On Mobile Agent Verifiable Problems

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On Mobile Agent Verifiable Problems

Evangelos Bampas, David Ilcinkas

To cite this version:

Evangelos Bampas, David Ilcinkas. On Mobile Agent Verifiable Problems. Latin American Theoretical

Informatics Symposium (LATIN 2016), Apr 2016, Ensenada, Mexico. pp.123-137,

�10.1007/978-3-662-49529-2_10�. �hal-01323114�

On Mobile Agent Veriable Problems

⋆

EvangelosBampasandDavidIl inkas

CNRS&Univ.Bordeaux,LaBRI,UMR5800,F-33400,Talen e,Fran e

{evangelos.bampas, david.il inkas}labri.fr

Abstra t. We onsiderde isionproblemsthataresolvedinadistributed fashionbysyn hronousmobileagentsoperatinginanunknown, anony-mousnetwork. Ea hagent hasa uniqueidentier and an input string andtheyhavetode ide olle tivelyapropertywhi hmayinvolvetheir inputstrings,the graphonwhi htheyare operating,and their parti -ularstartingpositions.Buildingonre entworkbyFraigniaudandPel [LATIN 2012, LNCS 7256, pp. 362374℄, we introdu e several natural new omputability lasses allowing for aner lassi ation ofproblems below

co

-

MAV

or

MAV

,the latterbeingthe lassof problemsthat are veriablewhentheagentsare providedwithanappropriate erti ate. Weprovidein lusionandseparationresultsamongallthese lasses.We alsodeterminetheir losure propertieswithrespe tto set-theoreti op-erations. Our main te hni al tool, whi h is of independent interest, is a new meta-proto ol that enables the exe ution of a possibly innite numberofmobileagentproto olsessentiallyinparallel,similarlytothe well-knowndovetailingte hniquefrom lassi al omputabilitytheory.

1 Introdu tion

1.1 Context and motivation

The last few de ades have seen a surge of resear h interest in the dire tion

of studying omputability- and omplexity-theoreti aspe ts for various

mod-els of distributed omputing. Signi ant examples of this trend in lude the

investigation of unreliable failure dete tors [5,6℄, as well as wait-free hierar- hies [14℄.A morere entlineof work studiestheimpa tof randomizationand

non-determinisminwhat on ernsthe omputational apabilitiesofthe

LOCAL

model [9,12℄, as well as the impa tof identiers in the same model [10,11℄. A dierentapproa h onsidersthe hara terizationofproblemsthat anbesolved

under various notions of termination dete tion or various types of knowledge

aboutthenetworkinmessage-passingsystems[1,2,3,4,17℄.Finally,are entwork

fo uses on the omputationalpowerof teams of mobile agents[13℄. Ourwork

liesinthislatterdire tion.

⋆

ThisworkwaspartiallyfundedbytheANRproje tsDISPLEXITY

(ANR-11-BS02-014) andMACARON(ANR-13-JS02-002). Thisstudyhasbeen arried outinthe

frameoftheInvestmentsforthefutureProgrammeIdExBordeauxCPU (ANR-10-IDEX-03-02).

that fa ilitates several fundamental networking tasks in luding, among others, faulttoleran e,networkmanagement,anddataa quisition[15℄,andhasbeenof signi antinteresttothedistributed omputing ommunity(see,e.g.,there ent surveys[7,16℄).Assu h,itishighlypertinenttodevelopa omputabilitytheory for mobile agents, that lassies dierent problems a ording to their degree of(non-) omputability, insofarasweareinterestedinreally understandingthe omputational apabilitiesofgroupsofmobileagents.

Inthispaper,we onsideradistributedsysteminwhi h omputationis

per-formed by oneormoredeterministi mobile agents,operatingin an unknown,

anonymousnetwork.Ea hagenthasauniqueidentierandisprovidedwithan

inputstring,andtheyhaveto olle tivelyde ideapropertywhi hmayinvolve theirinputstrings,thegraphonwhi htheyareoperating,andtheirparti ular starting positions. One may argueabout theusefulnessof developingatheory spe i allyfor mobileagentde isionproblems.Webelievethat, apartfrom its inherenttheoreti alinterest,su hastudyisboundtoyieldintermediateresults, tools, intuitions, and te hniques that will proveuseful when one moveson to

onsider from a omputability/ omplexity point of view other, perhaps more

traditional,mobileagentproblems,su hasexploration,rendezvous,pattern for-mation,et .Onesu htoolis theproto olthat wedevelopinthis paper,whi h enablestheinterleavingoftheexe utionsofapossiblyinnitenumberofmobile agentproto ols.

1.2 Related work

In[13℄,FraigniaudandPel introdu edtwonatural omputability lasses,

MAD

and

MAV

,aswellas their ounterparts

co

-

MAD

and

co

-

MAV

. The lass

MAD

, forMobileAgentDe idable,isthe lassofallmobile agentde isionproblems whi h anbede ided, i.e., for whi h there exists a mobile agent proto ol su h thatallagentsa eptinayesinstan e,whileatleastoneagentreje tsinano instan e.Ontheotherhand,the lass

MAV

,forMobileAgentVeriable,isthe lassofallmobileagentde isionproblemswhi h anbeveried.Morepre isely, in ayes instan e, there existsa erti atesu hthat ifea hagentre eivesits dedi atedpie eofit,thenallagentsa ept,whereasinanoinstan e,forevery possible erti ate,atleastoneagentreje ts.Certi atesareforexampleuseful in appli ations in whi h repeated veri ations of some property are required. Fraigniaudand Pel proved in [13℄ that

MAD

is stri tly in ludedin

MAV

, and theyexhibitedaproblemwhi his ompletefor

MAV

underanappropriatenotion ofora leredu tion.

