Logicin

Games

EricPacuit

ILLC,UniversityofAmsterdam

staff.science.uva.nl/∼epacuit

epacuit@science.uva.nl

November29,2006

Core

LogicLecture

Ifyouthinkthatyourpaper

isvacuous,

Use

the�rst-order

functionalcalculus.

Itthen

becomes

logic,

And,asifbymagic,

Theobviousishailed

asmiraculous.

(MosheVardi)

WhatLogicin

WhichGames?

�Gametheory

isabagofanalyticaltoolsdesigned

tohelp

usunderstandthephenomenathatweobservewhen

decision-m

akers

interact.�

OsborneandRubinstein.Introductionto

GameTheory.MIT

Press

.

WhatLogicin

WhichGames?

Gametheory

isfullofdeeppuzzles,andthereisoften

disagreem

ent

aboutproposedsolutionsto

them

.Thepuzzlementand

disagreem

entare

neither

empiricalnormathem

aticalbut,rather,

concern

themeaningsoffundamentalconcepts

(`solution',

`rational',`complete

inform

ation')andthesoundnessofcertain

arguments...Logicappears

tobeanappropriate

toolforgame

theory

both

because

theseconceptualobscurities

involvenotions

such

asreasoning,know

ledgeandcounter-factuality

whichare

part

ofthestock-in-tradeoflogic,andbecause

itisaprimefunctionof

logicto

establish

thevalidityorinvalidityofdisputedarguments.

M.O.L.Bacharach.Logic

andtheEpistemic

FoundationsofGameTheory..

(Modal)Logicin

Games

M.Pauly

andW.vander

Hoek.ModalLogic

form

GamesandInform

ation.

HandbookofModalLogic(2006).

G.Bonanno.Modallogiandgametheory:Twoalternativeapproaches.

Risk

DecisionandPolicy

7(2002).

J.vanBenthem

.Extensivegamesasprocess

models.JournalofLogic,Lan-

guageandInform

ation11(2002).

J.Halpern.Acomputerscientistlooksatgametheory.Games

andEconomic

Behavior45:1

(2003).

R.Parikh.SocialSoftware.Synthese132:3(2002).

Logicin

Games:

RelevantConferences

LOFT:Conference

onLogicandtheFoundationsofGameand

DecisionTheory

(Amsterdam:www.illc.uva.nl/LOFT2008/)

TARK:TheoreticalAspects

ofRationality

andKnow

ledge

(Brussels2007:www.info.fundp.ac.be/

pys/TARK07/)

GLoRiClass

Sem

inar:

www.illc.uva.nl/GLoRiClass/

New

perspectivesonGames

andInteraction,Feb.5-7,2007

(www.illc.uva.nl/KNAW-AC/)

WhatIWantto

Talk

About

•GameLogics

•Logicsforsocialinteractivesituations

•When

are

twogames

thesame?

•Epistemicprogram

ingametheory

•Aggregatingindividualjudgments

•Axiomatizationresultsin

SocialChoice

•(Form

ally)Verifyingthatasocialprocedure

iscorrect

•SocialSoftware

=(SocialChoice+

GameTheory+Com

puterScience)/Logic

•Develop(�well-behaved�)logical

languages

thatcanexpress

gametheoreticconcepts,such

astheNash

equilibrium

WhatIWillTalk

About

•GameLogics

•Logicsforsocialinteractivesituations

•When

are

twogames

thesame?

•Epistemicprogram

ingametheory

•Aggregatingindividualjudgments

•Axiomatizationresultsin

SocialChoice

•(Form

ally)Verifyingthatasocialprocedure

iscorrect

•SocialSoftware

=(SocialChoice+

GameTheory+Com

puterScience)/Logic

•Develop(�well-behaved�)logical

languages

thatcanexpress

gametheoreticconcepts,such

astheNash

equilibrium

WhatIWillActuallyTalk

About

•Aparadox

surroundingtheepistemicfoundationsofsolution

concepts

•An�Axiomatization�resultin

Social

ChoiceTheory

•Someim

possibilityresults

•A�logicalapproach�to

backwardsinduction

Goal:

Illustrate

wherelogicnaturallyshow

supin

thesocial

sciencesandpointto

somerelevantliterature.

EpistemicProgram

inGameTheory

FundamentalProblem:

Whatdoes

itmeanto

saythatthe

playersin

agamearerational,each

thinkseach

other

isrational,

each

thinkseach

other

thinkstheothersare

rational,etc.?

EpistemicProgram

inGameTheory

FundamentalProblem:

Whatdoes

itmeanto

saythatthe

playersin

agamearerational,each

thinkseach

other

isrational,

each

thinkseach

other

thinkstheothersare

rational,etc.?

EpistemicProgram

inGameTheory:

Anexplicitdescription

oftheplayers'beliefsispart

ofthedescripitonofagame.

