An introduction to Abstract Argumentation

Transcription

1 EPCL Basic Training Camp 2013 An introduction to Abstract Argumentation Pietro Baroni DII - Dip. di Ingegneria dell Informazione University of Brescia (Italy)

2 Roadmap PART I: Basics PART II: Advanced What s in the words? From the Big Bang to now Dung s framework A conflict calculus : argumentation semantics Basic semantics properties A catalogue of semantics Properties of semantics (so far) Taking topology seriously Differences vs commonalities: semantics agreement Comparing argumentation frameworks Skepticism and related properties A richer notion of justification status Skepticism-related criteria for semantics Satisfying all criteria Computational issues Beyond Dung s framework

3 Roadmap What s in the words?

4 Abstract (for good) General, independent of specific details,able to capture a variety of situations (MW: disassociated from any specific instance, theoretical) In computer science, abstraction is the process by which data and programs are defined with a representation similar in form to its meaning (semantics), while hiding away the implementation details. Abstraction tries to reduce and factor out details so that the programmer can focus on a few concepts at a time. A system can have several abstraction layers whereby different meanings and amounts of detail are exposed to the programmer (Wikipedia).

5 Abstract (for bad) Poorly related (or totally unrelated) with real problems (MW: insufficiently factual, difficult to understand, abstruse)

6 Argumentation (for what?) Argumentation is a multi-faceted word, with a variety of informal/intuitive and also formal meanings The abstraction process detaches the word from some/most/all of its meanings and properties, keeping only those required by the desired abstraction level (and possibly adding other ones) Abstract arguments are not arguments

7 Roadmap What s in the words? From the Big Bang to now

8 The Big Bang Less than 20 years ago Dung s 1995 paper on Artificial Intelligence Journal Theory of Abstract Argumentation (AA) Frameworks Basic notions related to Abstract Argumentation Semantics (conflict freeness, admissibility, 4 traditional semantics: complete, grounded, stable, preferred) Ability to capture a variety of other (less abstract) formal contexts:» default logic» logic programming with negation as failure» defeasible reasoning» N-person games» stable-marriage problem

9 The explosion Not an easy to read paper, but a very general (and basically simple) formalism with ideas and results regarded as extremely powerful Enormous impact on the literature (675 citations ISI, 1098 Scopus, 2041 Google Scholar) Probably the most cited paper in the computational argumentation literature Originated entire new research lines: many followers (and some criticisms)

10 The AA universe Abstract argumentation framework Traditional semantics notions Extensions/variations of the definition of the basic framework (and relevant semantics) New semantics Formalisms and properties for old and new semantics

11 A note on notation There is no universally adopted notation even for the most basic concepts The slides include excerpts taken directly from the original papers and reflect all these differences Should not be too mysterious some notes along the way for any doubt, please ask

12 Roadmap What s in the words? From the Big Bang to now Dung s framework

13 Dung s framework is (almost) nothing A directed graph (called defeat graph) where:» arcs are interpreted as attacks» nodes are called arguments by chance (let say historical reasons)

14 Dung s framework is (almost) nothing α β β α γ

15 Dung s framework is (almost) everything Arguments are simply conflictables Conflicts are everywhere Conflict management is a fundamental need with potential spectacular/miserable failures both in real life and in formal contexts (e.g. in classical logic) A general abstract framework centered on conflicts has a wide range of potential applications

16 Dung s framework is (almost) everything The pervasiveness of Dung s framework and semantics is witnessed by the correspondences drawn in the original paper with a variety of other formal contexts Many extensions and variations of Dung s framework allow a translation procedure back to the original framework to exploit its basic features

17 A conflict calculus: abstract argumentation semantics A way to identify sets of arguments surviving the conflict together given the conflict relation only In general, several choices of sets of surviving arguments are possible Two main styles for semantics definition: extensionbased and labelling-based These points will be discussed extensively after some user instructions

18 AA user manual 1. Identify an interesting application domain where conflict management plays a key role 2. Define a suitable formalization of problem instances in the selected domain 3. Define the notions of argument and attack in your formalization, i.e. a translation/abstraction method from problem instances to argumentation frameworks 4. Play at your will with conflict calculus at abstract level 5. Map back the results of conflict calculus (extensions or labellings) into entities at the problem level 6. Are they meaningful? Do they provide useful/original perspectives? Did you avoid to reinvent the wheel?

19 An example: the stable marriage problem Steps 1 and 2

20 An example: the stable marriage problem Step 3

21 An example: the stable marriage problem Steps 4 and 5

22 An example: the stable marriage problem Step 6

23 An example: the stable marriage problem Step 6 The notion of preferred extension provides a natural solution to a non traditional version of the marriage problem

24 Roadmap What s in the words? From the Big Bang to now Dung s framework A conflict calculus : argumentation semantics

25 Abstract argumentation semantics A way to identify sets of arguments surviving the conflict together given the conflict relation only Two main styles for semantics definition: extensionbased and labelling-based In general, several choices of sets of surviving arguments are possible (multiple-status semantics) but some semantics prescribe exactly one extension/labelling (single status semantics)

26 Extension-based semantics A set of extensions is identified Each extension is a set of arguments which can survive together or are collectively acceptable i.e. represent a reasonable viewpoint The justification status of each argument can be defined on the basis of its extension membership» skeptical justification = membership in all extensions» credulous justification = membership in one extension

27 Sets of extensions α β E 1 = {{α},{β}} E 2 = { } β E 1 = {{α},{β},{γ}} E 2 = { } α γ

28 Labelling-based semantics A set of labels is defined (e.g. IN, OUT, UNDECIDED) and criteria for assigning labels to arguments are given Several alternative labellings are possible The justification status of each argument can be defined on the basis of its labels