In[8℄, Daset al. fo us on the omplexity ofdistributed veri ation, rather than on its omputability. In fa t, theirmodel diers in several aspe ts. First ofall,thenetworksinwhi hthemobileagentsoperatearenotanonymous,but ea hnodehasauniqueidentier.Thisgreatlyfa ilitatessymmetrybreaking,a

entral issuein anonymous networks. On theother hand though, thememory

of the mobile agentsis limited. Indeed, in [8℄, the authors study the minimal

MAD

s

MAV

MAV

s

co-MAV

s

co-MAV

co-MAD

MAD

teamsize

degree

degree

degree

γ

treesize

degree

γ

allempty

allempty

mineven

Fig.1. Containments between lasses below

MAV

and

co

-

MAV

with orresponding illustrativeproblems.ClassandproblemdenitionsaresummarizedinTables1and2 , respe tively.

lassesofgraphproperties.Again,thestudiedpropertiesaredierentfromthe

ones studied here and in [13℄, sin e they do notdepend on themobile agents

ortheirstartingpositions.However,theymaydependonlabelsthatnodes an possessin additiontotheiruniqueidentiers.

1.3 Our ontributions

Weintrodu eseveralnewmobile agent omputability lasseswhi h playakey

roleinourendeavorforaner lassi ationofproblemsbelow

MAV

and

co

-

MAV

. The lasses

MAD

s

and

MAV

s

arestri tversionsof

MAD

and

MAV

, respe tively, inwhi hunanimityisrequiredinbothyesandnoinstan es.Furthermore,we onsiderthe lass

co

-

MAV

′

(andits ounterpart

MAV

′

)ofmobileagentde ision problemsthatadmita erti ateforno instan es,whileretainingthe

system-widea eptan eme hanismof

MAV

.

Weperformathoroughinvestigationoftherelationshipsbetweenthenewly

introdu ed and pre-existing lasses. As a result, we obtain a omplete Venn

diagram (Figure 1) whi h illustrates the tight inter onne tions betweenthem. We take areto pla e naturalde isionproblems (in themobile agent ontext) in ea h of the onsidered lasses. Among other results, we obtain a ouple of fundamental,previouslyunknown,in lusionswhi h on ernpre-existing lasses:

MAD

⊆ co

-

MAV

and

co

-

MAD

⊆ MAV

.

We omplementourresultswitha ompletestudyofthe losurepropertiesof these lassesunder thestandardset-theoreti operationsof union,interse tion,

and omplement.Thevarious lassdenitions togetherwith the orresponding

losurepropertiesaresummarizedin Table1.

Themainte hni altoolthatwedevelopanduseinthepaperisanew meta-proto olthatenablestheexe utionofapossiblyinnitenumberofmobileagent proto olsessentiallyin parallel.This anbeseenasamobileagent omputing analogueofthewell-knowndovetailingte hniquefrom lassi alre ursiontheory.

sureproperties.Thenotation

yes

(resp.

no

)meansthatallagentsa ept(resp.reje t). Similarly,

yes

ı

(resp.

no

Ù

)meansthatatleastoneagenta epts(resp.reje ts).

Denition ClosureProperties

yesinstan es no instan es Union Interse . Compl.

MAD

s

(

∀

erti ate:)

yes

(

∀

erti ate:)

no

✓

✓

✓

MAD

(

∀

erti ate:)

yes

(

∀

erti ate:)

no

Ù

✗

✓

✗

co

-

MAD

(

∀

erti ate:)

yes

ı

(

∀

erti ate:)

no

✓

✗

✗

MAV

s

∃

erti ate:

yes

∀

erti ate:

no

✓

✓

✗

co

-

MAV

s

∀

erti ate:

yes

∃

erti ate:

no

✓

✓

✗

MAV

∃

erti ate:

yes

∀

erti ate:

no

Ù

✗

✓

✗

co

-

MAV

∀

erti ate:

yes

ı

∃

erti ate:

no

✓

✗

✗

MAV

′

_∃

erti ate:

yes

ı

∀

erti ate:

no

✓

✓

✗

co

-

MAV

′

_∀

erti ate:

yes

∃

erti ate:

no

Ù

✓

✓

✗

2 Preliminaries

Thegraphsin whi h themobile agentsoperateareundire ted, onne ted, and

anonymous. The edges in ident to ea h node

v

(ports) are assigned distin t

lo al port numbers(also alledlabels)from

{1, . . . , d

v

}

,where

d

v

isthedegree ofnode

v

.Theportnumbersassignedtothesameedgeatitstwoendpointsdo nothavetobeinagreement.

We onventionallyx abinaryalphabet

Σ = {0, 1}

.In viewofthe natural bije tion between binarystringsand

N

whi h maps a stringto its rankin the quasi-lexi ographi order of strings(shorter stringspre ede longer strings,the rankoftheempty string

ε

being

0

),wewill o asionallytreatstringsand nat-uralnumbersinter hangeably.If

x

and

y

arestrings,then

hx, yi

standsfor any standarden odingas astringofthepairofstrings

(x, y)

.

