Depth-bounded Belief functions

Contents lists available atScienceDirect

International

Journal

of

Approximate

Reasoning

www.elsevier.com/locate/ijar

Depth-bounded

Belief

functions

✩

Paolo Baldi

∗

,

Hykel Hosni

DepartmentofPhilosophy,UniversityofMilan,Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received30September2019

Receivedinrevisedform12March2020 Accepted13May2020

Availableonline22May2020 Keywords:

Belieffunctions Uncertainreasoning Depth-boundedlogics Probability

This paper introduces and investigates Depth-bounded Belief functions, a logic-based representation of quantified uncertainty. Depth-bounded Belief functions are based on the framework of Depth-bounded Boolean logics [4], which provide a hierarchy of approximationsto classical logic. Similarly, Depth-bounded Belieffunctions give rise to ahierarchy of increasingly tighter lower and upper bounds over classical measures of uncertainty.Thishastheratherwelcomeconsequencethat“higherlogicalabilities”lead tosharperuncertaintyquantification.Inparticular,ourmainresultsidentifytheconditions underwhichDempster-ShaferBelieffunctionsandprobabilityfunctionscanberepresented asalimitofasuitablesequenceofDepth-boundedBelieffunctions.

©2020TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticleunderthe CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introductionandmotivation

The linkbetweentherulesofrationalreasoningandtheprobabilistic representationofuncertaintyisastrongandold one.SinceJacobBernoulli’s1713ArsConjectandi,anumberofargumentshavebeenputforwardtotheeffectthatdeparting fromaprobabilisticassessmentofuncertaintyleadstoirrational patternsofbehaviour, asfixedby thewellknownresults ofdeFinettiandSavage[7,25] ([16] and[22] providerecentintroductoryreviews).

Over thepast few decades,however,a numberofconcerns havebeenraised against theadequacyof probability asa norm ofrationalreasoninganddecision-making.Following thelead of[9], whomin turnfoundhimself onthe footsteps of[15] and[14], manydecisiontheorists tookissuewiththenormativeadequacyofprobability.As aresult, considerable formal andconceptualefforthasgoneintoextendingthescopeoftheprobabilisticrepresentationofuncertainty,asbriefly recalledinSection1.1below.

One keycommonality amongthose“non probabilistic”approachesistheconviction that probabilityfailson represen-tational grounds.On those grounds, they insist that therational representation ofuncertainty need not necessitate that all uncertaintybe quantifiedprobabilistically.Butthiswas preciselyakeyconcerntackledbythe“Mathematicaltheory of evidence” put forwardby GlennShaferand whichhasbecome known astheDempster-Shafer theory of Belieffunctions (see [8] for aretrospectiveonthe developmentofthe theoryandacomprehensivebibliography). Shafer’soriginal aimin [27] wasto provide a generaltheory ofevidence and uncertain reasoning: in his interpretation, givena Belief function

Bel, the value Bel

(θ )

stands for the degree ofsupport that a piece ofevidence provides for theevent orproposition

θ

, according tothe judgment ofa certain agent.Since then, variousinterpretations of Belieffunctionshavebeen advanced, especially concerningtheirrelationwithprobability.Ofparticularrelevanceforourpresentpurposes isShafer’sown later

✩ _{ThisresearchwasfundedbytheDepartmentofPhilosophy“PieroMartinetti”oftheUniversityofMilanundertheProject“DepartmentsofExcellence}

2018-2022”awardedbytheMinistryofEducation,UniversityandResearch(MIUR).

*

Correspondingauthor.

E-mailaddresses:[email protected](P. Baldi),[email protected](H. Hosni).

https://doi.org/10.1016/j.ijar.2020.05.001

0888-613X/©2020TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

work[28],inwhichhesuggeststotakedegreesofbelieftobedegreesofsupporttoanswersforagivenquestion,forwhich noprobabilitydistributionisknown,butthatnevertheless canbecomputedbyresortingtoanswerstoarelatedquestion, whichimplytheoriginal ones,andforwhichatthesametimeaprobability distributionisactuallyknown.Thisapproach isexemplifiedbyassessingthedegreeofsupportforacertainpropositionintermsofreliabilityofwitnesses(whichcanbe probabilisticallyevaluated)assertingthatproposition. AninstanceofsuchviewofBelieffunctions, relevantforuslater,is alsotheso-calledprobabilityofprovability interpretation,aspresentede.g.in[23] and[19].

Inspiredbythesettingof[19],theinvestigationreportedinthispapertacklestherepresentationalshortcomingsinthe probabilisticuncertainreasoningfromalogical pointofview.Weobservethattheseissuescanbetracedbacktoanalogous shortcomingsofclassicallogic,whichprovidesadecidedlypoorrepresentationofthekeynotionof“information”,cognate conceptof“evidence”.Armedwiththissimplebutveryimportantobservation,wetakeastepbackandrethinkthelogic in

thefirst place.Hence, developingideas firstdiscussedin [5],we will castthe question ofquantifyingrationaldegreesof beliefintheframeworkofDepth-boundedBooleanlogics,recalledinSection2below.

Thisapproach leads to adefinitionof Depth-boundedBelieffunctions, which are introducedin Section 3. Onthisbasis, in Section 4 we provide a novel representation ofclassical Belief functions, which we see as the limit of sequences of Depth-boundedBelieffunctions.Wewilltheninvestigate,inSection5and6interestingsubclassesofDepth-boundedBelief functions,whicharisebysomeprincipledrestrictionofthesetofformulaswhererationaldegreesofbeliefarequantified. Inparticular,onesuchsubclassleadalsotoarepresentationtheoremforclassicalProbabilityfunctions,whichareobtained asthelimitofsequencesof(suitablyconstrained)Depth-boundedBelieffunctions.

Before delvinginto the details ofour proposal, however,it will be usefulto motivate its relevance against the wider landscapeofrationalreasoninganddecision-makingunderuncertainty.

1.1. Uncertainty,ignoranceandinformation

Uncertaintyhastodo,ofcourse,withnotknowing,andinparticularnotknowingtheoutcome(s)ofaneventofinterest, orthevalueofarandomvariable.Ignorancehasmoresubtlefeatures,andisoftenthoughtofasourinabilitytoquantifyour ownuncertainty.In[15],Knightgavethisimpalpabledistinctionan operationalmeaninginactuarialterms.Hesuggested that the presence of ignorance is detected by the absence of a complete insurance market for the goods at hand. On the contrary, a complete insurance market provides an operational definition ofprobabilisticallyquantifiable uncertainty.

ContemporaryfollowersofKnightinsistthattheinevitabilityofignoranceimpliesthatnotalluncertaintyisprobabilistically quantifiable andseek to introduce more general norms of rational belief anddecision under “Knightian uncertainty”or “ambiguity”(seee.g.[12]).AtellingillustrationofthisargumentfrominformationisduetoDavidSchmeidler[26]:

The probability attached to an uncertain event doesnot reflect the heuristic amount of information that led to the assignmentofthat probability.Forexample,whentheinformationonthe occurrenceoftwo eventsissymmetricthey areassignedequalprobabilities.Iftheeventsarecomplementarytheprobabilitieswillbe 1

/

2 independentofwhether thesymmetricinformationismeagerorabundant.