Identify

foranygam

ethestrategiesthatare

chosenbyrationaland

intelligentplayerswhoknow

thestrucutreofthegame,the

preference

oftheother

players

andrecognizeeach

othersrationality

andbeliefs.

Literature

See,forexample,

R.Aumann.InteractiveEpistemologyI&

II.InternationalJournalofGame

Theory

(1999).

P.BattigalliandG.Bonanno.Recentresultsonbelief,

knowledgeandthe

epistemic

foundationsofgametheory.Researchin

Economics(1999).

B.deBruin.ExplainingGames.

Ph.D.Thesis,2004.

R.Stalnaker.BeliefRevisionin

Games:

Forward

andBackward

Induction.

Mathem

aticalSocialSciences(1998).

DescribingBeliefs

Fix

asetofpossiblestates(complete

descriptionsofasituation).

Twomain

approaches

todescribebeleifs:

•Set-theortical(K

ripkeStructures,AumannStructures):For

each

stateandeach

agent

i,specifyasetofstatesthat

i

considerspossible.

•Probabilistic(BayesianModels,HarsanyiTypeSpaces):For

each

state,de�nea(subjective)

probabilityfunctionover

the

setofstatesforeach

agent.

AQuestion

Whichmodelisthe�correct�

oneto

work

with?

AQuestion

Whichmodelisthe�correct�

oneto

work

with?

Itturnsoutthat�ndingtheconnectionbetweenrationality,what

agents

thinkaboutthesituationandwhat

actuallyhappens

dependsontheexistence

ofa�richenough�space

oftypes,i.e.,a

universaltypespace.

AQuestion

Whichmodelisthe�correct�

oneto

work

with?

Itturnsoutthat�ndingtheconnectionbetweenrationality,what

agents

thinkaboutthesituationandwhat

actuallyhappens

dependsontheexistence

ofa�richenough�space

oftypes,i.e.,a

universaltypespace.

Itis

noten

ough

[...]thatAnnshould

consider

each

of

Bob'sstrategies

possible.Rather,sheconsiderspossible

both

everystrategy

thatBobmightplayandeverytype

that

Bobmightbe.(L

ikew

ise,

Bobconsiderspossible

both

every

strategy

thatAnnmightplayandeverytype

thatAnnmight

be.)

Brandenburger,Friedenburg

andKeisler.Admissibilityin

Games.

2004.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?

∗Anassumption(orstrongestbelief)

isabeliefthatim

plies

all

other

beliefs.

A.Brandenburger

andH.J.Keisler.AnIm

possibilityTheorem

onBeliefsin

Games.

forthcomingin

Studia

Logica.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?Yes.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?Yes.

Then

accordingto

Ann,Bob'sassumptioniswrong.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?Yes.

Then

accordingto

Ann,Bob'sassumptioniswrong.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?Yes.

Then

accordingto

Ann,Bob'sassumptioniswrong.

So,accordingto

Ann,Bob'sassumptioniscorrect�

i.e.,Bob's

assumptionis

notwrong.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?Yes.

Then

accordingto

Ann,Bob'sassumptioniswrong.

So,accordingto

Ann,Bob'sassumptioniscorrect�

i.e.,Bob's

assumptionis

notwrong.

So,theanswer

must

beno.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?No.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?No.

Then

accordingto

Ann,Bob'sassumptioniscorrect.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?No.

Then

accordingto

Ann,Bob'sassumptioniscorrect.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?No.

Then

accordingto

Ann,Bob'sassumptioniscorrect.

Thatis,itiscorrectthatAnnbelievesthatBob'sassumptionis

wrong.

AParadox

AnnbelievesthatBobassumes∗

that

AnnbelievesthatBob'sassumptioniswrong.

Does

AnnbelievethatBob'sassumptioniswrong?No.

Then

accordingto

Ann,Bob'sassumptioniscorrect.

Thatis,itiscorrectthatAnnbelievesthatBob'sassumptionis

wrong.

So,theanswer

must

beyes.

Main

Result

BeliefModel:

asetofstatesforeach

player,andarelationfor

each

playerthatspeci�es

when

astate

ofoneplayer

considersa

state

oftheother

player

tobepossible.

Main

Result

BeliefModel:

asetofstatesforeach

player,andarelationfor

each

playerthatspeci�es

when

astate

ofoneplayer

considersa

state

oftheother

player

tobepossible.

Language:

thelanguageusedbytheplayers

toform

ulate

their

beliefs

Main

Result

BeliefModel:

asetofstatesforeach

player,andarelationfor

each

playerthatspeci�es

when

astate

ofoneplayer

considersa

state

oftheother

player

tobepossible.

Language:

thelanguageusedbytheplayers

toform

ulate

their

beliefs

Complete:

Abeliefmodeliscomplete

foralanguageifevery

statementin

aplayer'slangaugewhichispossible(i.e.truefor

somestates)

canbeassumed

bytheplayer.

Main

Result

BeliefModel:

asetofstatesforeach

player,andarelationfor

each

playerthatspeci�es

when

astate

ofoneplayer

considersa

state

oftheother

player

tobepossible.

