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2163
To cite this version:
Dubois, Didier
and Prade, Henri
and Schockaert, Steven
Generalized
possibilistic logic: Foundations and applications to qualitative reasoning about
uncertainty.
(2017) Artificial Intelligence, 252. 139-174. ISSN 0004-3702 .
Official URL:
https://doi.org/10.1016/j.artint.2017.08.001
Generalized
possibilistic
logic:
Foundations
and
applications
to
qualitative
reasoning
about
uncertainty
Didier Dubois
a,
Henri Prade
a,
Steven Schockaert
b,
∗
aToulouse University, Université Paul Sabatier, IRIT, CNRS, 118 Route de Narbonne, 31062 Toulouse Cedex 09, France bCardiff University, School of Computer Science & Informatics, 5 The Parade, Cardiff CF24 3AA, UK
a
b
s
t
r
a
c
t
Keywords: Possibilisticlogic Epistemicreasoning Non-monotonicreasoning
Thispaper introducesgeneralizedpossibilisticlogic(GPL),alogicforepistemicreasoning basedonpossibilitytheory.FormulasinGPLcorrespondtopropositionalcombinationsof assertionssuchas “itiscertaintodegreeλthatthepropositionalformula
α
istrue”.As itsnamesuggests,thelogicgeneralizespossibilisticlogic(PL),whichatthesyntacticlevel onlyallowsconjunctionsoftheaforementionedtypeofassertions.Atthesemanticlevel, PLcanonlyencodesetsofepistemicstatesencompassedbyasingleleastinformedone, whereasGPLcanencodeanysetofepistemicstates.ThisfeaturemakesGPLparticularly suitableforreasoningaboutwhatanagentknowsaboutthebeliefsofanotheragent,e.g., allowingtheformertodrawconclusionsabout whattheotheragentdoesnot know.We introduceanaxiomatizationforGPLandshowitssoundnessandcompletenessw.r.t. possi-bilisticsemantics.Subsequently,wehighlighttheusefulnessofGPLasapowerfulunifying frameworkforvariousknowledgerepresentationformalisms.Amongothers,weshowhow comparative uncertaintyand ignorancecanbemodelledinGPL.Wealsoexhibitaclose connectionbetweenGPLandvariousexistingformalisms,includingpossibilisticlogicwith partiallyorderedformulas,alogicofconditionalassertionsinthestyleofKraus,Lehmann andMagidor,answersetprogrammingand afragmentofthelogicofminimalbeliefand negationasfailure.Finally,weanalysethecomputationalcomplexityofreasoninginGPL, identifyingdecisionproblemsatthefirst,second,thirdandfourthlevelofthepolynomial hierarchy.1. Introduction
Possibilistic logic[1](PL)isa logic forreasoningwithuncertainpropositional formulas.FormulasinPLtake the form
(
α
,
λ)
whereα
isa propositionalformulaandλ
isacertaintydegree takenfromtheunitinterval,orfromanotherlinear scale. Contraryto probabilistic logics, possibilisticlogic models acceptedbeliefsin thesense thatif twopropositions are believedtoacertainlevel,soistheirconjunction. Inmanyapplications,aPLknowledgebaseencodestheepistemicstate ofanagent.Wethenassumethatall theagent knowsaretheformulascontainedintheknowledgebaseandtheirlogical consequences,withtheweightsreferringtothedegreeofepistemicentrenchment[2]orthestrengthofbelief.However,in itsstandardform,possibilisticlogichaslimitationsasatoolforepistemicreasoning,i.e.,reasoningabout uncertainty,inat leasttworespects.*
Correspondingauthor.First, giventhat aknowledge baseencodesa single epistemicstate, PLdoesnot allowusto encode incomplete infor-mation abouttheepistemic stateofan agent. Forexample,assumethat thisagent privatelyflips acoin andlooksatthe result withoutrevealingit. Theneither theagent knows thatthe resultwas tails,which could beencoded as
{(
tails,
1)
}
, where1indicatescompletecertainty,ortheagentknowsthattheresultwasheads,whichcouldbeencodedas{(¬
tails,
1)
}
. However,allanoutsideagentknowsisthatoneofthesetwosituationsholds,andinparticularthisotheragentknowsthat thefirst agentisnotignorantabouttheoutcomeofthecoinflip.ToexpressthissituationinPL,wewouldneedtowrite adisjunction(
tails,
1)
∨ (¬
tails,
1)
whichisnotallowedinthelanguage.Inthispaper,weproposeageneralized possibilis-tic logic (GPL) inwhich such disjunctions can be expressed.This brings PL syntax closerto the one ofmodal logics for epistemicreasoning,and,toemphasizethis,wewilluseaslightlydifferentnotationandwriteN1(
tails)
∨
N1(
¬
tails)
instead.Second, PLdoesnotallowustoexplicitlyencodeinformationabouttheabsenceofknowledge.Instead,inpractice,we must relyon a kindofclosed-world assumption,i.e., assume thatthe agent doesnot knowwhether
α
istrue ifneitherα
norits negationcan be derivedfromthe givenknowledgebaserepresenting whatisknown aboutthisagent’sbeliefs. Whenreasoningaboutbeliefsasrevealedbyanagent,thisassumptionishardtokeepandweneedtodistinguishbetween situations wherewe (the outsideagent) knowthat theagent isignorant aboutα
andsituations wherewe donot know whether theagent knowsα
ornot. InGPL, thiscan be achievedby puttinga negation infront ofPL formulas:¬
N1(
α
)
expresses that we know that the agent doesnot believe in thetruth of
α
,1 whereas situations wherewe have nosuch knowledgeareencodedbyGPLtheorieswhichhavemodelsinwhichN1(
α
)
istrueandmodelsinwhichN1(
α
)
isfalse.GPL is closelyrelatedto modallogics forepistemic reasoning such asKD45 andS5.However, it is essentiallya two-tiered propositional logic,and, instead ofusing Kripke frames, thesemantics we propose forGPL isbased on possibility distributions,whichexplicitlyrepresentepistemicstates.OurabilitytodirectlyinterpretthemodalityN as aconstrainton anecessitymeasureresultsfromthefactthatwedonotallowthemodalityN tobe nested.Furthermore,bynotallowing objectiveformulas,wecannaturallyinterpreteachGPLformulaasaconstraintonthepossibleepistemicstates(i.e., possi-bilitydistributions)ofan agent.Comparedtoexistingepistemicmodallogics[3],we thustradesomeexpressivenessfora moreintuitive wayofcapturingrevealedbeliefs.Amongothers,theuseofpossibilitydistributionshastheadvantagethat (strengthof)beliefcanbenaturally encodedasagradednotionandthatexistingconcepts frompossibilitytheorysuchas minimalspecificityandguaranteedpossibilitycanbeexploitedtomodelignoranceinanaturalway.Thiswillenableusto encode variousformsofnon-monotonic reasoninginGPL.Forinstance,we willshowhowGPL canbeusedto modelthe semanticsofanswersetprogramming[4](ASP)withoutrelyingonafixpointconstruction,unlikemostexisting characteri-zationsofASP,andhowdefaultrulesinthesenseofSystemP[5]canbemodelledbytakingadvantageofthefactthatGPL canexpresscomparativeuncertainty.
