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TheSynchronousModelofComputa6on

StavrosTripakisUCBerkeley

EE249Lecture–Sep15,2009

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Fundamentalcharacteris6csofthesynchronousMoC

•  No6onofsynchronous round (orcycle)•  Concurrency•  Determinism(mostofthe6me)

–  Same(sequenceof)inputs=>same(sequenceof)outputs

•  Contrastthisto:–  Concurrencywiththreads:

•  Non‐determinis6c:resultsdependoninterleaving–  ConcurrencyinKahnProcessNetworks:

•  Asynchronous(interleaving),buts6lldeterminis6c•  Needsunboundedbuffersingeneral,forcommunica6on

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Thesynchronousround

3

inputs

outputs

inputs

outputs

…

rounds

round1 round2

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Example:synchronousblockdiagram

A

C

rounds

A, B, C  A, C, B  … 

B

5

Example:synchronousblockdiagram

A

C

rounds

A,   … 

B

B C 

A,   B C 

determinis6cconcurrency

Example:FIRfilter

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Whereisthesynchronousroundhere?

Example:sequen6allogicdiagram

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Whereisthesynchronousroundhere?

Example:controlloop

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initialize state; while (true) do read inputs; compute outputs; update state; write outputs; end while;

Whereisthesynchronousroundhere?

Example:controlloop(v2)

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initialize state; while (true) do await clock tick; read inputs; compute outputs; update state; write outputs; end while;

Isthisanimportantmodelofcomputa6on?

•  Yes!– Extremelywidespread,bothintermsofmodels/languages,andintermsofapplica6ons

•  Examplesofapplica6ons:– Synchronousdigitalcircuits– 99%(?)ofcontrolsoeware

•  Read‐compute‐writecontrolloops

•  Nuclear,avionics,automo6ve,…

– Mul6media,…

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Isthisanimportantmodelofcomputa6on?

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CopyrightTheMathworks

EnginecontrolmodelinSimulink

HW SW++c.f.SimulinktoFPGA,

ortoHDL

Isthisanimportantmodelofcomputa6on?

•  Yes!–  Extremelywidespread,bothintermsofmodels/languages,andintermsofapplica6ons

•  Examplesofmodelsandlanguages:– Mealy/Mooremachines–  Verilog,VHDL,…–  (discrete‐6me)Simulink–  Synchronouslanguages–  (Synchronous)Statecharts–  Thesynchronous‐reac6ve(SR)domaininPtolemyII– …

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Mythsaboutsynchronousmodels

•  Synchronousmodelshavezero‐1meseman6cs–  Synchronousseman6csareessen6allyun0med:theydonothaveaquan6ta6veno6onof6me.

–  FamousEsterelstatements[Berry‐Gonthier‘92]:•  every 1000 MILLISEC do emit SEC end •  every 1000 MILLIMETER do emit METER end

–  Synchronousmodelscancaptureboth0me‐triggeredandevent‐triggeredsystems.E.g.:

•  Dosomethingevery20ms•  Dosomethingwheneveryoureceiveaninterruptfromtheengine

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Example:controlloop(v3)

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initialize state; while (true) do await clock tick

or any other interrupt; read inputs; compute outputs; update state; write outputs; end while;

Mythsaboutsynchronousmodels

•  But:–  Thesynchronouscyclescouldbeinterpretedasdiscrete6me:0,1,2,3,…,inwhichcasewehaveadiscrete‐6meseman6cs…

–  …andthiscanalsobeseenasanabstrac6onofreal‐6me:

–  C.f.6minganalysisofdigitalcircuits

–  C.f.WCETanalysisofsynchronouscontrolloops

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Mythsaboutsynchronousmodels

•  Synchronousmodelsarenon‐implementable(becausezero‐6meisimpossibletoachieve)– Hein?

