From asynchronous to synchronous specification for distributed program synthesis

(1)

HAL Id: hal-00339740

https://hal.archives-ouvertes.fr/hal-00339740

Submitted on 25 Nov 2008

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

From asynchronous to synchronous specification for distributed program synthesis

Julien Bernet, David Janin

To cite this version:

Julien Bernet, David Janin. From asynchronous to synchronous specification for distributed program

synthesis. SOFSEM, Jan 2008, Slovakia. pp.161-173. �hal-00339740�

(2)

Speiations for Distributed Program Synthesis

JulienBernetandDavid Janin

LaBRI,UniversitédeBordeauxI

351,oursdelaLibération

33405TaleneedexFRANCE

{bernet|janin}labri.fr

Abstrat. Distributed games[7℄ allowthe speiation ofvariousdis-

tributed program synthesis problems. In suh games, a team of Pro-

essplayersompeteagainstauniqueEnvironmentplayer,eahProess

playingonitsownarena,withoutexpliitommuniationswithitsteam-

mates.

Thestandarddenitionofdistributedgamesallowssomedegreeofasyn-

hrony :theEnvironmentanplayonlyonpartofthearenas,therefore

onealingtosomeProessplayersthattheplayisgoingon.Whilethis

is onvenient for modeling distributed problems(espeially those that

themselvesmakeuseofasynhrony),thesegamesare bynomeanseasy

tomanipulate,andtheexistingonstrutionsaremuhmoretedious.

In this paper, we provide a uniform redution of any nite state dis-

tributedgame,synhronousorasynhronous,toasynhronous onewith

thesamenumberofplayers.Thisredutionisshowntobeorretinthe

sensethat it preservestheexistene of nitestate distributed winning

strategiesfor theteamofProesses. Itisuniforminthesensethatitis

designedfor arbitraryinputdistributed gamewithout any priorknowl-

edge aboutitssatisability.Thesizeoftheresultingsynhronousgame

isalsolinearinthesizeoftheoriginalasynhronousoneand,moreover,

thereisnoblowup ofthe sizeof thewinningstrategies. Additionalex-

petedpreservationproperties(e.g.informationows)are studied.Sur-

prisingly, it seems that nosuh redution exists for arbitrary (innite

state)strategies.

Introdution

Asynhronyin adistributed environmentanbedened in anabstrat way as

theabilityofsomeproessesoragentstoperformsomeationinthedistributed

systemswhileotherproessesoragentsnotonlyperformnoationbutareeven

not aware that they are staying idle. In other words, in a model where every

event in a proess (e.g. a loal ation or an inomingor outgoing message)is

triggeredbythetikofaloallok,asynhronyourswhenthereisnoglobal

lokonwhihloalloksaresynhronized.

However,if thememory of adistributed systemhas nitely many possible

states,itisommonlyunderstoodthatsynhronousandasynhronousbehaviors

(3)

eventsbetweenanytwosynhronousevents anbe bounded bythe numberof

globalstates.Morepreisely,the(presumablyasynhronous)distributedsystem

anbemodeled(atamoreabstratlevel)byanequivalentsynhronousone.

For instane, within a distributed system with an asynhronous message

passing mehanism, if the number of messages that have been sent but not

yet reeived is bounded, there annot be an unbounded number of (relevant)

asynhronouseventsbetweenanytwosynhronizations.

Of ourse asynhrony makes a dierene at the implementation level. For

instaneommuniationprotoolsdohavetoopewithasynhronytobeorret.

However,atsomemoreabstrat level,provided theglobalmemoryofasystem

is nite, it seems that,in some sense,asynhronous and synhronous systems

havethesameomputationalpower.

Themain objetiveofthispaperisto substantiatethisintuition.Morepre-

isely,inthesettingofdistributedprogramsynthesisproblem,weaimatproving

that(1)whenthedistributedenvironmentisnitestateand(2)whenoneon-

sidersonly nitestateprograms, anysynhronousorasynhronousdistributed

programsynthesisproblem is equivalentto asynhronousdistributed program

synthesisproblem.

