Abstract Interpretation of FIFO Replacement

Abstract Interpretation of FIFO Replacement Daniel Grund

Jan Reineke

Saarland University, Saarbrücken, Germany

Static Analysis Symposium 2009

saarland university

computer science

saarland university

Outline 1

computer science

Introduction & Motivation Timing Analysis Cache Analysis

2

Abstract Interpretation of FIFO Challenge FIFO Replacement Domain Cooperation Must Analysis May Analysis

3

Evaluation Related Work Analysis Precision

4

Summary

Daniel Grund and Jan Reineke

Abstract Interpretation of FIFO


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saarland university

Outline 1

computer science


2


3


4

Summary




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saarland university

Frequency

Notions in Timing Analysis

computer science

Analysis-guaranteed timing bounds Input- and state-induced variance

LB

BCET

Overest.

WCET

UB

Execution time

Execution time depends on I I

program input initial hardware state

Bounds required for schedulability analysis of real-time systems




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saarland university

Static Timing-Analysis Framework Binary Executable

computer science

Legend: Data Action

CFG Reconstruction

Framework implemented by aiT of AbsInt Micro-architectural analysis

Control-flow Graph

models pipeline, caches, buses, etc. derives bounds on BB exec. times

Value Analysis

Loop Bound Analysis

Control-flow Analysis

Annotated CFG

Microarchitectural Analysis

Basic Block Timing Info


is an abstract interpretation with a huge domain is the computationally most expensive module

Global Bound Analysis



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saarland university

Caches and Replacement Policies

computer science

“hit”

[ab] CPU

Cache

Main Memory

Capacity: Latency:

32 KB 3 cycles

2 MB 100 cycles

Caches transparently buffer memory blocks




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saarland university


computer science

“hit”

[ab] CPU

Capacity: Latency:

a?

Cache

Main Memory

32 KB 3 cycles

2 MB 100 cycles





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saarland university


computer science

“hit”

[ab] CPU

Capacity: Latency:

a!

Cache

Main Memory

32 KB 3 cycles

2 MB 100 cycles





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saarland university


computer science

“miss” [ab] CPU

Capacity: Latency:

c?

Cache

Main Memory

32 KB 3 cycles

2 MB 100 cycles





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saarland university


computer science

“miss” [ab] CPU

Cache

Capacity: Latency:

32 KB 3 cycles

c?

Main Memory

2 MB 100 cycles





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saarland university


computer science

“miss” [ac ] CPU

Cache

Capacity: Latency:

32 KB 3 cycles

c!

Main Memory

2 MB 100 cycles

Caches transparently buffer memory blocks Replacement policy dynamically decides which element to replace LRU least recently used PLRU pseudo LRU FIFO first-in first-out Have great influence on abstraction and (obtainable) analysis precision Daniel Grund and Jan Reineke



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saarland university


computer science

“miss” [ac ] CPU

Capacity: Latency:

c!

Cache

Main Memory

32 KB 3 cycles

2 MB 100 cycles

Caches transparently buffer memory blocks Replacement policy dynamically decides which element to replace LRU least recently used PLRU pseudo LRU FIFO first-in first-out Have great influence on abstraction and (obtainable) analysis precision Daniel Grund and Jan Reineke



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saarland university

Cache Analysis: Motivation & Application

computer science

Cache performance has great influence on overall performance Need tight bounds on cache performance Otherwise derived timing bounds may be useless: I I

tasks are deemed not schedulable waste of hardware resources

Application: Buffers with transparent replacement I I I

Instruction- and data-caches Branch target buffers (BTB, BTIC) Translation lookaside buffers (TLB)




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saarland university

Static Cache Analysis

computer science

derives approximations to cache contents at each program point in order to classify memory accesses as cache hits or cache misses

Must-information Underapproximation of cache contents Used to soundly classify cache hits

May-information Overapproximation of cache contents Used to soundly classify cache misses




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saarland university

Outline 1

computer science


2


3


4

Summary




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saarland university

Concrete Semantics: What is FIFO?