Language:

thelanguageusedbytheplayers

toform

ulate

their

beliefs

Complete:

Abeliefmodeliscomplete

foralanguageifevery

statementin

aplayer'slangaugewhichispossible(i.e.truefor

somestates)

canbeassumed

bytheplayer.

Theorem

(Brandenburger

andKeisler)Nobeliefmodelcanbe

complete

foralanguagethatcontains�rst-order

logic.

OpenQuestion

Canwe�ndalogicLsuch

that

1.Complete

beliefmodelsforLexistforeach

game;

2.notionssuch

asrationality,beliefin

rationality,etc.

are

expressibleinL;and

3.theingredients

in1and

2canbecombined

toyield

various

well-know

ngame-theoreticsolutionconcepts.

AggregatingPreferences:

SomeNotation

•Suppose

thatthereare

nindividualsandtwoalternatives

x

and

y

•Let

xP

iydenote

that

iprefers

xto

yand

xI i

ydenote

that

iis

indi�erentbetween

xand

y

AggregatingPreferences:

SomeNotation

•Foreach

ithereisavariable

Di∈{−

1,0,

1}where

D=

−1

ifyP

ix

0if

xI i

y

1if

xP

iy

•f

:{−

1,0,

1}n→{−

1,0,

1}isthegroupdecisionfunction

Sim

pleMajority

Procedure

For

k∈{−

1,0,

1},let

Nk(D

1,.

..,D

n)

=|{

i|D

i=

k}|

Let

~ D=〈D

1,.

..,D

n〉

fisasimplemajority

decisionmethodi�

f(~ D

)=

−1

ifN

1(~ D

)−

N−

1(~ D

)<

0

0if

N1(~ D

)−

N−

1(~ D

)=

0

1if

N1(~ D

)−

N−

1(~ D

)>

0

Propertiesofgroupdecisionfuncitons

Agroupdecisionfunction

fis

•Decisiveifitisatotalfunction

•Symmetric

iff(D

1,.

..,D

n)

=f(D

j(1

),.

..,D

j(n

))forall

permutations

j.I.e.,fissymmetricin

allof

itsarguments.

•Neutralif

f(−

D1,.

..,−

Dn)

=−

f(D

1,.

..,D

n)

•Positively

Responsiveif

D=

f(D

1,.

..,D

n)

=1/

2or

1,and

D′ i=

Diforall

i6=

i 0,and

D′ i 0

>D

i 0,then

D′=

f(D

′ 1,.

..,D

′ n)

=1

May'sTheorem

Theorem

(May,1952)Agroupdecisionfunctionisthemethod

ofsimplemajority

decisionifandonly

ifitisalwaysdecisive,

symmetric,neutralandpositively

responsive

May'sTheorem

Theorem

(May,1952)Agroupdecisionfunctionisthemethod

ofsimplemajority

decisionifandonly

ifitisalwaysdecisive,

symmetric,neutralandpositively

responsive

Form

alMinim

alism

M.Pauly.OntheRoleofLanguagein

SocialChoiceTheory.Availableatthe

author'swebsite

(2005).

GeneralizingMay'sTheorem

InMay'sTheorem,theagents

are

makingasinglebinary

choice

betweentwoalternatives.Whataboutmore

generalsituations?

GeneralizingMay'sTheorem

InMay'sTheorem,theagents

are

makingasinglebinary

choice

betweentwoalternatives.Whataboutmore

generalsituations?

•Agents

choose

betweenbetweenmore

thantwoalternatives.

•Thereare

multipleinterconnectedpropositionsonwhich

simultaneousdecisionsare

tobemade.

CondorcetParadox

Suppose

thatthereare

threeagents

choosingbetweenthree

alternatives.

P1

a>

b>

c

P2

b>

c>

a

P3

c>

a>

b

Pairwisemajority

votingproducesanon-transitive

group

preference.

P1

a>

b>

c

P2

b>

c>

a

P3

c>

a>

b

•a

>b?

•b

>c?

•a

>c?

P1

a>

b>

c

P2

b>

c>

a

P3

c>

a>

b

•a

>b?

•b

>c?

•a

>c?

P1

a>

b>

c

P2

b>

c>

a

P3

c>

a>

b

•a

>b?

Yes

•b

>c?

•a

>c?

P1

a>

b>

c

P2

b>

c>

a

P3

c>

a>

b

•a

>b?

Yes

•b

>c?

Yes

•a

>c?

P1

a>

b>

c

P2

b>

c>

a

P3

c>

a>

b

•a

>b?

Yes

•b

>c?

Yes

•a

>c?

No

Arrow'sTheorem:SomeNotation

LetR

bethesetofallre�exive,transitive

andconnectedrelations

onasetofcandidates

X.

Asocialwelfare

function

Fisafunction

F:R

n→R

Suppose

that

R=

F(R

1,.