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Real‐TimeWorkshop

Benefitsofsynchronousmodels

•  Oeenmorelight‐weightthanasynchronous– Nointerleaving=>lessstateexplosion

•  Oeendeterminis6c–  Easiertounderstand,easiertoverify

•  SWimplementa6ons:– Noopera6ngsystemrequired–  Sta6cscheduling,nomemoryalloca6ons,nodynamiccrea6onofprocesses,…

•  Simple6ming/schedulabilityanalysis– OeensimpleWCETanalysisalso:noloops

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Asynchronousvs.SynchronousProduct

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Lectureplan

•  Part1:Single‐ratesynchronousmodels

•  Part2:Mul6‐ratesynchronousmodels

•  Part3:FeedbackandCausality

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Part1:Single‐ratesynchronousmodels

•  Moore/Mealymachines•  Synchronousblockdiagrams

–  Inspiredbydiscrete‐6meSimulink,andSCADE

•  Lustre•  Esterel

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MooreMachines

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•  States:{q0,q1,q2,q3}•  Ini6alstate:q0•  Inputsymbols:{x,y,z}•  Outputsymbols:{a,b,c}•  Outputfunc6on:

– Out:States‐>Outputs•  Transi6onfunc6on:

– Next:StatesxInputs‐>States

Whereisthesynchronousroundhere?

determinis0c

Mooremachine:acircuitview

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Next Out

clock

x(n)

y(n)

s(n)

MealyMachines

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•  States:{S0,S1,S2}•  Ini6alstate:S0•  Inputsymbols:{0,1}•  Outputsymbols:{0,1}•  Outputfunc6on:

– Out:StatesxInputs‐>Outputs•  Transi6onfunc6on:

– Next:StatesxInputs‐>States

Whereisthesynchronousroundhere?

Mealymachine:acircuitview

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Next Out

clock

x(n)

y(n)

s(n)

Isthisa“purelysynchronous”model?

Moorevs.Mealymachines

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MooreorMealy?

MooreorMealy?

Moorevs.Mealymachines

•  EveryMooremachineisalsoaMealymachine– Why?

•  IsitpossibletotransformaMealymachinetoaMooremachine?

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Synchronousblockdiagrams

•  Physicalmodelsoeendescribedincon6nuous‐6me

•  Controllerpart(e.g.,TransmissionControlUnit)isdiscrete‐6me

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Synchronousblockdiagrams

Example:FIRfilter

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WhatistheMealymachineforthisdiagram?

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Hierarchyinsynchronousblockdiagrams

A B

P

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Hierarchyinsynchronousblockdiagrams

Fundamentalmodularityconcept

P

Seman6csofhierarchicalSBDs

•  Canwedefinetheseman6csofacompositeSBDasaMealymachine?–  Inpar6cular,withapairof(Out,Next)func6ons?

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Problemwith“monolithic”seman6cs

A

B

P

P.out(x1, x2) returns (y1, y2) { y1 := A.out( x1 ); y2 := B.out( x2 );

return (y1, y2); }

x1

x2y2

y1

FalseI/Odependencies=>

Modelnotusableinsomecontexts

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Solu6on

•  Generalizefromasingle,toMANYoutputfunc6ons

P.out1( in1 ) returns out1 { return A.out( in1 ); }

P.out2( in2 ) returns out2 { return B.out( in2 ); }

A

B

P

[DATE’08,RTAS’08,POPL’09]

Lustre

•  TheFIRfilterinLustre:

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node fir (x : real) returns (y : real); var s1, s2 : real; let s1 = 0 -> pre x; s2 = 0 -> pre s1; y = x/3 + s1/3 + s2/3; tel

Lustre

•  TheFIRfilterinLustre:

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node fir (x : real) returns (y : real); var s1, s2 : real; let s1 = 0 -> pre x; s2 = 0 -> pre s1; y = x/3 + s1/3 + s2/3; tel

Lustre

•  TheFIRfilterinLustre:

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node fir (x : real) returns (y : real); var s1, s2 : real; let y = x/3 + s1/3 + s2/3; s2 = 0 -> pre s1; s1 = 0 -> pre x; tel

Whathaschanged?Isthiscorrect?