If

x

isalist,then

|x|

isthelengthof

x

and

x

i

isthe

i

-thelementof

x

.If

f

is afun tionthat anbeappliedtotheelementsof

x

,thenwewillusethenotation

f (x) = f (x

1

), . . . , f (x

|x|

)

.Inthesamespirit, if

x

and

y

areequal-lengthlists ofstrings,then

hx, yi

standsforthelist

hx

1

, y

1

i , . . . ,

x

|x|

, y

|y|



.

We denote by

Σ

0

1

the set of re ursivelyenumerable (or Turing-a eptable)

de ision problems,

Π

0

1

= co

-

Σ

0

1

, and

∆

0

1

= Σ

0

1

∩ Π

0

1

.

∆

0

1

is exa tly the set of Turing-de idableproblems.

2.1 Mobile agent omputations

A mobile agent proto ol is modeled asadeterministi Turing ma hine. Mobile agents are modeled as instan es of amobile agent proto ol (i.e., opies of the orrespondingdeterministi Turingma hine)whi hmoveinanundire ted,

on-ne ted, anonymous graphwith port labels. Ea h mobile agent is provided

isunique. Theexe ution ofagroupof mobileagentson agraph

G

pro eedsin syn hronoussteps.Atthebeginningofea hstep,ea hagentisprovidedwithan additionalinputstring,whi h ontainsthefollowinginformation:(i)thedegree of the urrentnode

u

,(ii) theportlabelat

u

throughwhi h theagentarrived at

u

(or

ε

iftheagentisinitsrststepordidnotmoveinthepreviousstep),and (iii) the ongurationof allother agentswhi hare urrently on

u

.Then, ea h agentperformsalo al omputationandeventuallyhaltsbya eptingor reje t-ing,oritmovesthroughoneoftheportsof

u

,orremainsatthesamenode.We assumethatalllo al omputationstakethesametimeandthatedgetraversals areinstantaneous.Therefore,theexe utionis ompletelysyn hronous.

Let

M

be a mobile agent proto ol,

G

be a graph,

id

be a list of distin t

IDs,

s

bealist ofnodes of

G

, and

x

bealist ofstringssu h that

|id| = |s| =

|x| = k > 0

. We denote by

M (id, G, s, x)

the exe ution of

k

opies of

M

, the

i

-th opy starting on node

s

i

and re eiving as inputs the ID

id

i

and the string

x

i

.Thetuple

(id, G, s, x)

is alledtheimpli itinput.Similarly,wedenote by

M (id, x; id, G, s, x)

thepersonal viewof theexe utionof

M

ontheimpli it input, asexperien ed bytheagentwith ID

id

andinput

x

. Wedistinguish be-tweentheexpli it input

(id, x)

,whi h isprovidedtotheagentat thebeginning

of the exe ution, andthe impli it input, whi h may ormay not be dis overed

bytheagentinthe ourseoftheexe ution.

Givenanimpli it input,wewrite

M (id, x; id, G, s, x) = yes

(resp.

no

)ifthe agentwithexpli itinput

(id, x)

a epts(resp.reje ts)during

M (id, G, s, x)

. Fur-thermore,wewrite

M (id, G, s, x) 7→ yes

(resp.

no

),if

∀i M (id

i

, x

i

; id, G, s, x) =

yes

(resp.

no

), and

M (id, G, s, x) 7→

ˆ

yes

(resp.

ı

no

), if all agents halt and for some

i M (id

i

, x

i

; id, G, s, x) = yes

(resp.

no

).

2.2 Mobile agentde isionproblems

Denition1 ([13℄). A mobile agent de ision problem on anonymous graphs

is a set

Π

of instan es

(G, s, x)

, where

G

isa graph,

s

is a non-empty list of nodesof

G

,and

x

isalist ofstringswith

|x| = |s|

,whi hsatisesthe following losureproperty:Forevery

G

andforeveryautomorphism

α

of

G

thatpreserves portnumbers,

(G, s, x) ∈ Π

ifandonly if

(G, α(s), x) ∈ Π

.

1

Wewill referto instan es whi h belong to aproblem

Π

as yes instan es of

Π

. Instan es that do not belong to

Π

will be alled no instan es of

Π

.

The omplement

Π

of amobile agentde isionproblem

Π

is theproblem

Π =

{(G, s, x) : |s| = |x|

and

(G, s, x) 6∈ Π}

. 2

Someexamplesof de isionproblems

areshownin Table2.

1

Notethat this losure propertyis synta ti ally dierent fromthe oneusedin[13℄ duetonotationaldieren es,butthetwoareequivalent.

2

Itiseasyto he kthatif

Π

isade isionproblem,then

Π

alsosatisesthe losure propertyofDenition1.Therefore,

Π

isalsoade isionproblem.

ofthepaper.

alone

= {(G, s, x) : |s| = 1}

allempty

= {(G, s, x) : ∀i x

i

= ε}

consensus

= {(G, s, x) : ∀i, j x

i

= x

j

}

degree

= {(G, s, x) : ∀i ∃v d

v

= x

i

}

degree

_γ

= {(G, s, x) : G

ontainsanodeofdegree

γ}

(for

γ

≥ 1

)

mineven

= {(G, s, x) : min

i

x

i

iseven

}

path

= {(G, s, x) : G

isapath

}

teamsize

= {(G, s, x) : ∀i x

i

= |s|}

treesize

= {(G, s, x) : ∀i G

isatreeofsize

x

i

}

Denition2 ([13℄). A de ision problem

Π

is mobile agentde idable if there

exists a proto ol

M

su h that for all instan es

(G, s, x)

: if

(G, s, x) ∈ Π

then

∀id M (id, G, s, x) 7→ yes

,whereasif

(G, s, x) /

∈ Π

then

∀id M (id, G, s, x) 7→

ı

no

. The lass ofallde idable problems isdenotedby

MAD

.