Gilboa [12] interprets Schmeidler’s observationas expressinga form of “cognitive unease”, namely a feeling that the theoryofsubjectiveprobability whichspringsnaturallyfromBayesianepistemologyissilentononefundamentalaspectof rationality,namelyhowtheavailableinformation,orthelackthereof,guidestheprocess ofuncertaintyquantification.But whyisitso?Supposethatsomematteristobedecidedbythetossofacoin.AccordingtoSchmeidler’slineofargument, Ishouldprefertossingmy own,ratherthansome oneelse’scoin,onthebasis,sayofthefactthatIhaveneverobserved signsof“unfairness” inmy coin, whilstIjust don’t knowanythingaboutthestranger’s coin. Thisqualitativeinformation shouldbereflectedinhowuncertaintyistoberationallyevaluated.

Similar considerations had been put forward in the foundations of statistics and later penetrated the broad field of uncertaintyinArtificialIntelligence.Asanticipatedabove,anearlyamendmentofprobabilitytheoryaimedatcapturingthe asymmetrybetweenuncertaintyandignoranceisthetheoryofBelieffunctions.Keytorepresentingthisasymmetryisthe relaxationoftheadditivityaxiomofprobability.Thisinturnmayleadtosituationsinwhichtheprobabilisticexcludedmiddle

doesnothold. Thatistosayan agentcould rationallyassignbelieflessthan1to theclassicaltautology

φ

∨ ¬φ

.Indeed, aswenowillustrate,theproblemwithnormalisingonthetautologiesofclassical logiccanbetakenasastartingpointfor moregeneralconsiderations.

1.2. Probabilityandclassicallogic

Itiswell-knownthateveryprobabilityfunctionarisesfromdistributingtheunitmassacrossthe2n atomsoftheBoolean (Lindenbaum)Algebrageneratedbythepropositionalvariables

{

p1

,

. . .

pn

}

ofalanguage

L

,andconversely,thataprobability functiononformulasover

L

iscompletelydeterminedbythevaluesittakesonsuchatoms(see,e.g.[19] forapresentation inthespiritofourwork).Sucharepresentationmakesexplicitthetwofoldroleplayedbyclassicallogicanditsextensions inthetheoryofprobability.

First,itprovides alanguageinwhichevents–the bearersofprobability –canbe expressed,combinedandevaluated. Theprecisedetailsdependontheframework.See[10] foracharacterisationofprobabilityonclassicallogic,and[11] forthe

generalcaseofDempster-ShaferBelieffunctionsonthemany-valuedextensionofclassicallogic.Incontrast,the measure-theoretic presentationsof probability identifies eventswithsubsets ofthe field generatedby a givensample space



. A popularinterpretationfor



isthatoftheelementaryoutcomesofsomeexperiment,aviewendorsedbyA.N.Kolmogorov, who insistedon the generality ofhis axiomatisation. More precisely, let

M

= (,

F,

P

)

be a measurespace where,



=

{

ω

1

,

ω

2

. . .

}

isthesetofelementaryoutcomes,

F =

2isthefieldofsets(

σ

−

algebra)over



.Wecallevents theelements

of

F

,andP

:

F → [

0

,

1

]

aprobabilitymeasure ifitisnormalised,monotoneand

σ

-additive,i.e. (K1) P

()

=

1

(K2) A

⊆

B

⇒

P

(

A

)

≤

P

(

B

)

(K3) If

{

E

}

i isacountablefamilyofpairwisedisjointeventsin

F

then P

(



iEi

)

=



iP

(

Ei

)

The Stone representationtheorem forBoolean algebrasandthe representationof probability functionsrecalledabove guaranteethatthemeasure-theoreticaxiomatisationofprobabilityisequivalenttothelogicaloneyieldedbyclassicallogic, which isobtainedby lettingafunction fromtheformulas F m_L ofalanguage

L

tothe realunit intervalbe aprobability function if

(PL1)

|= θ ⇒

P

(θ )

=

1

(PL2)

|= ¬(θ ∧ φ)

⇒

P

(θ

∨ φ)

=

P

(θ )

+

P

(φ)

.

Lessstraightforwardbutequallycompellingisthecaseyieldedbythemany-valuedextensionofclassicallogic,see[17]. Obviousasthelogical“translation”oftheKolmogorovaxiomsmaybe,ithighlightsasecondandcrucialroleforlogicin thetheoryofprobability,whichisbestappreciatedbyfocussingontheconsequencerelation

|=

.

In its measure-theoretic version, thenormalisation axiom (K1) isquite uncontroversial. Lessso, ifframed interms of classical tautologies, as in PL1. Indeed the arguments against the probabilistic representation of rational belief recalled above, emerge now formally. For

|=

interprets symmetrically “knowledge” and“ignorance” ascaptured by the fact that

|= θ ∨ ¬θ

isatautology.Indeedsimilarlybothersomeconsequencesfollowdirectlyfrom P L1 andP L2,namely

1. P

(

¬θ)

=

1

−

P

(θ )

2.

θ

|= φ ⇒

P

(θ )

≤

P

(φ)

Thoseexamplessufficetomakeakeypoint:Manyofthefeaturesofprobabilitywithwhichthecriticsrecalledabovetake issueclearlyhavetheirlogicalrootsinthesemanticsofclassicallogic.

Thelogicalframingofprobabilityallowsustorefinethisanalysis.Foritisthesemanticsofclassicallogicthatprovides the uncertaintyresolution device forthe evaluationofprobability.This isbest illustrated by thepiecemeal identification of “events”withthe“sentences” ofthelogic. Ontheone hand,an event,understood classically,eitherhappensornot. A sentenceexpressinganevent,ontheotherhandisevaluatedinthebinarysetasfollows

v

(θ )

=



1 if the event obtained 0 otherwise

.

Hence, theprobabilityofanevent P

(θ )

∈ [

0

,

1

]

measures theagent’sdegreeofbeliefthattheeventdidorwillobtain. Finding this out is,inmostapplications, relatively obvious.However, aspointedout in [10], ageneral theory ofwhat it meansfor“statesoftheworld”to“resolveuncertainty”isfarfromtrivial.

Amethodologically moreadequate wayofevaluating eventsarisesbytakinganinformation-based view onuncertainty resolution.Thekeydifferencewiththeprevious,classicalcase,liesinthefactthatthisleadsnaturallytoapartial evaluation

ofevents,thatis vi

(θ )

=

⎧

⎪

⎨

⎪

⎩

1 if I am informed that

θ

0 if I am informed that

¬θ

∗

if I am not informed about

θ .

Quiteobviouslystandardprobabilitylogicdoesnot applyhere,becausetheclassicalresolutionofuncertaintyhasnoway ofexpressingthe

∗

condition.TomodelthisweturntothetheoryofDepth-boundedBooleanLogics[4,3].