Lustre

•  TheFIRfilterinLustre(noexplicitstatevars):

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node fir (x : real) returns (y : real); let y = x/3 + (0 -> pre x)/3 + (0 -> (0 -> pre pre x))/3; tel

Esterel•  TheFIRfilterinEsterel:

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module FIR: input x : double; output y : double;

var s1 := 0 : double, s2 := 0 : double in loop await x ; emit y(x/3 + s1/3 + s2/3) ; s2 := s1 ; s1 := x ; end loop end var.

Esterel•  TheFIRfilterinEsterel:

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module FIR: input x : double; output y : double;

var s1 := 0 : double, s2 := 0 : double in loop await x ; emit y(x/3 + s1/3 + s2/3) ; s1 := x ; s2 := s1 ; end loop end var.

Whathaschanged?Isthiscorrect?

Esterel•  AspeedometerinEsterel:

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module SPEEDOMETER: input sec, cm; % pure signals output speed : double; % valued signal loop var cpt := 0 : double in abort loop await cm ; cpt := cpt + 1.0 end loop when sec do emit speed(cpt) end abort end var end loop.

Lustre

•  ThespeedometerinLustre:

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node speedometer(sec, cm: bool) returns (speed: real); var cpt1, cpt2 : int; sp1, sp2 : real; let cpt1 = counter(cm, sec); sp1 = if sec then real(cpt1) else 0.0; cpt2 = counter(sec, cm); sp2 = if (cm and (cpt2 > 0)) then 1.0/(real(cpt2)) else 0.0; speed = max(sp1, sp2); tel

Part2:Mul6‐ratesynchronousmodels

•  Synchronousblockdiagramswithtriggers–  Inspiredbydiscrete‐6meSimulink,andSCADE

•  Lustrewithwhen/current•  WhataboutEsterel?

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Triggeredand6medsynchronousblockdiagrams

•  Mo6vatedbySimulink,SCADE

Triggeredblock

Simulink/Stateflowdiagram Sample6me

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Triggeredsynchronousblockdiagrams

A

B C

P

TRIGGEREDBLOCK

TRIGGER

mul0‐ratemodels:

•  B executed only when trigger = true

•  All signals “present” always •  But not all updated at the

same time •  E.g., output of B updated only

when trigger is true need initial value in case trigger = false at n = 0 (initial round)

v

Ques6on:dotriggersincreaseexpressiveness?

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Triggerelimina6on

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Triggerelimina6on:atomicblocks

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Timeddiagrams

A

(3,1)

BC

P

“TIMED”BLOCKS

“sta0c”mul0‐ratemodels (2,0)

(period,phase)specifica6ons

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Timeddiagrams=sta0callytriggereddiagrams

A

(3,1)

BC

P

(2,0)A B C

P

(3,1) (2,0)

=

(2,0)where

produces: true, false, true, false, …

Mul6‐clocksynchronousprogramsinLustre

•  Thenwhenandcurrentoperators:

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node A(x: int, b: bool) returns (y: int); let y = current (x when b); tel

0 1 2 3 4 5 ... T F T F F T ... 0 2 5 ... 0 0 2 2 2 5 ...

x: b:

x when b: y:

Mul6‐clocksynchronousprogramsinLustre

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node A(x1,x2: int, b: bool) returns (y: int); let y = x1 + (x2 when b); tel

Whatisthemeaningofthisprogram?