The paperis structured asfollows. First, we recall some basic notions from possibility theory and possibilistic logic. In Section 3we definethe language ofGPLanda corresponding semanticsinterms ofpossibilitydistributions. Wethen provide anaxiomatizationwhichissoundandcompletew.r.t.thislattersemantics.InSection 4weanalyzehowGPL can be used to reason about the ignorance of another agent, focusing on the role of minimal specificity and an extension to the language of GPL related to the notion of “only knowing” [6]. In Section 5 we then focus on the ability of GPL to model comparative uncertainty (e.g.,
α
is more certain thanβ
), showing how GPL can be used to encode a variant of possibilisticlogic withpartially ordered formulas [7], andhow, asa result, a conditional logic basedon SystemP [5]can be embedded inGPL. Subsequently,inSection 6 we explainin moredetail howGPL relatesto anumberof existing formalismsfornon-monotonic reasoningthatarebasedonthenotionofnegationasfailure.Section7discussesanumber ofcomputationalissues,includingthecomplexityofthemainreasoningtasks.WealsoproposeareductiontoSAT,allowing forastraightforwardimplementationofthereasoningtasksatthefirstlevelofthepolynomialhierarchy.Finally,wepresent ourconclusions.
This paper aggregates and significantly extends parts of [8] and [9]. In particular, in [8] we introduced the syntax, semanticsandaxiomatizationofGPL,whereasin[9]westudiedmethodsformodellingignoranceinGPL,introducedanew proof ofthecompletenessoftheaxiomatization,anddiscussedsomeofthecomplexity resultsfromSection7.Theresults inSections5and6areentirelynew(althoughtheencodingsinSection6aresimilarinspirittotheencodingofequilibrium logicin[8]).
2. Preliminariesfrompossibilitytheory
Consideravariable X whichhasanunknownvaluefromsomefiniteuniverse
U
.Inpossibilitytheory[10–12],available knowledge about the value of X is encoded as a mappingπ
:
U → [
0,
1]
, which is called a possibilitydistribution. The intended interpretationofπ
(
u)
=
1 isthat X=
u isfullycompatiblewithallavailable information,whileπ
(
u)
=
0 means that X=
u can be excluded based on available information. Note that the special case where we have no information about X isencodedusingthevacuouspossibilitydistribution,definedasπ
(
u)
=
1 for allu∈
U
.Usually,we requirethatπ
(
u)
=
1 forsomeu∈
U
,whichcorrespondstotheassumptionthattheavailableinformationisconsistent.Ifthepossibility distributionπ
satisfiesthiscondition,itiscallednormalized.Ingeneral,thevalueof
π
(
u)
canbeinterpretedintermsofdegreesofpotentialsurprise:thesmallerthevalueofπ
(
u)
, themorewewouldbesurprisedtofindoutthat X=
u.ThisinterpretationgoesbacktoShackle[13]andsupportsapurely qualitativeinterpretationofthepossibilitydegreesπ
(
u)
.Insuchacase,wecouldreplacetheunitinterval[
0,
1]
byanother linear scale (althoughan involutive order-reversing mapping isalso needed). Other interpretations of possibility degrees relateapossibilitydistributiontoafamilyofprobabilitydistributions[14],toafamilyoflikelihoodfunctions[15],toShafer belieffunctions [16], orto Spohnordinal conditional functions[2,17] andthus to infinitesimal probabilities [18], among others.2.1. Setfunctionsinpossibilitytheory
Apossibilitydistribution
π
inducesapossibilitymeasure,definedfor A
⊆
U
as[10]:(
A)
=
maxu∈A
π
(
u).
AdualmeasureN,calledthenecessitymeasure,isdefinedforA
⊆
U
as[11]:N
(
A)
=
1− (
U
\
A)
=
minu∈/A
(
1−
π
(
u)).
Intuitively,
(
A)
reflectsto whatextentit ispossible,giventheavailableknowledge, thatthevalue of X isamongthose in A, while N(
A)
reflectsto whatextent theavailable knowledge entailsthatthe value of X must necessarilybeamong thoseinA.Twoothermeasuresthatcanbeintroducedaretheguaranteedpossibilitymeasureandthepotentialnecessity measure
∇
,definedfor A⊆
U
as[12]:(
A)
=
minu∈A
π
(
u)
;
∇(
A)
=
1− (
U
\
A)
=
maxu∈A
(
1−
π
(
u)).
Intuitively,
(
A)
reflectstheextenttowhichallvaluesinA areconsideredpossible,while∇(
A)
reflectstheextenttowhich somevalueoutsideA isimpossible.Notethatforall A= ∅
(
A)
≤ (
A)
;
N(
A)
≤ ∇(
A).
If
π
isnormalized,wehave(
A)
=
1 or N(
A)
=
0,andthusinparticular:N
(
A)
≤ (
A).
If
π
(
u)
=
0 forsome u∈
U
,wehave(
A)
=
0 or∇(
A)
=
1,andthus:(
A)
≤ ∇(
A).
Finally,notethat
andN aremonotonew.r.t.setinclusionwhile
and
∇
areantitone,i.e.,for A⊆
B wehave(
A)
≤ (
B)
;
N(
A)
≤
N(
B)
;
(
A)
≥ (
B)
;
∇(
A)
≥ ∇(
B).