..,R

n)

Arrow'sTheorem:Conditions

•UniversalDomain:

Fisatotalfunction

•WeakPareto

Principle:Foranytwocandidates

x,y

ifxR

iy

foreach

agent

ithen

xF

(~ R)y

•IndependenceofIrrelevantAlternatives:

Suppose

that

~ R

and

~ R∗are

twopreference

pro�lesand

xand

yare

two

candidatessuch

thatforallindividuals

i,if

xR

iyi�

xR∗ iythen

xF

(~ R)y

i�xF

(~ R∗ )

y.

•Non-Dictatorship:Theredoes

notexistanindividualisuch

thatforallpro�les

~ R∈R

n,if

xR

iythen

xF

(~ R)y.

Arrow'sTheorem

Theorem

(Arrow1951/1963)Thereexists

nosocialwelfare

functionwhichsatis�es

UniversalDomain,WeakPareto

Principle,

Independence

ofIrrelevantAlternativesandNon-Dictatorship.

Arrow'sTheorem

Theorem

(Arrow1951/1963)Thereexists

nosocialwelfare

functionwhichsatis�es

UniversalDomain,WeakPareto

Principle,

Independence

ofIrrelevantAlternativesandNon-Dictatorship.

Form

alizingArrow

'sTheorem

T.Agotnes,W.vander

Hoek

andM.Wooldridge.

TowardsaLogic

ofSocial

Welfare.ProceedingsofLOFT,COMSOC(2006).

TheDoctrinalParadox

P:�UvAteachersget

a10%

raise"

Q:�Thequality

ofeducationforallstudents

willincrease"

P→

Q:�IfUvAteachersget

a10%

raise,then

thequality

of

educationforallstudents

willincrease"

PP→

QQ

Individual1

True

True

True

Individual2

True

False

False

Individual3

False

True

False

Majority

True

True

False

ASecondParadox(K

ornhauserandSager1993)

P:avalidcontract

wasin

place

Q:thedefendant'sbehaviourwassuch

asto

breach

acontract

of

thatkind

R:thecourt

isrequired

to�ndthedefendantliable.

PQ

(P∧

Q)↔

RR

1yes

yes

yes

yes

2yes

no

yes

no

3no

yes

yes

no

Should

weaccept

R?

PQ

(P∧

Q)↔

RR

1yes

yes

yes

yes

2yes

no

yes

no

3no

yes

yes

no

Should

weaccept

R?No,asimplemajority

votesno.

PQ

(P∧

Q)↔

RR

1yes

yes

yes

yes

2yes

no

yes

no

3no

yes

yes

no

Should

weaccept

R?Yes,amajority

votesyesfor

Pand

Qand

(P∧

Q)↔

Risalegaldoctrine.

PQ

(P∧

Q)↔

RR

1yes

yes

yes

yes

2yes

no

yes

no

3no

yes

yes

no

ListandPettitIm

possibilityResult

Suppose

thereare

nagetnsandletLbeapropositionallanguage.

Personaljudgementsets:aconsisten

t,complete

anddeductively

closedsetofform

ulas�

amaxim

allyconsistentset.

Acollectivejudgementaggregationfunction:Let

M={Γ

|Γisamaxim

allyconsistentset}

then

acollective

aggregationfunctionisde�ned

asfollow

s:

F:M

n→

M

SomeConditions

UniversalDomain

Fisatotalfunction

AnonymityForall

~ Γ∈

Mn,F

(Γ1,.

..,Γ

n)

=F

(Γπ(1

),.

..,Γ

π(n

))

forallpermutations

π

SystematicityThereexists

afunction

f:{

0,1}

n→{0

,1}such

thatforany

~ Γ∈

Mn,

F(Γ

1,.

..,Γ

n)

={φ

∈X|f

(δ1(φ

),..

.,δ n

(φ))

=1},where,foreach

agent

iandeach

φ∈

X,δ i

(φ)

=1if

φ∈

Γiand

δ i(φ

)=

0if

φ6∈

Γi

Theorem

(ListandPettit,2001)Thereexists

nojudgem

ent

aggregationfunctiongeneratingcomplete,consistentand

deductivelyclosedcollectivesets

ofjudgem

ents

whichsatis�es

UniversalDomain,AnonymityandSystem

aticity.

BriefSurveyoftheLiterature

•See

personal.lse.ac.uk/LIST/doctrinalparadox.htmfora

detailed

overviewofthecurrentstate

ofa�airs.

Some

highlights:

•Other

impossibilityresults:

Pauly

andvanHees(2003),van

Hees(2004),Gärdenfors

(2004),andothers

•ListandPettit(2005)compare

theirim

possibilityresultwith

Arrow

'sTheorem

•Forageneralapproach

seeDaniëlsandPacuit(2006).

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

Backward

Induction

AB

A (2,1)

(1,4)

(4,3)

(3,6)

ALogicalCharacterizationofBackwardsInduction

Models:

Extensivegames

(labeled

treeswithpreference

relations

over

theendnodes)

Goal:

�ndalanguageandaform

ula

from

thatlanguagethat

�characterizes�thebackward

inductionrelation.