2. Aninformationalviewofpropositionallogic:Depth-boundedBooleanlogics

Depth-boundedBooleanlogicsarebasedontheideathatclassicalpropositionalconnectivesshouldbegivenan informa-tionalmeaning.Thisisachievedbyreplacingthenotionsof“truth”and“falsity”by“informationaltruth”and“informational falsity”,namelyholdingtheinformation thatasentence

ϕ

istrue,respectivelyfalse.Here,bysayingthatanagenta holds the

informationthat

ϕ

istrueorfalseitismeantthatthisinformationisavailable toa inthesensethata isreadytoactupon it.

Fig. 1. Informational tables for the classical operators.

Bothproof-theoreticandmodel-theoreticpresentationsofDepth-boundedBooleanlogics areavailable,andindeed ow-ing tocompleteness results,theyare provablyequivalent. Whilstour mainresultsbeloware castproof-theoretically,itis certainlyhelpful totheunfamiliarreaderifwestart bypresentingDepth-boundedBooleanlogics semantically.Indeed,as anticipated,theyarise naturallyby shiftingtheinterpretationofbooleantablesfrom“truth”to“informationaltruth”.This apparentlyinnocentshiftmakesundesirabletheclassicalprincipleofBivalence:itmaywellbethatforagiven

ϕ

,we nei-therholdtheinformation that

ϕ

istrue,nordowe holdtheinformationthat

ϕ

is false.So,theinformationalsemantics whichwearenowreadytorecall,hasabuilt-infeaturethatmarkstheasymmetrybetween“knowledge”and“ignorance”.

2.1. Semantics

Fortherestofthepaperwefixalanguage

L

,overafinitesetV ar

= {

p1

,

. . . ,

pn

}

ofpropositionalvariables.Welet F mL be theformulas builtfromthepropositional variablesby the usualclassical connectives

∧,

∨,

¬,

→

andtheconstant for falsum



.Foreach pi

∈

V ar wedenoteby

±

pianyoftheliteralspi and

¬

pi.Finally,foreachsetofformulas



wedenote by S f

()

thesubformulas oftheformulasin



.Weusethevalues1 and0 to represent,respectively, informationaltruth andfalsity.Whenasentencetakesneitherofthesetwo definedvalues,wesaythatitisinformationallyindeterminate.Itis technicallyconvenienttotreat informationalindeterminacyasathirdvalue that wedenoteby “

∗

”.1 _{The threevaluesare}

partiallyorderedbytherelation

suchthat v

w (“v islessdefinedthan,orequalto,w”)if,andonlyif, v

= ∗

orv

=

w

forv

,

w

∈ {

0

,

1

,

∗}

.

Notethattheoldfamiliarbooleantablesfor

∧,

∨

and

¬

arestillintuitivelysoundunderthisinformational reinterpreta-tionof1 and0.However,theyarenolongerexhaustive:theydonottelluswhathappenswhenoneoralloftheimmediate constituentsofacomplex sentencetake thevalue

∗

.Aremarkableconsequenceofthisapproachisthat thesemanticsof

∨

and

∧

becomes, asfirst noticed by Quine [24], non-deterministic. In some cases an agent a may accept a disjunction

ϕ

∨ ψ

astruewhileabstainingonbothcomponents

ϕ

and

ψ

.Thisisoftenthecasewhen,tryingtologintoawebsite,we are promptedwiththeerrormessage “eitheryourusername (

ϕ

)orpassword(

ψ

)are wrong”. Possessingthisdisjunctive pieceofinformationdoesnotgive usanydefiniteinformationabouteitherdisjunct, andthereforeitsseemsonlyrational to refrain to act asif eitherwas trueor false. Similarly, a may reject a conjunction

ϕ

∧ ψ

asfalse whileabstaining on bothcomponents.Supposea holdstheinformationthatAliceandBobarenotsiblings

¬(

ϕ

∧ ψ)

.Withthisinformationa

isclearlynotinapositiontoeitherassentordissenttosentenceslike“IsJohnAlice’sfather?”and“IsJohnBob’sfather?”. Howevera shouldcertainly dissentto “JohnisAlice’sfatherandBob’sfather”.Continuingwithinformalexamplesofthis sort,onecanalsoseethatdependingontheinformationactuallypossessedbytheagent,when

ϕ

and

ψ

arebothassigned thevalue

∗

,thedisjunction

ϕ

∨ ψ

maytake thevalue1 or

∗

,andtheconjunction

ϕ

∧ ψ

maytakethevalue 0 or

∗

.This motivatestheinformationaltablesintroducedinFig.1.

Asaconsequenceofthisinformationalinterpretation,theclassicalbooleantablesforthe

∨

,

∧

and

¬

shouldbereplaced bythe“informationaltables”inFig.1,wherethevalueofacomplexsentence,insomecases,isnotuniquelydeterminedby thevalueofitsimmediatecomponents.Anon-deterministictablefortheinformationalmeaningoftheBooleanconditional canbeobtainedintheobviousway,byconsidering

ϕ

→ ψ

ashavingthesamemeaningas

¬

ϕ

∨ ψ

[see3,p. 82].

2.2. Derivation

Theinferenceswhichareallowed bytakingtheclosureoftheinformationaltablesjust described canbecharacterised equivalentlyintermsofIntroduction(Table1)andEliminationrules(Table2).

TherulesinTables1and2determineanotionof0-depthconsequencerelationasfollows.

Definition1.Foranysetofformulas



∪ {

α

}

⊆

F m_L,welet





0

α

iffthereisasequenceofformulas

α

1

,

. . . ,

α

n suchthat

α

n

=

α

andeachformula

α

i iseitherin



orobtainedbyapplicationoftherulesinTable1andTable2ontheformulas

α

j with j

<

i.

The key feature of the consequence relation



0 is that only informationactually possessed by the agent is allowed

ina “0-depthdeduction”. This makes asharpdistinction withclassicaldeductionwhere unboundeduse canbe madeof

virtualinformation, i.e.informationnot actuallypossessed bythe agent atthetime ofcarryingout the deduction.Virtual informationisusedtofillout,inallpossiblecompletions,gapsintheagent’sinformation,asillustratedbyExample1below.

Table 1 Introductionrules. ϕψ ϕ∧ ψ(∧I) ¬ϕ ¬(ϕ∧ ψ)(¬ ∧ I1) ¬ψ ¬(ϕ∧ ψ)(¬ ∧ I2) ¬ϕ¬ψ ¬(ϕ∨ ψ)(¬ ∨ I) ϕ ϕ∨ ψ(∨I1) ψ ϕ∨ ψ(∨I2) ϕ¬ψ ¬(ϕ→ ψ)(¬ → I) ¬ϕ ϕ→ ψ(→ I1) ψ ϕ→ ψ(→ I2) ϕ ¬ϕ  (I) ϕ ¬¬ϕ(¬¬I) Table 2 Eliminationrules. ϕ∨ ψ ¬ϕ ψ (∨E1) ϕ∨ ψ ¬ψ ϕ (∨E2) ¬(ϕ∨ ψ) ¬ϕ (¬ ∨ E1) ¬(ϕ∨ ψ) ¬ψ (¬ ∨ E2) ϕ∧ ψ ϕ (∧E1) ϕ∧ ψ ψ (∧E2) ¬(ϕ∧ ψ) ϕ ¬ψ (¬ ∧ E1) ¬(ϕ∧ ψ) ψ ¬ϕ (¬ ∧ E2) ϕ→ ψ ϕ ψ (→ E1) ϕ→ ψ ¬ψ ϕ (→ E2) ¬(ϕ→ ψ) ϕ (¬ → E1) ¬(ϕ→ ψ) ¬ψ (¬ → E2) ¬¬ϕ ϕ (¬¬E)  ϕ(E)

While



0 allowsfornouseofvirtualinformation,thecentralideaofDepth-boundedBooleanLogicsconsistsinkeeping

trackoftheamountk ofvirtualinformationwhichagentsareallowedtouseintheirdeductions.Thisnaturallyleadstothe recursivedefinitionoftheconsequencerelation



k,fork

>

0,asfollows.

