By Tariq Bashir Ahmad Taylor Expansion Diagrams (TED) Adapted from the paper M. Ciesielski, P....

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?

By Tariq Bashir Ahmad Taylor Expansion Diagrams (TED) Adapted from the paper M. Ciesielski, P....

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Transcript of By Tariq Bashir Ahmad Taylor Expansion Diagrams (TED) Adapted from the paper M. Ciesielski, P....