ALogicalCharacterizationofBackwardsInduction

Whatdowewantto

express?

•�after

action

a,φistrue�:〈a〉φ

ALogicalCharacterizationofBackwardsInduction

Whatdowewantto

express?

•�after

action

a,φistrue�:〈a〉φ

•�agent

i'sturn

tomove�:

turn

i

ALogicalCharacterizationofBackwardsInduction

Whatdowewantto

express?

•�after

action

a,φistrue�:〈a〉φ

•�agent

i'sturn

tomove�:

turn

i

•�after

somemove

φistrue�:〈m

ove〉

φ

ALogicalCharacterizationofBackwardsInduction

Whatdowewantto

express?

•�after

action

a,φistrue�:〈a〉φ

•�agent

i'sturn

tomove�:

turn

i

•�after

somemove

φistrue�:〈m

ove〉

φ

•�φ

istrueafter

theagentchoosesin

itsbestinterest�:〈b

i〉φ

ALogicalCharacterizationofBackwardsInduction

Whatdowewantto

express?

•�after

action

a,φistrue�:〈a〉φ

•�agent

i'sturn

tomove�:

turn

i

•�after

somemove

φistrue�:〈m

ove〉

φ

•�φ

istrueafter

theagentchoosesin

itsbestinterest�:〈b

i〉φ

•�φ

istruein

apreferred

node�:

♦iφ

ALogicalCharacterizationofBackwardsInduction

Whatdowewantto

express?

•�after

action

a,φistrue�:〈a〉φ

•�agent

i'sturn

tomove�:

turn

i

•�after

somemove

φistrue�:〈m

ove〉

φ

•�φ

istrueafter

theagentchoosesin

itsbestinterest�:〈b

i〉φ

•�φ

istruein

apreferred

node�:

♦iφ

•�φ

istrueafter

agents

repeatedlychoose

intheirbestinterests�

〈bi∗〉φ

ALogicalCharacterizationofBackwardsInduction

PropositionTherelation

bicorrespondingto

auniqueoutcomeof

aBackward

Inductioncomputationistheonly

binary

relationona

gamemodelsatisfyingthefollow

ingprinciplesforallpropositions

φ:

1.〈m

ove〉>→

(〈bi〉¬

φ→¬〈

bi〉φ

)

2.Forallplayers

i,

(turn

i∧〈b

i∗〉(

end∧

φ))→

[mov

e]〈b

i∗〉(

end∧

♦iφ

)

J.vanBenthem

,S.vanOtterlooandO.Roy.Preference

logic,conditionals

andsolutionconceptsin

games.

ILLCPrepublications2005.

Conclusion

WhatcanLogicdoforGameTheory?

See

staff.science.uva.nl/∼epacuit/caputLLI.htmlformore

inform

ation.

Thankyou.

VerifyingSocialProcedures

M.Pauly

andM.Wooldridge.

Logic

forMechanism

Design�

AManifesto.

Availableatauthors

website

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q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLo

gic

(LT

L)

[Pnu

elli,

1977

]:R

easo

ning

abou

tco

m-

puta

tion

s:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nchi

ng-tim

eTem

pora

lLog

ic(C

TL

,CT

L! )

[Cla

rke

and

Em

erso

n,19

81,

Em

erso

nan

dH

alpe

rn,19

86]:

Allo

ws

quan

tific

atio

nov

erpa

ths:

!!!:

ther

eis

apa

thin

whi

ch!

isev

entu

ally

true

.

•A

ltern

atin

g-tim

eTem

pora

lLo

gic

(AT

L,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,19

97]:

Sele

ctiv

equ

anti

ficat

ion

over

path

s:

1

Rea

sonin

gab

out

coal

itio

ns

Apri

l29

,20

05

1B

ack

gro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,19

77]:

Rea

sonin

gab

out

com

-puta

tion

s:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,19

81,

Em

erso

nan

dH

alper

n,19

86]:

Allow

squ

anti

fica

tion

over

pat

hs:

!!!:

ther

eis

apat

hin

whic

h!

isev

entu

ally

true.

•A

lter

nat

ing-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nge

r,K

upfe

r-m

an,19

97]:

Sel

ecti

vequ

anti

fica

tion

over

pat

hs:

1

Rea

sonin

gab

out

coal

itio

ns

Apri

l29

,20

05

1B

ack

gro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,19

77]:

Rea

sonin

gab

out

com

-puta

tion

s:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,19

81,

Em

erso

nan

dH

alper

n,19

86]:

Allow

squ

anti

fica

tion

over

pat

hs:

!!!:

ther

eis

apat

hin

whic

h!

isev

entu

ally

true.

•A

lter

nat

ing-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nge

r,K

upfe

r-m

an,19

97]:

Sel

ecti

vequ

anti

fica

tion

over

pat

hs:

1

. . .

