Generalized possibilistic logic: Foundations and applications to qualitative reasoning about uncertainty

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Generalized possibilistic logic: Foundations and

applications to qualitative reasoning about uncertainty

Didier Dubois, Henri Prade, Steven Schockaert

To cite this version:

Didier Dubois, Henri Prade, Steven Schockaert. Generalized possibilistic logic: Foundations and

applications to qualitative reasoning about uncertainty. Artificial Intelligence, Elsevier, 2017, 252,

pp.139-174. �10.1016/j.artint.2017.08.001�. �hal-02640549�

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This is an author’s version published in:

http://oatao.univ-toulouse.fr/2

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

,

∗

a_{ToulouseUniversity,UniversitéPaulSabatier,IRIT,CNRS,118RoutedeNarbonne,31062ToulouseCedex09,France} b_{CardiffUniversity,SchoolofComputerScience&Informatics,5TheParade,CardiffCF243AA,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)

where

α

isa propositionalformulaand

λ

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 ifneither

α

norits negationcan be derivedfromthe givenknowledgebaserepresenting whatisknown aboutthisagent’sbeliefs. Whenreasoningaboutbeliefsasrevealedbyanagent,thisassumptionishardtokeepandweneedtodistinguishbetween situations wherewe (the outsideagent) knowthat theagent isignorant about

α

andsituations wherewe donot know whether theagent knows

α

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 than

β

), 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 mapping

π

:

U → [

0

,

1

]

, which is called a possibilitydistribution. The intended interpretationof

π

(

u

)

=

1 isthat X

=

u isfullycompatiblewithallavailable information,while

π

(

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,definedas

π

(

u

)

=

1 for allu

∈

U

.Usually,we requirethat

π

(

u

)

=

1 forsomeu

∈

U

,whichcorrespondstotheassumptionthattheavailableinformationisconsistent.Ifthepossibility distribution

π

satisfiesthiscondition,itiscallednormalized.

Ingeneral,thevalueof

π

(

u

)

canbeinterpretedintermsofdegreesofpotentialsurprise:thesmallerthevalueof

π

(

u

)

, themorewewouldbesurprisedtofindoutthat X

=

u.ThisinterpretationgoesbacktoShackle[13]andsupportsapurely qualitativeinterpretationofthepossibilitydegrees

π

(

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

)

=

max u∈A

π

(

u

).

AdualmeasureN,calledthenecessitymeasure,isdefinedforA

⊆

U

as[11]:

N

(

A

)

=

1

− (

U

\

A

)

=

min

u∈/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.Twoothermeasuresthatcanbeintroducedaretheguaranteedpossibilitymeasure



andthepotentialnecessity measure

∇

,definedfor A

⊆

U

as[12]:

(

A

)

=

min u∈A

π

(

u

)

;

∇(

A

)

=

1

− (

U

\

A

)

=

max

u∈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)

,where

λ

∈]

0

,

1

]

isa certainty degree and

α

is apropositional formula, builtfroma setofatomicformulas At usingthe connectives conjunc-tion

∧

,negation

¬

,disjunction

∨

,implication

→

,andequivalence

≡

intheusualway.Let



bethesetofallinterpretations

ofAt andlet

L

bethesetofallpropositionalformulasbuiltfromAt.Thesemanticsofpossibilisticlogicisdefinedinterms

ofpossibilitydistributionsover



.Specifically,apossibilitydistribution

π

over



satisfiestheformula

(

α

,

λ)

iffN

(

J

α

K)

≥ λ

, where

J

α

K

denotes thesetofall (classical)modelsof

α

.As

π

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 iff

π

isane-modelofeveryformulainK .K generally

has multiplee-models, butthey can be partially ordered by the specificityordering, whereby

π

1 isless specific than

π

2,

written

π

1



π

2,if

π

1

(

ω

)

≥

π

2

(

ω

)

forevery

ω

∈ 

.Itcanbeshownthatthesetofe-modelsofasetofPLformulas K has

auniqueleastelement

πK

w.r.t.



,whichiscalledtheleastspecifice-modelofK .Itcanbeexpressed,forall

ω

∈ 

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 by

ω

, the less plausible

ω

is consideredtobe.

