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Taylor Expansion Diagrams (TED): Verification

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EC667: Synthesis and Verification of Digital Systems

Spring 2011

Presented by: Sudhan

- 1. What is a TED?
- 2. What is the Motivation behind TED’s development?
- 3. How to create a TED?
- 4. How TED is used for Verification?

- Canonical representation for arithmetic expressions useful for equivalence checking of a design among others.
- Works at a higher level of abstraction.
- Built on Taylor series expansion.
(Representation of a function as a sum of infinite terms)

f(x) is a real, differentiable function corresponding to an algebraic expression F. F1(x=0) is the first derivative of the function w.r.t x and evaluated at x0=0.

F(x,y,..) = F0(x=0)+ x F1(x=0) + x2 F2(x=0) + …

- Size and complexity of digital designs steadily increasing.
- So does the need for verification tools capable of working at higher abstraction levels.
- Typically high level descriptions contain word-level arithmetic computations along with Boolean logic.
- This need for canonical representations working with symbolic, algebraic word-level computations was the motivation behind creating a new representation.

X + Y

4

2

1

4

0

0

1

1

x2

x1

x0

y0

y1

y2

Y

X

2

1

Symbolic

Bit level

- Logic of equality with uninterpreted functions. [J. Burch, ’94]
- Use of Propositional logic Formula. [R. Bryant, 01]
- Reduced, ordered Binary Decision Diagrams
(ROBDD) and its variants. [R.E. Bryant, ‘86]

- Binary Moment Diagrams. [R.E. Bryant, 95]
- Symbolic Algebra Methods.[G. DeMicheli, 03]
- Drawbacks:
- Requires bit level functions for word level signals.
- Algebra tools require canonical representations for equivalence checking.

- It represents a set of binary valued decisions in a rooted directed acyclic graph(DAG).
- It is based on a recursive Shannon decomposition and has a set of reduction rules.
- Advantages :
- Widely available software Packages and used for combinational equivalence checking.
- BDD based verification systems successful in verifying control dominated circuits.
- Drawbacks:
- Not successful for data path design.
- With complex circuits, BDD representations face size limits.

- Performs decomposition of a linear function based on its first two moments (linear moment) and (constant moment).
- Uses modified Shannon’s expansion, in which a binary variable is treated as a (0,1) integer.
- Drawbacks:
- For high degree polynomials defined over words with large bit-width, it becomes an expensive representation.
- It has a set of restrictions imposed to make it canonical and thus makes it difficult to construct.

- Notation:
- f(x=0) 0-child
- f’(x=0) 1-child
- f’’(x=0) 2-child

f(x)

x

1

x

x2

…

f’(x)

f(x)

f’’(x)

3. Creating a TED - Steps

- 1.Start with the algebraic Expression and ordering of variables.
- 2. Recursively, from F find the values for all the constant, first and second derivative terms w.r.t to each variable.
- 3. Form the TED with this information based on the notation.
- 4. Apply reduction rules and Normalization to get Normalized TED.

Arithmetic expression : F = (A+B) (A+2C) = A2 + AB + 2AC + 2BC

Consider ordering : A,B,C

Decomposition w.r.t A:

F(A=0) = 2BC

F’(A=0) = B + 2C

1/2F’’(A=0) = 1

Decomposition of G=F(A=0), H=F’(A=0), F’’(A=0) w.r.t B:

G(B=0) = 0 H(B=0) = 2C

G’(B=0) = 2C H’(B=0) = 1

Similarly, Decomposition w.r.t C:

F(C=0) = 0

F’(C=0) = 2

F

A

1

A

A2

G=2BC

B

B

H=B+2C

B

1

1

B

C

2C

1

C

0

2

1

From TED: F = A2+AB+2AC+2BC

Every path from root node to non zero vertex is a non zero term in the expression

0

u

3. Creating a TED – Reduction Rules

1. Eliminate redundant nodes:

- Nodes with all non zero edges connected to terminal 0

v

x

f = 0 v2 + 0 v + y(u) = y(u)

u

y

y

- 2. Merge Isomorphic Graph:
- Isomorphic if structure and attributes of the two TED’s match.

