Appearance based equivalence checking
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Appearance-Based Equivalence Checking. Speaker: Daw-Ming Li Advisor: Chun-Yao Wang 2009.02.10. Introduction. Traditional approaches of equivalence checking Building BDDs SAT solving Drawback Exponential growth of required memory and runtime.

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Appearance based equivalence checking

Appearance-Based Equivalence Checking

Speaker: Daw-Ming Li

Advisor: Chun-Yao Wang

2009.02.10


Introduction

Introduction

  • Traditional approaches of equivalence checking

    • Building BDDs

    • SAT solving

      • Drawback

        • Exponential growth of required memory and runtime


Appearance based approach

Appearance-based approach

  • Given two circuits N1,N2

  • Transform circuits N1 and N2 according to their appearance

  • Extract the same appearance of N1 and N2 to construct the circuit I


Terminologies

Terminologies

  • Given two circuits N1 and N2

  • Circuit I is an isomorphism between N1 and N2

  • Largest Common Connected Subgraph(LCCS)

  • Largest Common Connected PI Subgraph (LCCPIS)

  • Chun Chi’s work-Rewiring using IRredundancy Removal and Addition (IRRA)


Ideas

Ideas

  • Find the LCCS between N1 and N2

  • Apply IRRA, transform the LCCS to the LCCPIS

  • Delete the LCCPIS from circuits N1 and N2, and add the LCCPIS to circuit I

    • Goal

      • Reduce the size of circuits N1 and N2 as small as possible


Flow chart

Flow chart

Transform N1 and N2 to

NAND-NOT circuit

N1 and N2 are

equivalent

Find the LCCS between

circuits N1 and N2

True

Are both N1 and N2 empty?

True

Is N1 or N2 empty?

True

False

False

Apply the IRRA technique,

transform the LCCS to

the LCCPIS

Redundant?

False

Delete the LCCPIS from N1 and

N2, and add the LCCPIS to I

N1 and N2 are not

equivalent


Example

...

...

N1_0

N2_0

..

..

..

..

N2:

N1:

N1_1

N2_1

N1_2

N2_2

..

..

N1_3

N2_3

Example

Find the largest common connected subgraph


Example1

Example

...

...

N1_0

N2_0

..

..

..

..

N2:

N1:

N1_1

N2_1

N1_2

N2_2

..

..

N1_3

N2_3

N2_4

Apply the IRRA technique, transform the LCCS to

the LCCPIS


Example2

Example

...

...

N1_0

N2_0

..

..

..

..

N2:

N1:

N1_1

N’2_1

N1_2

N2_2

..

..

N1_3

N2_3

Put the LCCPIS to the isomorphism set


Example3

Example

...

...

N1_0

N2_0

..

..

..

..

..

N2:

N1:

N2_2

N1_1

N’2_1

I:

..

N2_3

The PO of LCCPIS == The new PI of N1 and N2


Example4

..

..

Example

...

...

N1_0

N2_0

..

..

..

I_1

N2:

N1:

N1_1

N’2_1

I:

..

I_2

Find the LCCPIS between N1 and N2,

and iterate the procedure until N1 or N2 is empty


Case i

Case I

...

I

N1:

N2:

I:

……

Both N1 and N2 are empty

Circuits N1 and N2 are equivalent


Case ii

Case II

...

...

N2:

N2

N1:

I

I:

……

……

N1 is empty, but N2 is not empty

Check whether N2 is redundant or not

If N2is redundant, circuits N1 and N2 are equivalent


Case iii

Case III

...

...

N2:

N2

N1:

I

I:

……

……

N1 is empty, but N2 is not empty

Check whether N2 is redundant or not

If N2is not redundant, circuits N1 and N2 are not equivalent


The case of applying irra 1 3

The case of applying IRRA (1/3)

...

...

N2_0

N2_0

..

..

..

..

N2:

N2:

N’2_1

N2_2

N2_1

N2_2

..

..

N2_3

N2_3

N2_4


The case of apply irra 2 3

The case of apply IRRA(2/3)

...

...

N2_0

N2_0

..

..

..

..

N2:

N2:

N’2_1

N2_2

N2_1

N2_2

..

..

N2_3

N2_3

N2_4


The case of apply irra 3 3

The case of apply IRRA(3/3)

...

...

N2_0

N2_0

..

..

..

..

N2:

N2:

N’2_1

N2_2

N2_1

N2_2

..

..

N2_3

N2_3

N2_4


Reference

Reference

  • “A Direct Algorithm to Find a Largest Common Connected Induced Subgraph of Two Graphs”, Graph-Based Representations in Pattern Recognition, Springer Berlin Heidelberg, pp 162-171, 2005.

  • “Rewiring using IRredundancy Removal and Addition”, Chun-Chi


Future work

Future work

  • Consider the situation when N1 or N2 cannot become empty set

  • Study more papers


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