Animmediate diultyforthis taskishowevertondtheappropriatefor-

malism.

Infat,there exists plentyof models of distributed systemsand behaviors;

onsider for instane the diversity of models of ommuniations between pro-

essesoragents:fromommuniationviaglobalsharedmemorymehanismsto

manytypesofmessagepassingsystems.Fordistributedprogramsynthesisspe-

iation,whereonenotonlyaimsatmodelingasinglebehaviorbut afulllass

of behaviors that may, or may not, fulll the design objetives, there may be

evenmoreformalisms.SeeforinstanePnueliandRosner'snotionofdistributed

arhiteture[10,4℄orWonham etal.'sdistributedontrol theory[5,6℄.

Inthispaper,wehoosetostudytherelationshipbetweensynhronousand

asynhronousdistributed synthesisproblemsinafairlyabstratthoughgeneral

model: themodelofdistributedgames[7℄.

Thesegames,designedafterPetersonandReif'spartialinformationgames[9℄,

relyon anexpliit modeling ofthe loal information- projetionofthe global

state-available toeveryagent(orproess)forguiding itsownbehaviors.Asa

matteroffat,noexpliitommuniationismodeled,heneforthnoommitment

has to be done in favor of suh or suh ommuniationmehanism.Moreover,

mostdistributedsynthesisproblemanbeenodedandsolvedwithindistributed

gamesproblem[2,7℄.

Inadistributed game, ateam of Proessplayersompete againstaunique

Environmentplayer,eahProessplayingonitsownarena-withitsownproje-

tionoftheglobalstateofthesystem-whileEnvironmentplayerhasaomplete

knowledgeoftheglobalstate.

Communiations are impliitly modeled as follows. In a distributed game

denition, one may restrit Environment player. From a given global state -

(4)

to reah someglobalstateswith moreinformativeloal projetionsthan other

globalstates.

Asynhrony,that indues anextra layerof partial information -absene of

globallok-ismodeledasfollows:inadistributedgame,theEnvironmentmay

playonlyonpartofthearenas,thereforeonealingtosomeProessplayersthat

theplayisgoingonelsewhere.

Inthispaperweprovethat, asfar asnitesystemsand programsare on-

erned,asynhronydoesnotyieldextraexpressivepower.

Morepreisely,weprovidearedutionof anyasynhronousnite statedis-

tributed game to a synhronous one with the same number of players. This

redutionisshowntobeorretinthesensethatitpreservestheexisteneof-

nitestatedistributedwinningstrategiesfortheteamofProesses.Thesizeofthe

resultingsynhronousgameisalsolinearinthesizeoftheoriginalasynhronous

oneand,moreover,thereisnoblowupofthesizeofthewinningstrategies.Addi-

tionalexpetedpreservationproperties(e.g.informationows)arealsostudied.

For strategies with unbounded memory, it is onjetured that there is no

suh a redution. Still; onemay ask whether suh a result ould be extended

to moregenerallassesofprogramswithinnitememorysuh as,forinstane,

pushdownstrategies.Thisremainsanopenproblem.However,evendeidability

questionsare yetnotsettledforthesemoregeneralstrategies.

1 Notations

ForanynitealphabetA^,â^word ôverAîsâ^funtionw:N→A^whose^domain

isaninitialsegmentofN.Theonlywordsuhwithemptydomainisalledthe emptyword,anddenotedbyǫ^.

Thefollowingnotations are used : A^∗ ^is ^the^set ^of ^nite ^words^(i.e. ^whose

domainis nite)overA^, A^ω îs^the^set ôfînnite^wordsôverA^, A^∞=A^∗∪A^ω

is the set of niteand innite words overA^, ândA⁺ =A^∗− {ǫ} îs ^the ^set ôf

nitenon-emptywordsoverA^.