29 Labelling-based semantics L 1 α ΙΝ α α OUT β OUT β β IN L 2 α UND β UND

30 Labelling-based semantics L 1 α IN β OUT γ UND α UND β IN γ OUT β UND α OUT γ IN

31 Labelling-based semantics L 2 β UND α UND γ UND

32 Labellings vs. extensions Labellings based on {IN, OUT, UNDEC} and extensions can be put in direct correspondence Given a labelling L, LabToExt(L) = in(l) Given an extension E, a labelling L=ExtToLab(E) can be defined as follows: in(l)=e out(l)=attacked(e) undec(l)=all other arguments

33 Labellings vs. extensions α β

34 Labellings vs. extensions α ΙΝ β

35 Labellings vs. extensions α ΙΝ β OUT

36 Labellings vs. extensions α ΙΝ β OUT α β

37 Labellings vs. extensions L 1 α ΙΝ β OUT α OUT β IN

38 Labellings vs. extensions α β

39 Labellings vs. extensions L 2 α UND β UND

40 Labellings vs. extensions L 1 α ΙΝ β OUT α OUT β IN L 2 α UND β UND

41 Labellings vs. extensions L 1 α IN β OUT γ UND α UND β IN γ OUT β UND α OUT γ IN

42 Labellings vs. extensions L 2 β UND α UND γ UND

43 Defining argumentation semantics Many different proposals in the literature, corresponding to different intuitions There are cases where each literature semantics gives a different outcome How (and in how many ways) would you colour this graph? η θ γ α β δ ε κ ζ ι

44 Analyzing argumentation semantics A catalogue of literature semantics is not immediately useful since different semantics definitions don t follow a common pattern and are not directly comparable To start, examining a set of general properties driving the definition (or marking the differences) of argumentation semantics is more useful Focus on extension-based definition We will look at the semantics catalogue after that

45 Roadmap What s in the words? From the Big Bang to now Dung s framework A conflict calculus : argumentation semantics Basic semantics properties

46 Conflict freeness The conflict free principle states that an attacking and an attacked argument can not stay together Notational remark usually denotes the set of extensions prescribed by semantics for the argumentation framework

47 Conflict freeness The conflict free principle states that an attacking and an attacked argument can not stay together Very basic idea, followed by all semantics Minimal use of the attack relation (in fact the attack direction does not count) The empty set is conflict free by definition

48 Conflict freeness α β α β α β α β

49 Conflict freeness β β β α γ α γ α γ β β β α γ α γ α γ

50 I-maximality A given criterion can select sets which are in a relation of proper inclusion (this holds for conflict freeness and for more articulated notions) One may consider as a constraint that no extension is a proper subset of another one Important for some properties (e.g. to avoid triviality in the notion of skeptical justification) but not fundamental and poorly related with other principles

51 Defense related principles: admissibility To survive the conflict you should be able to defend yourself, namely to reply to every attack with a counterattack A conflict free set is admissible if it defends itself The attack direction counts The empty set is admissible by definition

52 Admissibility α β α β α β α β

53 Admissibility β β β α γ α γ α γ β β β α γ α γ α γ

54 Admissibility from a labelling perspective ou should have good reasons to assign IN or OUT labels, but you are free to be undecided All arguments UNDECIDED is an admissible labelling, as the empty set is always an admissible set

55 Defense related principles: strong admissibility/defense Admissibility includes self-defense A stronger notion of defense requires that defense comes from other arguments which are in turn strongly defended by other arguments Rather strong requirement: defense chains rooted in unattacked arguments The empty set is strongly defended by definition

56 Strong defense α β α β α β α β

57 Strong defense β β β α γ α γ α γ β β β α γ α γ α γ

58 Strong defense α β γ ε δ

59 Strong defense α ΙΝ β OUT γ IN ε UND δ UND

60 Defense related principles: reinstatement Reinstatement concerns effectiveness (or altruism) of defense: if you defend some argument you should take it on board (include it in the extension) Completeness requirement: can not leave out your protected ones The empty set satisfies reinstatement only if there are no unattacked arguments

61 Reinstatement α β γ ε δ

62 Reinstatement α β γ ε δ

63 Reinstatement α β γ ε δ

64 Reinstatement α β γ ε δ

65 Reinstatement α β γ ε δ

66 Reinstatement α β γ ε δ

67 Defense related principles: weak reinstatement A weaker notion of reinstatement is obtained considering strong defense instead of defense

68 Defense related principles: CF reinstatement Reinstatement does not require explicitly conflict freeness with the defended argument. Adding this gives rise to CF reinstatement

69 Directionality Basic idea: arguments affect each other following the direction of attacks. An argument (or set of arguments) is affected only by its ancestors in the attack relation

70 Directionality In words the restrictions of the extensions to an unattacked part of the graph are the same whatever is the remaining part of the graph (if any) Local extensions (or labellings) are included in global extensions* *Directionality implies that you may compute the restrictions of extensions to unattacked sets without considering the rest of the graph, it does not however imply that there is a way to proceed incrementally to compute the extensions for the rest of the graph

71 Directionality α β γ ε δ

72 Directionality α α ε ε

73 Directionality α β γ ε δ α β γ ε δ

74 Directionality α β ε γ

75 Directionality α β α β ε γ ε γ α β ε γ

76 A hidden principle: language independence Extensions only depend on the attack relation, i.e. on the graph topology not on argument names or on other underlying properties Isomorphic frameworks have the same (modulo the isomorphism) extensions

77 Implicit equal treatment The language independence principle implies equal opportunity for arguments, i.e. that topologically equivalent arguments are treated in the same way No semantics can prescribe an asymmetric set of extensions for a symmetric framework α β

78 Implicit equal treatment If there are reasons to prefer α to β, they should be reflected in the topology α β

79 Roadmap What s in the words? From the Big Bang to now Dung s framework A conflict calculus : argumentation semantics Basic semantics properties A catalogue of semantics

80 Conflict-freeness + maximality= naive semantics The so-called naive semantics prescribes as extensions the maximal conflict free sets of a framework As already remarked, it ignores the attack direction an so does not reflect the whole amount of available information It blatantly violates several basic principles like admissibility and reinstatement (even for unattacked arguments) In a sense, it is not really a semantics and is not considered in Dung s paper

81 Naive semantics α β γ α β γ

82 Can naive semantics work? Naive semantics makes sense when the direction of the attacks does not count: symmetric frameworks In any symmetric framework maximal conflict freesets are reasonable extensions for any multiplestatus semantics If the underlying instantiated formalism gives rise only to symmetric frameworks more sophisticated semantics notions do not make any difference Limited utility of semantics notions or limited expressiveness of the underlying formalism?

