Con straint Systems used in Worst-Case Execution Time Analysis

ConConstraint Systems used straint Systems used in Worst-Case Execution in Worst-Case Execution

Time AnalysisTime Analysis

Andreas ErmedahlAndreas Ermedahlandreas.ermedahlandreas.ermedahl@@itit.uu.se.uu.se

Dept. of Dept. of Information TechInformation Technologynology

Uppsala UniversityUppsala University

September 2000 ASTEC/WCET 2

Definition of WCETDefinition of WCET

WCETWCET = Worst = Worst possible execution time fpossible execution time for a or a program running on target hardwareprogram running on target hardware

One program in One program in isolationisolation No interrupts or context switchesNo interrupts or context switches

Other estimates: Other estimates: Best Case ETBest Case ET = Best case = Best case, Inverse of WCET, hard, Inverse of WCET, hard AveraAverage ge Case ETCase ET = = Soft reaSoft real-time, not hardl-time, not hard

0 tighter tighter

safe BCETestimates

safe WCETestimates

actual BCET

actual WCET

possible execution times


Uses of WCET EstimatesUses of WCET Estimates

Hard real-time systemsHard real-time systems Scheduling Scheduling Schedulability analysisSchedulability analysis System dimensioningSystem dimensioning Formal verification Formal verification Program performanceProgram performance

tuningtuning

void foo(int j, int a[]){ int i; for(i=100, i>0; i--) { if(j>50) a[i] = 2 * i; else a[i] = i; }}

?


Static WCET AnalysisStatic WCET Analysis

Don’t run the program - analyze it!Don’t run the program - analyze it! Guaranteed safe WCETGuaranteed safe WCET Trying to be as tight as possibleTrying to be as tight as possible

Provided all input is correctProvided all input is correct

0 tighter tighter

safe BCETestimates

safe WCETestimates

actual BCET

actual WCET

possible execution times

Static analysisStatic analysis will will give give

results in tresults in the he safe areasafe area

Measurement wMeasurement will ill give resultgive resultss in th in the e

unsafe areaunsafe area


Static WCET AnalysisStatic WCET Analysis

Flow analysisFlow analysis Determine the dynamic Determine the dynamic

behavior of programbehavior of program

Low level analysisLow level analysis Determine execution time for Determine execution time for

program parts on the program parts on the hardwarehardware

CalculationCalculation Combine flow and low-level Combine flow and low-level

times to give a WCET times to give a WCET estimateestimate. .

Compiler

ObjectCode

Target Hardware

Reality

program

ActualWCET

Low levelanalysis

Calculation

Flow analysis

Analysis

WCETEstimate

ActualWCET

WCETEstimate


WCET Analysis ArchitectureWCET Analysis Architecture

Compiler

Program

Flow Analysis

IntermediateCode

Cache Analysis

Cache Info

Simulator

Pipeline Analysis

WCET

CalculationConstraint

ProblemFlowInfo

ObjectCode

Modularization of WCET analysisModularization of WCET analysis Several sepSeveral sepaarate analysis steps rate analysis steps Make the analysis more retargetableMake the analysis more retargetable

Flow FactConversion

All analysis results are All analysis results are converted to converted to constraints constraints


Example program

Flow Info CharacteristicsFlow Info Characteristics

do{ if(...) do { if(...) ... else ... if(...) ... else ... } while(...) else ...} while(...)...

Basic finiteness

Statically allowed

Actual feasible

paths

// A// A

// B// B// C// C

// D // D // E// E

// F// F

// G// G// H // H

// I// I// J// J

Structurally possible

flows (infinite)

Relation between possible executions and flow info

max = 10max = 10

max = 20max = 20

samepath(D,G)samepath(D,G)

WCET found here = WCET found here =

desired resultdesired result

A

B

C D

F G

H

E

Basic block graph

J

I

WCET found here =WCET found here =

overestimationoverestimation


Program model:Program model: Basic block graph: Basic block graph:

nnodes and edges odes and edges Execution count Execution count

variable (variable (xxentityentity)) holds holds number of times entity number of times entity gets executedgets executed

Foo()

C

A

B

D

E

F

G

end

Count Count VariablesVariables

XXAA

XXBB

XXCC XXDD

XXEE

XXFF

XXGG

xxnode node for nodesfor nodes

XXGAGA

XXABAB

XXBCBC XXBDBD

XXDEDE

XXEGEG

XXCECE

XXEFEF

XXFGFG

XXfooAfooA

xxedge edge for edgesfor edges


Constraints:Constraints: Foo()

