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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→
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
(2005).
Computationalvs.
BehavioralStructures
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
Rea
sonin
gabout
coaliti
ons
Apri
l29,2005
1B
ackground
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
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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 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
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
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
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
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.
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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:
1Rea
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 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
squan
tifica
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
vequan
tifica
tion
over
pat
hs:
1
Rea
soni
ngab
out
coal
itio
ns
Apr
il29
,20
05
1B
ackgr
ound
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
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
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,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.
Recommended