83 A bit less naive: stage semantics Stage semantics prescribes as extensions the conflict free sets with maximal range, i.e. those maximizing the union of the extension with the arguments it attacks (attack direction partially counts ) Some extremely naive behavior of naive semantics are avoided, but there are still cases where unattacked arguments are excluded from some extensions

84 A bit less naive: stage semantics α β γ α β γ not a stage extension α β γ α β γ

85 Defense based semantics Dung s paper introduces several notions and semantics all based on the notion of defense (admissibility principle) + conflict-freeness The direction of attacks counts!

86 Complete semantics Adds to admissibility the reinstatement principle: if you defend an argument you have to include it The empty set is complete only if there are no unattacked arguments Completeness does not imply I-maximality: a complete extension can be a proper subset of another one (obtained adding some self-defending set of arguments)

87 Admissibility and completeness α β γ ε δ Admissible sets: { {}, {α}, {ε}, {α,γ}, {α,δ}, {ε,β}, {α,γ,δ} } Complete extensions: { {}, {ε,β}, {α,γ,δ} }

88 Admissibility and completeness α β γ ε δ {α,γ} is admissible but not complete

89 Admissibility and completeness α β γ ε δ Admissible sets: { {}, {α}, {δ}, {ε}, {α,γ}, {α,δ}, {α,ε}, {γ,δ}, {α,γ,δ}, {α,γ,ε} } Complete extensions: {{α,γ}, {α,γ,δ}, {α,γ,ε} }

90 Admissibility and completeness α β γ ε δ {α,γ} is admissible and complete

91 Admissibility and completeness α β γ ε δ {α,γ,δ} is also admissible and complete

92 Grounded semantics: accepting only the unquestionable Given that complete extensions satisfy some basic properties, if one wants to be cautious s/he has to choose the smallest (wrt inclusion) complete extension Actually, it can be proved that there is a unique smallest complete extension, called grounded extension So grounded semantics belongs to the uniquestatus approach

93 Grounded semantics: unquestionable=strong defense Dung s definition of grounded semantics takes another route First, the characteristic function of a framework associates with a set S the arguments it defends

94 Grounded semantics: unquestionable=strong defense The characteristic function is well-behaved This entails that it has a least fixed point, which, by definition, is called grounded extension It can be proved that the least fixed point of the characteristic function coincides with the least complete extension

95 Grounded semantics: unquestionable=strong defense This has a rather intuitive counterpart: start computing the arguments defended by the empty set, repeat the computation (i.e. find the arguments defended by them), and so on, until you reach a fixed point For finitary frameworks the grounded extension can be obtained as: F AF (F AF ( F AF ( ) )

96 Computing the grounded extension α β γ ι ζ ε δ η θ

97 Computing the grounded extension α β γ ι ζ ε δ η θ

98 Computing the grounded extension α β γ ι ζ ε δ η θ

99 Computing the grounded extension α β γ ι ζ ε δ η θ

100 Horror vacui: stable semantics Stable semantics prescribes that any argument is either in the extension or attacked by the extension In the labelling version: no argument is undecided In general there are several stable extensions Any stable extension is admissible and complete (hence it includes the grounded extension)

101 A stable extension α β γ ι ζ ε δ η θ

102 Another stable extension α β γ ι ζ ε δ η θ

103 Being bold is not always possible The very strong requirement posed by stable semantics in some cases is not satisfiable The empty set is not the most basic extension (unless the framework is empty too) There are frameworks where no stable extensions exist The absence of odd-length β cycles is a sufficient condition for existence of stable extensions α γ

104 Being bold is not always possible A limited (possibly isolated) part of the framework may prevent the existence of extensions for the whole framework Stable semantics is not directional β α β γ γ δ α

105 Accept as much as you can defend: preferred semantics Admissibility + maximality = preferred semantics From a labelling perspective this corresponds to maximizing IN without banning UNDEC Preferred extensions always exist (in general many) Any preferred extension is complete (hence it includes the grounded extension) Stable extensions are preferred, not viceversa

106 Preferred vs. stable semantics δ α β γ ε

107 Preferred vs. stable semantics δ α β γ ε δ α β γ ε

108 Minimizing indecision: semi-stable semantics Maximizing acceptance does not mean minimizing indecision: to this purpose you should maximize the union of IN and OUT Stable semantics achieves this implicitly by banning UNDEC, semi-stable semantics does this explicitly without banning UNDEC

109 Minimizing indecision: semi-stable semantics Semi-stable extensions always exist (in general many) When stable extensions exist, semi-stable extensions coincide with stable extensions Semi-stable extensions are preferred, not viceversa Semi-stable semantics is not as fragile as stable semantics, but it is non directional too

110 Single status from multiple status A natural way to obtain a single status semantics from a multiple status semantics consists in using the intersection Int of the extensions For all the semantics seen so far Int is a (possibly strict) superset of the grounded extension Int is conflict-free but not admissible in general Adding the requirement of admissibility to the one of inclusion in all extensions gives rise to a generic scheme for semantics definition which has been explicitly considered in two cases

111 Ideal and eager semantics The ideal extension is the maximal (wrt inclusion) admissible set included in all preferred extensions The eager extension is the maximal (wrt inclusion) admissible set included in all semi-stable extensions Both are supersets of the grounded extension The eager extension is a superset of the ideal extension