2.2. Possibilisticlogic
Aformulainpropositionalpossibilisticlogic[1](PL forshort) isanexpression ofthe form
(
α
,
λ)
,whereλ
∈]
0,
1]
isa certainty degree andα
is apropositional formula, builtfroma setofatomicformulas At usingthe connectives conjunc-tion∧
,negation¬
,disjunction∨
,implication→
,andequivalence≡
intheusualway.Letbethesetofallinterpretations ofAt andlet
L
bethesetofallpropositionalformulasbuiltfromAt.Thesemanticsofpossibilisticlogicisdefinedinterms ofpossibilitydistributionsover.Specifically,apossibilitydistribution
π
oversatisfiestheformula
(
α
,
λ)
iffN(
J
α
K)
≥ λ
, whereJ
α
K
denotes thesetofall (classical)modelsofα
.Asπ
representsan epistemicstate (itis afuzzysetofclassical models), wecallit anepistemicmodel of(
α
,
λ)
,orane-model forshort.Fortheeaseofpresentation, we willwrite N(
α
)
insteadofN
(
J
α
K)
throughoutthispaper.Apossibilitydistribution
π
isane-modelofasetofPLformulasK iffπ
isane-modelofeveryformulainK .K generallyhas multiplee-models, butthey can be partially ordered by the specificityordering, whereby
π
1 isless specific thanπ
2,written
π
1π
2,ifπ
1(
ω
)
≥
π
2(
ω
)
foreveryω
∈
.Itcanbeshownthatthesetofe-modelsofasetofPLformulas K hasauniqueleastelement
π
K w.r.t.,whichiscalledtheleastspecifice-modelofK .Itcanbeexpressed,forallω
∈
as[1]:π
K(
ω
)
=
1−
max{λ | (
α
, λ)
∈
K,
ω
|=
α
}
where we assume max
∅
=
0. Intuitively, the more certain the formulas that are violated byω
, the less plausibleω
is consideredtobe.ThefollowinginferencerulesarevalidinPL:
if
(
α
, λ)
∈
K then KP L(
α
, λ)
(1)if
α
then KP L(
α
,
1)
(2)if
λ
1≥ λ
2and KP L(
α
, λ
1)
then KP L(
α
, λ
2)
(3) if KP L(
α
∨ β, λ
1)
and KP L(
¬
α
∨
γ
, λ
2)
then KP L(β
∨
γ
,
min(λ
1, λ
2))
(4)Letuswrite K
|=
P L(
α
,
λ)
ifeverye-modelofK isane-modelof(
α
,
λ)
.Ifthereisnocauseforconfusionwealsowrite|=
P Las
|=
andP L as.ItispossibletoshowthatthefollowingstatementsareallequivalentforasetofPLformulas K (seee.g.,[19]):
1. K
P L(
α
,
λ)
canbederivedfrom(1)–(4).2. K
|=
P L(
α
,
λ)
.3. Theleastspecifice-model
π
K of K isane-modelof(
α
,
λ)
.Inferenceinpossibilisticlogicthusremainsclosetoinferenceinpropositionallogic.Inparticular,letthec-cutKcofK bethe
propositionaltheory Kc
= {
α
|
(
α
,
λ)
∈
K andλ
≥
c}
.Thenwehavethat K|=
P L(
α
,
λ)
iffKλ∪ {¬
α
}
isunsatisfiable.Itfollows that entailmentcheckinginpossibilisticlogiciscoNP-completeandthatefficientreasonerscan easilybeimplementedon topofoff-the-shelfSATsolvers.Possibilisticlogiccanbeseenasatoolforspecifyingarankingonpropositionalformulas.Assuch,itiscloselyrelatedto thenotionofepistemicentrenchment[20],ashasbeenpointedoutin[2].ThismakesPLanaturalvehicleforimplementing strategiesforbeliefrevision[21]andmanaginginconsistency[22].Alongsimilarlines,therearecloseconnectionsbetween PL anddefaultreasoning inthe sense of SystemP[5],which can be exploitedto implementseveral formsof reasoning aboutruleswithexceptions[23].
Syntactically, propositional possibilistic logic is similar to the propositional fragment of Markov logic [24]. Semanti-cally,however,thecertaintyweightsinMarkovlogicareinterpretedprobabilistically.Inparticular,aset M
= {(
α
1,
w1),
...,
(
α
n,
wn)
}
of(propositional)Markovlogicformulasdefinestheprobabilitydistribution pM definedasfollows(ω
∈
): pM(
ω
)
=
1 Zexp n i=1{
wi|
ω
|=
α
i}
(5)where Z isanormalizationconstant.ThisprobabilisticsemanticsmakesMarkovlogicparticularlyusefulinmachinelearning settings.Notethatwecanequivalentlydefine pM asfollows
pM
(
ω
)
=
1 Zexp n i=1{−
wi|
ω
|=
α
i}
(6)wherethenewnormalizationconstant Zisgivenby Z
=
exp(Ziwi).Thisalternativeformulationhighlightstheclose
rela-tionshipbetweenthepropositionalfragmentofMarkovlogicandtheso-calledpenaltylogic[25].Thetwomaindifferences are that negative weights are not considered inpenalty logic2 andthat the penalty associated with an interpretation is
not normalized.Thislackofnormalizationmakespenaltylogic somewhatcloserinspirit topossibilisticlogic.Attachinga positiveweight w toaformula
α
inpenaltylogicissimilartoattachingadegreeofnecessity1−
exp(
−
w)
tothisformula in possibilisticlogic.Thus themaindifference betweenpenaltylogic andpossibilisticlogicisthat intheformer casethe productisusedtocombinecertaintydegreeswhileinthelattercasetheminimumisused.3However, we can also view Markov logic, penalty logic and possibilistic logic as equivalent frameworks for defining rankingsofpossibleworlds.Indeed,aswasshownin[27],givenaMarkovlogicknowledgebaseM,wecanalwaysconstruct a possibilisticlogic knowledgebase K suchthat M andK define thesamerankingof possibleworlds,andvice versa.In fact,anyrankingofinterpretationscanberepresentedbyapossibilisticknowledgebase.
3. Generalizedpossibilisticlogic
While PLis useful to encode a single epistemic state, our aim is to develop GPL as a logic for reasoning about the epistemicstateofanagentfromitsrevealedbeliefs.AGPLknowledgebasethenencodesthesetofepistemicstatesthatare compatiblewiththeserevealedbeliefs.TheaimofthissectionistodefinethesyntaxandsemanticsofGPL,andtointroduce
2 NotehoweverthatinMarkovlogic,wecanreplace
(α,w)by(¬α,−w)thankstotheuseofthenormalizationconstant,soallowingnegativeweights doesnotincreasetheexpressivityofpropositionalMarkovlogic.