ThefollowinginferencerulesarevalidinPL:

if

(

α

, λ)

∈

K then K



P L(

α

, λ)

(1)

if



α

then K



P L

(

α

,

1

)

(2)

if

λ

1

≥ λ

2and K



P L(

α

, λ

1

)

then K



P L(

α

, λ

2

)

(3)

if K



P L

(

α

∨ β, λ

1

)

and K



P L

(

¬

α

∨

γ

, λ

2

)

then K



P L(β

∨

γ

,

min

(λ

1

, λ

2

))

(4)

Letuswrite K

|=

P L

(

α

,

λ)

ifeverye-modelofK isane-modelof

(

α

,

λ)

.Ifthereisnocauseforconfusionwealsowrite

|=

P L

as

|=

and



P L as



.ItispossibletoshowthatthefollowingstatementsareallequivalentforasetofPLformulas K (see

e.g.,[19]):

1. K



P L

(

α

,

λ)

canbederivedfrom(1)–(4).

2. K

|=

P L

(

α

,

λ)

.

3. Theleastspecifice-model

πK

of K isane-modelof

(

α

,

λ)

.

Inferenceinpossibilisticlogicthusremainsclosetoinferenceinpropositionallogic.Inparticular,letthec-cutKcofK bethe

propositionaltheory Kc

= {

α

|

(

α

,

λ)

∈

K and

λ

≥

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

),

...,

(

αn

,

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(Z

iwi).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.3

However, 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}

anaxiomatizationforthislogic.Wewilluse

α

,

β

,etc.todenotepropositionsinstandardpropositionallogic,formedwith the connectives,

∧

and

¬

. As usual, we will also usethe abbreviations

α

∨ β = ¬(¬

α

∧ ¬β)

,

α

→ β = ¬(

α

∧ ¬β)

and

α

≡ β = (

α

→ β)

∧ (β →

α

)

.Let

L

bethelanguageofallpropositionalformulasoverafinitesetofatomicpropositions At. Unlessstatedotherwise,werestrictthesetofcertaintydegreestothefinitesubset



k

= {

0

,

1_k

,

2_k

,

...,

1

}

oftheunitinterval,

withk

∈ N

\ {

0

}

andlet



+_k

= 

k

\ {

0

}

.

3.1. Syntax

Wedefinethelanguage

L

k_{G P L} ofgeneralizedpossibilisticlogicwithk

+

1 certaintylevelsasfollows:

•

If

α

∈

L

and

λ

∈ 

+_k,thenNλ

(

α

)

∈

L

kG P L.

•

If



∈

L

k_{G P L} and

∈

L

k_{G P L},then

¬

and



∧

arealsoin

L

k_{G 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 to

emphasizethelinkwithpossibilitytheory.Furthermorenote thatweview

L

k_{G P L} asalanguagewithk differentmodalities

N1

k

,

...,

N1,ratherthanalanguagewithasinglemodalityandconstantsdenotingcertaintydegrees.

Inthefollowing,wewillalsousethefollowingabbreviation:



λ

(

α

)

= ¬

Nν(λ)

(

¬

α

)

(7)

wherewewrite

ν

(λ)

asanabbreviationfor1

− λ

+

_k1.Semanticallythemodality



λwillcorrespondtoalowerboundon apossibilitymeasure,namely(7)isthecounterpartofthedualitybetweenapossibilityandanecessitymeasureonafinite scale,wherewehavetoshiftfromonelevelformovingfromastrictinequalitytoaninequalityinthebroadsense.

Letusdefineameta-atom asanexpressionoftheformNλ

(

α

)

,andameta-literal asanexpressionoftheformNλ

(

α

)

or

¬

Nλ

(

α

)

.Ameta-clause isanexpressionoftheform



1

∨ ...

∨ 

nwitheach



i ameta-literal.Ameta-term isanexpression

oftheform



1

∧ ...

∧ 

nwitheach



iameta-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 Let

P

k bethe setofall

suchpossibilitydistributions.Ane-modelofaGPLformulaisanypossibilitydistribution

π

from

P

k,namely:

•

π

isane-modelofNλ

(

α

)

iffN

(

α

)

≥ λ

;

•

π

isane-modelof



1

∧ 

2iff

π

isane-modelof



1 andof



2;

•

π

isane-modelof

¬

1 iff

π

isnotane-modelof



1;

where N is the necessity measure induced by

π

.As usual,

π

iscalled an e-model ofa set of GPL formulas K ,written

π

|=

k

G P LK ,ifit isan e-model ofeach formulain K .It iscalleda minimally specific e-modelof K if there isnoe-model

π



=

π

of K suchthat

π



(

ω

)

≥

π

(

ω

)

foreach possibleworld

ω

.Wewrite K

|=

k_{G P L}

φ

,for K asetofGPLformulasand

φ

a

GPL formula,ifevery e-model of K isalso an e-modelof

φ

.Whenk is clearfromthecontext,we will sometimeswrite

|=

k

G P L as

|=

G P L;furthermore,ifthereisnocauseforconfusion,wewillalsowrite

|=

kG P L as

|=

.