112 A less cautious single status β γ δ α β The ideal/eager extension is {δ}, while the grounded extension is empty γ δ α

113 A less cautious single status α β The ideal/eager extension is {α}, while the grounded extension is empty α β γ δ ε The eager extension is {β,δ}, the ideal extension is empty

114 Roadmap What s in the words? From the Big Bang to now Dung s framework A conflict calculus : argumentation semantics Basic semantics properties A catalogue of semantics Properties of semantics (so far)

115 Universal and almost universal properties Conflict freeness and language independence are satisfied by all semantics I-maximality is satisfied by single-status semantics and is directly implied by several multiple-status definitions, only complete semantics is not I- maximal

116 Conflict-free only semantics Naive and stage extensions can exclude unattacked arguments and include undefended arguments hence they can not satisfy admissibility, strong defense, reinstatement, weak reinstatement, and directionality They only satisfy CF-reinstatement

117 Complete-based semantics Grounded, stable, preferred, semi-stable, ideal and eager extensions are complete extensions This implies that they satisfy admissibility and reinstatement (including weaker forms) Strong defense is only satisfied by grounded semantics

118 Directionality Disproving directionality is immediate for stable, semi-stable and eager semantics Proving directionality is less immediate (but possible) for the other complete-based semantics

119 A synopsis NA STA CO GR STB PR SST ID EAG CF-principle Language indip. I-maximality N Admissibility N N Strong defense N N N N N N N N Reinstatement N N Weak reinstatement N N CF-reinstatement Directionality N N N N N

120 Roadmap What s in the words? From the Big Bang to now Dung s framework A conflict calculus : argumentation semantics Basic semantics properties A catalogue of semantics Properties of semantics (so far) Taking topology seriously

121 Following the graph topology Directionality suggests that the graph topology encodes a form of dependency among arguments The intuitive algorithm for computing the grounded extension starts from unattacked arguments and proceeds orderly following attacks as far as possible but it immediately terminates if no unattacked arguments exists (does not manage cyclic dependences) Is there a topological order to follow in presence of cycles? (and useful for other semantics too?)

122 Strongly Connected Components In a graph the relation of mutual reachability is an equivalence relation (also called path equivalence) The relevant equivalence classes are called Strongly Connected Components (SCCs): intuitively they are islands (or clouds) of mutual dependence The graph obtained considering SCCs as single nodes is acyclic, i.e. there is an acyclic dependence relation among different SCCs The SCC decomposition of a graph can be computed in polynomial time

123 Strongly Connected Components δ α β γ ε

124 Strongly Connected Components δ α β γ ε

125 Strongly Connected Components δ α β γ ε

126 A partial order for evaluation

127 A partial order for evaluation S 3 S 6 S 1 S 4 S 2 S 5 S 7

128 A partial order for evaluation: level 1 S 3 S 6 S 1 S 4 S 2 S 5 S 7

129 A partial order for evaluation: level 2 S 3 S 6 S 1 S 4 S 2 S 5 S 7

130 A partial order for evaluation: level 3 S 3 S 6 S 1 S 4 S 2 S 5 S 7

131 A partial order for evaluation: level 4 S 3 S 6 S 1 S 4 S 2 S 5 S 7

132 SCC-recursiveness Basic idea: a semantics is SCC-recursive if its extensions can be built following the partial order of SCCs SCC-recursiveness requires two ingredients:» a semantics-specific base function to be applied to the single SCCs starting from the initial ones and then along the way» a generic SCC-recursive scheme for the application of the base function along the way Since the two ingredients are intertwined with some intricacies, let us proceed informally

133 SCC-recursiveness informally 1. Extensions are built following the SCC order 2. A semantics-specific base function BF is given, which determines the set of extensions prescribed for a single-scc argumentation framework 3. For any initial SCC S, the base function BF is applied to determine the possible values of E S for any extension E to be built by adding elements from other SCCs

134 SCC-recursiveness at work: initial SCCs Suppose that applying the base function BF on initial SCCs it emerges that S 1 There is exactly one choice for E S 1 S 2 There is one choice for E S 2 (actuallythe emptyset) (in general multiple choices are possible)

135 SCC-recursiveness informally 1. Extensions are built following the SCC order 2. A semantics-specific base function BF is given, which determines the set of extensions prescribed for a single-scc argumentation framework 3. For any initial SCC S, the base function BF is applied to determine the possible values of E S for any extension E to be built by adding elements from other SCCs 4. For any possible choice of E S, every subsequent SCC S is partitioned into three subsets

136 Attack, extensions and SCCs Given the choice of E ( ) in the preceding SCCs, the nodes of a SCC S are partitioned into D AF (S, E) : directly attacked by E U AF (S, E) : defended from outside attacks by E P AF (S, E) : not attacked by E nor defended from outside attacks by E S 1 S 3 S 6 S 4 S 2 S 5 S 7

137 SCC-recursiveness informally 1. Extensions are built following the SCC order 2. A semantics-specific base function BF is given, which determines the set of extensions prescribed for a single-scc argumentation framework 3. For any initial SCC S, the base function BF is applied to determine the possible values of E S for any extension E to be built by adding elements from other SCCs 4. For any possible choice of E S, every subsequent SCC S is partitioned in three subsets 5. Nodes in D AF (S, E) are suppressed and the construction of E proceeds by applying recursively the same steps on the remaining part of S. The distinction between P AF (S, E) and U AF (S, E) may be taken into account in this step.