Denition3 ([13℄). A de ision problem

Π

is mobile agentveriable if there

exists a proto ol

M

su h that for all instan es

(G, s, x)

: If

(G, s, x) ∈ Π

then

∃y ∀id M (id, G, s, hx, yi) 7→ yes

,whereasif

(G, s, x) /

∈ Π

then

∀y ∀id M (id, G, s,

hx, yi) 7→

ı

no

.The lass of allveriable problems isdenotedby

MAV

.

Whenthere is noroom for onfusion, wewill use theterm erti ate both for the string

y

provided to an agent and for the olle tion of erti ates

y

providedto thegroupof agents.If weneedto distinguishbetweenthetwo,we will refer to

y

as a erti ate ve tor. Finally, if

X

is a lass of mobile agent de isionproblems,then

co

-

X

= {Π : Π ∈ X}

.

Remark 1. Notethatin[13℄,onlyde idable(inthe lassi alsense)mobileagent de isionproblemsweretakeninto onsideration.Asaresult,itwasbydenition

the asethat

MAD

and

MAV

werebothsubsetsof

∆

0

1

. Forthepurposesof this work,wedonotimpose this onstraint.

3 Mobile Agent De idability Classes

A problem

Π

is in

co

-

MAD

ifand only ifit an bede ided by amobile agent proto olinasensewhi hisdualtothat ofDenition 2:Iftheinstan eisin

Π

, then at least oneagent must a ept,whereasif theinstan e is notin

Π

, then allagentsmustreje t.Wewill onsideronemoresu hvariantintheformofthe stri t lass

MAD

s

.Aproblembelongstothis lassifit anbesolvedinsu ha waythat everyagentalwaysoutputsthe orre tanswer.

Denition4. Ade isionproblem

Π

isin

MAD

s

ifandonlyifthereexistsa pro-to ol

M

su hthatforallinstan es

(G, s, x)

:if

(G, s, x) ∈ Π

then

∀id M (id, G, s,

x) 7→ yes

,whereasif

(G, s, x) /

∈ Π

then

∀id M (id, G, s, x) 7→ no

.

Bydenition,

MAD

s

isasubsetofboth

MAD

and

co

-

MAD

andit iseasyto he kthat

MAD

s

= co

-

MAD

s

. Moreover,allof these lassesaresubsets of

∆

0

1

, sin ea entralized algorithm,provided withan en oding of thegraphand the startingpositions,inputs,andIDsoftheagents, ansimulatethe orresponding mobile agent proto ol and de ide appropriately. As mentioned in [13℄,

path

is

an example of a mobile agent de ision problem whi h is in

∆

0

1

\ MAD

, sin e, intuitively, anagent annot distinguisha longpath from a y le. In fa t, this observationyields

path

∈ ∆

0

1

\ (MAD ∪ co

-

MAD)

.

Anontrivialproblemin

MAD

s

is

treesize

.Theproblemwasalreadyshownto bein

MAD

in [13℄. Forthe strongerproperty that

treesize

∈ MAD

s

, weneeda modi ationoftheproto olgivenin [13℄.

Proposition1.

treesize

∈ MAD

s

.

Wenowshowthat

MAD

and

co

-

MAD

arestri t supersetsof

MAD

s

.

Proposition2.

allempty

∈ MAD \ MAD

s

and

allempty

∈ co

-

MAD

\ MAD

s

.

As we mentioned,

MAD

s

is in luded in both

MAD

and

co

-

MAD

. In fa t,

MAD

s

= MAD ∩ co

-

MAD

.Westatethisasatheoremwithoutproof,sin eit an beobtainedasa orollaryofTheorems2and3,whi hwewillproveinSe tion5.

Theorem1.

MAD

s

= MAD ∩ co

-

MAD

.

ByTheorem1,if

allempty

wasin ludedin

co

-

MAD

,wewouldobtain

allempty

∈

MAD

s

,whi hweknowtobefalse.Thus,

allempty

∈ co

/

-

MAD

andweobtaina sep-arationbetween

MAD

and

co

-

MAD

.Symmetri ally,

allempty

∈ co

-

MAD

\ MAD

.

4 Interleaving Multiple Mobile Agent Proto ols

Itisimportanttohaveatoolthatenablestheexe utionofseveralmobileagent proto olsonthesameinstan e,andthatalsopermitsthemobileagentstomake de isionsbased on theout omes ofthese exe utions.Forexample, ifonewere togiveadire t proofofTheorem1above,onewouldneedawayfortheagents to oordinateinordertoexe uteboththe

MAD

andthe

co

-

MAD

proto olfora parti ularproblem,andthen,basedontheout omeoftheseexe utions,togive aunanimous orre tanswer(inthespiritof

MAD

s

).

In lassi al omputing, the well known dovetailing te hniquea hieves this interleavingof dierent omputations.Classi aldovetailingpro eeds inphases: inphase

T

,therst

T

stepsoftherst

T

programsareexe uted.Atthispoint,an auxiliaryfun tionisexe uted,whi hde ides,basedontheseexe utions,whether to a ept,reje t,or ontinuewiththenextphase.Correspondingly,themobile

agent meta-proto ol whi h we propose in this se tion, pro eeds in phases: in

phase

T

,theagentsexe utetherst

T

stepsoftherst

T

mobileagentproto ols

and then de ide whether to a ept, reje t, or pro eed to the next phase. In

fun tion alo al de ider.