. . .

¬∀♦

Px=

1

AlternatingTransitionSystems

Thepreviousmodelassumes

thereis

oneagentthat�controls�the

transitionsystem

.

AlternatingTransitionSystems

Thepreviousmodelassumes

thereis

oneagentthat�controls�the

transitionsystem

.

Whatifthereismore

thanoneagent?

AlternatingTransitionSystems

Thepreviousmodelassumes

thereis

oneagentthat�controls�the

transitionsystem

.

Whatifthereismore

thanoneagent?

Example:Suppose

thatthereare

twoagents:aserver

(s)anda

client(c).

Theclientasksto

setthevalueofxandtheserver

can

either

grantordenytherequest.

Assumetheagents

make

simultaneousmoves.

AlternatingTransitionSystems

Thepreviousmodelassumes

thereis

oneagentthat�controls�the

transitionsystem

.

Whatifthereismore

thanoneagent?

Example:Suppose

thatthereare

twoagents:aserver

(s)anda

client(c).

Theclientasksto

setthevalueofxandtheserver

can

either

grantordenytherequest.

Assumetheagents

make

simultaneousmoves.

den

ygra

nt

set0

set1

AlternatingTransitionSystems

Thepreviousmodelassumes

thereis

oneagentthat�controls�the

transitionsystem

.

Whatifthereismore

thanoneagent?

Example:Suppose

thatthereare

twoagents:aserver

(s)anda

client(c).

Theclientasksto

setthevalueofxandtheserver

can

either

grantordenytherequest.

Assumetheagents

make

simultaneousmoves.

den

ygra

nt

set0

q 0⇒

q 0,q 1⇒

q 0

set1

q 0⇒

q 1,q 1⇒

q 1

AlternatingTransitionSystems

Thepreviousmodelassumes

thereis

oneagentthat�controls�the

transitionsystem

.

Whatifthereismore

thanoneagent?

Example:Suppose

thatthereare

twoagents:aserver

(s)anda

client(c).

Theclientasksto

setthevalueofxandtheserver

can

either

grantordenytherequest.

Assumetheagents

make

simultaneousmoves.

den

ygra

nt

set0

q⇒

qq 0⇒

q 0,q 1⇒

q 0

set1

q⇒

qq 0⇒

q 1,q 1⇒

q 1

Multi-agentTransitionSystems

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackground

x=

0 x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLogic

(LT

L)

[Pnuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nchin

g-tim

eTem

pora

lLogic

(CT

L,C

TL!)

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

ltern

ating-tim

eTem

pora

lLogic

(AT

L,A

TL!)

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gab

out

coal

itio

ns

Apri

l29

,20

05

1B

ack

gro

und

x=

0 x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,19

77]:

Rea

sonin

gab

out

com

-puta

tion

s:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,19

81,

Em

erso

nan

dH

alper

n,19

86]:

Allow

squ

anti

fica

tion

over

pat

hs:

!!!:

ther

eis

apat

hin

whic

h!

isev

entu

ally

true.

•A

lter

nat

ing-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nge

r,K

upfe

r-m

an,19

97]:

Sel

ecti

vequ

anti

fica

tion

over

pat

hs:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

lter

nating-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

lter

nating-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ack

gro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ack

gro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Multi-agentTransitionSystems

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackground

x=

0 x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLogic

(LT

L)

[Pnuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nchin

g-tim

eTem

pora

lLogic

(CT

L,C

TL!)

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

ltern

ating-tim

eTem

pora

lLogic

(AT

L,A

TL!)

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gab

out

coal

itio

ns

Apri

l29

,20

05

1B

ack

gro

und

x=

0 x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,19

77]:

Rea

sonin

gab

out

com

-puta

tion

s:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,19

81,

Em

erso

nan

dH

alper

n,19

86]:

Allow

squ

anti

fica

tion

over

pat

hs:

!!!:

ther

eis

apat

hin

whic

h!

isev

entu

ally

true.

•A

lter

nat

ing-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nge

r,K

upfe

r-m

an,19

97]:

Sel

ecti

vequ

anti

fica

tion

over

pat

hs:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

lter

nating-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

lter

nating-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ack

gro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ack

gro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

(Px=

0→

[s]P

x=

0)∧

(Px=

1→

[s]P

x=

1)

Multi-agentTransitionSystems

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackground

x=

0 x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLogic

(LT

L)

[Pnuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nchin

g-tim

eTem

pora

lLogic

(CT

L,C

TL!)

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

ltern

ating-tim

eTem

pora

lLogic

(AT

L,A

TL!)

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gab

out

coal

itio

ns

Apri

l29

,20

05

1B

ack

gro

und

x=

0 x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,19

77]:

Rea

sonin

gab

out

com

-puta

tion

s:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,19

81,

Em

erso

nan

dH

alper

n,19

86]:

Allow

squ

anti

fica

tion

over

pat

hs:

!!!:

ther

eis

apat

hin

whic

h!

isev

entu

ally

true.