computer science

State of FIFO cache of size k : s ∈ S := T k last-in

first-in

[t0 , . . . , tk −1 ] Examples: c

[d , c , b, a] −− → [d , c , b , a ] hit e

[d , c , b, a] −miss −→ [e, d , c , b] Update: US : S × T → S

US ([t0 , . . . , tk −1 ], t ) :=


[t0 , . . . , tk −1 ] : ∃i : t = ti [t , t0 , . . . , tk −2 ] : otherwise


“cache hit” “cache miss”


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saarland university

Challenge: How to Predict Hits?

computer science

Consider a FIFO cache with unknown contents a

b

a

b

[?, ?, ?, ?] −− → [?, a, ?, ?] −− → [?, a, ?, b] hit hit [?, ?, ?, ?] −− → [?, ?, ?, a] −miss −→ [b, ?, ?, ?] hit If a may be a hit, then b may evict a

⇒ Can only predict hits for most recently accessed element Can one do better? a

b

[?, ?, ?, ?] −miss −→ [a, ?, ?, ?] −miss −→ [b, a, ?, ?] If a is a miss, then b cannot evict a

⇒ Can predict hits for a until k further misses might have happened




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saarland university

Challenge: How to Predict Hits?

computer science

Consider a FIFO cache with unknown contents a

b

a

b

[?, ?, ?, ?] −− → [?, a, ?, ?] −− → [?, a, ?, b] hit hit [?, ?, ?, ?] −− → [?, ?, ?, a] −miss −→ [b, ?, ?, ?] hit If a may be a hit, then b may evict a

⇒ Can only predict hits for most recently accessed element Can one do better? a

b

[?, ?, ?, ?] −miss −→ [a, ?, ?, ?] −miss −→ [b, a, ?, ?] If a is a miss, then b cannot evict a

⇒ Can predict hits for a until k further misses might have happened ⇒ Need may-information to obtain precise must-information Daniel Grund and Jan Reineke



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saarland

A Solution. Well, our Contributions

university

computer science

Framework for static cache analysis I I I

policy independent couples must- and may-analyses analyses cooperate via “update reduction”

FIFO must-analysis I I

can profit from may-information hence, also better must-information

FIFO may-analysis I I

utilizes order of hits and misses more precise than prior analyses




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saarland university

Framework and Classification

Domain:

Fifo := Must × May

Classification:

Class := {H, M}>

computer science

H

: cache hit : cache miss > : unclassified H M

> M

CFifo : Fifo × T → Class CFifo ((must , may ), t ) := CMust (must , t ) u CMay (may , t )




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saarland

Domain Cooperation via Update Reduction

(must , may )

S


Independent

university

computer science

(must 0 , may 0 )

S



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saarland


(must , may )

Independent

university

computer science

(must 0 , may 0 )

γ

α

US

S


S



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saarland university


(must , may )

γ

Independent

computer science

(must 0 , may 0 )

is imprecise

γ

α

α

US US

S


S



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saarland


(must , may )

S


Best possible

university

computer science

(must 0 , may 0 )

S



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saarland


(must , may )

γ

S


Best possible

university

computer science

(must 0 , may 0 )

γ

S



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saarland


(must , may )

γ

S


Best possible

university

computer science

(must 0 , may 0 )

γ

S



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saarland university


(must , may )

γ

Best possible

computer science

(must 0 , may 0 )

is infeasible

γ

α

α

US

S


S



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saarland


(must , may )

S


Using classification

CFifo ((must , may ), t ) = M

university

computer science

(must 0 , may 0 )

S



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saarland


(must , may )

γ

S




university

computer science

(must 0 , may 0 )

γ

S



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saarland


(must , may )

γ

S




university

computer science

(must 0 , may 0 )

γM (t ) = {s ∈ S | t 6∈ s}

S



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saarland


(must , may )

γ


CFifo ((must , may ), t ) = M γM (t ) = {s ∈ S | t 6∈ s}

university

computer science

(must 0 , may 0 )

α

US

S


S



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saarland university

Outline 1

computer science


2


3


4

Summary




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saarland university

Must-Analysis: Potential Misses

computer science

For FIFO, a newly inserted element is evicted after k misses

⇒ Maintain upper bound on number of misses: Potential misses Abstract must-domain closely resembles the concrete domain Must Fifok := [T0 , . . . , Tk −1 ] , where Ti ∈ P(T ), Ti ∩ Tj = ∅, and