Still,itmayhappenthatoneormoreagentshaltasaresultofexe utingthe lo al de ider, while othersde ide to ontinue. Insu h a ase, theexe ution of theproto olsinthenextphase ouldbe orruptedbe ausethehaltedagentsno

longer follow theproto ol. However, these halted agents an now beregarded

as xed tokens and the meta-proto ol uses them in order to reate a map of

the graph.In fa t,this is done in su h awayas to ensurethat all non-halted agentsobtainnotonlythemapofthegraphbuta tuallyfull knowledgeofthe impli itinput.Basedonthisknowledge,ea hagentde idesirrevo ablywhether toa eptorreje tbymeansofase ondfun tion whi hisgivenasaparameter tothemeta-proto ol,andwhi hwe allaglobal de ider.

4.1 Ingredients of the meta-proto ol

Weproposeageneri meta-proto ol

P

N ,f,g

, whi h isparameterizedby

N , f, g

.

The set

N

is a, possibly innite, re ursively enumerable set of mobile agent

proto ols.Let

N

i

,

i ≥ 0

,denote the

i

-th proto ol insu h anenumeration.The fun tions

f

and

g

are omputablefun tionswhi hrepresentlo al omputations withthefollowingspe i ations:

Globalde ider:Thefun tion

f

mapspairs onsistingofanexpli itandan im-pli itinput,i.e.,tuplesoftheform

(id, x; id, G, s, x)

,totheset

{accept, reject}

. Inthis ase,wesaythat

f

isaglobalde ider.Whenanagentexe utes

f

,ithalts bya eptingorreje tinga ordingtotheout omeof

f

.

Lo alde ider:Thefun tion

g

takesasinputanexpli itinput

(id, x)

andalist

(H

1

, . . . , H

σ

)

of arbitrarylength

σ

, where ea h

H

j

is thehistoryofthe partial exe ution of

N

j

(id, x; id, G, s, x)

for a ertain numberof steps and

(id, G, s, x)

is an impli it input ommon for allhistories

H

1

, . . . , H

σ

. The out ome of

g

is one of

{accept, reject, continue}

. When an agent exe utes

g

, it haltsin the orresponding stateifthe out omeis

accept

or

reject

,otherwise it ontinues withouthalting.

Iffor everyimpli it input

(id, G, s, x)

andfor every

T

0

, there exists a

T ≥

T

0

and some

i

su h that the lo al omputation

g(id

i

, x

i

, H

1

, . . . , H

min(T,|N |)

)

returnseither

accept

or

reject

,whereea h

H

j

isanen odingoftheexe ution of

N

j

(id

i

, x

i

; id, G, s, x)

for

T

steps,thenwesaythat

g

isalo al de iderfor

N

.

Themeta-proto olusesthefollowingpro eduresCreate-MapandRdv:

Pro edureCreate-Map

(R)

:Anagentexe utesthispro edureonlywhenit

isonanodewhi h ontainsatleastonehalted(oridle)agent.Startingfromthis node,andtreatingthehaltedagentasaxedmark,itattemptsto reateamap ofthegraphassumingthat thegraph ontainsatmost

R

nodes.Morepre isely, the agentrst reates a map onsisting in a singlenode orresponding to the markednode

r

,with

d

r

pendingedges with port numbersfrom

1

to

d

r

. Then, whilethereremainsomependingedgesandthereareatmost

R

explorednodes,

the agent explores somearbitrary pending edge as follows. The agent goes to

It then determines whether its urrentposition

v

orresponds to anode of its

map, as follows: For everynode

w

in its map, it omputesa pathin the map

going from

w

to

r

and follows the orrespondingsequen e of port numbersin

theunknowngraph,startingfrom

v

.Ifitleadstothemarkednode,then

v = w

andtheagentupdatesitsmapbylinkingthependingedgesof

u

and

w

withthe appropriateportnumbers.Otherwise,itretra esitsstepsto omeba kto

v

and testsanextnode

w

.Ifallnodesturnoutto bedierentfrom

v

,thentheagent

goesba ktothemarkednodethrough

u

,andupdatesitsmapbyaddinganew

node orrespondingto

v

, linkedto

u

,and withtheappropriatenumberofnew pendingedges.Attheendofthepro edure,theagenteitherhasa ompletemap

of thegraph,orknowsthat thegraphhasmorethan

R

nodes.Thispro edure

takesat most

4R

4

steps.

Pro edure Rdv

(R, id)

: This pro edure guaranteesthat a group of

k

agents

whi h(a)knowthesameupperbound

R

onthenumberofnodesin thegraph,

(b) havedistin t

id

's

{id

1

, . . . , id

k

}

, and ( ) start exe uting Rdv

(R, id

i

)

at the same time from dierent nodes

s

i

, will all meet ea h other after nite time.

Moreover,ea h agent knows when it has met all other agents exe uting Rdv,

evenwithoutinitialknowledgeof

k

.