3 Moreover,itisworthnoticingthat(6)definestheprobabilityofaninterpretationbyusingapossibilitydistributionwhichisrenormalizedbydividing eachpossibilitydegreebytheirsum.See[26]foradiscussionofthistypeofpossibility–probabilitytransformation.
anaxiomatizationforthislogic.Wewilluse
α
,β
,etc.todenotepropositionsinstandardpropositionallogic,formedwith the connectives,∧
and¬
. As usual, we will also usethe abbreviationsα
∨ β = ¬(¬
α
∧ ¬β)
,α
→ β = ¬(
α
∧ ¬β)
andα
≡ β = (
α
→ β)
∧ (β →
α
)
.LetL
bethelanguageofallpropositionalformulasoverafinitesetofatomicpropositions At. Unlessstatedotherwise,werestrictthesetofcertaintydegreestothefinitesubsetk
= {
0,
1k,
2k,
...,
1}
oftheunitinterval,withk
∈ N
\ {
0}
andlet+k
=
k\ {
0}
. 3.1. SyntaxWedefinethelanguage
L
kG P L ofgeneralizedpossibilisticlogicwithk+
1 certaintylevelsasfollows:•
Ifα
∈
L
andλ
∈
+k,thenNλ(
α
)
∈
L
kG P L.•
If∈
L
kG P L and∈
L
kG P L,then¬
and∧
arealsoinL
kG P L.The corresponding logicwill be referred toasGPLk.When k is clearfromthecontext we willalso referto thislogic as
GPL,andtothecorrespondinglanguageas
L
G P L.NotethatGPLisagradedversionofthelogiccalledMEL(Meta-Epistemic,oryet Minimal Epistemic, Logic),whichwas introduced in[28].The MELlanguage is aspecial caseofGPL wherek
=
1. WhereasMELusesastandardmodallogicsyntax(2
=
N1),weuseamodalitywhichrefers tothenecessitymeasure N toemphasizethelinkwithpossibilitytheory.Furthermorenote thatweview
L
kG P L asalanguagewithk differentmodalitiesN1
k
,
...,
N1,ratherthanalanguagewithasinglemodalityandconstantsdenotingcertaintydegrees.Inthefollowing,wewillalsousethefollowingabbreviation:
λ
(
α
)
= ¬
Nν(λ)(
¬
α
)
(7)wherewewrite
ν
(λ)
asanabbreviationfor1− λ
+
k1.Semanticallythemodalityλwillcorrespondtoalowerboundon apossibilitymeasure,namely(7)isthecounterpartofthedualitybetweenapossibilityandanecessitymeasureonafinite scale,wherewehavetoshiftfromonelevelformovingfromastrictinequalitytoaninequalityinthebroadsense.
Letusdefineameta-atom asanexpressionoftheformNλ
(
α
)
,andameta-literal asanexpressionoftheformNλ(
α
)
or¬
Nλ(
α
)
.Ameta-clause isanexpressionoftheform1
∨ ...
∨
nwitheachi ameta-literal.Ameta-term isanexpression
oftheform
1
∧ ...
∧
nwitheachiameta-literal. 3.2. Semantics
The semantics of GPL are defined in terms of normalized possibility distributions over propositional interpretations, encoding epistemicstates,wherepossibilitydegreesare, byduality,ofthe form1
− λ
,∀λ
∈
k.4 LetP
k bethe setofallsuchpossibilitydistributions.Ane-modelofaGPLformulaisanypossibilitydistribution
π
fromP
k,namely:•
π
isane-modelofNλ(
α
)
iffN(
α
)
≥ λ
;•
π
isane-modelof1
∧
2iffπ
isane-modelof1 andof
2;
•
π
isane-modelof¬
1 iffπ
isnotane-modelof1;
where N is the necessity measure induced by
π
.As usual,π
iscalled an e-model ofa set of GPL formulas K ,writtenπ
|=
kG P LK ,ifit isan e-model ofeach formulain K .It iscalleda minimally specific e-modelof K if there isnoe-model
π
=
π
of K suchthatπ
(
ω
)
≥
π
(
ω
)
foreach possibleworldω
.Wewrite K|=
kG P Lφ
,for K asetofGPLformulasandφ
a GPL formula,ifevery e-model of K isalso an e-modelofφ
.Whenk is clearfromthecontext,we will sometimeswrite|=
kG P L as
|=
G P L;furthermore,ifthereisnocauseforconfusion,wewillalsowrite|=
kG P L as|=
.Intuitively, N1
(
α
)
means that it is completely certain thatα
is true,whereas Nλ(
α
)
withλ
<
1 means that there is evidence which suggeststhatα
is true, and none that suggests that it is false. Note that we can distinguish between completeandpartialcertaintyonlyifk≥
2.Formally,anagent assertingNλ(
α
)
hasanepistemicstateπ
suchthatN(
α
)
≥
λ
>
0. Hence¬
Nλ(
α
)
stands for N(
α
)
< λ
, which means(
¬
α
)
≥
1− λ
+
1k. The abbreviation introduced in (7) thuscorresponds to a syntactic counterpart of the dualitybetween necessity and possibility measures. Note how the use of a finite scalemakes itpossible to expressstrict inequalities, even though we only useinequalities inthe wide sense in theinterpretationofgradedmodalities.Intuitively
1
(
α
)
meansthatα
isfullycompatiblewithouravailablebeliefs (i.e.,nothingprevents
α
frombeingtrue),whileλ
(
α
)
withλ
<
1 meansthatα
cannotbefullyexcluded((
α
)
≥ λ
).This formalism is similar to an autoepistemic logic [29,6]. However the latteraims to capture how an agent reasons aboutits ownbeliefs.Onecrucialdifference,whichhasbeenpointedoutin[30],isthatwhenreasoningaboutone’sown beliefs,itshould notbe possibleto stateN1
(
α
)
∨
N1(β)
withouteitherstatingN1(
α
)
or N1(β)
.Indeed,ifweaccept thatan agent isaware ofits epistemic state,the agent cantell,foreach propositional formula, whetherornot itis believed. Accordingly,instandardpossibilisticlogic,wecannotencode N1
(
α
)
∨
N1(β)
.Wecanjustencode N1(
α
)
orN1(β)
,ortheir4 Inourconventions,itcomesdowntousing
conjunction.However,wewillbeabletoovercomethislimitationinGPL.Moregenerally,inagradedsetting,iftheagentis awareofitsepistemicstate,itcantellwhichoftwopropositionalformulasitconsiderstobemostcertain.Thisisagainin accordancewithpossibilisticlogic,whereasinGPLwewillbeabletoencodethecasewhereweareignorantaboutwhich of two formulasis mostcertain foranexternalagent. Thissuggeststhat whilestandard possibilistic logicoffers a natural settingforreasoningwithone’sown beliefs,GPLnaturally lendsitselfto reasoningaboutanotheragent’sbeliefs.Forthis reason,wecouldsaythatGPLisan“alter-epistemic”logic.
As tothepossiblekindsofconclusions thatcanbe inferredfromaGPLbase K regardingapropositional formula
α
,ifk
=
2,onecandistinguishbetweenthefollowingfivecases:•
K|=
N1(
α
)
meansthatweknowthattheagentknowsthatα
istrue.•
K|=
N1(
¬
α
)
meansthatweknowthattheagentknowsthatα
isfalse.•
K|=
N1(
α
)
∨
N1(
¬
α
)
, K|=
N1(
α
)
and K|=
N1(
¬
α
)
meansthat we knowthat theagent knowswhetherα
istrueorfalse,butwedonotknowwhichitis.