P

| Ti | ≤ k .

t ∈ Ti ⇒ at most i misses since insertion of t Concretization example

γ([{f }, ∅, {a, c }, {b}]) = {[f , c , a, b] , [f , a, c , b]}




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saarland university

Must-Analysis: Update

computer science

UMust : Must × T × Class → Must   [∅, T0 , . . . , Tk −2 ∪ {t }] : cl = > : cl = H UMust ([T0 , . . . , Tk −1 ] , t , cl ) := [T0 , . . . , Tk −1 ]  [{t }, T0 , . . . , Tk −2 ] : cl = M Misses classified by may-analysis Last case only possible due to domain cooperation




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saarland university

Outline 1

computer science


2


3


4

Summary




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saarland university

May-Analysis: Definite Misses

computer science

How to predict misses?

⇒ Maintain lower bound on number of misses: Definite misses Initially, anything might be cached To classify a miss for an individual access, one needs to predict k other misses first

Lemma A newly inserted element is evicted after accesses to at most 2k − 1 pairwise different elements.




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saarland university

May-Analysis: Idea “Early Misses” Initial state: [x , c , b, a]

m

Sequences of different length: I

computer science

ha, b, c , e, f , g , hi

eviction M

3 M

M

M insertion

0



M

H

1

Common final state: [h, g , f , e]

M

H

2

M

H

M H

1

M H M H

2

3 h


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saarland university


m

Sequences of different length: I I

computer science

ha, b, c , e, f , g , hi he, f , g , hi

eviction M

3 M

M

M insertion

0



M

H

1


M

H

2

M

H

M H

1

M H M H

2

3 h


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saarland university


m

Sequences of different length: I I I

computer science

ha, b, c , e, f , g , hi he, f , g , hi ha, e, f , c , g , hi

eviction M

3 M

M

M insertion

0



M

H

1


M

H

2

M

H

M H

1

M H M H

2

3 h


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saarland university


m


computer science


eviction M

3 M

M

M

⇒ Sequences differ in number of hits



insertion

0

M

H

1


M

H

2

M

H

M H

1

M H M H

2

3 h


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saarland university


m


computer science


eviction M

3 M

M

M

⇒ Sequences differ in number of hits

insertion

0

M

H

1


M

H

2

M

H

M H

1

M H M H

2

3 h

“Early misses” I I

preclude hits to thereby evicted elements reduce number of possible accesses between insertion and eviction

⇒ Order of hits and misses is important




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saarland university

May-Analysis: Domain

computer science

May analysis approximates position in triangle Unclassified access → “take the longer way” dm 3 M

H,M,>

H,>

2 M

M

M

H,>

0

H,> H,M,>

M H,>

1

H,M,>

M

H,>

1

H,>

2

3 cw

For each element t, the analysis maintains: A Set of Potentially Accessed Elements dm Number of Definite Misses cw Number of Covered Ways (Covered Cache Positions) Daniel Grund and Jan Reineke



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saarland university

May-Analysis: Example

computer science

dm 3 M

H,M,>

H,>

2 M

M

M

H,>

0

H,> H,M,>

M H,>

1

H,M,>

M

H,>

1

H,>

2

3 cw

Assume sequence hx , a, b, c i and all accesses are unclassified Then for x one has: A = {a, b, c } a, b and c might have been accessed since the last insertion of x dm = 0 0 misses have definitely happened since the last insertion of x cw = 3 Assuming that all unclassified accesses were hits, then 3 elements of A must be cached

Consider next access to d Daniel Grund and Jan Reineke



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saarland university

Outline 1

computer science


2


3


4

Summary




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saarland university

Relative Competitiveness

computer science

How many misses would FIFO have if LRU has mLRU misses?