Definition2.Foreach k

>

0 and setof formulas



∪ {

α

}

⊆

F m_L, welet





k

α

iffthere isa finitesequence offormulas

α

1

,

. . . ,

α

n anda formula

β

∈

S f

(

∪ {

α

})

,suchthat

α

n

=

α

,andforeach

α

i,withi

<

n,we have

α

1

,

. . . ,

α

i−1

,

β



k−1

α

i and

α

1

,

. . . ,

α

i−1

,

¬β 

k−1

α

i.

In other words,wesuppose that

β

isa pieceof“virtualinformation”whichis notactually possessedby theagent at levelk

−

1 butwhichcan beseen tobe sufficienttoderive

α

through case-basedreasoning.2 Notethat theconsequence relation presentedin Definition2 isan instance ofa strong depth-boundedconsequencerelations, whichis transitive,in contrast to its weak counterpart, see [4] for an exhaustive discussion. Variants ofboth strongand weak depth-bounded consequencerelationsarisewhenallowingdifferentkindofformulastobeusedasvirtualinformation(virtualspaces inthe terminologyof[4]).

Example1.Considertheexcludedmiddleformulap

∨ ¬

p.Direct inspectionoftherules inTables 1and2showsthatthe formula isnot 0-depthderivable,i.e.



0 p

∨ ¬

p.However, ifwe allowthe useofvirtual information p, wefind outthat

both p



0p

∨ ¬

p and

¬

p



0p

∨ ¬

p.Fromthisfollowsthat



1p

∨ ¬

p.

LetusjustrecallasmallsetofpropertiesofDepth-boundedBooleanlogics,whichwillplayaroleinwhatfollows. First,itisshownin[4] thattherelation



0issoundandcompletewithrespecttotheconsequencerelationdefinedon

the basisofthenon-deterministicsemantics illustratedinSection 2.1.As animmediateconsequencewecanobservethat Example 1impliesthat it is not thecasethat

φ

∨ ¬φ

is a 0-depth tautology. Thismatches quite naturally theconcerns raised againsttheso-calledprobabilistic excludedmiddle.Intheframework ofDepth-boundedBooleanlogics, ifan agent hasnoinformationabout

φ

,thentheyshouldnotbeforcedtoassignitthehighestdegreeofbelief,foritisnotatautology. Second, therecursivedefinitionof



k ensuresthattheboundeduseofvirtualinformationismonotonicandeventually becomes“unbounded”,therebyprovidingahierarchyofconsequencerelationsapproximatingtheclassicalone.

Theorem1.[4] Therelations



kapproximatetheclassicalconsequencerelation



,thatis,limk→∞



k

= 

.

2 _{Thedefinitionof}

kcanbereformulatedintermsofcalculithataddtotherulesinTable1and2,abranchingrule,tobeappliedonlyinlimitedform.

Suchrule(seee.g.[6])bearssomesimilaritytotheeliminationrulefordisjunctioninnaturaldeduction,wherethevirtualinformationplaystheroleof formulastobedischarged.

Finally,asweshalldiscussintheconcludingsectionofthispaper,thehierarchyofdepth-boundedlogicshasaverynice computationalfeature:eachconsequencerelation



kispolynomial.

3. BelieffunctionsandDepth-boundedlogic

InthissectionwewillusethetoolsofDepth-boundedlogicsandtheinformationalperspectivetorethinktheveryidea ofdegreesofbelief,makingitsensitivetothedistinctionbetweenthemanipulationofactualandvirtualinformation[5].

First,foreachformula

α

∈

F m_L,wedefinethefunctionvk_α

:

F m_L

→ {

0

,

1

}

suchthatvk_α

(

ϕ

)

=

1 iff

α



k

ϕ

andvkα

(

ϕ

)

=

0

otherwise.

Weaddtothesethefunction vk_∗ (where

∗

isasymbolnotoccurringin F m_L),suchthat vk_∗

(

ϕ

)

=

1 iff



k

ϕ

.Notethat, sincethe 0-depthlogic



0 doesnothavetautologies[4], v0_∗

(

ϕ

)

=

0 foreach

ϕ

∈

F mL.Wewillalsohaveavaluation vk_,

assigning vk



(

α

)

=

1 iff





k

α

.In thefollowing, wecall all thefunctions vkα ,for

α

∈

F mL

∪ {∗}

the k-depthinformation states.

Acoupleofobservations areinorder. First,theevaluations vk_{α can}be thoughtofaslower approximationsofasetof partial evaluationssatisfying

α

, whichsend allthe undeterminedtruth valuesto0. Anupper approximation,sending all theundeterminedtruth valuesto 1 is just obtainedbyletting wk

α

(

ϕ

)

=

0 iff

α



k

¬

ϕ

and wkα

(

ϕ

)

=

1 otherwise. Clearly, wk

a

(

ϕ

)

=

1

−

vαk

(

¬

ϕ

)

.We canrelatethepair

(

vkα

,

wkα

)

tothenondeterministicsemantics(the indeterminatevalue should

correspondtothecasewherevk

α

(

ϕ

)

=

wkα

(

ϕ

)

).

Finally,notethatthefunctionassociatingtoeachformula

α

thecorrespondingvk_{α is}notinjective:twodistinctformulas

α

and

β

mightdeterminethesamefunction vk

α andvkβ.Wecanthusthinkofsuchfunctionsasdeterminingapartitionof

thesetofformulas F m_L.

Thefirst stepinourcharacterisation ofBelieffunctionsbasedonDepth-bounded Booleanlogics consistsindefining a basicassignment over the setof formulas F m_L. Indoing so, we assume that it isnonzero over a finitesubset of F m_L. Inadditiontobeingmathematicallyconvenient,thismatchesourgeneralconcernformaintainingtheasymmetrybetween knowledgeandignorance.For,contrarytowhathappensforprobability,assigningadegree0 totheevidenceforaformula

α

is not equivalent, in our setting,to assigning 1 to the negation

¬

α

. Assignment ofdegree 0, even to infinitely many formulas,comeinasenseatnocost.