•A

lter

nat

ing-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nge

r,K

upfe

r-m

an,19

97]:

Sel

ecti

vequ

anti

fica

tion

over

pat

hs:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

lter

nating-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

lter

nating-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ack

gro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!grant,

set0

"

!den

y,s

et0"

!grant,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ack

gro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

!gra

nt,

set0

"

!den

y,s

et0"

!gra

nt,

set1

"

!den

y,s

et1"

x=

0

x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

1

Px=

0→¬[

s]P

x=

1

Multi-agentTransitionSystems

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackground

x=

0 x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLogic

(LT

L)

[Pnuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nchin

g-tim

eTem

pora

lLogic

(CT

L,C

TL!)

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

ltern

ating-tim

eTem

pora

lLogic

(AT

L,A

TL!)

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gab

out

coal

itio

ns

Apri

l29

,20

05

1B

ack

gro

und

x=

0 x=

1

q 0 q 1 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,19

77]:

Rea

sonin

gab

out

com

-puta

tion

s:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,19

81,

Em

erso

nan

dH

alper

n,19

86]:

Allow

squ

anti

fica

tion

over

pat

hs:

!!!:

ther

eis

apat

hin

whic

h!

isev

entu

ally

true.

•A

lter

nat

ing-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nge

r,K

upfe

r-m

an,19

97]:

Sel

ecti

vequ

anti

fica

tion

over

pat

hs:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

lter

nating-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

Rea

sonin

gabout

coaliti

ons

Apri

l29,2005

1B

ackgro

und

x=

0 x=

1

q 0 q 0q 0

q 0q 1

q 0q 0

q 0

q 0q 0

q 1

q 0q 1

q 0

q 0q 1

q 1

•Lin

ear

Tim

eTem

pora

lLog

ic(L

TL

)[P

nuel

li,1977]:

Rea

sonin

gabout

com

-puta

tions:

!!:

!is

true

som

eti

me

inth

efu

ture

.

•Bra

nch

ing-

tim

eTem

pora

lLog

ic(C

TL

,C

TL! )

[Cla

rke

and

Em

erso

n,1981,

Em

erso

nand

Halp

ern,1986]:

Allow

squanti

fica

tion

over

path

s:

!!!:

ther

eis

apath

inw

hic

h!

isev

entu

ally

true.

•A

lter

nating-

tim

eTem

pora

lLog

ic(A

TL

,A

TL! )

[Alu

r,H

enzi

nger

,K

upfe

r-m

an,1997]:

Sel

ecti

ve

quanti

fica

tion

over

path

s:

1

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sonin

gabout

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ons

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l29,2005

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gro

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•Lin

ear

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eTem

pora

lLog

ic(L

TL

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nuel

li,1977]:

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sonin

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com

-puta

tions:

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true

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eti

me

inth

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.

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ons

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ear

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eTem

pora

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ic(L

TL

)[P

nuel

li,1977]:

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true

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eti

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ear

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eTem

pora

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ic(L

TL

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nuel

li,1977]:

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-puta

tions:

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eti

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inth

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ture

.

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ons

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ic(L

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li,1977]:

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true

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inth

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ear

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eTem

pora

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ic(L

TL

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nuel

li,1977]:

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sonin

gabout

com

-puta

tions:

!!:

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true

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eti

me

inth

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ture

.

1

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sonin

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ons

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l29,2005

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ear

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pora

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ic(L

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nuel

li,1977]:

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-puta

tions:

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true

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eti

me

inth

efu

ture

.

1

Px=

0→

[s,c

]Px=

1

From

TemporalLogicto

StrategyLogic

From

TemporalLogicto

StrategyLogic

•LinearTim

eTem

poralLogic:Reasoningaboutcomputation

paths:

♦φ:

φistruesometimein

thefuture.

A.Pnuelli.A

TemporalLogic

ofPrograms.

inProc.18th

IEEESymposium

onFoundationsofComputerScience

(1977).

From

TemporalLogicto

StrategyLogic

•LinearTim

eTem

poralLogic:Reasoningaboutcomputation

paths:

♦φ:

φistruesometimein

thefuture.

A.Pnuelli.A

TemporalLogic

ofPrograms.

inProc.18th

IEEESymposium

onFoundationsofComputerScience

(1977).

•BranchingTim

eTem

poralLogic:Allow

squanti�cationover

paths:

∃♦φ:thereisapath

inwhich

φiseventuallytrue.

E.M.ClarkeandE.A.Emerson.DesignandSynthesisofSynchronization

SkeletonsusingBranching-tim

eTemproal-logic

Speci�cations.

InProceedings

WorkshoponLogic

ofPrograms,LNCS(1981).

From

TemporalLogicto

StrategyLogic

•Alternating-timeTem

poralLogic:Reasoningabout(localand

global)grouppow

er:

〈〈A〉〉�

φ:Thecoalition

Ahasajointstrategyto

ensure

that

φ

willremain

true.