C

A

B

D

E

F

G

end

Constraints GeneratedConstraints Generated

XXAA

XXBB

XXCC XXDD

XXEE

XXFF

XXGG

XXGAGA

XXABAB

XXBCBC XXBDBD

XXDEDE

XXEGEG

XXCECE

XXEFEF

XXFGFG

XXfooAfooAXXfoofoo=1=1

XXend end =1=1Start and end Start and end conditioncondition

XXABAB=X=XAA

XXEE=X=XCECE+X+XDEDE

XXAA=X=XfooAfooA+X+XGAGA

XXBCBC+X+XBDBD=X=XBB

Program Program structurestructure

XXAA<=100 <=100 Loop boundsLoop bounds

XXCC+X+XFF<=X<=XAA

Other flow Other flow informationinformation


A

B

Pipeline AnalysisPipeline Analysis Time for a basic block:Time for a basic block:

Time from first instruction enters pipeline Time from first instruction enters pipeline to last instruction leaves pipelineto last instruction leaves pipeline

ttA A = 7= 7

B

IF

EXEX

M

F

1 2 3 4 5

ttB B = 5= 5

IF

EXEX

M

F

1 2 3 4 5 6 7

A IF

EXEX

M

F

1 2 3 4 5 6 7


A

B

Pipeline AnalysisPipeline Analysis Pipeline overlap between basic blocksPipeline overlap between basic blocks

Timing effect of going from A to B is Timing effect of going from A to B is negative to indicate pipeline overlapnegative to indicate pipeline overlap

We use a general purpose simulator to We use a general purpose simulator to extract times for nodes and edgesextract times for nodes and edges

ttA A = 7= 7

B

IF

EXEX

M

F

1 2 3 4 5

ttB B = 5= 5

ttAB AB = 10= 10

ABAB = = -2-2

AB AB = = ttABAB - - ttAA - - ttBB = 10 - 7 - 5 = = 10 - 7 - 5 = -2-2

1 2 3 4 5 6 7 8 9

IF

EXEX

M

F

10

IF

EXEX

M

F

1 2 3 4 5 6 7

A IF

EXEX

M

F

1 2 3 4 5 6 7


Pipeline Analysis ResultPipeline Analysis Result

Result:Result: Foo()

C

A

B

D

E

F

G

end

Each edge Each edge annotated with annotated with pipeline timing pipeline timing effect effect

ttGAGA=-1=-1ttABAB=-2=-2

ttBCBC=-3=-3 ttBDBD=-5=-5

ttDEDE=-2=-2

ttEGEG=-2=-2

ttCECE=-1=-1

ttEFEF=-3=-3

ttFGFG=-3=-3

ttfooAfooA=-4=-4

Each basic block Each basic block annotated with its annotated with its execution timeexecution time

ttAA=7=7

ttDD=2=2

ttBB=5=5

ttCC=12=12

ttEE=4=4

ttFF=8=8

ttGG=20=20


WCET=WCET=max max (x(xentityentity * t * tentityentity))

Where each Where each xxentityentity

satisfies all constraintssatisfies all constraints

Foo()

C

A

B

D

E

F

G

end

ttAA=7=7

ttDD=2=2

ttBB=5=5

ttCC=12=12

ttEE=4=4

ttFF=8=8

ttGG=20=20

IPET CalculationIPET Calculation

XXAA

XXBB

XXCC XXDD

XXEE

XXFF

XXGG

XXGAGA

XXABAB

XXBCBC XXBDBD

XXDEDE

XXEGEG

XXCECE

XXEFEF

XXFGFG

XXfooAfooA

XXfoofoo=1=1XXABAB=X=XAA

XXEE=X=XCECE+X+XDEDE

XXAA=X=XfooAfooA+X+XGAGA

XXBCBC+X+XBDBD=X=XBB

XXAA<=100<=100

XXCC+X+XFF=100=100


Solution methods:Solution methods: Integer linear programmingInteger linear programming Constraint satisfactionConstraint satisfaction

Solution:Solution: Counts for each Counts for each

individual node and edgeindividual node and edge The value of the WCETThe value of the WCET

Foo()

C

A

B

D

E

F

G

end

Calculation methodsCalculation methods

XXAA=100=100

XXBB=100=100

XXCC=100=100 XXDD=0=0

XXEE=100=100

XXFF=0=0

XXGG=100=100

WCET=4800WCET=4800

XXfoofoo=1=1

XXendend=1=1


Some flow information Some flow information complicates the picturecomplicates the picture

Examples: Examples: Local semantics:Local semantics:

“For each entry of “For each entry of inner loop...”inner loop...” Partially valid flow info: Partially valid flow info:

“During iterations 5 to 10 of “During iterations 5 to 10 of inner loop it holds that...”inner loop it holds that...”