Definition3(k-depthmassfunction).Letmk

:

F mL

∪ {∗}

→ [

0

,

1

]

besuchthat

Supp

(

mk

)

= {

α

∈

F mL

∪ {∗} |

mk

(

α

)

=

0

}

isfinite.Thenmkisak-depthmassfunction if

1.



_{α∈Supp(mk)}mk

(

α

)

=

1 2.

α

∈

/

Supp

(

mk

)

ifvkα

=

vk

Theideaisthatmk

(

α

)

expressestheportionofbeliefthatak-depthboundedagentwouldassignexclusively to

α

,based onitsactualinformation.Ontheotherhand,mk

(

∗)

standsjustfortheportionofbeliefnotassignedtoanyproposition.In casethereisaformula

γ

,suchthat

α



0

γ

foreach

α

∈

Supp

(

mk

)

,wewilldenotethesupportbySuppγ

(

mk

)

.

Massfunctionsmk whosesupportshavethelatterform, expressthemorerealisticsituationwhereanagent isjudging theevidence,when alreadyinpossessionofa backgroundinformation,representedby

γ

.Henceforth,ifnosuch formula exist,witha slightabuseofnotation,wewillsometimesdenotethesupportby Supp_∗

(

mk

)

,sothatwe canuniformlyuse thenotation Suppγ ,for

γ

inF m_L

∪ {∗}

.AdaptingfromtheterminologyinuseforclassicalBelieffunctions,wewillcallthe formulasin Suppγ

(

mk

)

focal formulas.

Wearenowreadytointroducethenotionofk-depthBelieffunction.

Definition4(k-depthBelieffunction).Let

γ

∈

F m_L

∪ {∗}

,andletmk be ak-depthmassfunction withsupport Suppγ .We defineacorrespondingk-depthBelieffunction asfollows:

Bk

(

ϕ

|

γ

)

:=

α∈Suppγ(mk)

mk

(

α

)

·

vkα

(

ϕ

)

forany

ϕ

∈

F m_L.Correspondingly,ak-depthplausibilityfunction is Plk

(

ϕ

|

γ

)

:=

α∈Suppγ(mk)

mk

(

α

)

·

wkα

(

ϕ

)

Itiseasytoseethat

Plk

(

ϕ

|

γ

)

=

1

−

Bk

(

¬

ϕ

|

γ

).

Givena formula

ϕ

andabackgroundinformation

γ

, thek-depthbeliefan agent possessesabout

ϕ

canbe framed in termsoftheinterval

[

Bk

(

ϕ

|

γ

),

Plk

(

ϕ

|

γ

)

]

.

Note the strong link betweenthe representationof uncertainty provided by the interval

[

Bk

(

ϕ

|

γ

),

Plk

(

ϕ

|

γ

)

]

and the nondeterministicsemanticsofk-depthlogics.OntheonehandBk

(

ϕ

|

γ

)

countsasevidencefor

ϕ

onlythe(syntactic repre-sentationsofthe)partialevaluationswhichsufficetoobtainthat

ϕ

istrue,deterministically; Plk

(

ϕ

|

γ

)

,ontheotherhand, takesintoaccountallthepartialevaluationswhichdonotsufficetoobtainthat

ϕ

isfalse,deterministically.Clearly,within thisrangeofevaluationsliealsoalltheevaluationswhichmightmake

ϕ

true,butnondeterministically.

Henceforth,foreaseofvisualization,givenaset A,wewillfindithelpfulattimestodenotesimplyby



A the expres-sion



_a∈Aa.Withalittleabuseofnotation,weletnow:

mk

(

vkα

)

:=

{

mk

(β)

| β ∈

Suppγ

(

mk

),

vkβ

=

vkα

}

and Ik_γ

:= {

vk_α

|

α

∈

Suppγ

(

mk

)}.

Weobtainthat Bk

(

ϕ

|

γ

)

=

α∈Suppγ(mk) mk

(

α

)

vkα

(

ϕ

)

=

vk α∈Ikγ mk

(

vkα

)

·

vkα

(

ϕ

).

(1)

Hence,wecanequivalentlythinkofk-depthmassfunctionsasactuallydefinedoverIk

γ ,whichisafinerframeofdiscernment

(seee.g.[27])thanF m_L

∪ {∗}

,sinceitidentifiesformulashavingthesamek-depthconsequences.

Notealsothat,inpassingfromm0 tomk,theframeofdiscernmentIkγ getsmuchcoarserthanI0γ :functionswhichwere

previouslydistinctturnouttobethesame.Asanexample,considerthat v0_∗andv0_p∨¬p boildowntothesamefunction,i.e.

v1_∗

=

v1_p∨¬p.

AfinalcommentonDefinition4,anditsequivalentreformulation(1), isinorder.Toquantifytheirdegreeofbeliefina formula

ϕ

,anagentcollectsand“sumsup”alltheactualinformationorevidencewhichpermitstoinfer

ϕ

.Whatourlogical analysisaddstothisclassicalviewisawayofdistinguishingtwofeaturesofthisbeliefformationprocesswhichareusually conflatedinset-theoreticmodelsofbelief:ontheonehandtherepresentationoftheevidencepossessedbyanagent,and ontheotherhandtheirinferentialability.Astothefirstfeature,ourmodelborrowstheconceptofinformationstates from

thetheoryofDepth-boundedBooleanlogics.Informationstatesarebuiltstartingfromanyformulaofthelanguage:foreach

α

,we considertheinformationstatewheretheagentholdsonlythat

α

anditslogicalconsequences(uptoafixed k)are true.Thisaccountsforthesecondfeatureofbeliefformation:(limited)inferentialability.Definition4accountstransparently fortheroleofboth.Confrontthiswiththeusual“possibleworlds”representationwhichistypicalofset-theoreticmodelsof belief:eachpossibleworldcorrespondstoaclassicalevaluation,henceitrequiresthatthetruthvalueofeachpropositional variablesissettledandthatagentsarecapableoffulldeductiveclosure.

Ourfirstpropositioncollectsthemainpropertiesofk-depthbelieffunctions,andindeedjustifiestheterminology. Proposition1.EachBkis(a-b) normalized,(c) monotoneand(d) totallymonotonefunction,i.e.foreachformulas

γ

,

ϕ

,

ϕ

1

,

. . . ,

ϕ

n, itsatisfies: (a)

γ



k

ϕ

impliesBk

(

ϕ

|

γ

)

=

1 (b)

γ



k

¬

ϕ

impliesBk

(

ϕ

|

γ

)

=

0 (c)

γ

,

ϕ



k

ψ

impliesBk

(

ϕ

|

γ

)

≤

Bk

(ψ

|

γ

)

(d) Bk

(

n i=1

ϕ

i

|

γ

)

≥



_∅=S

(

−

1

)

|S|−1Bk

(



i∈S

ϕ

i

|

γ

)

Proof. (a).BythedefinitionofSuppγ

(

mk

)

,wehave

α



0

γ

foreach

α

∈

Suppc

(

mk

)

.Hencesince



0

⊆

k,weobtain

α



k

γ

, andbythetransitivityof



k,

α



k

ϕ

,i.e.vkα

(

ϕ

)

=

1 foreach

α

∈

Suppγ

(

mk

)

.Finally,bythedefinitionofBk,weobtain

Bk

(

ϕ

|

γ

)

=

α∈Suppγ(mk) mk

(

α

)

vkα

(

ϕ

)

=

α∈Suppγ(mk) mk

(

α

)

=

1

.