•
K|=
1(
α
)
∧
1(
¬
α
)
meansthatweknowthattheagentisignorantaboutwhetherα
istrueorfalse.•
K|=
N1(
α
)
∨
N1(
¬
α
)
and K|=
1(
α
)
∧
1(
¬
α
)
means that we are ignorant about whetherthe agent is ignorantabout
α
.Thisisincontrastwiththeonlythreesituationsthatcanbedistinguished inclassicallogic(andinPL),i.e.,we knowthat
α
istrue,we knowthatα
isfalse,orwe donot knowwhetherα
istrueor false.Whenk>
2,we canconsider graded counterparts ofthe five aforementioned cases.Moreover, a GPL base can then also expresscomparative uncertainty. For example:•
K|=
ki=1Nik
(
α
)
∧ ¬
N ik
(β)
:weknowthattheagentismorecertainthatα
holdsthanthatβ
holds,noticingthatitisequivalentto
∃
i,
N(
α
)
≥
ki>
N(β)
.•
K|=
ki=1i
k
(
α
)
∧ ¬
ik
(β)
:weknowthattheagentwouldbelesssurprisedtolearnthatα
istruethantolearnthatβ
istrue,noticingthatitisequivalentto∃
i.
(
α
)
≥
ki> (β)
.•
K|=
ki=1(
Ni k(
α
)
∨
N i k(
¬
α
))
∧ ¬
N i k(β)
∧ ¬
N ik
(
¬β)
: we knowthat the agent is morecertain about thetruth or thefalsityof
α
thanaboutβ
,butwemaynotknowwithwhichcertaintydegreetheagentknowsthetruthvalueofα
,nor towhatextentthiscertaintydegreeisgreaterthanthecertaintydegreeaboutthetruthorthefalsityofβ
.•
K|=
ki=1(
Ni k(
α
)
∧ ¬
N i k(β))
∨ (
N i k(β)
∧ ¬
N ik
(
α
))
:weknowthattheagentconsidersoneofα
,β
morecertainthantheother,butwemaynotknowwhich.
•
K|=
ki=1(
Ni k(
α
)
→
Ni k
(β))
expressesthattheagentisatleastascertainabout
β
asaboutα
.Example1. Thesixnationschampionshipisa rugbycompetitionconsistingof5rounds. Ineach round,everyteamplays against one of the other 5 teams, so that over 5 rounds all teams have played once against each other. Let us write
playsi
(
x,
y)
to denote that x and y have played against each other in round i, andwoni(
x)
to denote that team x haswonits gameinroundi.LetT
= {
eng,
fra,
ire,
ita,
sco,
wal}
.Toexpressthatan agentknowstherulesofthechampionship, wecanconsiderformulassuchas,amongothers:N1
(
{
playsi(
x,
u)
|
u=
x,
u∈
T})
(8)wherex
∈
T .AformulasuchasN34
(
won1(
wal))
meansthattheagentstronglybelieves,butisnotfullycertain,that Wales(wal)haswonitsfirst roundgame,while
3
4
(
won1(
wal))
meansthat theagent doesnotexclude thatWaleshaswonitsfirst round game,withoutevidence asto thecontrary. Thefollowing formulaexpresses that theagent considers it more plausiblethatWaleshaswonitsfirstgamethanthatEngland(eng)haswonitsfirstgame
k
i=1i k
(
won1(
wal))
∧ ¬
i k(
won1(
eng))
(9)RecallthatthecertaintydegreesinGPLaretypicallyonlyassumedtohaveanordinalmeaning.Sayingthatthenecessity ofa formulais 34 then doesnothaveanyintrinsicmeaning,otherthanthefact thatthisformulaisconsiderede.g., more certainthanaformulawithnecessity 12 andlesscertainthanaformulawithnecessity 78.Theaboveexampleillustratestwo alternative waysinwhichapplicationscandealwithsuchordinalcertaintydegrees. Oneideaistouseasmallnumberof categoriesthataremeaningfultoauser,such ase.g.,‘completelycertain’,‘verycertain’,‘quitecertain’,‘somewhatcertain’, andmapthesecategoriestotheavailableelementsfrom
k (e.g.,‘verycertain’couldcorrespondtoanecessityof 34).The
secondideawouldbetoavoidassigningcertaintydegrees,andonlyexpresscertaintyinacomparativeway,asisillustrated in(9).ThissecondapproachwillbediscussedinmoredetailinSection5.
3.3. Axiomatization
Weconsiderthefollowingaxiomatization,whichcloselyparallelstheoneofMEL[28]:
(PL) Theaxiomsofclassicallogicformeta-formulas.
(K) Nλ
(
α
→ β)
→ (
Nλ(
α
)
→
Nλ(β))
.(N) N1
(
α
)
wheneverα
∈
L
isaclassicaltautology.(D) Nλ
(
α
)
→
1(
α
)
.(W) Nλ1
(
α
)
→
Nλ2(
α
)
,ifλ
1≥ λ
2.If
canbe derived froma setofGPLformulas K using theaxioms (PL), (K), (N), (D), (W) and modusponens,we write
K
G P L;ifthereisnocauseforconfusionwealsowrite K
.Noteinparticularthatwhen
λ
isfixedwegetafragmentofthemodallogicKD.Inparticular,theaxiomsentailthatNλ
(
α
∧β)
isequivalenttoNλ(
α
)
∧
Nλ(β)
.Itiseasytoseethatifα
andβ
arelogicallyequivalentformulas,thenNλ(
α
)
andNλ(β)
arealsoequivalent.Indeed,inthatcase,(
α
→ β)
∧ (β →
α
)
holds,andbyapplying (N),(W),(K),(D) wegetbothNλ(
α
)
→
Nλ(β)
andNλ(β)
→
Nλ(
α
)
.Alsonotethatfrom (N) and (W) wecanderiveagradedversionofthenecessitationrule,i.e.,ifα
thenG P LNλ(
α
)
foranyλ
∈
k.Finallynotethatinthecasewherek
=
1,GPLcoincideswiththelogicMEL.Inthislattercase,wehave1
(
α
)
= ¬
N1(
¬
α
)
whereasingeneralweonlyhave
1
(
α
)
= ¬
N1k
(
¬
α
)
.AswewillseeinSection6,theabilitytodifferentiatebetweenfullpossibilityfor
α
andthe lackoffullcertaintyfor¬
α
iscrucialwhenusingGPLtoprovideasemanticsfornegationasfailure.Proposition1(Soundnessandcompleteness).LetK beasetofGPLformulasand
aGPLformula.ItholdsthatK
|=
G P Liff KG P L
.
Proof. TheproofispresentedinAppendix A.