TheRdvpro edureusesasasubroutinethefollowingExplore-Ball

pro- edure:An agentexe utingExplore-Ball

(R)

attempts toexplore theballof

radius

R

around its starting node

s

i

, assuming an upper bound of

R

on the

maximum degree of the graph. This is a hieved by having the agent try

ev-ery sequen e of length

R

of port numbers from the set

{1, . . . , R}

, retra ing itsstepsba kwardafter ea h sequen eto returnto

s

i

.If aparti ularsequen e instru ts theagentto followaport numberthat does notexist at the urrent node(i.e.,theportnumberislargerthanthedegreeofthenode),thentheagent abortsthat sequen eandreturnsto

s

i

.Attemptingallpossiblesequen estakes at most

B(R) = 2R · R

R

steps. If an agent nishesearlier, it waits on

s

i

un-til

B(R)

steps are ompleted.Therefore, ateamof agentsthat startexe uting

Explore-Ball

(R)

atthesametimefromdierentnodesaresyn hronizedand

ba kattheirstartingpositions after

B(R)

steps.

Now,forea hbitof

id

i

,theRdvpro edureexe utesthefollowing:Ifthebit is

0

,theagentwaitsat

s

i

for

B(R)

stepsandthenexe utesExplore-Ball

(R)

,

whereas if the bit is

1

, the agent rst exe utes Explore-Ball

(R)

and then

waitsonitsstartingpositionfor

B(R)

steps.Afteritexhauststhebitsof

id

i

,the agentexe utestwi eExplore-Ball

(R)

.Thisguaranteesthat,ifthenumberof nodesisatmost

R

,thenafter

2 · (|id

i

| + 1) · B(R)

steps,ea hagent

i

islo ated at

s

i

andhasmetallotheragentsexe utingRdv.Notethataftereveryinteger multiple of

B(R)

steps,ea hagentislo atedat itsinitialnode

s

i

.

4.2 Des ription ofthe meta-proto ol

Themeta-proto ol

P

N ,f,g

worksinphases,whi h orrespondtoin reasingvalues

ofapresumedupperbound

T

onthenumberofnodesin thegraph,thelength

no

yes

no

no

yes

yes

yes

no

complete map

of

nodes

Execute

Execute

Dovetail protocols

Attempt map construction

assuming #nodes

and

exchange info

update flags

input is received

idle until implicit

synch

synch

synch

n

≤ T

g

f

accompanied

cautious

∨

neutralized

mapseeker

T

≤ T

Rdv

(n, id

i

)

accompanied

← false

Rdv

(2T, id

i

)

mapseeker

cautious

← false

← false

T

← T + 1

T

← 1

neutralized

← false

Fig.2.High-levelow hartofthemeta-proto olofSe tion4.

saythatanagentisidleifitiswaitingindenitelyonitsstartingnodeforsome other agentto provideitwith theknowledgeof thefullimpli itinput. Wewill say that anagent isparti ipating ifit is nothaltedandnot idle. Notethat an agentmayhaltonlyasaresultofexe utingoneofthede iderfun tions

f

and

g

. Inea h phase

T

,theagentsperformthefollowinga tions(seealsoFig.2):

Sear hfornearbystartingpositionsandsetags.Ea hparti ipatingagent

i

rstexe utesRdv

(2T, id

i

)

foratmost

2(T + 1)B(2T )

steps.BydesignofRdv, thisguaranteesthat agent

i

willexploreits

2T

-neighborhoodatleaston eand, in parti ular,if

T ≥ |id

i

|

, then for ea h otherparti ipating agent,agent

i

will exploreits

2T

-neighborhoodatleaston ewiththatagentstayingonitsstarting node.If,inthepro ess,theagentmeetsanyagent,thenitsetsits

accompanied

ag.Italsosetsits

neutralized

agiftheen ounteredagentisparti ipatingand it has alexi ographi ally largerID. If theen ountered agent is haltedor idle, theagentsetsits

mapseeker

ag.Finally,iftheagentndsanodewithdegree larger than

2T

or ifthe length ofits ID is greater than

T

, it sets its

cautious

ag.Allagentssyn hronizeatthispoint.

Mapseekeragentsattemptto reate amap ofthegraph.Next,ea hagent

i

with the

mapseeker

agset movestoahaltedoridle agentwhi h ithasfound previously,whileexe utingRdvinthe urrentphase.Then,itattemptsto reate amapofthegraphbyexe utingCreate-Map

(T )

andreturnsto

s

i

.Overall,this takesatmost

4T

4

_{+ 4T}

steps.Moreover,duringtheexe utionofCreate-Map ,

mapseeker

agents olle tstartingpositionandinputinformationfromallhalted

and idle agents that they en ounter. Meanwhile, non-

mapseeker

agents wait

for

4T

4

_{+ 4T}

steps.All agentssyn hronizeat thispoint.

Sofar,wehavea hievedthat,if

T ≥ n

,where

n

isthenumberofnodesin

G

, theneithernoagentisa

mapseeker

havingthefullmapof

G

,orallparti ipating

agentshavethe

mapseeker

agset andtheyhavethefull mapof

G

(Lemma1

below).Ifall

mapseeker

agentshavethefullmapof

G

and

T ≥ n

,thenea hsu h agent

i

exe utesRdv

(n, id

i

)

, whi h guarantees that, nally, it is lo ated at

s

i

Rdv pro edure,ea h

mapseeker

exe utes

f

with full knowledgeof theimpli it

input(Lemma2).