Foranynite wordw^, îts^length |w| îs ^theârdinal ôfîts ^domain. ^Forâny

wordsu∈ A^∗ ând v ∈A^ω^, ^the^wordu·v îs ^theonatenation ofwordsuând v^.^Forânyîntegerk^,^the^wordu^k îs^built^byonatenatingkôpiesôfu^.^When u6=ǫ^,^theînnite^wordu^ωîs^theînniteonatenationofu^withîtself.

Theseoperationsareextendedtothelanguages.GivenX ⊆A^∗^andY ⊆A^∞^,

weusethenotationsX·Y ={u·v : u∈X, v∈Y}^,X⁰={ǫ}^,X^k ={w·u : w∈X^k−1, u∈X}^for^anyk >0^,X^∗=S

k≥0X^k ^and,^whenǫ /∈X^,X^ω ^for^the

set of all wordsthat an be dened asaninnite produt ofwordsof X^. ^The

followingadditionalnotationsarealso used :X+Y =X∪Y^, X−Y =X\Y

andX^?={ǫ} ∪X^.

Givenn^sets X1, . . . , Xn^,^onsider ^their^diret^produtX =X1× · · · ×Xn^.

For every set I = {i1, . . . , ik} ⊆ {1, . . . , n}^, ônsider ^the ^natural ^projetion πI : X → Xi₁ × · · · ×Xik ^dened ^by πI((x1, . . . , xk) = (xi₁, . . . , xik)^. ^Fôr âny x∈X^,^weâlsoûse^the^moreônvenient^notationx[I] =πI(x)^.

(5)

Followingthesameidea,for anyset Y ⊆X^, ^denote^byY[I] ^the^set πI(Y)^.

If R ⊆ X^m îs â m^-ary ^relation ôver X^, ^we ^should âlso ^write ûse R[I] = {(x1[I], . . . , xm[I]) : (x1, . . . , xm)∈R}^.^WhenIⁱⁿânîntervalôfîntegersôf^the

form[i, j]^weûse^the^notationsx[i, j], Y[i, j], R[i, j]^.^WheneverIîs^redued^toâ

singleintegeri^,^these ^notations^simplify ^tox[i], Y[i], R[i]^.

Moreover, these notations are extended to words and languages : for any

wordw=x0·x1· · · ∈X^∞^,^thenw[I] =x0[I]·x1[I]· · ·^.

2 Distributed games

An^-proessdistributedarenaisagamearenabuiltfromsomeprodutofn^loal

standardgamearena.

Denition. Givenn^arenasGi =hPi, Ei, TP,i, TE,ii^fori∈[1, n]^with ^disjoint

sets of Proess position Pi^, ^sets ^of Environment position Ei^, ^sets ^of ^Proess

moves TP,i ⊆ Pi×Ei ând ^sets ôf Environment moves TE,i ⊆ Ei ×Pi^, ân n^-

Proess distributed game arena builtfrom theloal game arenas {Gi}i∈[1,n] ^is

anygamearenaG=hP, E, TP, TEi^suh^that:

Environmentpositions:E=Q

i∈[1,n]Ei^,

Proesses positions :P =Q

i∈[1,n](Ei∪Pi)−Q

i∈[1,n]Ei^,

Proesses moves : TP îs ^the ^set ôfâll ^pairs(p, e)∈ (P×E)^suh ^that,^for i∈[1, n]^:

• êitherp[i]∈Pi ând(p[i], e[i])∈TP,i ^(Proessi îsâtiveⁱⁿp^),

• ôrp[i]∈Ei ândp[i] =e[i]^(Proessiîsînativeⁱⁿ p^),

andEnvironmentmoves:TE îs^some ^subsetôf^the^set ôfâll^pairs(e, p)∈ (E×P)^suh^that,^fori∈[1, n]^:

• ^eitherp[i]∈Pi ^and(e[i], p[i])∈TP,i(EnvironmentativatesProessi^),

• ôrp[i]∈Ei ândp[i] =e[i](EnvironmentkeepsProessiînative).