Definition: Relative Competitiveness Policy P is (f , c ) miss-competitive relative to policy Q if mP (p, s) ≤ f · mQ (q , s) + c for all access sequences s and compatible cache states p, q. E.g. LRU(2k − 1) is (1, 0) miss-competitive vs. FIFO(k )

⇒ LRU(2k − 1) may-analysis can be used for FIFO(k ) may analysis




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saarland university

Evaluation Setup

computer science

Must-analysis: CM Canonical must-analysis (this paper)

May-analyses: No None RC Based on relative competitiveness EMX Early Miss eXploitation (this paper)

Instantiations of cache analysis framework: I I I

No+CM RC+CM EMX+CM

Synthetic benchmarks: I

Random access sequences and program fragments




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saarland university

Evaluation Results 100

computer science

% No+CM

90

RC+CM

80

EMX+CM

70

CollSem

60

100% ff

50

Misses

40

ff

30

ff

Unclassified Hits

20

0%

10 0

1

5

8

10

15

20

25

30

n

Average guaranteed hit- and miss-rates for a cache of size 8 n is number of pairwise different elements that are accessed Daniel Grund and Jan Reineke



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saarland university

Outline 1

computer science


2


3


4

Summary




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saarland university

Summary

computer science Using classification

(must , may )

Cache analysis framework I I


γ

Couple several analyses Cooperation via classifications

I

Potential misses

S

UMust ([T0 , . . . , Tk −1 ] , t , cl ) := 8 < [∅, T0 , . . . , Tk −2 ∪ {t }] : cl = > [T0 , . . . , Tk −1 ] : cl = H : [{t }, T0 , . . . , Tk −2 ] : cl = M

m

EMX FIFO may-analysis

eviction M

3 M

I I

Definite misses Early Miss eXploitation

M

0


M

H

1 insertion

M

H

2

M


α

US

S

Canonical FIFO must-analysis

(must 0 , may 0 )

M

H

H

1

M H

M

M H

2

3 h


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saarland university

Summary

computer science Using classification

(must , may )

Cache analysis framework I I


γ

Couple several analyses Cooperation via classifications

I

Potential misses

S

UMust ([T0 , . . . , Tk −1 ] , t , cl ) := 8 < [∅, T0 , . . . , Tk −2 ∪ {t }] : cl = > [T0 , . . . , Tk −1 ] : cl = H : [{t }, T0 , . . . , Tk −2 ] : cl = M

m

EMX FIFO may-analysis

eviction M

3 M

I I

α

US

S

Canonical FIFO must-analysis

(must 0 , may 0 )

Definite misses Early Miss eXploitation

M

M

0

M

H

1 insertion

M

H

2

M

H

H

1

M H

M

M H

2

3 h

Thank you for listening. Questions? Daniel Grund and Jan Reineke



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saarland university

Further Reading

computer science

R. Wilhelm et al. The worst-case execution time problem— overview of methods and survey of tools Transactions on Embedded Computing Systems, 7(3), 2008 J. Reineke and D. Grund Relative competitive analysis of cache replacement policies LCTES 2008




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saarland university

Related Work: LRU Analyses

computer science

Bounds on number of cache misses: Ghosh Cache Miss Equations, loop nests Chatterjee Exact model of cache behavior for loop nests Classification of individual accesses: Mueller By “static cache simulation” Li By integer linear programming Ferdinand By abstract interpretation Only for LRU caches




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saarland university

What About The Gap for n ≤ k ? 100

computer science

%

90 80 70 60 50 40 30 20 10 0

1

5


8

10

15

20


25

30

n


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saarland

Does the Initial Cache State Make a Difference?

university

computer science

Yes, a FIFO of size k , is (k ,k ) sensitive

Definition: Sensitivity Policy P is (f , c ) miss-sensitive if mP (p, s) ≤ f · mP (p0 , s) + c for all access sequences s and all cache states p, p0 .

⇒ For FIFO of size 4, execution time may differ by a factor of 3 R. Wilhelm et al. Memory Hierarchies, Pipelines, and Buses for Future Architectures in Time-critical Embedded Systems IEEE Transactions on CAD of Integrated Circuits and Systems 2009 Daniel Grund and Jan Reineke



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Abstract Interpretation of FIFO Replacement

Abstract Interpretation of FIFO Replacement

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