Performdovetailing.Atthispoint,ifnoagentisa

mapseeker

havingthefull map of

G

, the agents exe ute ea h of the proto ols

N

1

, . . . , N

min(T,|N |)

for at most

T

steps, and then retra e ba kward to

s

i

(agentsare syn hronizedafter exe uting ea h proto ol). If any of these proto ols instru ts an agentto halt, theagentinsteadwaitsuntilthe

T

-stepexe utionperiodhasnished,andthen returnsto

s

i

.Iftheagentdoesnothavethe

cautious

or

accompanied

agsset,it thenexe utes

g(id, x, H

1

, . . . , H

min(T,|N |)

)

,where

H

j

isthehistoryofthe

T

-step exe utionof

N

j

withexpli itinput

(id, x)

.Sin ethispro esstakesatmost

2T

2

steps,allagentsthat donothaltasaresultofexe uting

g

aresyn hronizedat theendofthe urrentphase.Itisguaranteedthatthehistoriesfed tothelo al de ider

g

orrespond to orre t exe utions of the orresponding proto ols for impli itinput

(id, G, s, x)

,eventhoughsomeoftheagentsmay havehaltedor be omeidleinearlierphases(Lemma3andCorollary1).

Neutralizedagentsbe omeidle.Finally,attheendofthephase,

neutralized

agents start waiting for the impli it input (i.e., they be ome idle), and when theyre eiveit(fromsome

mapseeker

agent),theyexe utetheglobalde ider

f

. Lemma1. In ea h phase, either all or none of the parti ipating agents (i.e., non-haltedandnon-idle)exe ute

f

.

Lemma2. Anyagent that exe utes

f

has full knowledge of the impli it input

(id, G, s, x)

.

Lemma3. Ifanagent

i

exe utes

g

duringphase

T

,thennootheragent's start-ingnodeisatdistan e

2T

orlessfrom

s

i

.

ByLemma3,weobtainfollowing orollary:

Corollary 1. Anyagent

i

thatexe utes

g

hashistorieswhi h orrespondtothe orre thistoriesof

N

j

(id

i

, x

i

; id, G, s, x)

for

T

steps(

1 ≤ j ≤ min(T, |N |)

),even thoughsomeof the agentsmayhave haltedorbe omeidleinearlier phases.

Inviewof Corollary1, we anshow that allagentsterminate and, in fa t, theyallterminate ontheirrespe tivestartingnodes.

Lemma4. Let

f

be aglobal de ider andlet

g

be a lo al de ider for

N

. Then, ea h agent halts under the exe ution

P

N ,f,g

(id, G, s, x)

by exe uting either

f

or

g

.Moreover,ea h agent

i

halts onitsstarting node

s

i

.

4.3 Appli ation of the meta-proto ol

Tosummarize, themeta-proto ol isageneri toolthat enablesusto interleave the exe utions of a possibly innite set of mobile agentproto ols. Eventually, ea hagenta eptsorreje ts,basedeitheronthehistoriesoftheexe utionsofa

oftheimpli itinput(bymeansoftheglobalde ider).

Weuse the meta-proto ol in order to pla e a parti ular problem in one of themobileagent omputability lassesofTable1.A ommonpartoftheproofs onsists in dening thelist of proto ols

N

and suitablede iders

f

and

g

, and in showingthat

f

and

g

indeed satisfytheglobal and lo alde iderproperties, respe tively.Thisisfollowedbyaparttailoredto ea h parti ularresult,where

weusethepropertiesof themeta-proto ol(Lemmas 14andCorollary1) and

theparti ulardenitions of

f

and

g

, inorder toshowthatagentsthatexe ute

P

N ,f,g

alwaysterminate in thedesired state.Thedesired stateis indi atedby the lassinwhi hwewishtopla etheproblem.Forexample,ifwewishtoshow thataproblemisin

MAD

s

,wewillhavetoshowthatallagentsgivethe orre t answerforallimpli itinputs.

5 Mobile Agent Veriability Classes

Denition5. Ade isionproblem

Π

isin

MAV

s

ifandonlyifthereexistsa pro-to ol

M

su hthatforallinstan es

(G, s, x)

:if

(G, s, x) ∈ Π

then

∃y ∀id M (id, G,

s, hx, yi) 7→ yes

,whereasif

(G, s, x) /

∈ Π

then

∀y ∀id M (id, G, s, hx, yi) 7→ no

.

Bydenition,

MAV

s

⊆ MAV

. Moreover,

MAV

⊆ Σ

0

1

, sin ea entralized al-gorithm an simulate the

MAV

proto ol for all possible erti ate ve tors(by lassi aldovetailing)and a eptifit ndsa erti atefor whi h allagents a - ept.Bytaking omplements,weobtainaswellthat

co

-

MAV

s

⊆ co

-

MAV

⊆ Π

0

1

. Thereexistseveralnontrivialproblemsin

MAV

s

and

co

-

MAV

s

(Proposition3).

Furthermore, we an show that

MAV

is a stri t superset of

MAV

s

and, as a

orollary,

co

-

MAV

isastri tsupersetof

co

-

MAV

s

(Proposition 4).

Proposition3. Foranyxed

γ ≥ 1

,

degree

γ

∈ MAV

s

.Furthermore,

consensus

∈

co

-

MAV

s

and

alone

∈ co

-

MAV

s

.

Proposition4.

degree

∈ MAV \ (MAV

s

∪ co

-

MAV

)

.