Intuitively, N1

(

α

)

means that it is completely certain that

α

is true,whereas Nλ

(

α

)

with

λ

<

1 means that there is evidence which suggeststhat

α

is true, and none that suggests that it is false. Note that we can distinguish between completeandpartialcertaintyonlyifk

≥

2.Formally,anagent assertingNλ

(

α

)

hasanepistemicstate

π

suchthatN

(

α

)

≥

λ

>

0. Hence

¬

Nλ

(

α

)

stands for N

(

α

)

< λ

, which means

(

¬

α

)

≥

1

− λ

+

1k. The abbreviation introduced in (7) thus

corresponds 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

)

meansthat

α

isfullycompatiblewithouravailablebeliefs (i.e., nothingprevents

α

frombeingtrue),while



λ

(

α

)

with

λ

<

1 meansthat

α

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 that

an 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)

,ortheir

4 _{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

,if

k

=

2,onecandistinguishbetweenthefollowingfivecases:

•

K

|=

N1

(

α

)

meansthatweknowthattheagentknowsthat

α

istrue.

•

K

|=

N1

(

¬

α

)

meansthatweknowthattheagentknowsthat

α

isfalse.

•

K

|=

N1

(

α

)

∨

N1

(

¬

α

)

, K

|=

N1

(

α

)

and K

|=

N1

(

¬

α

)

meansthat we knowthat theagent knowswhether

α

istrueor

false,butwedonotknowwhichitis.

•

K

|= 

1

(

α

)

∧ 

1

(

¬

α

)

meansthatweknowthattheagentisignorantaboutwhether

α

istrueorfalse.

•

K

|=

N1

(

α

)

∨

N1

(

¬

α

)

and K

|= 

1

(

α

)

∧ 

1

(

¬

α

)

means that we are ignorant about whetherthe agent is ignorant

about

α

.

Thisisincontrastwiththeonlythreesituationsthatcanbedistinguished inclassicallogic(andinPL),i.e.,we knowthat

α

istrue,we knowthat

α

isfalse,orwe donot knowwhether

α

istrueor false.Whenk

>

2,we canconsider graded counterparts ofthe five aforementioned cases.Moreover, a GPL base can then also expresscomparative uncertainty. For example:

•

K

|=



k_i=1Ni

k

(

α

)

∧ ¬

N i

k

(β)

:weknowthattheagentismorecertainthat

α

holdsthanthat

β

holds,noticingthatitis

equivalentto

∃

i

,

N

(

α

)

≥

_ki

>

N

(β)

.

•

K

|=



k_i=1



i

k

(

α

)

∧ ¬

i

k

(β)

:weknowthattheagentwouldbelesssurprisedtolearnthat

α

istruethantolearnthat

β

istrue,noticingthatitisequivalentto

∃

i

.

(

α

)

≥

_ki

> (β)

.

•

K

|=



k_i=1

(

Ni k

(

α

)

∨

N i k

(

¬

α

))

∧ ¬

N i k

(β)

∧ ¬

N i

k

(

¬β)

: we knowthat the agent is morecertain about thetruth or the

falsityof

α

thanabout

β

,butwemaynotknowwithwhichcertaintydegreetheagentknowsthetruthvalueof

α

,nor towhatextentthiscertaintydegreeisgreaterthanthecertaintydegreeaboutthetruthorthefalsityof

β

.

•

K

|=



k_i=1

(

Ni k

(

α

)

∧ ¬

N i k

(β))

∨ (

N i k

(β)

∧ ¬

N i

k

(

α

))

:weknowthattheagentconsidersoneof

α

,

β

morecertainthanthe

other,butwemaynotknowwhich.

•

K

|=



k_i=1

(

Ni

k

(

α

)

→

Ni k

(β))

expressesthattheagentisatleastascertainabout

β

asabout

α

.

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 has

wonits gameinroundi.LetT

= {

eng

,

fra

,

ire

,

ita

,

sco

,

wal

}

.Toexpressthatan agentknowstherulesofthechampionship, wecanconsiderformulassuchas,amongothers:

N1

(



{

plays_i

(

x

,

u

)

|

u

=

x

,

u

∈

T

})

(8)

wherex

∈

T .AformulasuchasN3

4

(

won1

(

wal

))

meansthattheagentstronglybelieves,butisnotfullycertain,that Wales

(wal)haswonitsfirst roundgame,while



3

4

(

won1

(

wal

))

meansthat theagent doesnotexclude thatWaleshaswonits

first round game,withoutevidence asto thecontrary. Thefollowing formulaexpresses that theagent considers it more plausiblethatWaleshaswonitsfirstgamethanthatEngland(eng)haswonitsfirstgame

k



i=1



i 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

)

whenever

α

∈

L

isaclassicaltautology.