2
Themainideabehind theproof isthatwecanseeformulasinGPLaspropositionalformulaswhicharebuiltfromthe setofatomicformulasof
L
kG P L.GivenaknowledgebaseK inGPL,weconstructapropositionalbaseK∗ madeofformulas
of K plus axiomsof GPL,viewed aspropositional formulasaswell. We then show that there exists abijection between theset ofpropositionalmodels of K∗ (seenasapropositional logicknowledge base)andtheset ofe-modelsof K (seen
asaGPLknowledgebase).Averysimilarstrategyhasbeenused,amongothers,in[31],[32]and[33,34],inthecontextof multi-valuedmodallogicsforreasoningaboutnecessity(seeSection3.4).
Proposition 1remainsvalidevenifthesetAt ofatomicpropositionsiscountablyinfinite.Ontheotherhand,the com-pletenessresultnolongerholdsifinfinitelymanycertaintydegreesareallowedinthelanguage,ase.g.
{
Nλ(
a)
|
λ
<
12}
|=
GPL N12
(
a)
,fora∈
At but{
Nλ(
a)
|
λ
<
12
}
GPLN12(
a)
.Thisisnotarealrestriction,sinceknowledgebasesonlyhavefinitelymanyformulasinpractice,whichmeansthatonlyfinitelymanycertaintylevelsactuallyneedtobeused,andsincethesemantics ofGPL isbasedonthe relativeordering ofthecertainty degrees,we canthen always mapthesecertainty degreesto
k
forsomek. InSection 5,however,wewill discussan extension ofGPLinwhichwe canexpresscomparativeuncertainty statements,whereitwillbedesirabletoallowanunboundednumberofcertaintydegreesatthesemanticlevel.
UsingProposition 1,andsomewell-knownpropertiesonnecessityandpossibilitymeasures,itfollowsthatthefollowing formulasaretheoremsinGPL:
Nλ
(
α
)
∧
Nλ(β)
≡
Nλ(
α
∧ β)
λ
(
α
∧ β) →
λ(
α
)
∧
λ(β)
Nλ(
α
)
∨
Nλ(β)
→
Nλ(
α
∨ β)
λ
(
α
)
∨
λ(β)
≡
λ(
α
∨ β)
NextisacounterparttothemodusponensruleinPL(4):
Nλ1
(
α
)
∧
Nλ2(
α
→ β) →
Nmin(λ1,λ2)(β)
(10)ToshowthatthisisatheoreminGPL,thankstoProposition 1,itsufficestonotethateverynecessitymeasure N satisfying N
(
α
)
≥ λ
1 and N(
¬
α
∨ β)
≥ λ
2 also satisfies N(β)
≥
min(λ
1,
λ
2)
,which is equivalent to theusual modus ponensin PL,a specialcaseof(4).Toseehow(10)canbederivedfromtheaxiomsofGPL,notethat thedeductiontheoremisvalidin GPL, anditthus suffices toshow that Nmin(λ1,λ2)
(β)
can be derived from{
Nλ1(
α
),
Nλ2(
α
→ β)}
.Starting fromthis lattersetofpremises,weapply (W) toobtainNmin(λ1,λ2)
(
α
)
andNmin(λ1,λ2)(
α
→ β)
.Applyingmodus ponensonaxiom(K) and Nmin(λ1,λ2)(
α
→ β)
,weobtainNmin(λ1,λ2)(
α
)
→
Nmin(λ1,λ2)(β)
.UsingmodusponensonthelatterformulaandNmin(λ1,λ2)(
α
)
weobtainNmin(λ1,λ2)
(β)
.Thefollowingtheoremisthecounterpartofahybridmodusponensruleintroducedin[35]:
Againadirectproofcanbegiven,usingthedeductiontheorem,byprovingNν(λ1)
(
¬
α
)
fromNλ2(
α
→ β)
andNν(λ1)(
¬β)
inthesameway(justrewriting
α
→ β
as¬β → ¬
α
).However,weneedtoassumeν
(λ
1)
≤ λ
2inordertoweakenNλ2(
α
→ β)
intoNν(λ1)
(
α
→ β)
.Andν
(λ
1)
≤ λ
2isequivalentto1− λ
1+
1k
≤ λ
2,i.e.,λ
2>
1− λ
1.5
Resolution rules inpossibilistic logic[35],extending(10) and(11), canbe proved likewisein GPLor, alternatively,by usingthedecomposabilityofNλ
(
·)
w.r.t.conjunction.3.4. Relatedwork
Althoughpossibilitytheoryhasbeenthebasisofanoriginaltheoryofapproximatereasoning[36],itwasnotintroduced as a logical settingfor epistemic reasoning,strictly speaking. Nonetheless, in the setting of his representation language PRUF [37], Zadeh discusses the representationof statements of the form“ X is A” (meaningthat the possible values of the single-valuedvariable X are fuzzilyrestrictedby fuzzyset A), linguisticallyqualified intermsoftruth,probability,or possibility.Interestingly,therepresentationofpossibility-qualifiedstatementsledtopossibilitydistributionsoverpossibility distributions,butcertainty-qualifiedstatements,firstconsideredin[38](seealso[11]),andusedasthebasicbuildingblocks ofpossibilisticlogic, werenot consideredatall,justbecausenecessitymeasuresasthedual ofpossibilitymeasures were playing almost no role inZadeh’s view (with the exception ofhalf a page in[39]). Possibility-qualified statementswere exploitedin[35]inrelationwithaweightedresolutionprinciple extendingtheinferencerule(11),whoseformal analogy withaninferenceruleexistinginmodallogicwasstressed.
Thesimilaritybetweenpossibilitytheory(includingnecessitymeasures)andmodallogicshouldnotcomeasasurprise sincetheanalogybetweenthedualitypropertyN
(
A)
=
1−( \
A)
inpossibilitytheoryandthedefinitionof3
p as¬2¬
pis striking, andhas been known fora long time [40].Likewise, theaxiom
2
p→ 3
p (axiomD
in modallogic systems) mayencode theinequality N(
A)
≤ (
A)
,andthecharacteristicaxiomofnecessitymeasures N(
A∩
B)
=
min(
N(
A),
N(
B))
correspondstothetheorem
(
2
p∧ 2
q)
↔ 2(
p∧
q)
whichisvalidinmodalsystemK
.Nevertheless,noformallyestablished connectionbetweenmodallogicandpossibilitytheoryexisteduntilthelate1980s.This strikingparallel betweenpossibilitytheory andmodal logiceventually led toproposals fora modalanalysisand encoding ofpossibility theory. Forinstance, L. Fariñas andA. Herzig [41] proposed such an encoding by heavily relying on Lewis’conditionallogicsofcomparativepossibility[42],asindeedtheonlynumericalcounterparts ofLewis possibility relations are possibilitymeasures [43].Anotherattemptwas later madebyBoutilier[44],inthe scopeofnon-monotonic inferencebasedonaplausibilityrelationoverpossibleworlds.Theideawastousethisordinalcounterpartofapossibility distributionasanaccessibilityrelationandtoconstructmodalitiesfromit.Another,moresemantically-orientedtrendwas tobuildspecificaccessibilityrelationsagreeingwithpossibilitytheory[45,46].