Loop dependencies:Loop dependencies: ”Number of iterations of inner loop ”Number of iterations of inner loop depends on current iteration of outer”depends on current iteration of outer”

Non-linear constraints: Non-linear constraints: ”If D was taken then K will be taken ”If D was taken then K will be taken once” once”

B

C

D E

G H

I

F

J

A

LK

Complicating Flow InformationComplicating Flow Information


scope foo

scope loop

Local semanticsLocal semantics

Create entry count variable: Create entry count variable: xxentry(scope) entry(scope) holds number of holds number of times loop is enteredtimes loop is entered

Local fLocal flow low info areinfo are raised raised to the global levelto the global level

For example: For example: loop:[ ]:Xloop:[ ]:XH H 1010

gets converted to:gets converted to: XXH H 10 * X10 * Xentry(loop)entry(loop)

B

C

D E

G H

I

F

J

A

Scope graph

““For each For each entry of the entry of the loop block Hloop block H

will bewill be executed at executed at

most 10most 10 times”times”

““Block H can not be executed Block H can not be executed

more than 10 * the number more than 10 * the number

of times the loop is entered”of times the loop is entered”

xxentry(loop)entry(loop)


scopeloop

Partial Flow Information Partial Flow Information

B

A

C

D

F

Facts can (partially) overlapFacts can (partially) overlap Virtual scopes: Virtual scopes: scope:rangescope:range

let facts be valid for complete let facts be valid for complete range of iterationsrange of iterations

G

E

Loop bound: 20loop : 1..5 : XC=5 (f1)loop : 3..10 : XC + XF 8 (f2)

loop:1..2{f1}

loop:3..5{f1,f2}

loop:6..10{f2}

loop:11..20 { }

XC = XC + XC + XC + XC

1..2 3..5

6..10 11..20

XC + XC =5 (f1) 1..2 3..5


scopeloop

Partial Flow Information Partial Flow Information

B

A

C

D

F

GGraph is unrolled according to raph is unrolled according to overlapping flow info overlapping flow info

Can generateCan generate l large graphsarge graphs Flow information often local (but Flow information often local (but

can stretch over loop borders) can stretch over loop borders) Dependent flow info can be used to Dependent flow info can be used to

consider subpart of graph in isolation consider subpart of graph in isolation

G

E

loop:1..2{f1}

loop:3..5{f1,f2}

loop:6..10{f2}

Fact f1 and f2 Fact f1 and f2 only overlaps only overlaps

iterations iterations 1..101..10


Some flow info generates Some flow info generates non-linear constraints non-linear constraints ””Number of iterations of inner Number of iterations of inner

loop depends on current iteration loop depends on current iteration of outer” of outer” outer:outer: XXB B 55 55 or or outer:outer: XXA A * X* XA A X XBB

””If D was taken then K will be If D was taken then K will be taken once” taken once” ifif XXD D >> 0 then X0 then XK K >> 0 and 0 and

ifif XXD D = 0 then X= 0 then XK K = 0= 0

More powerful solver needed?More powerful solver needed?

B

C

D E

G H

I

F

J

A

LK

Non-linear ConstrainsNon-linear Constrains


WCET Tool PrototypeWCET Tool Prototype Prototype tool implemented Prototype tool implemented

Works over whole program or use flow info to work bottom up over Works over whole program or use flow info to work bottom up over smaller program partssmaller program parts

Fast solution times when testing with CPLEX or similar Fast solution times when testing with CPLEX or similar solver solver (a network flow problem?)(a network flow problem?)

We use integer linear programming (ILP) and lp_solve()We use integer linear programming (ILP) and lp_solve() Not all flow information can be handledNot all flow information can be handled Rather fast calculation timesRather fast calculation times

Flow information sometimes generates large graphs Flow information sometimes generates large graphs and solution timesand solution times

Other WCET research: Expressing hardware effects Other WCET research: Expressing hardware effects using constraints generates huge constraint systems using constraints generates huge constraint systems and large solution timesand large solution times


Experimental ResultsExperimental Results Small programs: fast calculation Small programs: fast calculation

(even with complex structure and flow)(even with complex structure and flow)

Larger programs: Flow information slows Larger programs: Flow information slows down but increase precision down but increase precision

Example program: NsichneuExample program: Nsichneu Automatically generated program with massive amount Automatically generated program with massive amount

of if-statements (> 250)of if-statements (> 250)Benchmark Flow WCET Graph gen Solving Variables ConstraintsProgram Facts estimate Time (s) Time (s)Nsichneu 0 97662 3.34 1.54 2143 1524Nsichneu (simple facts) 126 51133 3.48 1.44 2143 1650Nsichneu (range facts) 129 51101 233.64 90.16 10943 7945Nsichneu (more facts) 130 51085 279.80 202.37 15222 10971


QuestionsQuestions

What constraint solver should What constraint solver should we use?we use?

Are the generated constraints of Are the generated constraints of a certain type?a certain type?

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Con straint Systems used in Worst-Case Execution Time Analysis