WhenthesetTEôfEnvironmentmovesismaximal,weallsuhanarenathefree asynhronousprodutofarenas{Gi}_i∈[1,n]ândîtîs^writtenG1⊗G2⊗ · · · ⊗Gn^.

Remark. The essential idea behind this denition is to get a denition of

a multiplayer game in whih a team of Proesses ompete against a unique

Environmentto ahievesomeinnitary goal. Thefollowing point isimportant

: thisdenition allowstheEnvironmentto playonlyonasubsetofthearenas,

therefore hidingthat the play is going on to theProesses on whih arenas it

doesnotplay.Thiswill bereferredin thefollowingasasynhronousmove,and

allowstoenodeneatlymanydistributedsynthesisproblemsfromtheliterature,

suhas[10,4℄or[5℄.

Indistributed games, asynhronyours when Environment playerdeides

to keeponeormoreProess playersinative.A synhronousdistributed arena

anthusbedenedasfollows.

Denition. Ann^-proessdistributedarenaG=hP, E, TP, TEi^is^a^synhronous

distributedarena whenTE⊆E×Q

i∈[1,n]P[i]^.

Sineadistributedarenaisbuiltuponn^simple^arenas,^we^need^a^denition

tospeakaboutitsloalomponents:

Denition. Given a distributed arena G = hP, E, TP, TEi âs âbove, ^given â

non empty set I⊆[1, . . . , n] ^we^dene ^theânonial ^projetion G[I] ôf GônI

(6)

asthearenaG[I] =hP^′, E^′, T_P^′, T_E^′i^given^by:P^′ =P[I]−E[I]^(possibly^smaller

thanP[I]^!),E^′=E[I]^,T_P^′ =TP[I]∩(P^′×E^′)^,^andT_E^′ =TE[I]∩(E^′×P^′)^.

A distributed game arena is, at rst sight, a partiular ase of standard

disrete and turn base two player game arena. Standard notions of plays and

strategiesarestilldened.However,in ordertoavoidonfusionwithwhat may

happenintheloalarenaadistributed gameisbuild upon,weshallspeaknow

ofaglobalplayandglobalandloalstrategy.Partialinformationisthenaptured

bymeansofthenotionofloalview of play anddistributedstrategy.

Denition. Given an n^-proess distributed arena G^, ^a ^global ^play ^from ^an

initial position e ∈ E îs ^dened âs â ^path ⁱⁿ G ^(seen âs â ^bipartite ^graph)

emanatingfrompositione^that^is^builtalternativelybytheEnvironmentplayer andtheProessteam.

Morepreisely, from aurrentposition x∈ P ∪E^, êither x ∈ E ând ît îs

Environment player turn to play by hoosing some position y ∈ P ^suh ^that (x, y) ∈ TE ôr x ∈ P ând ît îs ^Proess ^team ^turn ^to ^play ^by ^hoosing ^some

positiony∈E ^suh^that(x, y)∈TP^.

Aordingly,aglobal strategy fortheProessteam isapartial funtion σ : (E.P)⁺ → E ^suh ^that ^for êvery ^play ôf ^the ^form w.p ∈ dom(σ) ^with w ∈ E.(P.E)^∗ ândp∈P ône^has(p, σ(w.p))∈TP^.

A play w îs ^said ômpatible ^with ^strategy σ ^when, ^for êveryîntegern ≥0

suhthat w[n]∈P ône^hasw[n+ 1] =σ(w[0, n]) ^wherew[0, n] îs^the ^prexôf wôf^lengthn+ 1^.

Denition. A gameG=hP, E, TE, TP, e0,Wiîs â^game ârenahP, E, TE, TPi

equipped with an extra initial position e0 ∈ E ând â distinguished set W ⊆ (E+P)^ωâlled înnitaryôndition^for^the^Proess^team.

Wesaythatglobalstrategyσ^is^a^winning^strategy ^for^the^Proess^team^from

positione0∈E ^withônditionW ^whenêvery^maximal^plays^startingⁱⁿe0 ând

ompatible withstrategyσîsêither ^niteândêndsⁱⁿ ânenvironmentposition or isinniteandbelongstoW^.

