Optimal Redundancy Removal without Fixedpoint Computation

Mike Case†, Jason Baumgartner, Hari Mony, Bob Kanzelman IBM System and Technology GroupFMCAD 2011

† Now with Calypto Design Systems

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Outline

Introduction

The Proof Graph

Proof Graph + Induction

Proof Graph + TBV

Experimental Results

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Redundancy Removal

Redundancy Removal

Synthesis for verification

A B

Combinational Simplification

RetimingRedundancy

RemovalInterpolation

Commonly calleda “merge”

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Proof Methods

Induction

TBV (Transformation-Based Verification)

Base Case: (A=B)0

Inductive Step: (A=B)t implies (A=B)t+1

AB

miter Any Model Checking Flow

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Fixed-Point Flow

function redundancyRemoval() {

conjecture that all gates are equivalent

random simulation + refine equiv. classes

bounded model checking + refine equiv. classes

while (proveEquivalences() returns a counterexample) {

simulate the counterexample + refine equiv. classes

}

merge the remaining equivalences

}

Using induction or TBV

Problem: Repeatedly tests the same

equivalences until all are proved

Problem: Merge only at the end. Timeouts?

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Our Contributions Track dependencies between equivalences

– Which subset(s) of equivalences are soundly proved, despite the existence of counterexamples?

Benefits:

– Partial results

– Don’t re-test soundly proved equivalences

– Skip SAT calls that cannot lead to merges

Induction: up to 75% less runtime 97% reduction in SAT calls

TBV: up to 90% less runtime 80% fewer calls to interpolation

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Outline

Introduction

The Proof Graph

Proof Graph + Induction

Proof Graph + TBV

Experimental Results

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Dependencies

Suppose induction cannot prove A=Bbut can prove (A=B)^(C=D)

Then A=B depends on C=D

“Proof graph” used to track such dependencies

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The Proof Graph

RepresentsEquiv Class 1

RepresentsEquiv Class 2

Represents:Equiv Class 3

Node 3

Node 2

Node 1 Node 4

Represents:Equiv Class 4

Nodes: equivalence classes, edges: “depends on”

Observation: something is soundly proved iff all of its dependencies are soundly proved

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Redundancy Removal w/ Proof Graph

while (there are suspected equivalences) {

setup the current iteration + build the proof graph

while (candidates = getProofCandidates()) {

switch (proveEquivalences(candidates)) {

case proved:

update the proof graph + try early merging

case counterexample:

simulate the counterexample + refine equivalences

update the proof graph

}}}

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Outline

Introduction

The Proof Graph

Proof Graph + Induction

Proof Graph + TBV

Experimental Results

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Proof Dependencies

(A=B)^(C=D)

[(A=B)^(C=D)]T implies [(A=B)^(C=D)]T+1

(A≠B)T+1 such that (A=B)T and (C=D)T

...

(A≠B)T+1 is unsatisfiable, and the proof depends on (C=D)T

Suspected equivalences:

Inductive step:

Boolean Satisfiability problems:

Results:

(A=B) (C=D)New edge:

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Structural Dependencies

Speculative reduction

A1 B1

A2 B2=?

≡

A2 B2=?

=?

C2 D2

A1 B1

C1 D1

≡≡

Dependencies

(A=B) (C=D)(A=B) (A=B)New edges:

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Proof Graph Algorithms By Example

proved = 0soundlyProved = 0falsified = 0

Node 3

Node 2

Node 1

proved = 0soundlyProved = 0falsified = 0

Node 4

proved = 0soundlyProved = 0falsified = 0

proved = 0soundlyProved = 0falsified = 0

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Proof Graph Algorithms By Example

proved = 0soundlyProved = 0falsified = 0

Node 3

Node 2

Node 1

proved = 0soundlyProved = 0falsified = 0

Node 4

proved = 0soundlyProved = 0falsified = 0

proved = 0soundlyProved = 0falsified = 0

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Proof Graph Algorithms By Example

proved = 0soundlyProved = 0falsified = 0

Node 3

Node 2

Node 1

proved = 0soundlyProved = 0falsified = 0

Node 4

proved = 1

proved = 0soundlyProved = 0falsified = 0

soundlyProved = 0falsified = 0

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Proof Graph Algorithms By Example

proved = 0soundlyProved = 0falsified = 0

Node 3

Node 2

Node 1

proved = 0soundlyProved = 0falsified = 0

Node 4

proved = 1

falsified = 0

proved = 1

Merge

Merge

soundlyProved = 1

falsified = 0

soundlyProved = 1

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Proof Graph Algorithms By Example

proved = 0soundlyProved = 0falsified = 0

Node 3

Node 2

Node 1

proved = 0soundlyProved = 0falsified = 0

Node 4

proved = 1soundlyProved = 1falsified = 0

proved = 1soundlyProved = 1falsified = 0

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Proof Graph Algorithms By Example

Node 3

Node 2

Node 1

proved = 0soundlyProved = 0

Node 4

proved = 1soundlyProved = 1falsified = 0

proved = 1soundlyProved = 1falsified = 0

falsified = 1

proved = 0soundlyProved = 0falsified = 0

Skip

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Outline

Introduction

The Proof Graph

Proof Graph + Induction

Proof Graph + TBV

Experimental Results

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TBV Review

Netlist

Suspected Redundancies

Speculative Reduction

+Miter Creation

Spec Reduced

Netlist

LogicSynthesis

Interpolation

Problems

–Long runtime

–Frequent restarts

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TBV Dependencies

No inductive hypothesis no proof dependencies

Structural dependencies:

A B=?=?

=?=?

C D(A=B) (C=D)

New edge:

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Outline

Introduction

The Proof Graph

Proof Graph + Induction

Proof Graph + TBV

Experimental Results

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SixthSense Experiment Setup 1300 Benchmarks

– IBM property checking and SEC benchmarks

– Hard HWMCC 2010

– Largest AIG is 5.3M Ands and 330k registers

Redundancy removal w/ k=1 induction

Redundancy removal w/ TBV(combinational simplification + interpolation)

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Experimental Results : Runtime

Induction TBV

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Experimental Results : Number of Proofs

Induction TBV

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Conclusion

Major enhancement to redundancy removal

–Minor book-keeping overhead

–Reduces the runtime of our engine

–Reduces the number of SAT calls

–Provides partial results

Used everyday within IBM