(D) Nλ

(

α

)

→ 

1

(

α

)

.

(W) Nλ1

(

α

)

→

Nλ2

(

α

)

,if

λ

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

isfixedwegetafragment

ofthemodallogicKD.Inparticular,theaxiomsentailthatNλ

(

α

∧β)

isequivalenttoNλ

(

α

)

∧

Nλ

(β)

.Itiseasytoseethatif

α

and

β

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.,if



α

then



G P LNλ

(

α

)

forany

λ

∈ 

k.Finallynotethatinthe

casewherek

=

1,GPLcoincideswiththelogicMEL.Inthislattercase,wehave

1

(

α

)

= ¬

N1

(

¬

α

)

whereasingeneralwe

onlyhave

1

(

α

)

= ¬

N1

k

(

¬

α

)

.AswewillseeinSection6,theabilitytodifferentiatebetweenfullpossibilityfor

α

andthe lackoffullcertaintyfor

¬

α

iscrucialwhenusingGPLtoprovideasemanticsfornegationasfailure.

Proposition1(Soundnessandcompleteness).LetK beasetofGPLformulasand



aGPLformula.ItholdsthatK

|=

G P L



iff

K



G P L



.

Proof. TheproofispresentedinAppendix A.

2

Themainideabehind theproof isthatwecanseeformulasinGPLaspropositionalformulaswhicharebuiltfromthe setofatomicformulasof

L

k

G 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

N1

2

(

a

)

,fora

∈

At but

{

Nλ

(

a

)

|

λ

<

1

2

}



GPLN1₂

(

a

)

.Thisisnotarealrestriction,sinceknowledgebasesonlyhavefinitelymany

formulasinpractice,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 latter

setofpremises,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)

in

thesameway(justrewriting

α

→ β

as

¬β → ¬

α

).However,weneedtoassume

ν

(λ

1

)

≤ λ

2inordertoweakenNλ2

(

α

→ β)

intoNν(λ1)

(

α

→ β)

.And

ν

(λ

1

)

≤ λ

2isequivalentto1

− λ

1

+

1

k

≤ 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

)

inpossibilitytheoryandthedefinitionof

3

p as

¬2¬

p

is striking, andhas been known fora long time [40].Likewise, theaxiom

2

p

→ 3

p (axiom

D

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

)

whichisvalidinmodalsystem

K

.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 guaranteedpossibilitymeasure



andpotential necessitymeasure

∇

.Counterparts of these measures can be introduced as abbreviations in the language, by noting that

(

α

)

=

minω∈JαK

(

{

ω

})

. For a propositionalinterpretation

ω

letuswrite_{conjω for} theconjunctionofall literalsmadetrueby

ω

,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

)

,if

λ

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

forwhich

π

(

ω

)

=

0 foratleastone propositionalinterpretation

ω

.Similarly,fortheseaxiomstobe complete, we needto dropthe requirementthat

π

(

ω

)

=

1 for atleastone interpretation.In fact,the soundnessandcompleteness resultfromProposition 1 canstraightforwardlybe adaptedtoa logiccenteredonthe



modality,by takingadvantage of thefollowingduality:

π

|=

Nλ

(

α

)

iff

π

|= 

λ

(

¬

α

)

(15)

wherethepossibilitydistribution

π

isdefinedas

π

(

ω

)

=

1

−

π

(

ω

)

forall

ω

∈ 

.Thisdualitycanbereadilyverifiedusing thedefinitionsoftheN and



measuresinpossibilitytheory(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)

and



λ1

(

α

∧ β) ∧ 

λ2

(

¬

α

∧

γ

)

→ 

min(λ1,λ2)

(β

∧

γ

)

(16)

∇

λ1

(

α

∧ β) ∧ 

λ2

(

¬

α

∧

γ

)

→ ∇

λ1

(β

∧

γ

)

, if

λ

2

≥

ν

(λ

1

)

(17)

Notethat(16)isthecounterpartofabasicinferenceruleofthelogicofaccumulateddata[50].