Astrategywith nitememory fortheProessteamisgivenasatupleM= hM, m0, µ: M ×(P∪E) → M, h: M ×P → Ei^, ^where M îs â ^nite ^set ôf

memory states, m0 îs ^the înitial ^memory, µ îs ^the ûpdate ^funtion, ând h îs

the hint funtion.The indued strategy σM : (E·P)⁺ → E ^is ^then ^dened,

foranyplayw·p∈(E·P)⁺^,^byσM=h(µ^∗(m0, w), p)^(whereµ^∗ îs ^dened^by µ^∗(m, ǫ) =m^,ândµ^∗(m, w·x) =µ(µ^∗(m, w), x)^forêverym∈M^,w∈(E∪P)^∗^, x∈(E∪P)^).

In distributed games, it is intended that, within the Proess team, every

proesshasonlyapartialviewofaglobalplay.Notonlyeveryproessonlysees

itsownprojetionofeveryglobalpositions,but,whenidle,aproessisevennot

awarethat theplayisgoingon.Thisintentionisformallydenedasfollows.

Denition. Theloalview Proessi^hasôfâ^global^playⁱⁿâdistributedgame

G ^is ^given^by ^the ^map viewi : (E·P)^∗·E^? → (Ei·Pi)^∗·E_i^? ^dened ⁱⁿ ^the

followingway:

viewi(ǫ) =ǫ

viewi(x) =x[i]

(7)

viewi(w·x·y) =

viewi(w·x)^ifx[i] =y[i]

viewi(w·x)·y[i]^otherwise.

A play w ∈ (E·P)⁺ îs ^said ^to ^be âtive ^for ^Proess i ^when w ênds ⁱⁿ â

positionp∈P ^suh^that p[i]∈P[i]^.

Remark. Observe that in a synhronousdistributed arena, asexpeted, for

every play w ∈ (E.P)^∗.E^? ône ^has viewi(w) = w[i]^, î.e. ^the ^loal ^view ôf â

globalplayis justtheprojetionofthisplay.

Denition. A globalstrategy σ^for ^the^Proess ^teamîs âdistributedstrategy when,foreveryi∈[1, n]^,^thereîsâ^proess^strategyσi : (E[i].P[i])⁺→E[i]ⁱⁿ

the loalgame G[i]^, ^from ^now^on^alled ^loal ^strategy ^for^Proess i^, ^suh^that,

foranyplayoftheform w·p∈(E·P)⁺^, ^given^the^setI ⊆ {1, . . . , n}^of ^ative

proessesintheglobalProessespositionp^,σ(w·p) =eîfândônlyîf

e[i] =σi(viewi(w)·p[i])^fori∈I

e[i] =p[i]^fori∈ {1, . . . , n} −I

Inthisase,wewriteσ1⊗σ2⊗ · · · ⊗σn ^for^thedistributedstrategyσ^.

Remark. Observethat when Gîs synhronous, thedistributed strategy σ= σ1⊗ · · · ⊗σn ân^simply^be^dened^forâny^global^playw∈(E·P)^∗ ^by^:

σ(w) = (σ1(w[1]), . . . , σn(w[n]))

Remark. Globalstrategiesarenotalwaysdistributed.Inpartiular,thereare

distributed games where Proess team has a winning global strategy, but no

winning distributedstrategy.Formoredetails,thereaderanreferto[7℄.

3 From asynhronous game to synhronous game

Weproveherethatevery(asynhronous)distributedgameisequivalentinsome

senseto asynhronousdistributed game.Morepreisely:

Theorem1. There exists a mapping that maps every distributed game G ^to

existsadistributedgameGe ^suh^that^the^Proess^team^haveâdistributedwinning strategy with nite memory in G îf ând ônly îf ^they ^have ône ⁱⁿ Ge^. ^Moreover,

game Ge ^has ^the ^same ^numberôf ^Proess ^players âs^game G^with ônly â^linear

inreaseofnumberof positions.Moreover,thismappingisdeneduniformlyon

distributedgames, bethemwinning for theproessteamor not.

