Equality Saturation: A New Approach to Optimization - PowerPoint Presentation

Slide1

Equality Saturation:A New Approach to Optimization

Ross Tate

, Mike

Stepp

, Zach

Tatlock

,

Sorin

Lerner

University of California, San DiegoSlide2

Traditional Optimization

Phase

Ordering

Problem

Local

Profitability

Heuristics

Original Program

Optimized Program

OptimizationsSlide3

Traditional OptimizationSlide4

Exploring EquivalencesSlide5

Exploring EquivalencesSlide6

Our Approach

CFG→PEG Conversion

P

rogram

E

xpression

G

raphSlide7

Our Approach

Equality SaturationSlide8

Our Approach

Global Profitability HeuristicSlide9

Our Approach

PEG→CFG ConversionSlide10

Mitigates Phase Ordering ProblemNon-destructive updates allow exponential searchGlobal Profitability Heuristic

Explore first, then decide

Translation ValidationVerify translations using equality saturation

BenefitsSlide11

sum = 0;

for (

i

= 0; i < 10; i++)

sum += 4 * i;return sum;

ExampleSlide12

sum = 0;

for (

i

= 0;

i

< 10;

i

++)

sum += 4 *

i

;

return sum;

Representing Loops

1

θ

1

*

4

+

0

Complete Representation

Referentially Transparent

No Intermediate VariablesSlide13

Identify Equalities

PEG Node Granularity

Equality Axioms

∀X. 4*X = (X≪2)

Equality Analyses1

θ

1

*

4

+

0

2

<<

∀

X. 4

*

X = (X≪2)Slide14

Equality Inference

1

θ

1*

4

+

0

+

*

4

1

*

4

θ

1

*

4

*

4

0

∀

X,Y,Z. X

*

(Y+Z) = X

*

Y+ X

*

Z

∀

X. X

*

0 = 0

2

<<

∀

X. 4

*

X = (X≪2)

∀

X,Y. 4

*

θ

(X, Y) =

θ

(4

*

X, 4

*

Y)

∀

X. X

*

1 = XSlide15

E-PEG

1

θ

1

*

4

+

0

*

+

4

1

θ

1

*

4

*

4

0

2

<<Slide16

PEG Selection

+

4

θ

1

0

1

θ

1

*

4

+

0

*

1

*

4

*

4

2

<<

Global Profitability HeuristicSlide17

Optimized PEG

4

θ

1+

0Slide18

sum = 0;

for(j = 0; j < 40; j += 4)

sum += j;

return sum;

Optimized Program

4

θ

1

+

0Slide19

sum = 0;

for(j = 0; j < 40; j += 4)

sum += j;

return sum;

Optimized Program

sum = 0;

for(

i

= 0;

i

< 10;

i

++)

sum += 4

*

i

;

return sum;

Loop Induction Variable Strength ReductionSlide20

Optimizations composed from simple rulesLoop Induction Variable Strength Reduction

Loop-Operation Factoring

Loop-Operation DistributingInter-Loop Strength ReductionTemporary Object RemovalPartial

InliningEmergent OptimizationsSlide21

ImplementationSlide22

Algorithm provided in the Technical ReportModel heap with values having linear types

ImplementationSlide23

Tarjan’s Union-Find Algorithmtracks equivalence classes in the E-PEG

Rete

Pattern Matching Algorithmincrementally finds significant nodes in the E-PEGEquality Analyses:

PEG Operator AxiomsLanguage-Specific AxiomsDomain-Specific AxiomsImplementationSlide24

Pseudo-Boolean SolverAssign a cost to each operation in the E-PEG

Impose constraints for a well-formed PEG

Minimize the cost of the selected PEGImplementationSlide25

1499

msec

14

msec

88 msec43 msec

Observed Emergent Optimizations

Traditionally need to be explicitly implementedDomain-Specific Analyses:7% runtime improvement on Java ray tracerCompilation of

SpecJVM (per method):1030 programs found in less than 200MB memoryAverage compilation time per stage:

Peggy: Java Bytecode Optimizer

1499

msecSlide26

Validation of Soot optimizer on

SpecJVM

:

98% of optimized methods successfully validated

Optimization bug found within remaining 2%

Translation

Validator

CFG→PEG Conversion

CFG→PEG Conversion

Equality Saturation

?Slide27

PowerfulSimultaneous Exponential Search

Emergent Optimizations

ExtensibleCooperative Equality AnalysesDomain-Specific Axioms

GeneralOptimizationTranslation ValidationConclusionsSlide28

E-GraphsDenali: Basic Block Assembly Superoptimizer

Simplify: Theorem

ProverRepresentationsThinned-Gated SSA

Lucid programming languageValue Dependence GraphDependence Flow GraphProgram Dependence Graph/WebRewrite-Based OptimizersTAMPRASF+SDFStratego

Related Work