Foranypossibilitydistribution

π

over



,we caneasilydefine aGPLknowledgebasewhichhas

π

asitsonlye-model, usingthemodality



.Inparticular,let

α

1

,

...,

αk

bepropositionalformulassuchthat

J

αi

K

= {

ω

|

π

(

ω

)

≥

ki

}

.Thenwedefine



π

=

k

i=1 N_ν(i k)

(

α

i)

∧ 

ki

(

α

i). (18)

Aformulaoftheform



π definesaGPLbasewhichiscompleteinthefollowingsense.

Proposition2.

∀

α

∈

L

,

λ

∈ 

,



π



Nλ

(

α

)

or



π

 ¬

Nλ

(

α

)

.

Proof. InEquation (18),the degreeofpossibilityofeach

ω

∈ J

αi

K

isdefinedbyinequalities fromaboveandfrombelow. Indeed,



i

k

(

αi

)

meansthat

π

(

ω

)

≥

i

k forall

ω

∈ J

αi

K

,whereas,Nν(_ki)

(

αi

)

means

π

(

ω

)

≤

i−1

k forall

ω

∈ J

/

αi

K

.Itfollowsthat

π

(

ω

)

=

0 if

ω

∈ J

/

α

1

K

,

π

(

ω

)

=

_ki if

ω

∈ J

αi

K

\ J

αi

+1

K

(fori

<

k)and

π

(

ω

)

=

1 if

ω

∈ J

αk

K

.Inotherwords,

π

isindeedthe

onlye-modelof



π .SinceweclearlyhaveN

(

α

)

≥ λ

or

¬(

N

(

α

)

≥ λ)

foranynecessitymeasure,itfollowsthat



π



Nλ

(

α

)

or



π

 ¬

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 wehave

π

(

ω

)

=

1 for

ω

∈ JβK

and

π

(

ω

)

=

k−_kj for

ω

∈ JβK

/

.Thismeansthatinthenotationof(18),

αk

₋j+1

= ...

αk

= β

and

weobtain



π

=

1

(β)

∧

N_ν(k− j+1

k )

(β)

∧ 

k− j

k

(

)

.Inthecasewhenk

=

1,Equation(18)readsN1

(

α

)

∧ 

1

(

α

)

andisolatesa

single crispe-modelcorrespondingtothesetofclassicalmodelsof

α

asalreadypointedout in[28].Itexpressesthatwe preciselyknowtheepistemicstateoftheexternalagent,namelythat(s)heonlyknowsthat

α

istrue.

In practice,we willoftenhave incompleteknowledge abouttheepistemic state ofthisagent. Supposewe only know thattheepistemicstateisamongthosein S

⊆

P

k.ThiscanbeencodedasaGPLknowledgebase



S

=



_π∈S



π 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,andallowingthemodality



inthelanguageisthusessentialifwewanttocaptureour 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

))

,as

thatwouldentaile.g.,

¬

N1

(

won2

(

wal

))

,whichisnotwarranted.

Toencodelimitationsontheknowledgeoftheagentonaparticulartopic,understoodasasetofpropositionalvariables

X ⊆

At,weproposethefollowingvariantofthe



modality:



X_λ

(

α

)

=

ω∈JαK



λ

(

conjXω

)

where conjX_ω is the restriction of _{conjω to} those literals about variables in

X

, i.e., conjX_ω

=



{

x

|

x

∈

X ,

ω

|=

x

}

∧



{¬

x

|

x

∈

X ,

ω

|= ¬

x

}

. Notethat

|=

G P L



λ

(

α

)

≡ 

Atλ

(

α

)

. Forexample,inthe scenario fromExample 2,instead of assert-ing

1

(

won3

(

wal

))

,wecanassert



X1

(

won3

(

wal

))

,with

X = {

plays3

(

x

,

y

)

|

x

,

y

∈

T

}

∪ {

won3

(

x

)

|

x

∈

T

}

thesetofallatomic

formulasaboutround3ofthechampionship.AswewillseeinSection7,allowingthisrefinementofthe



modalityleads toafurtherincreaseincomputationalcomplexity.

4.3. Ignoranceasminimalspecificity

The lessspecificthan relation



definesapartialorderonthe setofe-models ofaGPLknowledge baseK in anatural

way,whichallowsustointroducetwonon-monotonicentailmentrelations:

•

Wesaythat



isabraveconsequenceofK ,written K

|=

b



iff



issatisfiedbyaminimallyspecifice-modelofK .

•

We say that



isa cautious consequence of K ,written K

|=

c



iff



issatisfied by all minimallyspecific e-models