Theremainingofthissetionisdevotedtotheproofofthisresult.

Let G = hP, E, T, e0,Wi ^be â n^-proesses distributed game. First, we are goingtodesribethesynhronousgameGe^,^then^we^will^show^thatîtîsêquivalent

toGⁱⁿ ^terms^ofdistributed strategieswithnitememory.

For any set Ei^, ^dene cEi âs ân êquipotent ^set, ^suh ^that Ei ∩cEi = ∅^;

for any e ∈ Ei^, ^denote ^by ebi ^its ^image ⁱⁿ cEi^. ^Let Eb = Q

i∈{1,...,n}cEi ^and ^let

(8)

Pb=Q

i∈{1,...,n}Pbi^(i.e.^we^restrit^to^relevant^proess^positionsⁱⁿ^a^synhronous

game:proesspositionswhereeveryproessisative).

ConsiderGe=hP ,e E,e T , ee 0,Wif^,^whose^positions^are:

Pei=Pi ∪ Eci ; Efi=Ei ^(for^alli∈ {1, . . . n}⁾

For any position x ∈ P ∪E^, ^denote ^by xb ^the ^position ^of Pe ^obtained ^by

replaingin eêahômponent^from Ei ^with^theirîmageⁱⁿEci ^,î.e.^:

b x[i] =

dx[i]^ifx[i]∈Ei

x[i]^ifx[i]∈Pi

The funtion that maps any x ∈ P ∪E ^to îts îmage xb ⁱⁿ Pe îs ^triviallyâ

bijetion.ThemovesofGe^are^dened ^as^follows:

Tf_i^P =Ti^P∪ {(be, e) :e∈Ei}

Tf^E={(e,p)b ∈Ee×Pe|(e, p)∈T^E}

∪ {(e,be) :e∈E}

Thefuntionanel(Ee·P)e ^∗→(E·P)^∗êrasesânyasynhronousmovefroma globalplay:anel(ǫ) =ǫ^,ânel(w·e·p) =ânel(w)·e·p^,ândânel(w·e·be) =

anel(w)^(wherep∈P^,e∈E^).

This funtion is generalized to innite words by: anel(x0 ·x1 · · · ·) = limi→∞ânel(x0· · ·xi)^(it îs âônverging ^sequene, ⁱⁿ ^the ^sense ôf ^the ^prex

topologyoverwords,asdenedforinstanein [8℄).

Thewinning onditionofGe ^is^then^dened ^as^follows:

Wf=^anel⁻¹(W) ∪ (Ee·Pe)^∗·(E·E)b ^ω

Remark. Theunderlying bipartitegraphofGîsêmbedded înto^theône^from Ge^. ^The ^graph ôf ^the ârena ôf Ge îs âtually ^nothing ^more ^than ^the ^subgraph

induedbythisembedding,whereoneahpositionoftheenvironmentaloopof

size2hasbeenadded,orrespondingtoatotallyasynhronousmove.Moreover,

thewinning onditionWfîs^not^muh ^moreômpliated ^thanW ^:âmongst^the

usualinnitary winningonditions(reahability,safety,parity,Muller,et....),

onlythesafetyonditioninnotpreservedbythis onstrution.

Lemma1 (From asynhronousto synhronous). Forany distributedand

winningstrategyfortheProessteaminG^,^there^is^a^winningdistributedstrategy for the Proess teamin Ge^.

TheideaisatuallytoopythisstrategyonGe^,^ensuringⁱⁿ^the^proess^that

theProessplayersdonottaketheasynhronousmovesplayedontotheirarena

intoaount.

Foranyi∈ {1, . . . , n}^,^letûs^deneâ^funtionâneli: (fEi·Pei)^∗→(Ei·Pi)^∗

that erases the loal asynhronous moves: aneli(ǫ) = ǫ^, ^aneli(w·e·p) =