138 SCC-recursiveness at work: propagating choices S 3 S 6 S 1 S 4 S 2 S 5 S 7

139 SCC-recursiveness at work: recursion on restricted SCCs S 3 restricted S3 restricted S 3 The restriction of S 3 consists of a single SCC: the base function is applied and there is exactly one choice for E S 3 S 5 restricted S 5 restricted S 5 The restriction of S 5 consists of three SCCs: the SCC recursive scheme is applied starting from the first one. The difference between green and yellow nodes may also be taken into account. Assume that it turns out that there is exactly one choice for E S 5

140 SCC-recursiveness at work: reaching final SCCs S 3 S 6 S 1 S 4 S 2 S 5 S 7 or... S 7

141 SCC-recursiveness formally The set of extensions is given by a generic function GF parametric with respect to the base function BF GF is recursive with respect to SCC decomposition Single-SCC argumentation frameworks are the base of recursion The second parameter of BF and GF identifies the green nodes with respect to yellow nodes

142 SCC-recursiveness is well-founded Defining a SCC-recursive semantics simply amounts to define a base function BF Simple requirements on the base function BF are sufficient to ensure reasonable results: If BF is conflict free the resulting extensions are conflict free If BF treats correctly an argumentation framework consisting of a single non self-defeating node, the resulting extensions are all supersets of the grounded extension

143 SCC-recursiveness in the wild The four traditional Dung s semantics (complete, grounded, stable, preferred) are SCC-recursive SCC-recursiveness + universal existence of extensions implies directionality Stable semantics is SCC-recursive since stable extensions may not exist Other non directional semantics are not SCCrecursive The reverse does not hold: ideal semantics is directional but not SCC-recursive

144 Ideal semantics is not SCC-recursive β Ideal extension = β α γ δ ε Ideal extension = {ε} γ δ ε The base function should give different results with the same input for the same SCC α

145 SCC-recursiveness for easy semantics definition Defining a semantics simply amounts to define a base function for single SCCs Four semantics were defined in the SCCrecursiveness paper Among them CF2 semantics has the most interesting properties

146 CF2 semantics The base function is very simple: it selects the maximal conflict free sets within the SCC It departs from the notion of admissibility Symmetric treatment of odd- and even-length cycles 2 extensions 3 extensions

147 A motivation for symmetric treatment of cycles Unreliable witnesses example (taken from Pollock) Witness A says that witness B is unreliable, witness B says that witness C is unreliable, witness C says that witness A is unreliable. All witnesses say that the statement S is false. Witness A says that witness B is unreliable, witness B says that witness C is unreliable, witness C says that witness D is unreliable, witness D says that witness A is unreliable. All witnesses say that the statement S is false.

148 A motivation for symmetric treatment of cycles Should the number of witnesses make any difference on the justification status of an argument involving the statement S? In fact an odd or even number makes a difference according to stable, semi-stable, and preferred semantics (and to any admissibility-based multiplestatus semantics) It might not be what you want: different semantics are appropriate for different contexts and needs

149 A motivation for symmetric treatment of cycles Unreliability claims among witnesses Consonant witnesses vs. claims A dissonant claim

150 A motivation for symmetric treatment of cycles Unreliability claims among witnesses Consonant witnesses vs. claims A dissonant claim The dissonant claim is skeptically accepted with three witnesses

151 A motivation for symmetric treatment of cycles The dissonant claim is NOT skeptically accepted with four witnesses

152 CF2 semantics When AF consists of a single SCC, CF2 extensions coincide with maximal conflict free sets of AF In general, it turns out that a CF2 extension is a maximal conflict free set of AF but not vice versa. S 2 S 1

153 CF2 semantics First choice in the first SCC

154 CF2 semantics Three extensions based on the first choice in the first SCC

155 CF2 semantics Second choice in the first SCC Only one extension based on the second choice A maximal CF set which is not a CF2 extension

156 Evaluating CF2 semantics Like naive or stage semantics CF2 is mainly based on conflict-freeness and trades off the symmetric treatment of odd- and even-length cycles with admissibility (and reinstatement) However thanks to the SCC-recursive scheme it has other properties (directionality, weak reinstatement) that naive or stage semantics fail to achieve

157 Evaluating CF2 semantics N N N N N STA Language indip. N N N CF2 N N CO GR N N STB N PR N N SST N N Directionality N Weak reinstatement N EAG CF-principle CF-reinstatement N Reinstatement N N Strong defense N Admissibility I-maximality ID NA

158 A natural development Recently stage2 semantics has been proposed It embeds the notion of stage extension in the SCCrecursive scheme Same gains, as in CF2 semantics, and more maximization It fixes some counterintuitive behavior of CF2 semantics for instance with long cycles

159 Stage2 vs. CF2 Stage2 and CF2 extensions Other CF2 extensions

160 Roadmap What s in the words? From the Big Bang to now Dung s framework A conflict calculus : argumentation semantics Basic semantics properties A catalogue of semantics Properties of semantics (so far) Taking topology seriously Differences vs commonalities:semantics agreement

161 Why are they so many? Different intuitions Different needs How to patch a counterintuitive behavior in a previous semantics? Just define a new semantics! How to get some papers published? Just define a new semantics! Parametric definition schemes offer new opportunities of exploration of the semantics design space

162 Why are they all different? If it is not different it does not get published Many questions in semantics design/behavior admit reasonable alternative answers e.g. Is self-defense sufficient? (Grounded NO, Others ES) Is undecision admitted? (Stable NO, Others ES) Are multiple alternatives possible? (Grounded, Ideal, Eager NO, Others ES) Should odd- and even-length cycles be treated equally? (Grounded, CF2, Stage ES, Others NO)

163 Why are they all similar? Different semantics are not behaving differently in all cases: they can t, because of» very basic constraints» common (possibly implicit) underlying principles

164 Obvious commonalities The empty argumentation framework: AF = <, > Of course, for any semantics S, E S (AF) = For any pair of semantics S 1 and S 2 there is at least one argumentation framework where S 1 and S 2 agree

165 Obvious commonalities The monadic argumentation framework: AF = <α, > α Of course, for any semantics S, E S (AF) = {α} A valid argument not interacting with any other argument can not be rejected (basic reinstatement)