A majordifferencewithGPL isthatthe semanticsoftheabove logicsrelieson accessibilityrelations.GPLcan be em-beddedintoamultimodallogic,butitisactuallyjustatwo-levelpropositionallogicsinceitssemanticsisbasedongraded epistemic states, viewed as higher-order interpretations, not relying on accessibility relations. This point was discussed in[47]:relationalsemanticsofepistemiclogicsmaymakesenseinthescopeofintrospectivereasoning,butappearsmore difficulttojustifyformodellingpartialknowledgeabouttheepistemicstateofanexternalagent.InGPL,anyagent is sup-posedtobeawareofitsownepistemicstate,soitcanmodelitsownbeliefsusingacompleteGPLbase(seeSection4on thispoint).Also, formally,GPLisacomplexificationofpropositionallogic,addingweighted modalitiesinfrontof proposi-tionalformulasonly, and,atthesemanticlevel,moving fromusual interpretationstosets thereof,whilesimpleepistemic logics like S5 orKD45 are constructed asa simplification ofa complex logic allowing nested modalities naturally inter-pretedviaaccessibilityrelations,andneedintrospectionaxiomstosimplifycomplexformulasintoequivalentonesofdepth atmost 1.SobeyondtheformalanalogiesbetweenmodallogicandGPL,themotivationsandtheconstructionmethodare radicallydifferent.
AproposalclosertoGPListheoneofHájek[31],wherepossibilitytheoryiscastintoamany-valuedlogicsetting,using many-valuedmodalformulas.ThemaindifferencewithGPL,fromaformal pointofview,isthatnecessityisexpressedas asinglemulti-valued modality,ratherthanasetofclassicalmodalitiesinGPL.Thisimpliesthatnecessitystatementsneed tobecombinedusingafuzzylogic,ratherthanclassicalpropositional logicinGPL.Anumberofrelatedlogicsarestudied in[33,34],whichareusingvariantsofŁukasiewiczlogicbothfortheformulasinsidethemodalitiesandforcombiningthe multi-valuedmodalities.IncasethesevariantsofŁukasiewiczarefinite-valued(ore.g.,includetheBaaz
connective[48]), it iseasytoseethat GPLcanbe framedasafragment ofsucha multi-valuedmodallogic.Ageneralcompletenessresult forsuch two-tiered(multi-valued)modellogics hasbeenintroducedin[32].LiauandLin[49] havealsostudied amodal logic which is very similar to GPL, albeit using
[
0,
1]
as a possibility scale (which forces them to introduce additional multimodal formulasto deal withstrict inequalities). Their tableau-based proof methods could be of interestto develop inferencetechniquesforGPL.Whilefromaformalpointofview,GPLisclosetosomeoftheseaforementionedlogics,ourfocusinthispaperisrather different. Specifically,ourmainaimistostudywhatisgained,intermsofthekindsofepistemicreasoningscenariosthat can be modelled,fromthe increasein syntacticfreedom compared tostandard possibilisticlogic. Among others,we will analyseseveralwaysinwhichpartialignorancecanbemodelled,studytherelationbetweenGPLandlogicsofcomparative
5 Ifν(λ
uncertainty,andshowhow differentformsofnon-monotonicreasoningcan naturallybe modelledusingGPL.Tothebest ofourknowledge, theselinkswithpossibilistic logic(orthe relatedmulti-valuedmodal logics)havenotbeen studiedin previouswork.
4. ReasoningaboutignoranceinGPL
Possibilitytheoryoffers anumberoftoolsformodellinglimitationsonwhatisknown.ThesetoolscanbeusedinGPL toexplicitlymodelwhatweknowthatanexternalagentdoesnotknow.Inparticular,Section4.1proposesamethodbased onthe guaranteedpossibilitymeasure,which issubsequentlyrefinedin Section4.2.InSection 4.3,wethen analysehow theprincipleofminimalspecificitycanbeappliedtoreasonaboutwhatanexternalagentdoesnotknow.
4.1. Ignoranceasguaranteedpossibility
UsingthemodalitiesN and
wecanmodelconstraintsoftheformN
(
α
)
≥ λ
,N(
α
)
≤ λ
,(
α
)
≥ λ
and(
α
)
≤ λ
.Sofar, however,we havenot consideredthe guaranteedpossibilitymeasureandpotential necessitymeasure
∇
.Counterparts of these measures can be introduced as abbreviations in the language, by noting that(
α
)
=
minω∈JαK(
{
ω
})
. For a propositionalinterpretationω
letuswriteconjω for theconjunctionofall literalsmadetruebyω
,i.e.,conjω=
ω|=aa∧
ω|=¬a¬
a.Thenwedefine:λ
(
α
)
=
ω∈JαK
λ
(
conjω)
∇
λ(
α
)
= ¬
ν(λ)(
¬
α
)
(12) Infact,since(
α
)
=
maxω∈JαK(
{
ω
})
,anotherstrategywecouldhavetakenistoaxiomatizealogicbasedonguaranteed possibility,andtodefine themodalityN asan abbreviation.Inparticular,such alogic couldbe axiomatizedbyusingthe followinggradedversionofthedatalogicofDubois,HájekandPrade[50]:(PL) Theaxiomsofclassicallogicformeta-formulas.
(K)
λ
(
α
∧ ¬β)
→ (
λ(
¬
α
)
→
λ(
¬β))
. ()
1
(
α
)
whenever¬
α
∈
L
isatautology. (D)λ
(
α
)
→ ∇
1(
α
)
. (W)λ1
(
α
)
→
λ2(
α
)
,ifλ
1≥ λ
2;andthemodusponensrule.Wecouldthenalsointroducethefollowingabbreviations:
λ
(
α
)
=
ω∈JαK
λ
(
conjω)
(13)Nλ
(
α
)
= ¬
ν(λ)(
¬
α
)
(14)TheresultinglogicisverysimilartoGPL.However,for (D) tobesound,weneedtorestricte-modelstopossibility distri-butions
π
forwhichπ
(
ω
)
=
0 foratleastone propositionalinterpretationω
.Similarly,fortheseaxiomstobe complete, we needto dropthe requirementthatπ
(
ω
)
=
1 for atleastone interpretation.In fact,the soundnessandcompleteness resultfromProposition 1 canstraightforwardlybe adaptedtoa logiccenteredonthemodality,by takingadvantage of thefollowingduality:
π
|=
Nλ(
α
)
iffπ
|=
λ(
¬
α
)
(15)wherethepossibilitydistribution
π
isdefinedasπ
(
ω
)
=
1−
π
(
ω
)
forallω
∈
.Thisdualitycanbereadilyverifiedusing thedefinitionsoftheN andmeasuresinpossibilitytheory(seeSection2.1).