Proposition 4 also separates

MAV

from

co

-

MAV

. In order to separate

Σ

0

1

from

MAV

and

Π

0

1

from

co

-

MAV

, weobservethat the

teamsize

problem, whi h is learlyin

∆

0

1

= Σ

0

1

∩ Π

0

1

,isneitherin

MAV

norin

co

-

MAV

.

Proposition5.

teamsize

∈ ∆

0

1

\ (MAV ∪ co

-

MAV)

.

De ision problems with no erti ates In lassi al omputability, the

lass

Π

0

1

= co

-

Σ

0

1

an be seen as the lass of problems that admit a no er-ti ate, i.e.:forno instan es,there existsa erti ate thatleadsto reje tion, whereasforyes instan es, no erti ate anleadtoreje tion. Inthis respe t,

while

MAV

an ertainly be onsidered as the mobile agent analogue of

Σ

0

1

,

co

-

MAV

is not quite the analogueof

Π

0

1

. Problems in

co

-

MAV

indeed admit a no erti ate, but the a eptan e me hanismis reversed:for no instan es,

deneandstudy

co

-

MAV

′

,the lassofmobileagentproblemsthatadmitano

erti ate while retaining the

MAV

a eptan eme hanism,aswellas its

om-plement

MAV

′

.Wegivethedenition of

MAV

′

below.

Denition6. Ade isionproblem

Π

isin

MAV

′

ifandonlyifthereexistsa pro-to ol

M

su hthatforallinstan es

(G, s, x)

:if

(G, s, x) ∈ Π

then

∃y ∀id M (id, G,

s, hx, yi) 7→

ˆ

yes

,whereasif

(G, s, x) /

∈ Π

then

∀y ∀id M (id, G, s, hx, yi) 7→ no

.

Bydenition,itholdsthat

MAV

s

⊆ MAV

′

and

co

-

MAV

s

⊆ co

-

MAV

′

.Toshow

MAV

′

= MAV

s

(andthus

co

-

MAV

′

_{= co}

-

MAV

s

),weneedtoboostthe

MAV

′

pro-to olsothattheagentsanswerunanimouslyeveninyesinstan es.Wea hieve thisbysupplyinganextra erti ate,whi hisinterpretedasthenumberofnodes ofthegraph.Thisenablestheagentstomeetandex hangeinformationinyes

instan es, and therefore rea h a unanimous de ision. The meta-proto ol from

Se tion 4 essentially provides for free the ne essary subroutines for meeting

andex hanginginformation.

Theorem2.

MAV

′

_{= MAV}

_s

and

co

-

MAV

′

_{= co}

-

MAV

s

.

InviewofTheorem2,itfollowsthat

MAV

s

⊆ MAV ∩ co

-

MAV

and

co

-

MAV

s

⊆

MAV

∩ co

-

MAV

.Weseparate

MAV

∩ co

-

MAV

frombothofthese lasseswiththe problem

mineven

:

Proposition6.

mineven

∈ (MAV ∩ co

-

MAV) \ (MAV

s

∪ co

-

MAV

s

)

.

Conne tions with the de idability lasses We explore the relationships

amongthede idability lassesofSe tion3andthe lassesdenedinthisse tion.

Fromthedenitions weknowthat

MAD

⊆ co

-

MAV

′

, therefore,byTheorem 2,

MAD

⊆ co

-

MAV

s

.Similarly,

co

-

MAD

⊆ MAV

s

.Therefore,sin e

MAD

s

⊆ MAD ∩

co

-

MAD

,wealsohavethat

MAD

s

⊆ MAV

s

∩ co

-

MAV

s

.

WeshowinTheorem3that,infa t,

MAD

s

= MAV

s

∩ co

-

MAV

s

.Furthermore,

from the denitions and Theorem 2, we have

MAD

⊆ MAV ∩ co

-

MAV

s

and

co

-

MAD

⊆ MAV

s

∩ co

-

MAV

. We show that these a tually hold asequalities in

Theorem4below.TheproofofTheorem3(resp.Theorem4)isbasedontrying

allpossible ombinationsof erti atesforthe

MAV

s

(resp.

MAV

)and

co

-

MAV

s

proto ols.Here,weusethefullpowerofthemeta-proto olofSe tion4inorder tointerleaveandsyn hronizethisinnitenumberofexe utions.

Theorem 3.

MAD

s

= MAV

s

∩ co

-

MAV

s

.

Theorem 4.

MAD

= MAV ∩ co

-

MAV

s

and

co

-

MAD

= MAV

s

∩ co

-

MAV

.

Notethatitwasshownin[13℄that,ifwe onsiderde isionproblemsthatare de idable or veriableby asingle agent(thus givingrise to the lasses

MAD

1

and

MAV

1

),thenitholdsthat

MAD

1

= MAV

1

∩ co

-

MAV

1

.Theorems3and4 an beseenasgeneralizationsofthatresulttomultiagent lasses.

Proposition7. Forany xed

γ ≥ 1

,

degree

γ

∈ MAV

s

\ co

-

MAD

and

degree

γ

∈

co

-

MAV

s

\ MAD

.

In view of Theorem 4, Proposition 7 yields a separation between

MAV

s

and

co

-

MAV

, as

degree

γ

∈ MAV

s

\ co

-

MAV

,and aseparationbetween

co

-

MAV

s

and

MAV

,as

degree

γ

∈ co

-

MAV

s

\ MAV

.

By ombining the results of this se tion with the results of Se tion 3, we obtainapi tureoftherelationshipsamongthe lassesbelow

MAV

and

co

-

MAV

, asillustratedinFigure 1.

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