166 Obvious commonalities The multi-monadic argumentation framework: AF = <Α, > α β γ For any semantics S, E S (AF) = A = {α,β,γ,...} A valid argument not interacting with any other argument can not be rejected (and conflicts consist only in explicit attack relations): let say the unattacked-in principle (basic reinstatement)

167 Obvious commonalities Attack chain argumentation framework α β γ For any semantics S, E S (AF) = {α, γ} Several principles come into play with this very simple example: unattacked-in, admissibility, reinstatement (and conflicts consist only in explicit attack relations)

168 Obvious commonalities Acyclic argumentation framework ε δ γ β α

169 Obvious commonalities? Acyclic argumentation framework ε δ γ β α For any semantics S, E S (AF) = {α, δ, ε} NO! The grounded prudent extension (we did not introduce) is {δ, ε}. α is not included due to indirect conflict with δ conflicts consist only in explicit attack relations does not hold in prudent semantics

170 Why caring about agreement? Theory:» Identification of universally accepted behaviors» Confirmation of already identified principles and reverse engineering of further principles Practice» Actual relevance of choice among semantics in specific contexts» Classification of application contexts according to their agreement properties

171 Formalizing agreement Two semantics S 1 and S 2 are in agreement about an argumentation framework AF if E S 1 (AF) = E S2 (AF) We require that both S 1 and S 2 admit extensions for AF, namely E S 1 (AF) and E S2 (AF) In other words, for a semantics S, agreement is evaluated only about argumentation frameworks where S admits at least one (possibly empty) extension Only stable semantics may not admit extensions on a given AF

172 Two main research lines Identifying topological families of argumentation frameworks ensuring agreement among some semantics Systematic identification of all possible agreement classes (given a reference set of semantics) using general set-theoretical properties of semantics extensions rather than topological properties of argumentation frameworks

173 Simple topological families: acyclic frameworks If AF is well-founded (i.e. acyclic when AF is finite) grounded, stable and preferred semantics agree The absence of cycles enforces a single-status behavior of multiple-status semantics: grounded extension is the natural and univocal extension in these cases Cycles as generators of alternative extensions Almost out of discussion (except prudent semantics) Further results extend this agreement to other semantics

174 Simple topological families: symmetric frameworks Naive, stable, stage, semi-stable, preferred, and CF2 semantics are in agreement about AF if AF is symmetric (all attacks are mutual) Symmetry of attacks (in absence of self-defeating arguments) prevents any difference among multiple-status semantics: all maximal conflict-free sets satisfy any requirement Symmetry leaves open only the choice between single-status and multiple-status

175 Simple (to state) topological families: frameworks without odd-length cycles Stable and preferred semantics are in agreement about AF if AF is limited controversial (i.e. free of odd-length cycles when AF is finite) [Dung 95] Odd-length cycles justify the distinction between stable and preferred semantics Odd-length cycles are the reason for non existence of stable extensions in some cases or for missing stable extensions in other ones The treatment of odd-length and even-length cycles is unequal in Dung s multiple status semantics

176 Simple (to compute) families: determined frameworks A framework is determined when its grounded extension is also stable (this can be checked in polynomial time) There are no undecided ( provisionally defeated ) arguments according to grounded semantics Determined Not determined

177 A topological characterization of determined arg. frameworks An argumentation framework AF is determined if and only if it is initial acyclic, namely AF is empty or the initial SCCs of AF are monadic the restriction of AF to the nodes not included in and not attacked by initial SCCs is in turn initial acyclic Dung s sufficient condition of acyclicity is included S 2 S 3 S 1 S 1 S 2

178 A topological characterization of determined arg. frameworks An argumentation framework AF is determined if and only if it is initial acyclic, namely AF is empty or the initial SCCs of AF are monadic the restriction of AF to the nodes not included in and not attacked by initial SCCs is in turn initial acyclic S 1 S 1 S 2 S 3 Initial acyclic S 2

179 A topological characterization of determined arg. frameworks An argumentation framework AF is determined if and only if it is initial acyclic, namely is is empty or the initial SCCs of AF are monadic the restriction of AF to the nodes not included in and not attacked by initial SCCs is in turn initial acyclic S 1 S 1 Initial acyclic S 2 Not initial acyclic

180 Single-status agreement Determined frameworks are a superset of acyclic frameworks Exploiting SCC-recursiveness, agreement with grounded semantics is extended to a larger family of frameworks and a larger set of semantics

181 Single-status agreement If AF is determined any SCC-recursive semantics with grounded semantics provided that: is conflict free treats reasonably any monadic AF agrees α A monadic AF The requirement on Uniform single status behavior of a large family of reasonable semantics (including classical ones) on determined argumentation frameworks

182 Single-status agreement Combining this result with some easy observations on well-known properties of inclusion relation and coincidence among extensions of various semantics it turns out that the following semantics agree on determined argumentation frameworks: Grounded, Complete, Preferred, Stable, Semistable, Stage, Ideal, CF2, Resolution-based grounded* *to be introduced later

183 Simple (to compute) families: almost determined frameworks α AF is not determined because Unability to prescribe extensions is an apparent cause of disagreement Any semantics able to prescribe extensions should agree with grounded semantics also in this kind of case

184 Simple (to compute) families: almost determined frameworks An argumentation framework is almost determined if any argument not included in and not attacked by the grounded extension is self-defeating Any SCC-recursive semantics which: is universally defined is conflict free reasonably treats monadic argumentation frameworks is in agreement with grounded semantics on any almost determined argumentation framework Larger class of argumentation frameworks providing a necessary and sufficient condition for agreement with grounded semantics of universally defined SCC-recursive semantics

185 Single-status agreement (reprise) Combining this result with some easy observations on well-known properties of inclusion relation and coincidence among extensions of various semantics it turns out that the following semantics agree on almost determined argumentation frameworks: Grounded, Complete, Preferred, Semi-stable, Ideal, CF2, Resolution-based grounded Stable semantics is missing for obvious reasons Stage semantics is missing for less obvious reasons: in presence of self-defeating arguments it may exclude unattacked arguments

186 Exploiting symmetry with SCCs A SCC-recursive semantics is *-symmetric if for any AF which is symmetric and free of self-defeating arguments coincides with the set of maximal (wrt inclusion) conflict free sets of AF Several reasonable multiple-status semantics (including preferred, stable and CF2) are *-symmetric Not extremely interesting per se but...

