A0•B1*2 +A0•B0

A[n:0], B[n:0],C[2n:0]

C = A*B

F (*)A

B

C

We can design at a higher level of abstraction, Can we verify at a higher level of abstraction ?


Boolean functions ( f : B B ) Truth table, Karnaugh map SoP, PoS etc. Binary Decision diagrams (BDD)

Arithmetic functions ( f : B Int ) Binary Moment Diagrams (BMD)

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

Common Representations


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

Binary Decision Diagrams (BDD)


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)

Application to Verification


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)

Binary Moment Diagrams (BMD)


Efficiently models word-level operators

4

10

x0

x1

x2

12

4

y0

y1

y2

2

1

X Y=X2x1x0 . y2y1y0

=0 1 1 . 1 0 1 = 15

X + Y=X2x1x0 + y2y1y0

=0 1 1 + 1 0 1 = 8

10

x0

x1

x2

y0

y1

y2

12

4

24

1

• Limitation: requires bit-level representation of a word

BMD Example


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

2

1

X + Y

10

X

Y

Symbolic

X Y

10

X

Y

Symbolic

10

x0

x1

x2

y0

y1

y2

12

4

24

1

Word level

Symbolic Representation


x

F0(x) F1(x) F2(x) …

F(x)

F(x) = F0(x) + x F1(x) + x2 F2(x) + …

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 ======… So


F = A2B + 2C + 3 with order A<B<C

A 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

B

B

(without normalization)

Construction of TED example


Few more examples

(A+B)C +1

10

B

C

A

1

(A+B)(A+2C)

10

B

C

A

B

1

2


a

b 0

f

g

b 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)

TED Reduction Rules


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)

TED Normalization


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 = (+, - , •)

TED Composition


Nodes indexed by same variable

• Nodes indexed by different variable (x > y)

u

u0 u1

x v

v0 v1

x+ =

u

u0 u1

xv

v0 v1

y+ =

u + v

u0+v0 u1+v1

x

u + v

u0+ v

x

u1

COMPOSE Operator – ADD/SUB


Compose Operation Example

2A+B+2C

C

A

B

1

2

2

1

+ =3+5

4+6A

0+5

B

0+0 2 1

C

1+1

A+B

0 1

4

3

A

BC

A+2C

0 1

6

5

A

2


Property BDD TED

Decomposition

Shannon Taylor

Composition And/OR Multiply/Add

CanonicityYes Yes

Reduction rules

Reduction rules +

normalization

Satisfiability Yes No

Comparison of BDD and TED


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


Application 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]


Result: RTL Verification

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

1

22k+2

2k+

2

-2k+

2

2k

-22k+2

-1-1

F1 = F2

Alo

2k

2 k+10


Questions?

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By Tariq Bashir Ahmad