AbducingthroughNegationasFailure: Stablemodelswithin theindependentchoicelogic

David PooleDepartment�

of ComputerScienceUni�

versityof British Columbia2366Main Mall

V�

ancouver, B.C.,CanadaV6T 1Z4poole@cs.ubc.ca�

http://www.cs.ubc.ca/spider/poole

May�

21,1998

Abstract

The�

independentchoicelogic (ICL) is partof aprojectto combinelogicand� decision/gametheoryinto a coherentframework. The ICL hasa sim-ple� possible-worldssemanticscharacterisedby independentchoicesandanac� yclic logic programthatspecifiestheconsequencesof thesechoices.Thispaper� gives an abductive characterizationof the ICL. The ICL is definedmodel-theoretically, but weshow thatit is naturallyabductive: thesetof ex-planations� of apropositiong� is aconcisedescriptionof theworldsin whichg� is true. We give analgorithmfor computingexplanationsandshow it issoundandcompletewith respectto thepossible-worldssemantics.Whatisunique aboutthisapproachis thattheexplanationsof thenegationof g� can� bederi�

ved from theexplanationsof g� . Theuseof probabilitiesover choicesinthis

framework andgoingbeyondacyclic logic programsarealsodiscussed.

1 Intr oduction

This paperis part of a projectaimedat combininglogic anddecisionor gametheory�

into a coherentframework [23]. This project follows from the work on

1

�������������������� "!��#���%$�&'�)(+*,�-!#�.(+/)01(���23�4 "&2

probabilistic5 Horn abduction[19], which showed how a combinationof inde-pendent5 probabilistichypothesesanda restrictedacyclic logic program(withoutnegationasfailure)thatgives theconsequencesof thechoices,canrepresentanyprobability5 distribution in muchthesameway as do Bayesiannetworks[17].

Theindependentchoicelogic (ICL) [23] buildsonprobabilisticHornabduc-tion�

toextendthelogic to includearbitraryacyclic logicprograms(thatcanincludene6 gationasfailure)andallow differentagentsto make choices.Thegeneralideais to have a structuredhypothesisspace,and an acyclic logic programto givethe�

consequencesof hypotheses.Thehypothesesarepartitionedinto alternatives7 .The setof all alternatives is a choicespace. Thereis a possibleworld for eachselectionof oneelementfrom eachalternative; the logic programspecifieswhatis true in that world. The semanticsof negationas failure is given in termsofstablemodels.This framework is definedmodel-theoreticallybut, aswe show inthis�

paper, it is naturallyabductive: the setof explanationsof a propositiong8 is9a concisedescriptionof theworlds in which g8 is true. We give analgorithmforcomputingexplanationsandshow it issoundandcomplete.Whatis uniqueaboutthis�

approachis thattheexplanationsof thenegationof g8 canbederived from theexplanationsof g8 ; thealgorithmis basedonReiter’s [27] hitting setalgorithm.

In:

otherwork [23], weconsiderhow thisframework canbeusedfor modellingmultipleagentsunderuncertaintyin awaythatextendsdecisionandgametheory.By;

allowingdifferentagentstomakeindependentchoices,thesemanticframeworkextendsthenotionof thestrategic or normalform of agame[29]by allowing for alogic<

programto modelthedynamicsof theworld andthecapabilitiesof agents.Oneof theagentscanbenature;in thiscasewehaveprobabilitydistributionsoveralternatives,with thealternatives correspondingto independentrandomvariables[19].

The goal of abduction[25, 12] is to explain why someobserved propositionis9

true;we wanta descriptionof whatthe(real)world maybelike to accountforthe�

observation. Theinput is a setof assumables(possiblehypotheses),a logicaltheory�

thataxiomatiseswhatfollowsfromtheassumables,andanobservationtobeexplained.Givenanobservationwewanttheexplanationstobedescriptionsof howthe�

world couldbeto producetheobservation: formally we wanta consistentsetof assumablesthatlogically impliestheobservations.Thusabductionis inherentlyaboutpartial information;beforewe make an observationwe don’t know whichassumptionswewill make.

In contrast,negationasfailure[5] is aboutcompleteknowledge.If someatomcannotbeproved, its negationis inferred. In combiningnegationasfailurewithabduction,we have completeknowledgeaboutsomepredicateseven thoughwe

=�>�?�@�A�B�C�D�E�F�G"H�@#D�F%I�J'D)K+L,B-H#C.K+M)N1K�B�O3@4G"J3

may only have partial knowledgeaboutothers[6, 20, 9]. Therearemany areaswherewewantnegationasfailureto meanthatall of thecasesfor apredicatehavebeenP

coveredeventhoughwe maynot have completeknowledgeaboutall of theatomsthat make up the definition. For example,thereis a neatsolution to theframeproblemusinglogic programmingandnegationasfailure[14,1,28],whichcanstill beusedevenif wedon’t havecompleteknowledgeaboutall of theatomsin9

thebodies.Considerthefollowing example:

Example1.1 Considerasimpledomainwherethereis a robotandakey, andtherobotQ canpick up or put down thekey, andmove to differentlocations. We canwrite rulessuchas,therobot is carryingthekey after it has(successfully)pickedit up1:

carryingR keS y T sU V T W�WYXdoZ [

pic\ kup] keS ŷ+_ T `baatc d robote Pof sg T hbiatc j keS y k Pof sl T mbnpic\ kup_succeedsU o T p+q

Togetherwith this rule thatspecifieswhencarryingcommences,weneedaframeruleQ tospecifywhencarryingpersists.Thegeneralformof aframeaxiomspecifiesthat�

a fluent is trueaftera situationif it weretruebefore,andtheactionwerenotonethatundidthefluent,andtherewasno othermechanismthatundidthefluent.Fr

or example,anagentis carryingthekey aslongastheactionwasnotto putdownthe�

key or pick up thekey, andtheagentdid not accidentallydropthekey whilecarryingoutanotheraction:2

s

carryingt keS y u sU v T w�wYxcarryingy keS y z T {b|} doZ ~ putdown\ keS y# T b doZ pic\ kup keS y# T b drZ ops keS y T #

1W

e areusingProlog’s conventionwith variablesin uppercase,but with negationwritten as“ ”, and conjunctionas “ ”. This axiomatisationis similar to a situationcalculusdefinition,but what whetheraction is attemptedat any time is a proposition. This is closerto the event

calculus[15], whereweareexplicitly interestedin narratives [28].2Note

that do pic kup ke y� T is anelementof thebodyof this rule becausewe don’t wantboth

rulesto beapplicablewhentheagentis carryingthekey andtriesto pick it up. In thatcase,for

thesakeof makingachoice,weassumetheactionis like thecasewhenit justpicksupthekey.

 �¡�¢�£�¤�¥�¦�§�¨�©�ª"«�£#§�©%¬�'§)®+¯,¥-«#¦.®+°)±1®�¥�²3£4ª"4³

Thesetwo rules cover all of the caseswhen the robot is carrying the key. Bythese�

rules,we really meanthe completion: the robot is carryingthe key if andonly if oneof thesetwo casesoccurs.However, wedon’t wantto globallyassumecompleteknowledge. For example,we may not know whetherpickup succeedsor whethertherobotdropsthekey (theICL [23] allows uncertaintyexpressedasprobabilities5 for theseatoms;seeSection5).

Supposewe have someexplanationsfor why the robot may drop the key3´.

Eachµ

suchexplanationwill serve to describeconditionsin which theagentdropsthe�

key. In all othersituations,theagentwon’t dropthekey, andso,assumingtheagentis carryingthekey anddoesn’t putdown or pickupthekey, shouldserve forexplanationsfor theagentcarryingthekey inthenext state.Whatdistinguishedthiswork is theinteractionof abductionandnegationasfailure. A setof explanationsfor¶

drZ

ops· keS y ¸ T ¹ will induceanothersetof explanationsfor º drZ ops» keS y ¼ T ½ , whichthen�

canbeusedto find explanationsfor carrying¾ keS y ¿ sU À T ÁÂÁ . These,in turn,couldbeP

usedtoderiveexplanationsfor à carryingÄ keS y Å sU Æ T ÇÂÇ . In theindependentchoicelogic, thisabductivecharacterizationis aconsequenceof anindependentlydefinedmodel-theoreticsemanticsfor thelogic.

WÈ

efirst overview theICL andgiveamodeltheoreticsemantics[23]. Themaincontribution of this paperis to provide anabductive characterisationof the logicin termsof operationsonsetsof assumptions(in termsof compositechoices)andhoÉ

w theassumptionsinteractwith thelogicprograms.Weshow how theabductivemachinerycanbeusedfor probabilisticreasoningandfor Bayesiandecisiontheory,andfinally discussgoingbeyondacyclic logic programs.

2 Background: Acyclic Logic Programs

WÈ

e usethe Prologconventionswith variables startingan uppercaseletter andconstants, function

Êsymbols, and prË edicate symbols startingwith lower case

letters.A termÌ

is eitheravariable,aconstant,of is of theform fÍ Î

t1 Ï�Ð#Ð+Ð+Ï tmÑ wherefÍ

is9

a function symboland t1 Ò�Ó+Ó+Ó+Ò tmÔ are terms. An atomic formula (atom) iseithera predicatesymbolor is of the form p\ Õ t1 Ö�×+×+×#Ö tmØ wherep\ is a predicatesymbolandt1 Ù�Ú+Ú+Ú+Ù tm areterms.A fÊ ormula is9 eitheranatomor is of theform Û fÍ ,

3To contrastthis with otherwork on abductive logic programming[12, 11], considerthecasewhereÜ dr opÝ ke y Þ Tß à isn’á t a logicalconsequenceof theclausesandnoexplanationsfor dr opâ ke y ã Tß äinvolveproofsthatusenegationasfailure. In standardabductivelogicprogramming,nohypothesesneedå to be addedto explain æ dr opsç ke y è Tß é . Any explanationsfor dr opê ke y ë Tß ì serví e only todisalloî

w othercombinationsof assumptions.

ï�ð�ñ�ò�ó�ô�õ�ö�÷�ø�ù"ú�ò#ö�ø%û�ü'ö)ý+þ,ô-ú#õ.ý+ÿ��1ý�ô��3ò4ù"ü5

fÍ �

g� or fÍ � g� wherefÍ andg� areformulae.A clauseis eitheranatomor is a ruleof theform ac � fÍ whereac is9 anatomandfÍ is9 a formula(thebody� of theclause).A logic program is asetof clauses.

A

ground term� isatermthatdoesnotcontainany variables.A groundinstanceof a term/atom/clausec is a term/atom/clauseobtainedby uniformly replacinggroundtermsfor the variablesin c. The Herbrand base is

9the set of ground

instances9

of theatomsin the language(inventinga new constantif the languagedoesnotcontainany constants).A Herbrand interpretation is anassignmentoftrueÌ

or falseÊ

to�

eachelementof theHerbrandbase.If Pf

is9

aprogram,let gr� Pf � bePthe�

setof groundinstancesof elementsof P.

Definition 2.1([10]) InterpretationM is a stable model4 of logic programF�

if9

for every groundatomh

, h

is truein M if andonly if eitherh

�

gr� � F � or thereisa rule h

�b in gr� � F � suchthatb is truein M . ConjunctionfÍ � g� is truein M if

bothP

fÍ

andg� aretruein M� . DisjunctionfÍ � g� is9 truein M� if9 eitherfÍ or g� (or both)aretruein M . Negation � fÍ is truein M if andonly if fÍ is not truein M .Definition 2.2([1]) A logic programF is acyclic if thereis an assignmentof anatural6 number(non-negative integer)to eachelementof theHerbrandbaseof F�suchthat,for every rule in gr� � F � the� numberassignedto theatomin theheadofthe�

rule is greaterthanthenumberassignedto eachatomthatappearsin thebody.

Ac

yclic programsaresurprisinglygeneral.Notethatacyclicity doesnot pre-cluderecursive definitions. It just meansthatall suchdefinitionshave to bewellfounded. They have very nice semanticproperties,including the following thatareusedin thispaper:

Theorem2.3([1]) Ac

yclic logic programshave thefollowing properties:

1. Thereis auniquestablemodel.

2. Clark’s completion[5] characteriseswhatis truein thismodel.

Section7 discusseswherewemaywantto gobeyondacyclic programs.

4Thisisaslightgeneralizationof thenormaldefinitionof astablemodelto includemoregeneralbodies

in clauses.This is doneherebecauseit is easierto describethe abductive operationsinterms�

of thestandardlogical operators.Note thatunderthis definitionb� ��� �

a! isá thesameasb� "

a! .

#%$'&)('*,+�-'.0/21'354)(6.'187%9:.�;=+?46-@;=A�BC;,+�DE(F3596

3 IndependentChoiceLogic

InG

thissectionwedefinethesemanticsof theindependentchoicelogic (ICL) wherethe�

baselogic is thesetof acyclic logicprogramsunderthestablemodelsemantics.Section4 givesanabductivecharacterisationof thelogic.

DefinitionH

3.1 An

independentI

choicelogic theory is9

apair J C K F� L , whereC, thechoicespace, is a setof non-emptysetsof groundatomicformulae,such

that�

if M 1 N C, O 2 P C and Q 1 RSUT 2 then� V 1 WYX 2 Z\[^] . An elementof Cis9

calledan alternative. An elementof an alternative is calledan atomicchoice.

F, the facts, is an acyclic logic programsuchthatno atomicchoiceunifieswiththe�

headof any rule.

InG

thispaperweassumethateachalternative is finite, andthattherearecountablyman_ y alternatives.

Thesemanticsis definedin termsof possibleworlds. Thereis apossibleworldfor¶

eachselectionof oneelementfrom eachalternative. Theatomswhich followfrom theseatomstogetherwith F aretruein thispossibleworld. Thisis formalisedin9

thenext two definitions.

DefinitionH

3.2 Given independentchoicelogic theory ` C a F� b , aselectorfunctionis a mapping ced C f g C suchthat hjilknmporq for all s\t C. The range ofselectorfunction u , writtenR v^wyx is9 theset zl{j|^}~ C . Therangeof aselectorfunction¶

is calleda totalÌ

choice.

Definition 3.3 SupposewearegivenICL theory C F . Foreachselectorfunction , weconstructapossibleË world w . If fÍ is9 aclosedformula,andw is9 apossibleworld, f

Íis trueÌ

in world w basedP on C F , written w @ C F fÍ , if fÍ is trueinthe�

(unique)stablemodelof F�

R ^ . fÍ is9 falseÊ in9 world w otherwise.WhenÈ

understoodfrom context, the C   F� ¡ is9 omittedasa subscriptof ¢ £ . Theuniqueness¤ of themodelfollows from theacyclicity of F� ¥ R ¦l§y¨ .

Note©

that, for eachalternative ª¬« C andfor eachworld w , thereis exactlyoneelementof ® that� is truein w ¯ . In particular, w °²± ³µ´j¶l·n¸ , andw ¹²º »µ¼½ for¶all ¾À¿²ÁÃÂrÄlÅjÆ^ÇnÈ,É .

Ê%Ë'Ì)Í'Î,Ï�Ð'Ñ0Ò2Ó'Ô5Õ)Í6Ñ'Ó8Ö%×:Ñ�Ø=Ù>Ï?Õ6Ð@Ø=Ú�ÛCØ,Ï�ÜEÍFÔ5×7

4 AbductiveCharacterisation of the ICL

TheÝ

semanticframework here,with probabilitiesonthechoices5 andacyclic logicprograms5 withoutnegationasfailure(with someotherrestrictionsthatarerelaxedhere;É

seeSection8.3)was thebasisfor probabilisticHornabduction[19]. Oneofthe�

mainresultsof [19] (proven in theappendixof thatpaper)was that thesetofminimalexplanationsof g� is aconcisedescriptionof thepossibleworldsin whichg� is9 true.

In thispaperwegiveanabductivecharacterisationof theabovesemanticdef-inition9

of consequencethat allows for negationasfailure. In particular, the ex-planations5 of a propositionwill be a descriptionof the setof possibleworlds inwhich thepropositionis true. This canberelatedto previouswork on abductivelogic<

programming[12], but thereis adifferentinteractionbetweenabductionandnegationasfailure. If g� hassomesetof explanations,then Þ g� alsohasa setofexplanationswhicharethedual

Zof theexplanationsof g� (Section4.2). Weinterpret

negationquitedifferently from the interpretationas“f ailure to prove” [5] that isappropriatewhenthereis completeknowledgeof all propositions. Negation isinterpreted9

with respectto eachworld; ß g� is9 true in a world if g� is9 falsein thestablemodeldefiningthatworld. Thisis in contrastto theview that à g� meansthatg� cannotbe proved—theremay be many proofsfor g� (eachrelying on differentassumptions)but thisdoesn’t meanwecan’t alsoexplain á g� . Section8.2discussesthe�

relationshipwith abductive logic programmingin moredetail.

4.1â

CompositeChoices

TheÝ

notion of a compositechoice is9

definedto allow us to partition the worldsaccordingto whichatomicchoicesaretrue. This formsthebasisfor theabductivecharacterisation.

DefinitionH

4.1 A

setã of atomicchoicesisconsistentwith respecttochoicespaceC if thereis no alternative which containsmorethanoneelementof ä . WhereC is understoodfrom context, we just saythat å is consistent.A consistentsetof atomicchoicesis calleda compositechoice. If A

æis9

a setof alternatives,acompositechoiceonA isasetof atomicchoicesthatcontainsexactlyonememberof eachelementof A

æ(andnoothermembers).

5çThereisaprobabilitydistributiononeachalternative(afunctionP0è éëê Cì íïî 0ð ñ 1ò suchí thatfor

alló ôöõ Cì ,÷ øöùëúëû Pü 0è ýÿþ���� 1), wherethedifferentalternativesareprobabilisticallyunconditionallyindependentá

(seeSection5).

�������������������������������� �!�"���#� $�%&���'()���8

A consistentsetof atomicchoicesis satisfiable:it canalwaysbeextendedtoa total choice,which is truein apossibleworld.

Thefollowing lemmashows therelationshipbetweencompositechoicesandtotal*

choices.

Lemma 4.2 A setof atomicchoicesis a totalchoiceif andonly if it is amaximalcompositechoice.

This+

is becauseeachtotal choiceis acompositechoice,andthereis nocompositechoicethatis a (strict) superset.Eachmaximalcompositechoiceis therangeof aselectorfunctionandsois a total choice.

Theelementsof acompositechoiceareimplicitly conjoined:compositechoice, is- truein world w . , written w /10 243 , if 576 R 8: 9 . Thusa setof compositechoicescanbeseenasa DNFformulamadeupof atomicchoices.

Definition 4.3 Two compositechoicesarecompatible if theirunionis consistent.A

setK of compositechoicesis mutually? incompatible if- for all @ 1 A K , B 2s C K ,D1 EFHG 2 implies I 1 J�K 2 is inconsistent.

Given thesyntacticdefinitionsof incompatibleandmutuallyincompatible,wecangiveanequivalentsemanticcharacterisation:

LemmaL

4.4 T+

wo compositechoicesarecompatibleif andonly if thereis aworldin whichthey arebothtrue. A setK of compositechoicesismutuallyincompatibleif-

andonly if thereis noworld in whichmorethanoneelementof K is-

true.

WÈ

eusesetsof compositechoicesasdescriptionsof setsof worlds. Thesearetypi-cally muchmoreconcisedescriptionsthandescribingthepossibleworldsdirectly(seesection6). Thisis usedto developanabductivecharacterisationof theICL. Inorderto developthetheory, wedefinesomeoperationsoncompositechoices.Thefirst is finding thecomplementof a setof compositechoices(a setof compositechoicesthat describethe complementarysetsof worlds to the original set),andthe*

notionof adual,which is asyntacticoperationto find acomplementarysetofcompositechoices.We thengive anabductive characterisationof theICL, whereexplanationsof aformulacorrespondto compositechoicesthatentailtheformula.This characterisationshows theinteractionbetweenthechoicesandtherules. InparticularM , negation asfailure is handledusingthe dualsof setsof explanations.

N�O�PQ�RS�T�U�V�W�X�YQ�U�W�Z�[�U�\ ]!S"Y�T#\ ^�_&\S�`(Q)X�[9

Thefinal operationis splittingcompositechoiceswhich is usefulfor makingmu-tually*

incompatiblesetsof compositechoicesthatdescribethesamesetof possibleworldsastheoriginal.

4.2â

Dualsand Complements

IfG

K is-

a setof compositechoicesdescribinga setof worlds, we often want adescriptionof all of theotherworlds. This will becrucial in definingabductionthrough*

negationasfailure;if somesetof compositechoicesdescribestheworldsin which ga is true,thenga is falsein all of theotherworlds,so b ga is truein theseotherworlds.

Definition 4.5 If K is asetof compositechoices,thenacomplementof K is asetK c of compositechoicessuchthatfor all worldswd e , wd fhg i K j if f wd kmln o K .Thenotionof a dualwill bedefinedto give a way to constructa complementofa setof compositechoices.Theideais thata dualof K containschoicesthatareincompatible-

with everyelementof K :

Definition 4.6 If K is a setof compositechoices,thencompositechoice prq is adual of K with respectto choicespaceC if sutwv K x�y{z}|~�{&� , & ,{}

C suchthat :�& . A dualis minimal if nopropersubsetis alsoadual.Let duals

C K   be¡ thesetof minimal dualsof K with respectto C. (Usually the

choicespaceC is-

implicit from thecontext.)

Example¢

4.7 SupposeC £¥¤¤ a¦ § b ¨ c©�ª�« d ¬ e f® ¯°¯ , thenduals

C ±²°² a¦ ³ d ´�µ�¶ b · ȩ°¸°¹»º¥¼¼ c½ ¾¿ f® À�Á b à d Ä�Å�Æ a¦ Ç eÈ° È�ÉThe+

following lemmashows the relationshipbetweendualsandnegation. Thedualsof asetK of compositechoicesis adescriptionof thecomplementarysetofpossibleM worldsthatis describedby K :LemmaL

4.8 IfÊ

K is-

asetof compositechoices,duals Ë

K Ì is- acomplementof K .InÊ

otherwords,for everyworld wd Í anelementof K is- truein wd Î if- f no elementofduals Ï

K Ð is truein wd Ñ . For aproofof this lemmaseeAppendixA.

Ò�Ó�ÔÕ�Ö×�Ø�Ù�Ú�Û�Ü�ÝÕ�Ù�Û�Þ�ß�Ù�à á!×"Ý�Ø#à â�ã&à×�ä(Õ)Ü�ß10

4.2.1 Computing Duals

There+

is a strongrelationshipbetweentheideaof a dualof K andthenotionof ahitting set[27]. Insteadof hitting onememberof every elementof theset,a dualhitsacomplementof oneof themembersof eachelementof K .

Definition 4.9 [27, Definition 4.3] SupposeC is a collectionof sets.A hittingsetf

åorCi sas et H

æ çSè é

C Sê

suchthatH ë Sê ìíïî:ð for eachSê ñ C.Definition 4.10 If ò is anatomicchoice,thecontrary to* ó with respectto choicespaceC, writtencontC ô:õ&ö is ÷1øúùüû&ý whereþ is thealternativein C whichcontainsÿ . Thisiswell definedas� is- in auniquealternative. If � is- acompositechoice,thecontrary of � with respecttochoicespaceC, writtencontC ����� , is ��� contC ���� .Example4.11 Considerexample4.7,whereC ����� a¦ � b � c����� d � e� f® ��� , then

contC �! a¦ " d #�$&% ' b ( c ) e* f® +contC ,�- b . e/�021 3 a¦ 4 c 5 d 6 f® 7

8:9 is- a dualof K , meansthatfor every element;=< K , >:? containsanelementthat*

is contraryto oneelementof @ . But this is thatsameasA�B containsanelementof contC CED:F . Thuswehave:Theorem4.12 G:H is- a dualof K with respectto C if- f I�J is- a consistenthitting setof K contC LEM:NPORQTS K U .

Reiter’shittingsetalgorithm[27,Section4.2]isdirectlyapplicablefor comput-ingduals.Wecanpruneany branchesin thehittingsettreewherethecorrespondingsetof atomicchoicesis inconsistent.This is shown in Figure1.

Example¢

4.13 ContinuingExample4.11, the set of consistentminimal hittingsetsof V�V b W c X eY f® Z\[^] a¦ _ c ` d a f® b�b is c�c cd�e�f f® g\h�i b j d k\l^m a¦ n eo�o . Althoughtheset p b q a¦ ris-

a hitting set it is inconsistent,and so not consideredin the set of duals. Itis easyto checkthat eachof the 9 worlds is coveredby either s�s a¦ t d u\v^w b x ey�y orz�z

c{\|^} f® ~\^ b d \� a¦ e� , but notboth.

^�^�^^��\^:�\� :¡��¢£¤�11

pr¥ ocedureduals ¦ K §Input: K — asetof compositechoicesOutput: thesetof dualsto K

SupposeK ¨ª©�« 1 ¬����¬¤® n¯ ° .Let±

D²

0 ³�´�´�µ�µ ; % Di¶ is thesetof dualsof ·E¸ 1 ¹^º�º\º�¹\» i¶ ¼for i ½ 1 to¾ n¿ do

Let±

D²

i À�Á d ÂÄà cÅPÆ d Ç D² i¶ È 1 É c Ê contC ËEÌ i¶ Í�Î ;remove inconsistentelementsfrom Di ;remoÏ veany Ð fromå D² i if- ÑÓÒ:ÔÖÕ D² i¶ suchthat ×:ØÚÙÜÛ

endfor;return Dn.

Figure1: Findingthedualof asetof compositechoicesK

4.3 Entailment

Definition 4.14 If Ý andÞ arepropositions,ß entails à with respecttoindependentchoiceframework theory á C â Fã ä if- å is- truein all worldsin which æ is- true.Example¢

4.15 IfÊ

C çéè�è a¦ ê bë� ë andFã ìîí c ï a¦ ð d ñ bò then* ó d entailsc anddentails ô c. Therearetwo worldshere: onewith a¦ õ c ö\÷ b ø\ù d true* andonewithb ú d û¤ü a¦ ý¤þ c true.*Entailmentcanbecontrastedwith theconsequencerelationof a logic program:

Definition 4.16 If ÿ is acompositechoice,wewrite � � ��� if � is truein thestablemodel� of Fã �� .If �� �� then� � entails � . The following exampleshows how entailmentin thesenseof Definition4.14is richerthanconsequenceby � � , even whentheleft-sideis�

acompositechoice:

Example4.17 SupposeC ����� a¦ � b����� c d !�! andthefactsare:ga 1 " a¦ # c $ga 1 % b & c '

c entailsga 1 bu( t c ) * + ga 1. In every world in which c is true,eithera¦ is trueor b istrue,�

andsoga 1 is alsotrue.

,.-0/�1024365078:90;=.@A7CBEDF3G

. 0¡�¢0£4¤6¥0¦§:¨0©=ª�¢�¦0¨?«.¬A¦CE®F¤Gª�¥HE¯C°K4¤6±M¢N©=¬13

4.4²

Explanations

DefinitionP

4.21 IfÊ

g³ is� a groundpropositionalformula,anexplanation of g³ is� acompositechoicethatentailsg³ . A minimal explanation is anexplanationsuchthat�

no subsetis anexplanation.A covering setof explanationsof g³ is� a setofexplanationsof g³ suchthatoneelementof thesetis truein all worldsin which g³is true.

A coveringsetof explanationsof g³ will betruein exactly theworldsin whichg³ istrue.�

Thiswill form aconcisedescriptionof theworldsin whichg³ is� true.Definition 4.22 If K1 andK2 aresetsof compositechoices,definetheconjunctionof K1 andK2´ to� bethesetof compositechoices:

K1 µ K2 ¶¸·g¹ 1 º¼» 2 ½¿¾ 1 À K1 Á� 2 à K2 Ä consistentÅWÆ 1 ÇÈ 2 É4Ê�ËIt is easyto seethatK1 Ì K2 definesthoseworldswherebothK1 andK2 aretrue.WO

eusethesymbol“ Í ” as theconjunctionis like thecrossproduct,but whereweareunioningthepairsandremoving inconsistentsets.

ThisQ

operationis usedin thefollowing recursive procedureto computeexpla-nations:ÎDefinitionP

4.23 IfÊ

G is�

agroundpropositionalformula,expl Ï GÐ is� thesetof com-posites choicesdefinedrecursively asfollows:

expl Ñ GÒÔÓ

minsÕ Ö expl × AØ`Ù expl Ú BÛ�Û if G Ü A Ý BminsÕ Þ expl ß Aà á`â expl ã Bä å4å if� G æ Aà ç Bäduals è

expl é Aê�ê if G ëíì Aî4îGï�ï if� G ð�ñ CòW ó

if G ô gr³ õ F öminsÕ ÷ i expl ø Bä i¶ ù4ù if� G úûýü C þ G ÿ Bä i � gr³ � Fã �

whereminsÕ � Sê ������� Sê �������� Sê ������������ . duals is� definedin Figure1. expl is�well definedasthetheoryis acyclic.

Note!

thatthesettheexplanationsof aformulaiscompositionalontheexplanationson theatomicformulaethatmake up theformula. In particular, theexplanationsof " Aà arecomputedfrom theexplanationsof Aà , andtheexplanationsof Aà # Bä arederived from theexplanationsof A

àandtheexplanationsof B

ä.

expl canbe useddirectly asa recursive procedureto computeexplanations,eithertop-down or bottom-up.Thefollowing theoremestablishesthecorrectnessof theexpl procedure:s

$&%�')(�*,+.-�/1032�465)(�/�287&9:/�;=+?5�-@;=A�B�;,+.CD(E46914

Theorem4.24 Groundformulag³ is truein world wF G if� f thereis someHJI expl K g³ Lsuchthat MON R PQSR . Moreover expl T g³ U is� afinite setof finite sets.F

or aproof seeAppendixA.

Corollary 4.25 expl V GW is� acoveringsetof explanationsof G.Note!

that expl X GY is� not necessarilythe setof minimal explanations,as thefollowing exampleshows.

Example4.26 SupposeC Z\[,[ a] ^ b_�`�a c b d c=d)e ef fg h,h andthefactsare:g³ 1 i a] j c kg³ 1 l b m c ng³ 2 o a] p c qg³ 2 r b s et

Inu

this caseexpl v g³ 1wyx{z|z a] } c~� ) b c| . Thereis oneminimal explanationfor g³ 1,namely c .

Also

expl g³ 2´ , a] c� � b e| , but the set of minimal explanationsof g³ 2is , a] c=) b e�� c e| . c e is an explanation,becauseif c and e were true,whichever of a] or b were true in a possibleworlds would make g³ 2 true� in thatpossibles world.

ThisQ

shows that thesetof minimal explanationsof a goal is not necessarilyaminimalcoveringsetof explanations.This ideashouldbecomparedto theideaofkerneldiagnosesandanirredundantsetof kerneldiagnoses[8].

Thesetof minimal diagnosescanbecomputedusinga notionof generalisedresolution  of explanations:Definition¡

4.27 Ifu ¢¤£¦¥¨§

1 ©)ª«ª=ª=¬E k ®°¯ C, and ± L² 1 ³)´«´=´=µ L² k ¶ is� asetof explanationsof g³ suchthat · i¶ ¸ Li¶ for eachi ¹¦º 1 ») ¼= ¼« ¼= ½k¾ ¿ , the generalisedresolution ofthe�

explanations À L² 1 Á�Â=Â=«à L² k] Ä with respectto alternative Å is� L² 1 ÆÈÇ=Ç=ÇÉÆ L² k Ê˨Ì1 Í)ΫÎ=Î=ÏEÐ k Ñ .

Figure2givesanalgorithmtorepeatedlyresolveclauseswith respecttoalternativesin�

thechoicespaceandremoveredundantclauses.It is similar to theuseof binaryresolution  to computetheprimeimplicatesof asetof clauses[13].LemmaÒ

4.28 TheQ

setof all minimal explanationsof g³ is� thesetK resulting  fromtermination�

of thealgorithmof Figure2.

Ó&Ô�Õ)Ö�×,Ø.Ù�Ú1Û3Ü�Ý6Þ)Ö�Ú�Ü8ß&à:Ú�á=â>Ø?Þ�Ù@á=ã�ä�á,Ø.åDÖEÝ6à15

K æèç minsÕ é expl ê g³ ë,ë ;while ìîíðï CñJòSó

iô õ÷öùø Liô ú K suchthat û iô ü Liý consistentþ L1 ÿ�������ÿ Lk � ���� E� � K suchthatE� L² 1 ��������� L² k] ���

do K ��� minsÕ � K ��� L1 ��������� Lk ���� "!

Figure#

2: Findingall minimal explanationsof g³

F#

or aproofof this lemmaseeAppendixA.

Example$

4.29 ConsidertheICL theoryof example4.26.To find the minimal explanationsof g³ 1 we start off with K % expl & g³ 1 ')(*"*

a] + c,�-/. b 0 c121 . As 3 a] 4 b576 C, we canresolve 8 a] 9 c: and ; b < c= resulting> in ? c@ .minsÕ A K B�C"C cD"D"EGF minsÕ H2I"I a] J cK�L/M b N cO�P/Q cR"R2SUTWV"V cX" X .

TY

o find the minimal explanationsof g³ 2 we start off with K Z expl [ g³ 2 \)]^"^a] _ c̀�a/b b c ed"d . As e a] f bgih C, we canresolve j a] k cl and m b n eo resultingin p c q er .

minsÕ s K tvu"u c w ex" x2y{z minsÕ |2}"} a] ~ c� b c� c e"") 2 a] c� b c� c e" . Nomore resolutionscanbecarriedout,andtheprocedurestops.

4.5 Splitting CompositeChoices

Thefinal operationon (setsof) compositechoicesis splitting a compositechoiceinto

a numberof compositechoices.This will beusedto make setsof mutuallyincompatiblecompositechoices.Thiswill beusedfor usingexplanationsto com-pute probabilities(Section5),but canbeusedwhenever wedo notwantredundantproofs. Recallthatweareassumingthateachalternative is finite.Definition 4.30 If �W 1 /����  k ¡ is analternativeand ¢ is acompositechoicesuchthat £¥¤§¦©¨«ª¬ , thesplit of on ® is thesetof compositechoices

¯°¥±�²³1́�µ¶�¶�¶�µ�·¥¸�¹º k »"»

Itu

is easyto seethat ¼ andasplit of ½ describethesamesetof possibleworlds:Lemma¾

4.31 Ifu

wF ¿ is apossibleworld, wF ÀÂÁ ÃÅÄ if f thereis someÆ iô ÇÉÈ suchthatwF ÊÂË ÌÎÍ¥Ï�ÐÑ i Ò .

ÓÕÔÖ/×Ø2ÙÛÚܧÝßÞàâá/×�ÜÞiãÕäåÜçæ�è�Ùéá�Úêæ�ëçìíæ2ÙÛîï×�àâä16

prð oceduredisjointñ ò K óInput: K — setof compositechoicesOutput:mutuallyincompatiblesetof compositechoicesequivalentto K

repeatifô õ÷ö

1 ø�ù 2ú û K and ü 1 ýÿþ 2úthen�

K ��� K ����� 2 elseif �� 1 �� 2 � K , suchthat � 1 ��� 2ú is consistent

then�

choose����� 1 ��� 2 and ��� C suchthat �!#"let$

K2 be%

thesplit of & 2 on 'K (�) K *�+�, 2 -/. K2

elseexit andreturnKforever

Figure#

3: Makesetof compositechoicesK mutually incompatible

Ifu

thereis afinite numberof alternatives,startingfrom setK of compositechoices,repeatedsplittingof compositechoicescanproducethesetof totalchoices(andsopossible worlds) in which K is true. Suchanoperationis not, however, of muchuse.0

TheY

mainusefor splittingis,given asetof compositechoicesto constructasetof mutuallyincompatiblecompositechoicesthatdescribesthesamesetof possibleworldsastheoriginal set.SupposeK is a setof compositechoices,therearetwooperationsweconsiderto form anew setK 1 of compositechoices:

1. removingdominatedelements:if 2 1 3�4 2 5 K and6 1 798 2, letK :A@ 2 B .2. splittingelements:if C 1 D�E 2 F K , suchthat G 1 H�I 2 is consistent(andneither

is

asupersetof theother),thereis a J�K�L 1 MON 2 andPRQ C suchthat S�T�U .WV

e replaceW 2 by% thesplit of X 2 on Y . Let K2 be% thesplit of Z 2 on [ , andK \

egfihkjilnmpoiq�rtsiuwvkj�qisyxgz{q}|~�mv�o|}|nmpjuwz17

Example4.32 Suppose

C a 1 a 2 a 3 �i b1 b2 b3 �i c1 c2ú c3

K a 1 b1 �¡i¢ a 1 £ c1 ¤¤�¥Theelementsof K arenot mutually incompatible(thereis a world in which theyarebothtrue— namelytheworld with total choice ¦ a 1 § b1 ¨ c1 © ). We cansplit thesecondelementof K on ª b1 « b2 ¬ b3 , resultingin

K ®

\^]�_�`�a�bdc�egfih�jlk�`�e�hnm^operq:sDbtk�cuq:vrwxq�bdyz`{jlo18

1. How many splitsmaybeneeded?

2. How many compositechoicesarein theresultingsetof mutuallyincompat-ible

compositechoices?

3. Is therea heuristicthat tells us on which elementwe shouldsplit? In thesecondoperationwe can split on | 1 or on } 2; doesone result in fewercompositechoices?

TheY

first thing to noticeis that to make a setof compositechoicesmutuallyincompatible,it maybethecasethatwe have to considereachpair of compositechoices.Given apairof compositechoiceswecananalyzeexactly thenumberofsplitsandthenumberof resultantcompositechoices.

Supposewearetrying to makecompositechoices~ 1 and 2 incompatible. LetK2 be

%the split of 2 on (where 1 containsan elementof ). We musthave

K2 n . All but oneof the elementsof K2 are incompatiblewith 1 (thusthered

are n 1 of thesecompositechoicesincompatiblewith 1). Let 3 be% theelementof K2 compatiblewith 1 ( 3 will be 1 2). Either   1 is asubsetof ¡ 3 (thisoccursiff ¢£ 1 ¤¦¥ 2ú §�¨ 1), or wehavehaveto repeattheloopto make © 1and ª 3 incompatible.

Suppose« 1 ¬® 2ú ¯ °± 1 ²� ³:³:³:²{´ kµ ¶ , where · i ¸º¹ i» . Either we are going toeventuallysplit ¼ 2 on ½ 1 ¾�¿�¿:¿:¾�À k, or elsewe aregoing to have to split Á 1 on theanalogousset, in orderto make the resultantsetof compositechoicesmutuallyincompatible. It makesno senseto split both  1 and à 2 in order to make themincompatible.

If werepeatedlysplit Ä 2 onthe Å i wewill needkÆ ÇHÈÉ 1 ÊÌË 2 Í splits.TheY

resultingsetof compositechoiceswill contain Î kµi» Ï 1 ÐÒÑ i ÓÔÖÕ kÆ × 1 elements.Thusif all of thealternativeshave thesamelength,to minimisethenumberof

compositechoicesin themutuallyincompatiblesetweshouldrepeatedlysplit thelargerof apair of compositechoices.

4.6Ø

An Example in Detail

ContinuingExample1.1,supposewe alsohave thattheagentoftendropsthekeyif it is slipperyandif the key isn’t slippery, it sometimesfumblesanddropsthekeyÙ

:

drñ

opsÚ keÆ y Û T ÜÞÝslipperyß à keÆ y á T âäãdrñ

op_slipperyß _keÆ y å T æ�ç

è^é�ê�ë�ì�ídî�ïgðiñ�òló�ë�ï�ñnô^õpïrö:÷Dító�îuö:ørùxö�ídúzë{òlõ19

drñ

opsû keÆ y ü T ýÞþÿ slipperyß � keÆ y � T ���fumblesê

_keÆ

y � T ���Supposethat,independentlyateachtime,theagenteitherdropsorholdsaslipperykey andeitherfumblesor retainsanunslipperykey. This is specifiedby:

T drñ op_slipperyß _keÆ y � T �� holds� _slipperyß _keÆ y � T ����� C�T � fumblesê _keÆ y � T ��� retains_keÆ y � T ����� C

Supposethat the key could start slipperyandsubsequentlybecomeunslippery.(We could modelthe key becomingslipperyby addingan extra clause,but thismakestheexamplemorecomplicated.)

slipperyß � keÆ y sß ! T "�"�#slipperyß $ keÆ y % T &�'staysß _slipperyß ( T )+*

slipperyß , keÆ y - 0.�/initially_slipperyß 0 keÆ y1+2

WhetherV

thekey remainsslipperyat eachstepandwhetherit is initially slipperyarebothchoices:

3T 4 staysß _slipperyß 5 T 6�7 stopsß _being_slipperyß 8 T 9�: initially_slipperyß ? keÆ y@+A initially_unslipperyB keÆ yC�D

lnmporqps�tvupwyx{zp|~}rq�wpznw+t}�u+�tvq|~20

at robot Pos sß T �� gotoï _action T �at robot Pos T +

at keÆ y Pos  T ¡

�������� �!#"�$&%('�)+*��,$�'.-�/0$21435!6*,"714829:1�!#; slippery ? keÆ y @ s A s B 0C�C�C . Thesecanbemadedisjoint giving theexplanations:

Dinitially_slippery E keÆ yF4G stays _slippery H 0I4J stops _being_slippery K s L 0M�M�NOinitially_slippery P keÆ yQ,R stops _being_slippery S 0T�UVinitially_unslipperyW keÆ yX�Y

Example4.36 Considertheexplanationsfor drZ

ops[ keÆ y \ s ] s ^ 0_�_�_ . Thefirst clausefor dr

Zopsresultsin oneexplanation:`drZ

op_slippery _keÆ y a s b s c 0d�d�d4e stays _slippery f s g 0h�h4i stays _slippery j 0k,linitially_slippery m keÆ yn�o

The�

secondclausefor drZ

opsresultsp in threemoreexplanationsfor drZ opsq keÆ y r s s s t 0u�u�u ,namely:

vfumblesê

_keÆ

y w s x s y 0z�z�z4{ initially_slippery | keÆ y},~ stays _slippery 04stops _being_slippery s 0��

fumblesê

_keÆ

y s s 0��, initially_slippery keÆ y4 stops _being_slippery 0�fumblesê

_keÆ

y s s 0��, initially_unslippery keÆ y�Example

4.37 There�

arefour explanationsfor drZ ops keÆ y s s   0¡�¡�¡ :¢initially_slippery £ keÆ y¤,¥ stays _slippery ¦ 0§4¨ retains_keÆ y © s ª s « 0¬�¬�¬,

stops _being_slippery ® s ¯ 0°�°�±²initially_slippery ³ keÆ ý4µ retains_keÆ y ¶ s · s ¸ 0¹�¹�¹,º stops _being_slippery » 0¼�½¾retains_ke

Æy ¿ s À s Á 0Â�Â�Â4à initially_unslipperyÄ keÆ yÅ�ÆÇ

initially_slippery È keÆ yÉ4Ê stays _slippery Ë 0Ì,Í stays _slippery Î s Ï 0Ð�Ð4Ñholds�

_slippery _keÆ y Ò s Ó s Ô 0Õ�Õ�Õ�ÖExample

4.38 Considerexplaining carrying× keÆ y Ø s Ù s Ú s Û 0Ü�Ü�Ü . Using the secondclausefor carrying,weneedtocombinethesefourexplanationsfor Ý drZ opsÞ keÆ y ß s à s á 0â�â�âwith explanationsfor carryingã keÆ y ä s å s æ 0ç�ç�ç .

Therearefour explanationsof carryingè keÆ y é s ê s ë s ì 0í�í�í�í :îinitially_slippery ï keÆ yð,ñ stays _slippery ò 0ó4ô retains_keÆ y õ s ö s ÷ 0ø�ø�ø,ù

stops _being_slippery ú s û 0ü�ü4ý picØ kup_succeeds þ s ÿ 0����� goto� _succeeds � 0����initially_slippery keÆ y�� retains_keÆ y � s s � 0������� stops _being_slippery � 0���

�������������� "!$#�%'&��� �#)(�*+ -,�./�0&��1,�2-34,���56�7%'*22

picØ kup_succeeds 8 s 9 0:�:�; goto� _succeeds < 0=�>?retains_ke

Æy @ s A s B 0C�CDC�E initially_unslipperyF keÆ yG�H picØ kup_succeeds I s J 0K�K�L

goto� _succeeds M 0NDOPinitially_slippery Q keÆ yR�S stays _slippery T 0U�V stays _slippery W s X 0YDY�Z

holds�

_slippery _keÆ y [ s \ s ] 0̂D̂�̂�_ picØ kup_succeeds ` s a 0b�b�c goto� _succeeds d 0eDfExampleg

4.39 Fromh

the above explanationsof carryingi keÆ y j s k s l s m 0n�nDn , we canderiveexplanationsfor o carryingp keÆ y q s r s s s t 0u�u�u , threeof whichare:

vgoto� _failsw x 0y�z{goto� _succeeds | 0}�~ picØ kup_failsw s 0�Dgoto� _succeeds 0� picØ kup_succeeds s 0�� dr op_slippery _keÆ y s s 0�D�

initially_slippery keÆ y� stays _slippery 0� stays _slippery s 0�DThere�

aresix otherexplanations.

What

is importanttonoticeabouttheseexamplesisthatwecanwritedeclarativeclausesdefiningthedynamicsof theworld, forgettingaboutthefactthatsomeoftheF

conditionswill be uncertain. The explanationsof a groundformula are adescriptionof exactly thoseworldsin which it is true. Themutualincompatibilitymeansthateachworld is only describedby oneexplanation.

5 Probabilities

In

many applicationswewouldliketoassignaprobabilityover thealternatives[19,23]. This letsususethelogic programmingrepresentationfor standardBayesianreasoning.p Therulestructuremirrorstheindependenceof Bayesiannetworks,andpro videsa form of contextual independencethatcanbeexploitedin probabilisticinference

[24].Supposewe aregiven a functionP0 from atomicchoicesinto   0 ¡ 1¢ suchthat£¥¤§¦ Pº 0 ¨ª©4«¬ 1 for all alternatives ®°¯ C. That is, Pº 0 is a probability distri-

b±ution on eachalternative. We assumethat the alternatives areunconditionally

probabilistically independent.Intuiti

vely, wewouldliketheprobabilitymeasureof any worldtobetheproductof the probabilitiesof the atomicchoicesthat make up the total choicedefiningtheF

world. That is, ²´³ wµ ¶¸·º¹ »½¼ R ¾À¿§Á P0 ÂÄÃ4Å . Theprobabilityof any propositionis thesumof theprobabilitiesof theworldsin which propositionis true. That is,

Æ�Ç�È�É�Ê�Ë�Ì�Í"Î$Ï�Ð'Ñ�É�Í�Ï)Ò�Ó+Í-Ô�Õ/Ë0Ñ�Ì1Ô�Ö-×4Ô�Ë�Ø6É7Ð'Ó23

P ÙÄÚ4ÛÝÜ wÞ ß�à áãâåä´æ wµ ç¸è . Thisonlyworkswhenthechoicespaceisfinite. However,whenthealternativesareparametrizedandtherearefunctionsymbols(asin Section4.6), thereareinfinitely many possibleworlds, eachwith measurezero,andweneedé a moresophisticatedconstruct.Thegeneralideais to definea measureoversetsof possibleworlds.

Letê

Wë Cì Fí îðïòñ wµ óõô÷ö is

aselectorfunctiononC ø . ThusWù C ú Fí û is

thesetof all

possible worlds. Wedefinethealgebraof subsetsof Wü C ý Fí þ thatF

canbedescribed

by±

finite setsof finite compositechoices.ÿ��

C � F������� W� C F����� finite setof finite compositechoicesKsuchthat � wµ � wµ ��� if f wµ � � K ����

C � Fí � is closedunderfinite unionsandcomplementation.That is, if � 1 "! 2# $%�&C ' F ( thenF ) 1 *,+ 2 -/.�0 C 1 F 2 andW3 C 4 F 57698 1 :/;�< C= F > .As?

shown in Section4.5, every set of compositechoicesis equivalent to amutually incompatiblesetof compositechoices. Thus, for every @BADC�E CF Fí G ,thereF

exists a mutually incompatiblesetof compositechoicesK suchthat wH IJ if f wH K L K .LemmaM

5.1 IfN

K andK O arebothmutuallyincompatiblesetsof compositechoicessuchthatK P K Q , then R�S K TVUXW P0 YZ\[�] ^`_ba K c dfebgXh"i P0j klnmpo .This lemmais thesameasLemmaA.8 in [19].

Wq

ecanthendefineaprobabilitymeasurertsvu�w Cx Fí y�z { 0 | 1} by:±~ �

K VX P0 nwhere K is a mutually incompatibleset of compositechoicessuch that wH if f wH K . Lemma5.1 impliesthatit doesn’t matterwhichK is chosen.LemmaM

5.2 satisfiestheaxiomsof probability,7 namely:é 1 9 ¡¢ where£ is thecomplementof ¤ and,¥ if ¦ 1 and § 2 aredisjoint sets,̈©ª 1 «,¬ 2# �®°¯±² 1³µ´·¶¸¹ 2# º .7»Note¼

thatwe don’t require ½ -additivity (thesumrule for infinite disjuncts)as,by acyclicity,each¾ groundformulahasafinite setof finite explanations.

¿ÁÀÃÂ�ÄÃÅÇÆÉÈÃÊ,ËÍÌÃÎXÏ�Ä"ÊÃÌ ÐÁÑÒÊÔÓÖÕ×ÆØÏ"ÈÙÓÖÚÔÛnÓÇÆÉÜÝÄ`ÎXÑ24

Wq

ecannow definetheprobabilityof groundformulagÞ by± P ß gÞ àáãâäÇå wH æ wH ç è gÞ éëê .Thisì

is well definedasfor any gÞ , by the acyclicity of the facts,thereis a finitecovering set of finite explanationsof gÞ . That is, í wH î wH ï ð gÞ ñ/òôó�õ Cö Fí ÷ . Inparticularø , wehave thefollowing:Prù

oposition5.3 IfN

K isú

acoveringandmutuallyincompatiblesetof explanationsof gÞ , then

P û gÞ üý þ�ÿK � ���

P0j �����Wq

ecandefineaconditionalprobabilityin thenormalmanner:If P �

���� 1,Pº ����������

defP �������

Pº !�"# $

FromthiswecanseethatBayesianconditioningcorrespondsto abduction.Whenweobserve % andconditionon it, thismeansthatwefind theexplanationsfor it.

Section4.4 shows how to constructa setof coveringexplanationsof gÞ . Theproblemø is to generatea coveringandmutually incompatiblesetof explanationsof gÞ . Therearethreeapproachesthancanbeused:

1. build thefactbasetoguaranteethatonlymutuallyincompatibleexplanationsarereturnedby expl & gÞ ' .

2. constructa coveringandmutually incompatiblesetof explanationsfrom acoveringsetof explanations.

3. computetheprobabilitiesdirectly from thesetof explanationsgeneratedbyexpl ( gÞ ) .

Theseì

arediscussedin thenext threesections.

5.1*

Disjointed Rule Bases

Poole+

[19] shows therelationshipto abductionin theacyclic definiteclausecase(without negationas failure) underthe constraintthat the bodiesof the groundinstancesú

of the rules for any atom are incompatible. That is, if a, - b1 anda, . b2 aretwo groundinstancesof rulesfor a, , thereis nopossibleworld in whichb1 andb2# isú true. Underthis restriction,theprobabilityof gÞ canbeobtainedbyaddingthe probabilitiesof the explanationsof gÞ [19]. In this sectionwe extendthis/

ideato themoregeneralrule formulationgiven in thispaper.

0�132543687:93;=3@BA54C;3>ED�FG;IHKJL7MAC9NHKOIP�H87:QR4S@BF25

Definition 5.4 A rulebaseF is disjointed if

• for every pair of differentruleshT

1 U b1 andhT 2 V b2 inú Fí andfor everypairø of groundingsubstitutionsW 1 and X 2, if hT 1 Y 1 Z hT 2 [ 2, thereis no worldin whichb1 \ 1 ] b2 ^ 2 isú true,

• for everyrulehT _

b inú

Fí

andfor everypairof groundingsubstitutions̀1 anda2 suchthath

T b1 c hT d 2 andbe 1 fg bh 2, thereis noworld in whichbi 1 j bk 2

isú

true,and

• whenever b1 l b2 appearsin thebodyof any groundrule, thereis no worldin whichany groundinstanceof b1 m b2 is true.

Supposewe have a disjointedknowledgebase.We canusea variantof explto/

computea mutually incompatibleand covering set of explanationsof gÞ . Inparticularø , noneof theminsn functionso areneeded(nosubsetof anexplanationwillever begeneratedasanexplanation). Theonly time thatmutually incompatiblecompositechoicescanbe generatedis in computingduals

(Section4.2.1). We

canmake thesedisjoint usingthe algorithmof Figure3. This is summarisedinalgorithmexpldp :

expld q Gr�s

t�uif G v true

expldp w Ax y{z expldp | B} ~ ifú G Ax B}expldp A{ expldp B if G A Bdisjoint

duals

expld Ax 88 ifú G AxG8 if G Ci expld B} i ifú G � C G B} i grÞ Fí

Prù

oposition5.5 IfN

Fí

isú

adisjointedrulebase,expld   gÞ ¡ will alwaysreturnaminimalmutually¢ incompatibleandcoveringsetof explanationsof gÞ .Theì

coveringnessis adirectconsequenceof Theorem4.24.Themutuallyincom-patibilityø is due to the disjointednessof the rule base; £ preservø es the mutualincompatibilityú

, andunionis only usedon pairwiseincompatiblesets.Themini-mality is aconsequenceof themutuallyincompatibility¤ — asubsetis compatiblewith its supersetif thesupersetis consistent.

Note¥

thatthetermminimaln heremeansthatthesetof explanationsis minimal;i.e., no subsetalsohasthis property. It doesnot meanthat the explanationsareminimal.¢

¦�§3¨5©3ª8«:¬3=®?¯3°B±5©C3¯E²�³GI´KµL«M±C¬N´K¶I·�´8«:¸R©S°B³26

This ideaof exploiting propertiesof rulesto gainefficiency is a powerful andgeneralidea. From experience,it seemsthat obeying the discipline of writingdisjointedrulesetshelpsto debugknowledgebasesandto write clearerandmoreconciseknowledgebases.

5.2*

Making Explanationsmutually incompatible

Given a coveringsetof explanationsof gÞ , as produced,for exampleby theuseofexpl, wecanusetherepeatedsplittingalgorithmof Section4.5to createacoveringandmutuallyincompatiblesetof explanationsof gÞ .5.3*

Computing probabilities fr om arbitrary setsof compositechoices¹

Wq

edonotneedtocreateamutuallyincompatiblesetof explanations.Probabilitiescanbecomputedfrom anarbitrarycoveringsetof explanations.Thegeneralideaisú

whenaddingprobabilitiesof disjunctionswe have to subtractthepartwe havedoublecounted.

Thefollowing formulais truewhetheror not º 1 and » 2 areincompatible.P¼ ½�¾

1 ¿�À 2 Á� P¼ Ã�Ä 1 Å{Æ P¼ Ç�È 2 ÉËÊ P¼ Ì�Í 1 Î�Ï 2 ÐTheì

generalcaseismorecomplicated.If wehave Ñ�Ò 1 Ó5ÔKÔKÔKÓSÕ n Ö asacoveringsetof explanationsof gÞ , wecanusethefollowing formulato computetheprobabilityof gÞ :

P¼ ×�Ø

1 ÙÛÚKÚÜÚSÙÞÝ nß àâán

jã ä

1 i1 åKåÜå i jã1 æ i1 çèKèKèêé i jã ë nì

íî1ï jã ð 1P¼ ñ�ò i 1 óÛôKôKôCó�õ i jö ÷

Thesecondsumis summingover all subsetsof ø�ù 1 ú3ûKûKûCúCü n ý that/ containexactly jþelements.

P¼ ÿ��

i1����������

i jö isú easyto compute.It is 0 if � i 1 �������� i jö isú inconsistent,andotherwiseit is �����

i1 ����������� ijö P0 ��� � .Theì

mainproblemwith this sumis thatwe aresumming2nß !

1 probabilities,wherenì isú thenumberof explanationsof gÞ . This canoftenbereducedaswe donotneedto considerany supersetsof aninconsistentcompositechoice.

"$#�%'&�(�)+*�,.-0/�132'&4,�/65$78,:9�;�):?@9�)+AB&C13727

6 Combinatorics

InN

this sectionwe explorehow muchtheabductive view cansave over themodeltheoretic/

view, andprovidesomeanswersto thequestion:how muchsmallerwillasetof coveringexplanationsbethanthesetof possibleworlds?Thisis importantasit is thesetof thesecoveringexplanationsthatweneedto sumover to determineprobabilities.ø

InN

thegeneralcasewith infinitely many finite alternatives,thereareinfinitelymany possibleworlds, but any groundatomalwayshasa finite setof coveringexplanations,eachof which is finite. This is guaranteedby theacyclicity of therulebase.

Whenq

therearefinitely many finitealternatives,thereareexampleswheretherearethe samenumberof covering explanationsof somegÞ asthereareworlds inwhich gÞ is true. This occurswhen (and only when) the explanationsare totalchoices.

It is interestingto considerthecasewheretheICL theoryis theresultof trans-formingo

a Bayesiannetwork asin [19]. Although any probabilisticdependencecanbemodelledwith independentchoices(hypotheses)in theICL, this is doneatthe/

costof greatlyincreasingthenumberof worlds. However, aswe seebelow,the/

abductive characterisationcan be usedto (often more than) counteractthiscombinatorialexplosion.

Supposethereare nì (binary) randomvariablesthat we want to model, andthere/

are no independenciesthat we can exploit. The Bayesiannetwork [17]representationfor this is acompletegraph.Thereare2n

ß D1 independentnumbers

that/

canbeassignedto specifythejoint distribution(wecanassignanon-negativerealnumberto eachof the2n worlds,but thisis over constrainedby onenumber—weneedto divideby thesumin orderto getaprobability). In orderto modelthiswith independentchoices,we have 2n E 1 binaryalternatives.This is exactly thenumberF of alternatives(or disjointdeclarationsin theterminologyof [19]) createdin the translationof the Bayesiannetwork into a probabilisticHorn abductiontheory/

[19]. This,however creates22nG H 1 possibleø worldsin theindependentchoice

logic.I

This combinatorialargumentwould seemto indicatethatthemodellingbyindependentchoicescanberuledoutoncombinatorialgrounds.

HoJ

wever, considerthe sizeof the minimal explanationsof any proposition.Eachminimal explanationhasat most nì assumptions;they at most assignonevK alueto eachof the original propositions.At the extreme,for the root variable(with nì L 1parents),thereare2nM 1 rules,eachwith itsownalternative. Onlyoneofthese/

rulesandonly oneof theatomicchoiceswill bein any minimalexplanation,

N$O�P'Q�R�S+T�U.V0W�X3Y'Q4U�W6Z$[8U:\�]�S

g$h�i'j�k�l+m�n.o0p�q3r'j4n�p6s$t8n:u�v�l

Ð$Ñ�Ò'Ó�Ô�Õ+Ö�×.Ø0Ù�Ú3Û'Ó4×�Ù6Ü$Ý8×:Þ�ß�Õ

�����������! �"$#&%�')(��*"�%,+�-."0/213�4(* 5/26078/��!9:�;')-31

Example7.6 Considera logic program,thatcontainstherules

ad < c = bb > ? c @ ad

baut isotherwiseacyclic wheretheassignmentof numbersis suchthatc is assigned

a lower numberthanboth ad andb. Thereis still a uniquemodel for eachtotalchoice,aseachtotal choice,togetherwith thefacts,entaileitherc or A c. Whichit entailsmaydependon thetotal choice.

Extendingthedefinitionto cover suchcaseswould not cover theclassof all pro-gramswith auniquemodelfor eachtotalchoice,asthefollowing exampleshows:

Example

7.7 HereJ

is anexample,which is not contingentlyacyclic:

F BDC someoneE wins F int G NH IKJ winsL M NH N2Oint P 0Q2Rint S sE T NH U�UWV int X NH YZ

C [D\]\ winsL ^ NH _*` losesa NH b�ced NH isæ a termfThis hasthe propertyof a uniquemodel for eachselection,but is not acyclicbecausea

someoneg winshash to beassigneda numberbiggerthanany integer. Thisprogrami canbe interpretedaccordingto our semanticsgiven in Section3; thereare2j 0k possiblei worlds,wheresomeoneg wins isæ truein all but oneof them.Whenwe considerthis probabilistically, unlessthereareonly finitely many alternativeswith a non-zeroprobability of a win, the world wheresomeoneg wins is falsehasmeasurel zero.

Onecouldthink of extendingthenotionof (contingently)acyclic programstoinclude,for example,limit ordinals,but whetherthiswouldeithercover all of thenaturallym occurringcasesorbeneededfor realapplicationsisstill anopenquestion.

Themotivation for restrictingto acyclic programswas to ensuretherewas auniquen modelfor eachtotal choice.Otherusesfor acyclic logic programsaretheability to prove termination[16]. Thesearerelated,andit is interestingto notethatí

the terminatingprogramsof Marchiori [16] resultin uniquemodels,but theconverseis not true. Onecomplicationis thatwe don’t want to have to, for eachtotalí

choice,prove thereis a uniquemodel,astherecanbe infinitely many totalchoices.

o�p�q�r�s�t!u�v$w&x�y)z�r*v�x,{�|.v0}2~3t4z*u5}208}�t!:r;y)|32

8 Comparisonwith other Formalisms

8.1

PropositionalSatisfiability

Whenî

the alternatives are binary, the operationson setsof compositechoicescorrespondto operationson DNF formulae. The main point of this paper, is toshow how the compositechoicesinteractwith the rulesthat includenegationasfailure.

Considerabinaryalternative ad b . Inany worldwheread is selected,b is false.In�

any world whereb isæ

selected,ad isæ false.Thusb ad . Whenever b appearsintheí

factsit canbereplacedby ad , withoutaffectingwhatis entailedby thetheory.Wî

ecouldeventhingof thisalternativeas ad * ad .For thebinarycase,acompositechoicecorrespondstoaconsistentconjunction

of literals. A setof compositechoicescorrespondsto a DNF formulamadeup ofliterals

of atomicchoices.The dual operationcorrespondsto negatingthe DNFandconvertingtheresultto DNF. Theoperation (Definition 4.22)correspondstoí

conjoiningtheDNF formulaeanddistributingbackinto DNF. ThealgorithmofFigure2correspondstotheuseof binaryresolutiontocomputetheprimeimplicatesof asetof clauses[13].

Da

vydov andDavydova [7] haveextendedtheaboveBooleanlogic notionstoallow morethatoneelementin eachalternative (asin thispaper).Theirnotionofa dualcorrespondsto thehitting setin this paper. They have providedanalgebraof operationsonthesedualstructures.Theirwork is orthogonalto thework in thispaperi . What is importantaboutthis paperis how the setsof compositechoicesinteractwith therules. If we considertheconstraintson theDNF formulae,thenwe realisethat any compositechoicesis consistentwith the facts; the factsbythemselví

esimposeno constraintson the compositechoices.The constraintsontheí

compositechoicesis provided by the interactionbetweenthe factsand theobservations.Thusthis paperis presentinga particularway to provide suchcon-straints(onethatcorrespondstoBayesianconditioning).Basedonthisinteraction,we have presentedanabductive characterizationof the logic. We have provideda limited setof operationson thesedualstructuresthatareapplicablefor eviden-tialí

reasoning.Davydov andDavydova usedthedualstructuresfor optimization,wherethey wantto selectthebesttotal choiceratherbasedonanevaluationfunc-tion,í

ratherthantheevidentialreasoningtaskof thispaper, wherewewantto sumtheí

measuresof theconsistenttotal choices.For decisionproblems[23], wewanttoí

both sum over choicesby natureand optimize over choicesby the decisionmakingagent. The combinationof thesetechniquesis an intriguing possibility,

�����!�$&�)�*�,�.0 2¡34*5 2¢0£8 �!¤:;)33

b¥ut beyondthescopeof thispaper.

8.2

AbductiveLogic Programming

The combinationof abductionand logic programminghasa long history (seeKakaset al. [12] for a goodsurvey). Thecombinationproposedin this paperisquitedifferentfrom otherproposalsmainlybecausetheabductivecharacterisationisæ

aconsequenceof anindependentlydefinedsemantics.Thenormaldefinitionofstablemodels[10] is usedto definenegationasfailure— thereis no alternativenotion of negationasfailure that needsto be definedandmotivated. Thereis amuchl closertie betweennegationasfailureusedin thispaperandso-called“real”negation; ¦ ad is truein aworld if andonly if ad is not truein theworld.

In�

abductivelogicprogramming,theminimalityof explanationshasasemanticsignificance;if E is anexplanationfor somegï , it doesnot imply thatE §©¨ ad ª , evenifæ

internally consistent,is an explanation. However, in the ICL, any consistentsupersetof anexplanationis anexplanation:if E

«isæ

anexplanationfor gï , anda¬ isæanatomicchoicethatis consistentwith E, thenE ¯® a¬ ° is anexplanationfor gï .

One of the things uniqueabout the work reportedin this paperis that theexplanationsof ± gï area functionof theexplanationsof gï . In otherframeworksfor

abductivelogic programming,if thereis anexplanationfor gï , andnegationwasnot usedto prove gï , thereareno explanationsfor ² gï (all explanationsof ³ gï areobtainedfrom negationasfailureusedto provegï ).

The

mainsemanticdifferenceisthatweinterpretfailure-to-provein eachworld,ratherthanfailuregiven thewhole theory. This meansthatequivalencesthataretrueí

for eachworld, suchasClark’s completionfor nonassumables,hold for thewholetheory. Ratherthanforcing thismeaning,it is anaturalconsequenceof theframe

work.Much´

of thepowercomesfrom having astructuredhypothesisspace.It is thisstructurethat allows us to give sucha cleansemantics,uponwhich it is easytoimposeæ

a probabilitymeasure(Section5), anduponwhich it is easyto extendtomultipleagentsmakingchoices[23].

The

useof a rule basethatis a completedefinition,evenif all of theelementsof the body of rulesarenot completelydefined,is similar to the completionofnon-abduciblepredicatesin the completionsemanticsfor abduction[6, 20], buttheseí

don’t allow for negationasfailure. It is alsosimilar to the motivation forOLP-FOL [9], but in the ICL the aim is to handleall uncertaintyin termsofBayesianµ

decisiontheory(or gametheorywhenthereis morethanoneagent)[23],asopposedto handlinguncertaintyin thenon-definedpredicatesusingfirst order

¶�·�¸�¹�º�»!¼�½$¾&¿�À)Á�¹*½�¿,Â�Ã.½0Ä2Å3»4Á*¼5Ä2Æ0Ç8Ä�»!È:¹;À)Ã34

logic (asis donein OLP-FOL).NotethatOLP-FOLtakesa differentapproachtobadly¥

definedpredicates(suchaspÉ inæ pÉ Ê Ë pÉ ). Whereas,Deneckerisquitehappytoí

haveathree-valuedsemanticsfor thesepredicates(butnotfor predicatesdefinedinæ

theFOL), wedon’t allow thesebecausewewantto haveanormalprobabilisticsemantics.Ratherthanallowingathree-valuedsemantics,werestrictthelanguagetoí

beacyclic to ensurewedon’t havesuchbadlydefinedpredicates.It alsoseemsthatí

writing acyclic programsis good programmingpractice;a cyclic programusuallyn indicatesabug.

8.3

Probabilistic Horn abduction

ProbabilisticHorn abduction[19, 18] is a pragmaticframework for combininglogic andprobabilitywith independenthypothesesanddefiniteclausesgiving theconsequencesof thehypotheses.Thereis a closerelationshipbetweenBayesiannetworks[17] andprobabilisticHornabduction[19].

The

independentchoicelogic extendsthe logical part of probabilisticHornabductionin allowing for negationasfailurein thebodyof rules,andin allowingfor

non-disjointrules. The modelling languageis thusmuchexpandedwithoutlosing

semanticsimplicity or elegance. A complementarypaper[23] considersallowing differentagentsto chooseassumptions,andexplorestherelationshiptonotionsm in gametheoryandstochasticdynamicalsystems.Thatpaperusesonly themodel-theoreticsemanticsandnot theabductivecharacterizationexploredhere.

9 Conclusion

This paperhaspresenteda mix of abductionandlogic programming(includingnem gationasfailure) that allows for a cleanmix of logic programmingandprob-ability. This was definedin termsof a semanticframework that allows for theindependentæ

choices.This framework allowsusto importdirectly thestablemod-elssemanticsfor our logic programs(or any othersemanticsthat is definitive ontotalí

choices).Theabductive characterisationis a consequenceof thesemantics— the setof explanationsof a formula is a concisedescriptionof the worlds inwhichtheformulais true. Theresultof this is acleanandusefulmix of abduction,logic

programmingandprobabilisticreasoning.This hasbeenimplementedandusedfor applicationsin decisiontheory[23].

Thecodeis availablefrom my website.

Ì�Í�Î�Ï�Ð�Ñ!Ò�Ó$Ô&Õ�Ö)×�Ï*Ó�Õ,Ø�Ù.Ó0Ú2Û3Ñ4×*Ò5Ú2Ü0Ý8Ú�Ñ!Þ:Ï;Ö)Ù35

A Proofs

Lemma�

4.8 If�

K isæ

asetof compositechoices,dualsß à

K á isæ acomplementof K .Proof: Let wL â be¥ aworld. To prove: an elementof K is truein wL ã if f no elementof dualsß ä

K å is truein wL æ . Therearetwo casesto consider.

Case1: There

isanelementçéè K thatí is truein wL ê . Toshow thereisnoelementof dualsß ë

K ì thatí is truein wL í . If î0ïKð dualsß ñ K ò then,í by definitionof dualsß , thereisæ

an óõô÷ö and ø8ù8ú÷û0ü suchthat ýþ ÿ*þ ����� A� � C. As � isæ truein wL , mustl betrueí

in wL � (i.e., � selects from A). So ��� is falsein wL � . So ��� is falsein wL � .

Case2: Thereis no elementof K that�

is true in wL � . This meansthat for everyelement��� K , thereis anelement����� suchthat doesn’t select! (andinsteadselectssome"�# ). Let $�%&% be¥ therangeof selectorfunction ' . Then (*)+) satisfiesallof the conditionsfor membershipin duals

ß ,K - exceptperhapsminimality. Then

there�

is somesubset.*/ of 0*1+1 that� is minimal andsoin dualsß 2 K 3 . 4�5 is truein wL 6 .Sothereis anelementof duals

ß 7K 8 that� is truein wL 9 .

Q.E.D.

Theorem4.19 : entails; with respectto independentchoiceframework theory<C = F> if f ?A@ B logically follows from thecompletionof C C D F E .

Proof: If M is a modelof the completionof F C G F H , thenoneelementof eachalternative is true in M

I, as for eachalternative JLK 1 MONPNPNPMRQ kS TVU C, thecompletion

containstheformula WYX 1 Z\[P[P[RZ^] k _a` i bc jd e�fLg ih i^j jd k . Eachof theseselectionsisl

consistentwith Clark’s completion(aswe only completedthe predicatesthatwerenotatomicchoices).ThuseachM is amodelof atotalchoice,andeachtotalchoicehasa modelM. The total choicetogetherwith Clark’s completionhasauniquen stablemodel,asthetheoryis acyclic [1].m entailsn with respectto independentchoiceframework theory o C p Fq meansr

isl

truein all possibleworldsin which s isl true. This is thatsameasfor all totalchoicestvu w is true in the stablemodelsof the total choicetogetherwith F,whichis trueiff for all totalchoicesxAy z logically{ followsfrom thetotalchoicetogether�

with Clark’scompletionof F|

[1], which is equivalentto }A~ logically{follows from thecompletionof C F .Q.E.D.

OAO*P+P*� ¡R36

Theorem4.24 Groundformulag¢ is truein worldwL £ ifl f thereissome¤^¥ expl ¦ g¢ §suchthat ¨^© R ªL«¬ . Moreover expl ® g¢ ¯ isl afinite setof finite sets.

Proof: Theproof is by inductiononstructureof theformulaandonthethelevelassignedby theacyclicity of F

°. Acyclicity is neededto make surethe inductive

proofs± groundout.

Basecase: Thebasecasefor theinductionis unstructuredformulae(atoms)thatareminimal in theacyclicity ordering.Theseareatomicchoicesandatomicfacts.Thetheoremis trivially truefor theatomicfacts,where²´³¶µY· .

Supposȩ isl

anatomicchoice. ¹ isl truein wL º ifl f »½¼ R ¾L¿À . expl ÁYÂ�ÃÅÄÇÆÈÆYÉ�ÊÊthus� Ë�ÌÎÍYÏ�Ð

, and Ñ^Ò R ÓYÔÖÕ is thesameas ×�Ø R ÙYÚÛ .

Structural Induction: Supposeg¢ isl a structuredformula,andthatthetheoremis truefor every substructurefor g¢ , to show it is truefor g¢ . g¢ is eitherof theformfÜ Ý

hÞ, fÜ ß

hÞ

or à fÜ wherefÜ andhÞ areformulae(for which thetheoremholds).Supposeg¢ isl of theform fÜ á hÞ . g¢ isl truein world wL â if f both fÜ andhÞ aretrue

in wL ã .• Supposeg¢ isä truein worldwL å , thusfÜ andhÞ aretruein wL æ , thenbytheinduction

thereç

is a è 1 é expl ê fÜ ë suchthat ì 1 í R îYïÖð anda ñ 2ò ó expl ô hÞ õ suchthatö2 ÷ R øLùú . ThenconsistentûLü 1 ýÿþ 2 � (asthey areboth true in wL � ), asso�1 ��� 2 � expl � g¢ � (or asubsetof 1 �� 2ò isä in expl � g¢ ), and � 1 ��� 2 � R ����� .

• Supposeg¢ is falsein world wL � , thenoneof fÜ or hÞ is falsein wL � . Suppose(without lossof generality)thatf

Üis falsein wL � . By theinductionargument,

thereç

is no ��� expl � fÜ � suchthat �! R "$#�% , andaseveryelementof expl & g¢ 'is a supersetof the elementsof expl ( fÜ ) , thereis no *,+ expl - g¢ . suchthat/�0 R 1�243 .

The5

proofwheng¢ isä of theform fÜ 6 hÞ isä similar.Supposeg¢ is of the form 7 fÜ . g¢ is true in wL 8 if f fÜ is falsein wL 9 if f thereis

no: element;=< expl > fÜ ? suchthat @=A R B$C�D (by theinductive assumption)whichholdsif andonly if thereis someEGFIH dualsJ K expl L fÜ MNM suchthat OQPSR R T$U�V (byTheorem5

4.8),but then WGXZY expl [ g¢ \ .

]_^a`cbadfehgaikjmlanpocbqialsr_tuiGvxwyezoqg{vx|G}~vfehbnpt37

Acyclicity ordering induction Finally supposeg¢ is a groundatom, and thetheoremç

holdsfor all atomslower in theacyclicity ordering,andfor all structuredformulaebuild from atomslower in theacyclicity ordering.Suppose g¢ bi istheç

setof all groundinstancesof rulesin gr¢ F with g¢ asthe head. g¢ isä true inw if f somebi is truein w which (by theinductive assumption)holdsiff thereissome! expl bi suchthat � R �� . But then � expl g¢ or asubsetof isä inexpl g¢ ; in eithercasethetheoremfollows.Q.E.D.

Lemma

4.28 The5

setof all minimalexplanationsof g¢ isä thesetK resulting fromterminationç

of thealgorithmof Figure2.

Pr

oof: It

is easyto seethatonly explanationsarein K . Moreover, if all of theminimal explanationsarein K then

çbecauseof theuseof mins  , therewill beno

non-minimal: explanationsin K . Theonly thing remainingto show is thatif ¡ isä aminimal¢ explanationof g¢ , thenit is in K .

Theproof of this will mirror proofsof thecompletenessof binaryresolution,with thesplitting treeplayingthepartof thesemantictree(see,for example,[4]).

A splitting treeis a treewith nodeslabelledwith compositechoices.A leafnode: is a nodesuchthata subsetof thelabel is in expl £ g¢ ¤ . If a nodeis not a leafnodethenthechildrenof thenodelabelledwith ¥ arelabelledwith thesplitsof ¦onalternate§ (wherë!©«ª¬¯®�° ).

If

the root of the treeis an explanationof g¢ , thenno matterwhich choiceismadefor thealternative to split on, therecanbeno branchesthatdo not leadtolea±

ves,aseventuallywe will endup with nodeslabelledwith total choices,andasubsetof atotalnodeis in expl ² g¢ ³ asexpl ´ g¢ µ is acoveringsetof explanationsof g¢ .

Suppose¶ isä a minimal explanationof g¢ . Considera minimal (in thenumberof nodesonthetree)splittingtreewith aroot labelledwith · . Claim: thissplittingtreeç

canbe convertedinto sequenceof resolutionsthat will derive ¸ . This willbe¥

carriedout bottomup. Replaceeachleaf nodeby theelementof expl ¹ g¢ º thatçit covers. For eachnon-rootnode,whenall of its childrenhave beenreplaced,thenç

we canreplaceit by theresolutionof its replacedchildrenon thealternativeon which it was split. Theonly thing we needto demonstrateis thateachof thereplacedchildrencontainoneelementof the splitting alternative (andso canberesolv edtogether).Supposeonechild doesnotcontainanelementof thesplittingalternative, thenthis split canbe replacedby subtreeat that node,andwe get asmallersplitting tree,whichcontradictstheminimality of thesplitting tree.

»_¼a½c¾a¿fÀhÁaÂkÃmÄaÅpÆc¾qÂaÄsÇ_ÈuÂGÉxÊyÀzÆqÁ{ÉxËGÌ~ÉfÀh;ÅpÈ38

Q.E.D.

Acknowledgements

Thiswork wassupportedby Institutefor RoboticsandIntelligentSystems,ProjectB5Î

andNaturalSciencesandEngineeringResearchCouncilof CanadaOperatingGrantOGPOO44121.Thanksto the constructive commentsof the anonymousreferees; thispaperis now muchmorereadablethanit was.

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