- All 3 subgraphs are isomorphic i.e. one to one mapping between vertex sets and edge sets.

‘Taylor Expansion Diagrams: A Canonical Representation for Verification of Data Flow Designs’ – Maciej Ciesielski, Priyank kalla, Serkan Askar

- Applied to reduce the graph.
- Move numeric values from nonzero terminal nodes to terminal edges.
- A reduced ordered TED representation
is normalized when:

- The weights assigned to edges spanning
Out of a node are prime.

- Numeric value 0 appears only in the terminal nodes.
- Graph contains no more than two terminal nodes.
Applying Normalization to example

- The weights assigned to edges spanning

F

A

B

B

C

2

0

1

2

‘Taylor Expansion Diagrams: A Canonical Representation for Verification of Data Flow Designs’ – Maciej Ciesielski, Priyank kalla, Serkan Askar

Like the apply operator in BDD, TED’s can be similarly composed, following specific rules.

Starting from root node of the two TED’s, the final TED is constructed by recursively:

- constructing all the non zero terms from two functions.
- combining the terms according to a given operation.
Rules:

Case 1: Nodes w and y with same index:

- 0-child is obtained by pairing 0-child of w,y.
- 1-child is obtained by the sum of two cross products of 0 and 1 children.
- 2-child is obtained by pairing the 1-children of w,y.
Case 2: Nodes w and y with different indices:

The node with the lower index is added to the 0-child of the node corresponding to variable

with higher index.

Case 3: Both the nodes are terminal nodes: val= val(w) + val(y) if add operation, val = val(w)*val(y) for

multiplication.

w

y

x

z

1

1

0

0

Apply composition and verify if the final TED is the same as the one constructed by expansion of the expression.

F = (A+B) (A+2C) = A2 + AB + 2AC + 2BC

Ordering : A,B,C

0-child is obtained by pairing 0-child of w,y.

w.y

A

A+B

A+2C

A

A

B

1

x.z

B

C

*

B

C

C

0.z

1.z

C

1.z

x.1

2

2

+

2

0

0

0

1

1

0

1

0

For terminal nodes,

val = val(u)*val(v)

1-child is obtained by the sum of two cross products of 0 and 1 children.

B

C

B

The node with the lower index is added to the 0-child of the node corresponding to variable with higher index.

x.1

C

0.z

2

1

0

0

w.y

The two TED’s are equal

A

On replacing

w.y

A

1

x.z

B

B

B

x.z

1.z

1.z

x.1

x.1

x.1

C

0.z

1.z

+

+

C

2

2

C

1.z

2

+

2

0

0

1

1

1

1

1

0

0

0

0

0

1

0

Given RTL design,

- Build TED’s for Primary Inputs
- Expansion of word level signals needed
- When one or more bits from i/p used in other parts of design.
- Then, TED’s constructed for all components of the design.
- TED’s for the primary outputs are generated by composing the TED’s in topological order from PI to PO.
- After constructing normalized TED for each design, the test for functional equivalence is performed by checking for isomorphism of the resulting graphs.

- Partitioning of word level signals into subvectors creates an issue for polynomial representation.
- Only finite Taylor expansionspermitted.
- They cannot represent relational operators in symbolic form.
- TEDs cannot represent modular arithmetic.

Salient features of TED:

- Represents arithmetic (word-level) operators and Boolean logic.
- Canonical(Based on Taylor’s theorem which states the uniqueness of the Taylor’s series representation of a function, evaluated at a particular point)

Applications

Useful for High-level synthesis, equivalence checking, symbolic simulation.

- Priyank Kalla, Maciej Ciesielski, Emmanuel Boutillon, Erric Martin, ‘High-Level Design Verification using Taylor Expansion Diagrams: first Results’
- Maciej Ciesielski, Priyank Kalla, Serkan Askar, ‘Taylor Expansion Diagrams: A Canonical Representation for Verification of Data Flow Designs’