R.Alur,T.Henzinger

andO.Kupferm

an.Alternating-tim

eTemproalLogic.

JouranloftheACM

(2002).

From

TemporalLogicto

StrategyLogic

•Alternating-timeTem

poralLogic:Reasoningabout(localand

global)grouppow

er:

〈〈A〉〉�

φ:Thecoalition

Ahasajointstrategyto

ensure

that

φ

willremain

true.

R.Alur,T.Henzinger

andO.Kupferm

an.Alternating-tim

eTemproalLogic.

JouranloftheACM

(2002).

•CoalitionalLogic:Reasoningabout(local)grouppow

er

(fragmentof

AT

L).

[C]φ

(equivalently〈〈C

〉〉©

φ):

coalition

Chasajointstrategy

tobringabout

φ.

M.Pauly.A

ModalLogic

forCoalitionPowers

inGames.

JournalofLogic

andComputation12(2002).

AnExample

Twoagents,Aand

B,must

choose

betweentwooutcomes,pand

q.

Wewantamechanism

thatwillallow

them

tochoose,whichwill

satisfythefollow

ingrequirem

ents:

1.Wede�nitelywantanoutcometo

result,i.e.,either

por

qmust

beselected

2.Wewanttheagents

tobeableto

collectivelychoose

and

outcome

3.Wedonotwantthem

tobeableto

bringaboutboth

outcomes

simultaneously

4.Wewantthem

both

tohaveequalpow

er

AnExample

Twoagents,Aand

B,must

choose

betweentwooutcomes,pand

q.

Wewantamechanism

thatwillallow

them

tochoose,whichwill

satisfythefollow

ingrequirem

ents:

1.Wede�nitelywantanoutcometo

result,i.e.,either

por

qmust

beselected:

[∅](

p∨

q)

2.Wewanttheagents

tobeableto

collectivelychoose

and

outcome

3.Wedonotwantthem

tobeableto

bringaboutboth

outcomes

simultaneously

4.Wewantthem

both

tohaveequalpow

er

AnExample

Twoagents,Aand

B,must

choose

betweentwooutcomes,pand

q.

Wewantamechanism

thatwillallow

them

tochoose,whichwill

satisfythefollow

ingrequirem

ents:

1.Wede�nitelywantanoutcometo

result,i.e.,either

por

qmust

beselected:

[∅](

p∨

q)

2.Wewanttheagents

tobeableto

collectivelychoose

and

outcome:

[A,B

]p∧

[A,B

]q

3.Wedonotwantthem

tobeableto

bringaboutboth

outcomes

simultaneously

4.Wewantthem

both

tohaveequalpow

er

AnExample

Twoagents,Aand

B,must

choose

betweentwooutcomes,pand

q.

Wewantamechanism

thatwillallow

them

tochoose,whichwill

satisfythefollow

ingrequirem

ents:

1.Wede�nitelywantanoutcometo

result,i.e.,either

por

qmust

beselected:

[∅](

p∨

q)

2.Wewanttheagents

tobeableto

collectivelychoose

and

outcome:

[A,B

]p∧

[A,B

]q

3.Wedonotwantthem

tobeableto

bringaboutboth

outcomes

simultaneously:¬[

A,B

](p∧

q)

4.Wewantthem

both

tohaveequalpow

er

AnExample

Twoagents,Aand

B,must

choose

betweentwooutcomes,pand

q.

Wewantamechanism

thatwillallow

them

tochoose,whichwill

satisfythefollow

ingrequirem

ents:

1.Wede�nitelywantanoutcometo

result,i.e.,either

por

qmust

beselected:

[∅](

p∨

q)

2.Wewanttheagents

tobeableto

collectivelychoose

and

outcome:

[A,B

]p∧

[A,B

]q

3.Wedonotwantthem

tobeableto

bringaboutboth

outcomes

simultaneously:¬[

A,B

](p∧

q)

4.Wewantthem

both

tohaveequalpow

er:¬[

x]p∧¬[

x]qwhere

x∈{A

,B}

AnExample

Consider

thefollow

ingmechanism:

Thetwoagentsvote

ontheoutcomes,i.e.,they

chooseeither

por

q.

Ifthereisaconsensus,then

theconsensusisselected;ifthereisno

consensus,then

anoutcome

por

qisselected

non-deterministically.

Pauly

andWooldridgeuse

theMOCHAmodelcheckingsystem

to

verify

thattheaboveprocedure

satis�es

thepreviousspeci�cations.

See,forexample,

M.Pauly.A

ModalLogic

forCoalitionPowers

inGames.

JournalofLogic

andComputation12(2002).

GorankoandJamroga.ComparingSemanticsofLogicsfroMulti-AgentSys-

tems.

See

thewebsite.

Conclusion

WhatcanLogicdoforGameTheory?

See

staff.science.uva.nl/∼epacuit/caputLLI.htmlformore

inform

ation.

Thankyou.