(b).FromthedefinitionofBk,we haveBk

(

ϕ

|

γ

)

=

0 ifandonlyifthereisatleastaformula

α

∈

Suppγ

(

mk

)

,suchthat

α



k

ϕ

.Ontheotherhand,since

α

∈

Suppγ

(

mk

)

,

α



0

γ

,hence

α



k

¬

γ

.Thelattertogetherwiththeassumption

γ



k

¬

ϕ

, givesusbytransitivitythat

α



k

¬

ϕ

.Henceweobtain

α



k



,thatis,vkα

=

vk.ByDefinition3thismeansthatmk

(

α

)

=

0, whichisincontradictionwith

α

∈

Suppγ

(

mk

)

.

(c)GiventhedefinitionofBk

(

ϕ

|

γ

)

,itsufficestoshowthat,whenever

α

issuchthatvkα

(

ϕ

)

=

1 thenvkα

(ψ)

=

1.Assume

that vk

α

(

ϕ

)

=

1, i.e.

α



k

ϕ

. Since

α

∈

Suppγ

(

mk

)

, we also have

α



k

γ

. On the other hand we get, by our hypothesis,

γ

,

ϕ



k

ψ

,hencebythetransitivityof



k,weobtain

α



k

ψ

,i.e.vkα

(ψ)

=

1.

(d).WeadaptasimilarproofinTheorem4.1in[19].Letusrecall(seeequation(1))thatmkcanbeseenasaprobability distributionoverthesetIk_{γ ,}andhencestraightforwardlydefinesaprobabilitymeasure(inusual,set-theoreticalterms)over thefiniteset

P(

Ik

γ

)

.Wethusobtain: Bk

(

n



i=1

ϕ

i

|

γ

)

=

vk α∈Ikγ mk

(

vkα

)

vkα

(

ϕ

1

∨ · · · ∨

ϕ

n

)

≥

vk α∈Ikγ mk

(

α

)

max

(

vkα

(

ϕ

1

), . . . ,

vkα

(

ϕ

n

))

=

mk

(

{

vkα

|

for some i

,

vkα

(

ϕ

i

)

=

1

})

=

∅=S⊆ {1,...,n}

(

−

1

)

|S|−1mk

(

{

vαk

|

vkα

(

ϕ

i

)

=

1 for all i

∈

S

})

=

∅=S⊆{1,...,n}

(−

1

)

|S|−1Bk

(

i∈S

ϕ

i

).

Theinequalityinthesecondlinefollowsfromtheobviousfactthat,forallthevk_{α such}thatvk_α

(

ϕ

)

=

1,i.e.

α



k

ϕ

i,wehave thatvk_α

(

ϕ

1

∨ · · · ∨

ϕ

n

)

=

1,i.e.

α



k

ϕ

1

∨ · · · ∨

ϕ

n,bythe

∨

I

ruleinTable1.Thethirdlineisareformulationoftheprevious one,usingtheadditivityofmkasaprobabilitymeasureover

P(

Ikγ

)

.

Thefourthlinefollowsfromtheapplicationoftheinclusion-exclusionprincipleforprobabilitymeasures.Finally,thelast equalityfollowsfromtheintroductionandeliminationrules forconjunction.Such rulesdetermineindeedthat, givenany

∅

=

S

⊆ {

1

,

. . . ,

n

}

,wehavethat

α



k

ϕ

iforeachi

∈

S,iff

α



k



i∈S

ϕ

i.



Example2.Assumethata0-depthboundedagentjudgesthattheactualinformationitpossessesprovidesastrongsupport onlyfortheformula p

∨

q,andnoneforits disjuncts.Wecanrepresentsucha situation,forinstancebym0

(

p

∨

q

)

=

0

.

8,

m0

(

∗)

=

0

.

2 and m0

(

α

)

=

0, for any other

α

∈

F m_L. Note inparticular that p

∨

q



0q

∨

p and p

∨

q



0

(

q

∧

p

)

∨ (

q

∧

¬

p

)

∨ (¬

q

∧

p

)

, hence one has B0

(

p

∨

q

)

=

0

.

8 and B0

(

q

∨

p

)

=

B0

((

q

∧

p

)

∨ (

q

∧ ¬

p

)

∨ (¬

q

∧

p

))

=

0, although both

q

∨

p and

(

q

∧

p

)

∨ (

q

∧ ¬

p

)

∨ (¬

q

∧

p

)

classicallyarelogically equivalentto p

∨

q.Consider nowa 1-depthagent, using the samepiece of evidence,i.e.m1

(

p

∨

q

)

=

m0

(

p

∨

q

)

=

0

.

8 and m1

(

∗)

=

m0

(

∗)

=

0

.

2. Forsuch agent v1p∨q

=

v1q∨p and v1_p∨q

=

v1_{(q∧p)∨(q∧¬p)∨(¬q∧p)}.Hence eventhough it gave nodirect support fortheinformation state on theright ofboth equalities,itwillassignB1

(

p

∨

q

)

=

0

.

8

=

B1

(

q

∨

p

)

andB1

(

p

∨

q

)

=

0

.

8

=

B1

(

q

∧

p

)

∨ (

q

∧ ¬

p

)

∨ (¬

q

∧

p

)

.

4. ThehierarchyofDepth-boundedBelieffunctions

Sofar,incorrespondencetoeachk-depthboundedlogic,wehavedefinedanotionofk-depthBelieffunction,basedona verygeneraldefinitionofk-depthmassfunction.WhilstthisrespondstothenaturalquestionofgroundingBelieffunctions onDepth-bounded Booleanlogics, itstill leavesimportantrepresentational desiderata unaddressed.Toseethis, note that while each k-depthlogical consequence (k

>

0) is recursively defined in terms of logical consequences at lower depth, no analogousrestrictionhas yetbeen imposed on themass functions, aswe move acrossdifferent depths.In particular, givenaformula

α

,ourDefinition3wouldinprincipleallowanagenttoassigncompletelyunrelatedvaluestom0

(

α

)

and

mk

(

α

)

.Addressingthisissuewillprovideanovel,tothebestofourknowledge,presentationofBelieffunctionintermsof ahierarchyofapproximationsthereof.

LetusrecallthatmassfunctionsinDempster-Shafertheory[27] aremeanttorepresentanagent’sjudgmentofevidence: inthissection,we takesuchevidencetobe determinedonlyata“shallow”level,correspondingtothemassfunctionm0.

Wecannot,however,justassumethateachmkequalsm0,sinceweneedtotakeintoaccounttheincreasedlogicalcapacity oftheagent.