However it isstraightforward toshow that (K), (
) and (W) are validin GPL.We can furthermoreshow that the followingformulasarevalidinGPL:
λ
(
α
)
∧
λ(β)
≡
λ(
α
∨ β)
∇
λ(
α
∨ β) → ∇
λ(
α
)
∧ ∇
λ(β)
λ
(
α
)
∨
λ(β)
→
λ(
α
∧ β)
∇
λ(
α
)
∨ ∇
λ(β)
≡ ∇
λ(
α
∧ β)
andλ1
(
α
∧ β) ∧
λ2(
¬
α
∧
γ
)
→
min(λ1,λ2)(β
∧
γ
)
(16)∇
λ1(
α
∧ β) ∧
λ2(
¬
α
∧
γ
)
→ ∇
λ1(β
∧
γ
)
, ifλ
2≥
ν
(λ
1)
(17)Notethat(16)isthecounterpartofabasicinferenceruleofthelogicofaccumulateddata[50].
Foranypossibilitydistribution
π
over,we caneasilydefine aGPLknowledgebasewhichhas
π
asitsonlye-model, usingthemodality.Inparticular,let
α
1,
...,
α
k bepropositionalformulassuchthatJ
α
iK
= {
ω
|
π
(
ω
)
≥
ki}
.Thenwedefineπ
=
ki=1 Nν(i k)
(
α
i)
∧
ki(
α
i).
(18)Aformulaoftheform
π definesaGPLbasewhichiscompleteinthefollowingsense.
Proposition2.
∀
α
∈
L
,λ
∈
,πNλ
(
α
)
orπ
¬
Nλ(
α
)
.Proof. InEquation (18),the degreeofpossibilityofeach
ω
∈ J
α
iK
isdefinedbyinequalities fromaboveandfrombelow.Indeed,
i
k
(
α
i)
meansthatπ
(
ω
)
≥
i
k forall
ω
∈ J
α
iK
,whereas,Nν(ki)(
α
i)
meansπ
(
ω
)
≤
i−1
k forall
ω
∈ J
/
α
iK
.Itfollowsthatπ
(
ω
)
=
0 ifω
∈ J
/
α
1K
,π
(
ω
)
=
ki ifω
∈ J
α
iK
\ J
α
i+1K
(fori<
k)andπ
(
ω
)
=
1 ifω
∈ J
α
kK
.Inotherwords,π
isindeedtheonlye-modelof
π .SinceweclearlyhaveN
(
α
)
≥ λ
or¬(
N(
α
)
≥ λ)
foranynecessitymeasure,itfollowsthatπNλ
(
α
)
orπ
¬
Nλ(
α
)
.2
If we view the epistemic state ofan agent asa possibilitydistribution, thismeans that every epistemic state can be modelledusingaGPLknowledge base.Conceptually,theconstructionof
π relatestothenotionof“onlyknowing”from
Levesque[6].Forexample,assumethatwewanttomodelthatalltheagentknowsisthat
β
istruewithcertainty kj.Then wehaveπ
(
ω
)
=
1 forω
∈ JβK
andπ
(
ω
)
=
k−kj forω
∈ JβK
/
.Thismeansthatinthenotationof(18),α
k−j+1= ...
α
k= β
andweobtain
π
=
1(β)
∧
Nν(k− j+1k )
(β)
∧
k− jk
(
)
.Inthecasewhenk=
1,Equation(18)readsN1(
α
)
∧
1(
α
)
andisolatesasingle crispe-modelcorrespondingtothesetofclassicalmodelsof
α
asalreadypointedout in[28].Itexpressesthatwe preciselyknowtheepistemicstateoftheexternalagent,namelythat(s)heonlyknowsthatα
istrue.In practice,we willoftenhave incompleteknowledge abouttheepistemic state ofthisagent. Supposewe only know thattheepistemicstateisamongthosein S
⊆
P
k.ThiscanbeencodedasaGPLknowledgebaseS
=
π∈Sπ with
π
definedasabove.Asaconsequence,anyGPLknowledgebaseissemanticallyequivalent toaformulaoftheform
S,and
anysubsetofepistemicstatescanbecapturedbyaGPLknowledgebase.
Since themodality
was introducedasan abbreviation, allowing thismodality hasnoimpact onthe expressiveness of the language or on the completeness ofthe axiomatization.However, the formula
λ
(
α
)
abbreviates a GPL formula whichmaybeofexponentialsize,andallowingthemodalityinthelanguageisthusessentialifwewanttocaptureour knowledgeaboutanagent’sepistemicstateinacompactway.AswewillseeinSection7,thisisreflectedinanincreasein computationalcomplexity.
4.2. Contextualignoranceasrestrictedguaranteedpossibility
The modality
allows us to express limitationson what an agent knows.However, it doesnot readily allow usto explicitlyencodetheignoranceofanagentonaparticulartopic.
Example2.ConsideragainthescenariofromExample 1andsupposewewanttoencode that“all theagentknowsabout the gamesin round3isthat Wales haswonits game”.We cannotrepresentthisasN1
(
won3(
wal))
∧
1(
won3(
wal))
,asthatwouldentaile.g.,
¬
N1(
won2(
wal))
,whichisnotwarranted.Toencodelimitationsontheknowledgeoftheagentonaparticulartopic,understoodasasetofpropositionalvariables
X ⊆
At,weproposethefollowingvariantofthemodality:
Xλ
(
α
)
=
ω∈JαK
λ
(
conjXω)
where conjXω is the restriction of conjω to those literals about variables in
X
, i.e., conjXω=
{
x|
x∈
X ,
ω
|=
x}
∧
{¬
x|
x∈
X ,
ω
|= ¬
x}
. Notethat|=
G P Lλ
(
α
)
≡
Atλ(
α
)
. Forexample,inthe scenario fromExample 2,instead of assert-ing1
(
won3(
wal))
,wecanassertX1
(
won3(
wal))
,withX = {
plays3(
x,
y)
|
x,
y∈
T}
∪ {
won3(
x)
|
x∈
T}
thesetofallatomicformulasaboutround3ofthechampionship.AswewillseeinSection7,allowingthisrefinementofthe
modalityleads toafurtherincreaseincomputationalcomplexity.
4.3. Ignoranceasminimalspecificity
The lessspecificthan relation
definesapartialorderonthe setofe-models ofaGPLknowledge baseK in anatural way,whichallowsustointroducetwonon-monotonicentailmentrelations: