Algorithms for computational argumentation in artificial intelligence

149 Pages·2011·1.06 MB·English

by  Efstathiou, V.

Algorithms for computational argumentation
in artificial intelligence
Vasiliki Efstathiou
Adissertationsubmittedinpartialfulfillment
oftherequirementsforthedegreeof
DoctorofPhilosophy
of
UCL.
DepartmentofComputerScience
UniversityCollegeLondon
2010
2
I,VasilikiEfstathiou,confirmthattheworkpresentedinthisthesisismyown. Whereinformationhas
beenderivedfromothersources,Iconfirmthatthishasbeenindicatedinthethesis.
Abstract
Argumentationisavitalaspectofintelligentbehaviourbyhumans. Itprovidesthemeansforcomparing
informationbyanalysingprosandconswhentryingtomakeadecision. Formalisingargumentationin
computational environment has become a topic of increasing interest in artificial intelligence research
overthelastdecade.
Computationalargumentationinvolvesreasoningwithuncertaintybymakinguseoflogicinorder
toformalizethepresentationofargumentsandcounterargumentsanddealwithconflictinginformation.
A common assumption for logic-based argumentation is that an argument is a pair Φ,α where Φ is
(cid:104) (cid:105)
aconsistentsetwhichisminimalforentailingaclaimα. Differentlogicsprovidedifferentdefinitions
forconsistencyandentailmentandhencegivedifferentoptionsforformalisingargumentsandcounte-
rarguments. The expressivity of classical propositional logic allows for complicated knowledge to be
represented but its computational cost is an issue. This thesis is based on monological argumentation
using classical propositional logic [12] and aims in developing algorithms that are viable despite the
computational cost. The proposed solution adapts well established techniques for automated theorem
proving,basedonresolutionandconnectiongraphs. Aconnectiongraphisagraphwhereeachnodeis
aclauseandeacharcdenotesthereexistcomplementarydisjunctsbetweennodes. Aconnectiongraph
allows for a substantially reduced search space to be used when seeking all the arguments for a claim
fromagivenknowledgebase. Inaddition,itsstructureprovidesinformationonhowitsnodescanbelin-
kedwitheachotherbyresolution,providingthiswaythebasisforapplyingalgorithmswhichsearchfor
argumentsbytraversingthegraph. Thecorrectnessofthisapproachissupportedbytheoreticalresults,
while experimental evaluation demonstrates the viability of the algorithms developed. In addition, an
extensionofthetheoreticalworkforpropositionallogictofirst-orderlogicisintroduced.
Acknowledgements
I would like to thank my supervisor Tony Hunter for his guidance on my work and his continuous
encouragement and advice during my PhD. I am grateful to Tony for introducing me to the area of
argumentationandgivingmethechancetoworkwithhimontheArgumentationFactoryproject. Ialso
wishtothankRobinHirschandEmmanuelLetierforassessingmyprogressduringmyPhD.Thanksto
NikosGorogiannisforbeingalwayswillingtohelpwithtechnicalissuesandMattWilliamsforproviding
actualmedicaldataforexperimentation. ThankstomyexaminersPaulKrauseandOdinaldoRodrigues
forreadingthisthesisandtakingpartinmyexam. Ithankmyfamilyandfriendsfortheirmoralsupport,
special thanks to Anna without whose help at earlier stages of my studies I might had not managed to
startthisPhD.Finally,IwouldliketoexpressmygratitudetoEPSRCforfundingthisresearch.
Contents
1 Introduction 11
1.1 Logicalargumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Problemstatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Proposedsolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Scopeofthisthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Contributionofthiswork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.1 Algorithmsforreducingthesearchspaceforarguments . . . . . . . . . . . . . 14
1.5.2 Algorithmsforproducingarguments . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.3 Algorithmsforgeneratingcounterarguments . . . . . . . . . . . . . . . . . . . 14
1.5.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.5 Extensiontofirst-orderlogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Listofpublications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Structureofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Background 17
2.1 Existingargumentationsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Abstractargumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.2 Argumentationbasedondefeasiblelogic . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 Assumptionbasedargumentation . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Argumentationbasedonclassicallogic . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Pollock’sproposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 AmgoudandCayrol’sproposal . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 BesnardandHunter’sproposal . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.4 Computationalissuesinargumentation . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Existingimplementationsofargumentationsystems . . . . . . . . . . . . . . . . . . . . 32
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Reducingthesearchspaceforarguments 34
3.1 Thelanguageofclauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
C
3.1.1 Definitionof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
C
3.1.2 Relationsin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
C
Contents 6
3.2 Resolutionforsatisfiabilitychecks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Theresolutionproofprocedure. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Linearresolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 ConnectionGraphsforpropositionalclauses . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 Connectiongraphdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 ConnectiongraphsforargumentsinCNF . . . . . . . . . . . . . . . . . . . . . 43
3.4 Algorithmsforproducingconnectiongraphs . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.1 Algorithmforthefocalgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.2 Algorithmforthequerygraph . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.3 Algorithmforzones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Experimentalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Searchingforarguments 53
4.1 Argumentsin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
C
4.1.1 Propertiesofdeductionsandargumentsin . . . . . . . . . . . . . . . . . . . . 55
C
4.1.2 TheSupportbasein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
C
4.2 Definitionsforprooftreesforarguments . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.1 Thepresupporttree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.2 Thecompletepresupporttree. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.3 Thesupporttree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.4 Theminimalitycheck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Algorithmsforproducingprooftrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.1 Algorithmforproducingcompletepresupporttrees . . . . . . . . . . . . . . . . 78
4.3.2 Algorithmforselectingthesupporttrees. . . . . . . . . . . . . . . . . . . . . . 80
4.4 Experimentalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Searchingforcanonicalundercuts 85
5.1 Reducingthesearchspaceforcanonicalundercuts . . . . . . . . . . . . . . . . . . . . 85
5.1.1 Thesetofstrongresolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.2 Usingthestrongresolventstofindcanonicalundercuts . . . . . . . . . . . . . . 87
5.2 Usingasupporttreetogeneratecanonicalundercuts . . . . . . . . . . . . . . . . . . . 91
5.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.1 Algorithmforgeneratingresolvents . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3.2 Algorithmsforgeneratingcounterarguments . . . . . . . . . . . . . . . . . . . 95
5.3.3 Algorithmforargumenttrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.4 Algorithmforthewarrantcheck . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Contents 7
6 Implementation 100
6.1 Systemarchitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Usingthesystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.1 Givingtheinput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.2 Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3 Experimentalevaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7 Extendingtofirst-orderlogic 110
7.1 Argumentationforalanguageofquantifiedclauses . . . . . . . . . . . . . . . . . . . . 110
7.2 Relationsonfirst-orderclauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.3 ConnectionGraphsforfirst-orderclauses . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.4 Prooftreesforfirst-orderclauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.5 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5.1 Algorithmforproducingcompleteassignmenttrees . . . . . . . . . . . . . . . . 126
7.5.2 Algorithmsforselectingtheminimalandconsistentassignmenttrees . . . . . . 130
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8 Conclusions 132
8.1 Argumentationoverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.2 Contributionofthiswork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.3 Assessmentandfurtherwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A Exampleknowledgebases 136
List of Figures
3.1 Focalgraphsizevariationwiththeclauses-to-variablesratio . . . . . . . . . . . . . . . 51
4.1 ApplyingalgorithmSearchTree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.1 DiagramwiththemajorclassesofJArgue . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Comparisonintimeperargumenttreenodeforknowledgebaseswith2-placeand3-place
clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.1 ApplyingalgorithmFirstOrderSearchTree . . . . . . . . . . . . . . . . . . . . . . . . 129
List of Tables
4.1 Experimentaldataongeneratingpresupporttrees . . . . . . . . . . . . . . . . . . . . . 83
6.1 Experimentaldataongeneratingargumenttreeswithknowledgebasesof1and2-place
clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Experimentaldataongeneratingargumenttreeswithknowledgebasesof1and3-place
clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
List of Algorithms
3.1 GetFocal(∆,φ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 GetQueryGraph(Φ,ψ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 RetrieveZones(Φ,ψ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1 SearchTree(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 IsMinimal((N,A,f)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1 GetSResolvents((N,A,f)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 GetCounterarguments((N,A,f)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 ArgumentTree((N ,A ,f )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
0 0 0
5.4 Mark((N,A)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.1 FirstOrderSearchTree(v) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 IsMinimal(N,A,e,f,g,h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.3 IsConsistent(N,A,e,f,g,h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130