187 Simple (to compute) families: SCC-symmetric frameworks An argumentation framework is SCC-symmetric if its restriction to any of its SCCs is symmetric In other words, non mutual attacks are possible, but any cycle of AF involves mutual attacks only S 2 S 2 S 1 S 1 Not SCC-symmetric SCC-symmetric

188 Multiple-status agreement All *-symmetric semantics are in agreement on any argumentation framework SCC-symmetric and free of self-defeating arguments In particular, they are in agreement with the stable semantics In particular, preferred and CF2 semantics are in agreement The class of SCC-symmetric argumentation frameworks is equivalent to other classes considered in the literature: symmetric conflict + transitive preference rebutting (à la Pollock) defeat only This suggests that this class may be relevant in practical applications

189 SCC-symmetric vs. limited controversial While the result on determined argumentation frameworks subsumes the original result on acyclic frameworks the result on SCC-symmetric frameworks complements the original result on limited controversial frameworks

190 SCC-symmetric vs. limited controversial SCC-symmetric and limited controversial SCC-symmetric not limited controversial Limited controversial not SCC-symmetric Not SCC-symmetric not limited controversial

191 Summing up on topologies ensuring agreement Determined (and almost determined) argumentation frameworks: are probably not sufficient for most practical applications provide a strong (single-status) reference behavior for any new semantics proposal SCC-symmetric argumentation frameworks: may be relevant in practice in application domains where only SCC-symmetric argumentation frameworks are present one should not bother about (multiple-status) semantics differences

192 Agreement classes Given a set of argumentation semantics, the set of argumentation frameworks where all semantics in agree will be denoted as E.g. denotes the set of argumentation frameworks (agreement class) where preferred, stable and semi-stable semantics agree Identifying agreement classes provides the reference points for investigations on agreement Agreement classes may admit, but do not necessarily have, a topological characterization

193 Basic properties E.g. It may be (and it is) the case that for some different sets of semantics and it holds that

194 Agreement classes: how many? The literature results I am aware of concern 7 semantics (naive, stage, eager are not included) Considering the set of 7 argumentation semantics any subset of such that gives rise, in principle, to an agreement class (120 classes in total) It is proved that most of these 120 classes are not actually different: only 14 distinct classes exist

195 An example of agreement class coincidence A* = AGR({GR,ST}) When stable extensions exist they coincide with semi-stable extensions A* = AGR({GR, ST, SST}) Every stable extension is a preferred extension (maximal complete extension) and the grounded extension is the least complete extension A* = AGR({GR, ST, SST, CO, PR}) The ideal extension is a superset of the grounded extension and is included in any preferred extension A* = AGR({GR, ST, SST, CO, PR, ID}) Every CF2 extension is a superset of the grounded extension and is a maximal conflict-free set + every stable extension is a maximal conflict-free set A* = AGR({GR, ST, SST, CO, PR, ID,CF2})

196 Agreement classes: which kind of analysis? Agreement classes are denoted as Σ 1 Σ 14 We proceed by partial order of inclusion: if Σ i Σ j then j > i For each class Σ i three main steps have been carried out: 1. identifying which classes Σ k, with k<i, are included in Σ i 2. for each of these classes Σ k, showing that Σ i \ Σ k 3. for any Σ h with h<i and Σ h Σ i, examining Σ i Σ h For any set of semantics not directly corresponding to any of Σ 1 Σ 14 it is shown that coincides with one of them

197 Agreement classes: which (known) properties can be exploited? Several kinds of inclusion relations between (sets of) extensions: for any argumentation framework AF

198 Agreement classes: which (known) properties can be exploited? Several kinds of inclusion relations between (sets of) extensions: for any argumentation framework AF Inclusion of the whole set of extensions

199 Agreement classes: which (known) properties can be exploited? Several kinds of inclusion relations between (sets of) extensions: for any argumentation framework AF The grounded extension is included in many kinds of extensions

200 Agreement classes: which (known) properties can be exploited? Several kinds of inclusion relations between (sets of) extensions: for any argumentation framework AF Any extension of one kind is included in an extension of another kind

201 Agreement classes: further properties we can prove Several properties based on the inclusion relationships can be derived

202 Agreement classes: which kind of properties we prove? Several lemmata based on the inclusion relationships can be derived Implications of cardinality

203 Agreement classes: which kind of properties we prove? Several lemmata based on the inclusion relationships can be derived Implications of inclusion in the set of extensions

204 Agreement classes: which kind of properties we prove? Several lemmata based on the inclusion relationships can be derived Some agreements imply others

205 Agreement classes: which kind of properties we prove? Several lemmata based on the inclusion relationships can be derived Implications of extension properties

206 Agreement classes: coincidence Coincidence of agreement classes follows (almost directly) from the lemmata. Examples are:

207 Agreement classes: unique-status behavior PR=SST=ID Σ 7 GR=PR=SST =ID=CO Σ 3 AF 3 AF 7 PR=CF2=SST=ID GR=PR=CF2=SST=ID=CO Σ 5 AF 5 Σ 2 AF 2 PR=ST=SST=ID PR=CF2=ST=SST=ID GR=ST=PR=CF2=SST=ID=CO AF 6 Σ 6 Σ 4 AF 4 Σ 1 AF 1 AF 8 Σ 8 GR=ID

208 Agreement classes: GR unique-status behavior PR=SST=ID Σ 7 GR=PR=SST =ID=CO Σ 3 AF 3 AF 7 PR=CF2=SST=ID GR=PR=CF2=SST=ID=CO Σ 5 AF 5 Σ 2 AF 2 PR=ST=SST=ID AF 6 Σ 6 PR=CF2=ST=SST=ID Σ 4 AF 4 GR=ST=PR=CF2=SST=ID=CO Σ 1 AF 1 Σ 1 is the class where all AF 8 Σ 8 GR=ID the considered semantics agree (in particular with GR) We already know it : determined argumentation frameworks

209 Agreement classes: GR unique-status behavior PR=SST=ID Σ 7 GR=PR=SST =ID=CO Σ 3 AF 3 AF 7 PR=CF2=SST=ID Σ 5 AF 5 GR=PR=CF2=SST=ID=CO Σ 2 AF 2 PR=ST=SST=ID PR=CF2=ST=SST=ID GR=ST=PR=CF2=SST=ID=CO AF 6 Σ 6 Σ 4 AF 4 Σ 1 AF 1 AF 8 Σ 8 Σ 2 is the class where all the considered semantics agree (but ST may be undefined) GR=ID We already know it : almost determined argumentation frameworks

210 Agreement classes: GR unique-status behavior PR=SST=ID Σ 7 GR=PR=SST =ID=CO Σ 3 AF 3 AF 7 PR=CF2=SST=ID Σ 5 AF 5 GR=PR=CF2=SST=ID=CO Σ 2 AF 2 PR=ST=SST=ID PR=CF2=ST=SST=ID GR=ST=PR=CF2=SST=ID=CO AF 6 Σ 6 Σ 4 AF 4 Σ 1 AF 1 AF 8 Σ 3 is the class where all but CF2 semantics agree (while ST may be undefined) Σ 8 GR=ID

211 Agreement classes: GR unique-status behavior PR=SST=ID Σ 7 GR=PR=SST =ID=CO Σ 3 AF 3 AF 7 PR=CF2=SST=ID Σ 5 AF 5 GR=PR=CF2=SST=ID=CO Σ 2 AF 2 PR=ST=SST=ID PR=CF2=ST=SST=ID GR=ST=PR=CF2=SST=ID=CO AF 6 Σ 6 Σ 4 AF 4 Σ 1 AF 1 AF 8 Σ 8 GR=ID Σ 3 \ Σ 2 as it includes AF 3 = β α γ

212 A note on complete semantics The grounded extension belongs to the set of complete extensions (is the least complete extension) For all the considered semantics, except CO, it holds that no extension can be a proper subset of another extension For all the considered semantics, any extension is a superset of the grounded extension It follows that agreement with CO is possible for a multiplestatus semantics only if also agreement with GR holds As a consequence CO only appears in agreement classes Σ 1,Σ 2,Σ 3

213 Agreement classes: GR unique-status behavior PR=SST=ID Σ 7 GR=PR=SST =ID=CO Σ 3 AF 3 AF 7 PR=CF2=SST=ID Σ 5 AF 5 GR=PR=CF2=SST=ID=CO Σ 2 AF 2 PR=ST=SST=ID PR=CF2=ST=SST=ID GR=ST=PR=CF2=SST=ID=CO AF 6 Σ 6 Σ 4 AF 4 Σ 1 AF 1 Σ 8 is the only other class of agreement involving GR Examples of argumentation frameworks in Σ 8 \ Σ 7 will be given later AF 8 Σ 8 GR=ID

214 Agreement classes: ID unique-status behavior Grounded semantics specifies a single-status behavior based on strong defense (self-defense is not enough) Ideal semantics specifies a single-status behavior based on defense (admissibility) and inclusion in all preferred extensions (no attacks from other admissible sets) A simple topological fragment realizes this difference α β Complete semantics is outside these agreements as already seen

215 Agreement classes: ID unique-status behavior PR=SST=ID Σ 7 GR=PR=SST =ID=CO Σ 3 AF 3 AF 7 PR=CF2=SST=ID Σ5 AF 5 GR=PR=CF2=SST=ID=CO Σ 2 AF 2 PR=ST=SST=ID PR=CF2=ST=SST=ID GR=ST=PR=CF2=SST=ID=CO AF 6 Σ 6 Σ 4 AF 4 Σ 1 AF 1 Σ 4 is the class where all but CO and GR semantics agree Σ 4 \ Σ 1 as it includes AF 4 = α β AF 8 Σ 8 GR=ID

216 Agreement classes: ID unique-status behavior PR=SST=ID Σ 7 GR=PR=SST =ID=CO Σ 3 AF 3 AF 7 PR=CF2=SST=ID Σ 5 AF 5 GR=PR=CF2=SST=ID=CO Σ 2 AF 2 PR=ST=SST=ID PR=CF2=ST=SST=ID GR=ST=PR=CF2=SST=ID=CO AF 6 Σ 6 Σ 4 AF 4 Σ 1 AF 1 Σ 5 is the class where all but CO and GR semantics agree, while ST may be undefined Σ 5 \(Σ 4 Σ 2 ) as it includes AF 5 = α AF 8 β Σ 8 γ GR=ID

217 Agreement classes: ID unique-status behavior PR=SST=ID AF 7 Σ 7 PR=CF2=SST=ID GR=PR=SST =ID=CO GR=PR=CF2=SST=ID=CO Σ 3 AF 3 Σ 5 AF 5 Σ 2 AF 2 PR=ST=SST=ID PR=CF2=ST=SST=ID GR=ST=PR=CF2=SST=ID=CO AF 6 Σ 6 Σ 4 AF 4 Σ 1 AF 1 Σ 6 is the class where all but AF 8 Σ 8 GR=ID CO, GR, and CF2 semantics agree (excluding CF2 requires a 3-length cycle) Σ 6 \(Σ 4 Σ 5 ) as it includes AF 6 = β α γ