Indeed,m0 isrequiredto assignvalue 0 only tothe

α

such that v0_α

=

v0_. Itmightbe thecase, however,that incon-sistenciesare recognizedassuch onlyatacertain depthk

>

0.Eachmk isobtainedthen fromm0 by distributingamong noncontradictoryformulasallthemassesoftheformulaswhichcanbeshowntobecontradictoryatdepthk.Thismotivates thefollowingdefinition.

Definition5.Let

γ

∈

F m_L

∪ {∗}

andm0

:

F m_L

∪ {∗}

→ [

0

,

1

]

bea0-depthmassfunctionwithsupport Suppγ

(

mk

)

.Wesay thatmkisanm0-basedk-depthmassfunction ifitsatisfiesthefollowing:

•

Foreach

α

∈

Suppγ

(

mk

)

,suchthat vkα

=

vk

mk

(

α

)

=

m0

(

α

)

1

−



{

m0

(β)

|

vk_β

=

vk_

}

.

Moreover,ifforeachk

>

0, Suppγ

(

m0

)

=

Suppγ

(

mk

)

,wesaythatthemkareconstantm0-basedk-depthmassfunctions. The k-depthm0-basedBelieffunctions arethen just obtainedasinDefinition 4, onthebasis ofm0-basedk-depthmass

functions.

The m0-basedBelief functionsarejust specialcasesofthose inDefinition 4,hencetheresultsofthe previoussection

stillapply.

We canthink ofm0-basedk-depthmass functionsasarising fromapeculiarkindofbelief revision,whichisnot due to newinformation,butonlyto(higher-depth)logicalreasoning.Inparticular,Definition 5expressesanagent’supdateof degreesofbelief,onlyuponthefollowingnewpieceofinformation:someoftheformulastowhichsheoriginallyassigned non-zeromassare, infact,contradictory. Suchinformationcouldalsobe encoded,forinstance,asasimplemass-function

m_k,suchthat m_k

(

α∈F m_L vk α=vk

¬

α

)

=

1

andm_k

(β)

=

0 foranyotherformula. Thenthemk inDefinition5canalsobeobtainedbyjustupdatingm0 withsuch a m_k viatheDempster-Shafer[27] ruleofcombination.

Notethat,byEquation(1),wecanalsoexpressmk,intermsoftheapparentlylooserconstraint:

mk

(

vkα

)

=



{

m0

(β)

|

vk_β

=

vk_α

}

1

−



{

m0

(β)

|

vk_β

=

vk_

}

(2) resultinginthesameclassofk-depthm0-basedBelieffunctions.

OurnextpropositionshowsthatBelieffunctionswhicharebasedonconstantm0-basedk-depthmassfunctions, deter-mineintervals

[

Bk

(

ϕ

|

γ

),

Plk

(

ϕ

|

γ

)

]

foreach

ϕ

∈

F mL,whichgetsmallerask increases.Thishasthewelcomeconsequence totheeffectthathigherlogicalabilities,asmeasuredby



k,leadtosharperuncertaintyquantification.

Proposition2.Let

γ

∈

F orm_L,m0bea0-depthmassfunction,mkbeaconstantm0-basedk-depthmassfunctionwithsupport Suppγ

(

mk

)

,and Bkand Plkthecorrespondingbeliefandplausibilityfunctions.Foreachk

≥

0,wehaveBk

(

ϕ

|

γ

)

≤

Bk+1

(

ϕ

|

γ

)

; Plk

(

ϕ

|

γ

)

≥

Plk+1

(

ϕ

|

γ

)

Proof. For each

α

∈

Suppγ

(

mk

)

,clearly

α



k

ϕ

implies

α



k+1

ϕ

,hence vkα

(

ϕ

)

≤

vkα+1

(

ϕ

)

foreach

ϕ

∈

F mL.Ontheother

hand,byDefinition5,itfollowsthatforeach

α

∈

Suppγ

(

mk

)

=

Suppγ

(

mk+1

)

suchthatvkα

=

vk,wehavemk

(

α

)

=

mk+1

(

α

)

.

HencefromDefinition4,wehaveBk

(

ϕ

|

γ

)

≤

Bk+1

(

ϕ

|

γ

)

.Asaneasyconsequenceofthisfact,weget Plk

(

ϕ

)

≥

Plk+1

(

ϕ

)

.



Letusconsidernowtheconnectionbetweenthek-depthBelieffunctionsdefinedabove andtheBelieffunctionsinthe classicalsetting.AcustomarypresentationofBelieffunctionsisassetfunctions,seee.g.[27,13],satisfyingtheset-theoretic counterpartofthe(a-d)inProposition1.

Inalogicalsetting[19],theycanberepresentedasfunctionsoverthebooleanLindenbaumalgebraofclassicallogic(see e.g. [19]).Recall that elements ofsuchalgebra arethe equivalenceclassesofthe form

[

α

]≡

γ,determined by therelation

≡

γ definedby

α

≡

γ

β

iff

γ

,

α

 β

and

γ

,

β



α

.

Let usnowintroduce, for

α

∈

F m_L,a function vα,such that vα

(

ϕ

)

=

1 if

α



ϕ

and vα

(

ϕ

)

=

0 otherwise. Notethat distinctformulas

α

and

β

maygiverisetothesamevaluation:wehavedistinct vα sonlyfordistinctclassesofthe Linden-baum algebraLindγ .Hence,inanalogytowhatwedidforDepth-boundedBelieffunctions,wecanequivalentlyformulate classicalBelieffunctionsintermsofaprobabilitydistributionoverthesetIγ

= {

vα

|

α

∈

F m_L

,

α



γ

}

.Moreprecisely,given anyclassicalBelieffunctionB

:

F m_L

→ [

0

,

1

]

wecanfindamassfunctionm (alsocalledMoebiustransform)m

:

Iγ

→ [

0

,

1

]

suchthat

vα∈Iγ

m

(

vα

)

=

1

,

m

(

v_

)

=

0,andtheBelieffunction B isobtainedby

B

(

ϕ

|

γ

)

=

vα∈Iγ

foranyformula

ϕ

.ThisshowsthatBelieffunctionsareobtainedasconvexcombinationsoffunctions(whicharenotclassical evaluations) vα

:

F m

→ {

0

,

1

}

.

Againstthisbackgroundwecannowstatethemainresultofthissection.

Theorem2.Let

γ

∈

F m_L

∪ {∗}

andB

:

F m_L

→ [

0

,

1

]

beaBelieffunction.ThenthereisasequenceofDepth-boundedBelieffunctions Bksuchthat,foreach

ϕ

wehave

B

(

ϕ

|

γ

)

=

lim

k→∞Bk

(

ϕ

|

γ

).

Proof. Bytheargumentabovewehave

B

(

ϕ

|

γ

)

=

vα∈Iγ

m

(

vα

)

vα

(

ϕ

),

soitsufficestotakeanymassfunctionm0suchthat,foreachvα

∈

Iγ



m0

(β)

| β ∈

F mL

,

vα

=

vβ



=

m

(

vα

).

Clearly,since



_v α∈Iγm

(

vα

)

=

1 wealsoget

α∈Suppγ(m0) m0

(

α

)

=

1

.