ForbiddeninLustre

Mul6‐clocksynchronousprogramsinLustre

•  InLustre,everysignalhasaclock=“temporal”type

•  Theclock‐calculus:asortoftypechecking– Onlysignalswithsameclockcanbeadded,mul6plied,…

– Howtocheckwhethertwoclocks(i.e.,booleansignals)arethesame?•  Problemundecidableingeneral•  InLustre,checkissyntac6c

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Mul6‐rateinEsterel

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every 1000 MILLISEC do emit SEC end || every 1000 MILLIMETER do emit METER end

MILLISEC

MILLIMETER

SEC

METER

Part3:FeedbackandCausality

•  Vanillafeedback:–  Cyclicdependencies“broken”byregisters,delays,…

•  Unbrokencyclicdependencies:–  Lustre/SBDsolu6on:forbidden–  Esterel/HWsolu6on:forbiddenunlessifitmakessense

•  Malik’sexample

•  Construc6veseman6cs

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FeedbackinLustre

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node counter() returns (c : int); let c = 0 -> (pre c) + 1; tel

node counter() returns (c : int); let c = 0 -> c + 1; tel

OK

Rejected

FeedbackinSynchronousBlockDiagrams

•  SameasLustre:

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A B

Rejected,unlessAorBisMooremachine

Whataboutthis?

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Cycliccombina0onalcircuit.Useful:equivalentacycliccircuitisalmost2xlarger

[Malik’94]

z = if c then F(G(x)) else G(F(x))

Canwegivemeaningtocyclicsynchronousmodels?

•  Thinkofthemasfix‐pointequa6ons:– x = F(x)

•  Whatisthemeaningofthese:– x = not x – x = x

•  Isuniquesolu6onenough?– x = x or not x

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Construc6veseman6cs

•  Reasoninconstruc6velogicinsteadofclassicallogic

•  “xornotx”notanaxiom

•  Thenwecannotprovex=1from:– x = x or not x

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Construc6veseman6cs

•  Fix‐pointanalysisinaflatCPO:–  Startwith“boxom”(undefined),iterateun6lfix‐pointisreached:

•  Guaranteedinfinitenumberofitera6ons,becauseno.signalsandno.valuesarebothfinite

–  Ifsolu6oncontainsnoundefinedvalues,thencircuitisconstruc6ve

•  Inourexample:– x = x or not x –  Boxomisthefix‐point–  Circuitnotconstruc6ve

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True

Т

False

Т

Т

Construc6veseman6cs:theore6calbasis

•  Kleenefixedpointtheorem:– Let L be a CPO and f : L → L be a con5nuous (and therefore monotone) func5on. Then f has a least fixed point equal to sup{bot,f(bot),f(f(bot)),…}

•  InourflatCPO,con6nuous=monotone:– Non‐monotone:f(bot) > f(a), whereaisnotbot – Notarealis6cfunc6on

•  InoutflatCPO,termina6onisguaranteed.

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Construc6veseman6cs

•  Anotherexample:– x = a and not y – y = b and not x

•  Herewehaveexternalinputs,musttryforallpossibleinputcombina6ons

•  Exercise!

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Summary

•  Synchronousmodelofcomputa6on:– Widespread,manylanguages,manyapplica6ons–  Easiertounderstand,easiertoverify(thanasynchronousinterleaving)

–  Interes6ngseman6cally

•  Togofurther:–  Interes6ngimplementa6onproblems:howtopreservetheproper6esthatthesynchronousabstrac6onprovides(determinism,values,…)duringimplementa6on?

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Ques6ons?

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References•  Statemachines(Moore,Mealy,…):

–  SwitchingandFiniteAutomataTheory.ZviKohavi,McGraw‐Hill,1978.•  Synchronousblockdiagrams:

–  LublinermanandTripakispapersonmodularcodegenera6on:availablefromhxp://www‐verimag.imag.fr/~tripakis/publis.html

•  Synchronouslanguages:–  “Thesynchronouslanguages12yearslater”,Proc.IEEE,Jan2003,and

referencestherein.•  Construc6veseman6cs:

–  SharadMalik.Analysisofcycliccombina0onalcircuits.ICCAD1993.–  GerardBerry.TheConstruc0veSeman0csofPureEsterel.Draebook,1996,

downloadable,googleit.•  General,overview:

–  P.Caspi,P.RaymondandS.Tripakis.SynchronousProgramming.InI.Lee,J.Leung,andS.Son,editors,Handbook of Real‐Time and Embedded Systems.Chapman&Hall,2007.Availablefromsiteabove.

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