Moreover, by the definition ofm0,since m

(

v0_

)

=

0, we get m0

(

α

)

=

0 for each vα

=

v_. Hence m0 isa 0-depthmass function and, lettingmk be the corresponding m0-based k-depthmass function (see Definition 5 and the reformulation thereafterinEquation(2)),wehave

mk

(

vkα

)

=



m0

(β)

|

vkβ

=

vkα



.

ByTheorem1weknowthatlimk→∞vk_α

(

ϕ

)

=

vα

(

ϕ

)

,henceweget lim k→∞mk

(

v k α

)

=

lim k→∞

{

m0

(β)

|

vkβ

=

vkα

}

=

{

m0

(β)

|

vβ

=

vα

}

=

m

(

vα

).

Wefinallyobtain lim k→∞Bk

(

ϕ

|

γ

)

=

klim→∞

α∈Suppγ(mk) vk_α

(

ϕ

)

mk

(

vkα

)

=

vα∈Iγ vα

(

ϕ

)

·

m

(

vα

)

=

B

(

ϕ

|

γ

).



Tosumup,theconditionsonthemassfunctionpinneddownbyDefinition 5leadtotheconstructionofahierarchyof BelieffunctionswhichapproximateDempster-ShaferBelieffunctions.EachelementinthehierarchyidentifiedbyTheorem2

can thus be interpreted asan approximation ofthe Dempster-Shafer degree of beliefof a realistic agent,i.e.one whose logicalabilitiesareboundedby



k.

5. Restrictingthefocalformulas:atomsandsubatoms

Letus now investigatesome classes ofk-depth Belief functionsarising fromrestrictions ofthe set offocal formulas. Throughoutthissection,wefixV ar_L

= {

p1

,

. . . ,

pn

}

andassume,forsimplicity,nobackgroundinformation,or,whichisthe same,that thebackgroundinformationamountsto

∗

.

WefirstconsiderthecaseofSupp

(

m0

)

⊆

At_L,where

At_L

= {

±

pi

|

pi

∈

V arL

}.

Intheclassicalsetting,thisreducesBelieffunctionstoprobabilities.Atomscorrespondindeedtobooleanevaluations:itis easy toseethat foreach booleanevaluation v

:

F m

→ {

0

,

1

}

thereisaunique atom

α

suchthat forall

ϕ

∈

F m,we have

TherestrictiontoatomsproducesalsoacollapseofthehierarchyoftheDepth-boundedlogics:atomscorrespondindeed to evaluations where (informational) truth and falsity is assignedto all propositional variables, and the tables in Fig. 1

coincidewiththeclassicalones,whentheindeterminatevalue

∗

isnotassigned.Asaproof-theoreticalcounterpartofthis fact,itiseasytocheckthat,foranyatom

α

∈

At_L,foreachk

≥

0,wehave

α



k

ϕ

iff

α



ϕ

forevery

ϕ

∈

F mL,i.e.vkα

=

vα.

In lightofsuch considerations,weobtain thatforany0-depthmassassignmentm0,with Supp

(

m0

)

⊆

At_L,the corre-sponding m0-based k-depthBelieffunction is such that Bk

(

ϕ

)

=

Bk+1

(

ϕ

)

foreach k

≥

0 and foreach

ϕ

∈

F mL.Each Bk thussatisfies,inadditiontothepropertiesofLemma1,alsofiniteadditivity,andthereforeisjustasaprobabilityfunction.

Amoreinterestingexamplearises,whenletting

Subat_L

= {

i∈I

±

pi

|

I

⊆ {

1

, . . . ,

n

}} ∪ {∗}

we assume that Supp

(

m0

)

⊆

Subat_L.While atomsare inone-to-one correspondencewithbooleanevaluations,each sub-atom

α

determines a corresponding three-valuednon-deterministic evaluation, i.e. theone which assigns the value

∗

to eachpropositionalvariablenotoccurringin

α

.

Giventhiscorrespondence,wecantakem0

(

α

)

torepresentthebeliefthatanagentassignsexclusivelytoasingle (three-valued)evaluation,i.e.theonecorrespondingtotheformula

α

.Thisamountstotheagentbeinginpossessofinformation aboutthetruthorfalsityofsomepropositionalvariables,butnotnecessarilyallofthem.Thiscaseisintermediatebetween the generalsettingpresentedinSection 6,wheretheagent expressesbeliefover arbitraryformulas(whichwe canthink ofasbeliefcommittedtosetsofpartialevaluations) andthecasejust discussedabove,reducingtoprobability,wherethe agent expressesbeliefoveraclassicalevaluation,i.e.abeliefconcerningthetruthorfalsityofevery propositionalvariable inthelanguage.

Letusobserveafewinterestingfeaturesofthisfamilyofk-depthBelieffunctions.

First, unlike the previous casebased on atoms, the k-depthBelief functions Bk willbe ingeneral distinct fromeach other.

Second,the B0 Belieffunctionsturnouttobefinitelyadditive,asweshowinthefollowing.

Proposition3.Letm0bea0-depthmassfunction,suchthatSuppγ

(

m0

)

⊆

Subat_L.Foreachformulas

ϕ

,

ψ

∈

F m_L,wehave B0

(

ϕ

∨ ψ) =

B0

(

ϕ

)

+

B0

(ψ )

−

B0

(

ϕ

∧ ψ).

Proof. Bydefinition B0

(

ϕ

∨ ψ) =

α∈Supp(m0) m0

(

a

)

v0α

(

ϕ

∨ ψ) =

α∈Subat_L m0

(

a

)

v0α

(

ϕ

∨ ψ).

Note that



0 has the disjunction property, i.e.

ϕ

∨ ψ

is derivable iff either

ϕ

or

ψ

are derivable. This means that, if

α

∈

Subat_L,then v0_α

(

ϕ

∨ ψ)

=

1 iff v0_α

(

ϕ

)

=

1 orv0_α

(ψ)

=

1.Weobtainthen

v0_α

(

ϕ

∨ ψ) =

v0_α

(

ϕ

)(

1

−

v0_α

(ψ ))

+

v0_α

(ψ )

· (

1

−

v0_α

(

ϕ

))

+

v0_α

(

ϕ

)

·

v0_α

(ψ ).

Hence,usingthelatter,weget:

B0

(

ϕ

∨ ψ) =

α∈Subat_L m0

(

v0α

)

v0α

(

ϕ

∨ ψ)

=

α∈Subat_L m0

(

α

)

[

v0α

(

ϕ

)

+

vα0

(ψ )

−

v0α

(

ϕ

)

·

v0α

(ψ )

]

=

α∈Subat_L m0

(

α

)

v0α

(

ϕ

)

+

α∈Subat_L m0

(

α

)

v0α

(ψ )

−

α∈Subat_L m0

(

α

)

v0α

(

ϕ

∧ ψ)

=

B0

(

ϕ

)

+

B0

(ψ )

−

B0

(

ϕ

∧ ψ).



Remark1.Note that theresultfor B0 is alsoan immediateconsequence ofTheorem 5in [20], sincethe evaluations v0_α

satisfyT2andT3,intheterminologyof[20].