A Sound and Complete Dialogue System for Handling Misunderstandings Andreas Xydis1,† , Christopher Hampson1 , Sanjay Modgil1 and Elizabeth Black1 1 Department of Informatics, King’s College London, London, United Kingdom Abstract A common assumption for argumentation dialogues is that any argument exchanged is complete, i.e. its premises entail its claim. But in real world dialogues, agents commonly exchange enthymemes — arguments with incomplete logical structure. This paper introduces a dialogue system that handles misunderstandings that may occur when there is a mismatch between what a participant excludes from an argument and what her interlocutor assumes to be the missing information. We also introduce a mechanism for determining the acceptability status of the dialogue moves so that, under certain conditions, the status of moves made during a dialogue conforming to our system, corresponds with the status of arguments in the Dung argument framework instantiated by the contents of the moves made at that stage in the dialogue. Keywords Enthymemes, locutions, dialogue, framework, ASPIC+ , argumentation 1. Introduction In approaches to structured argumentation, arguments typically consist of a conclusion deduc- tively and/or defeasibly inferred from some premises [1]. However, in practice, human agents typically assert ‘incomplete’ arguments known as enthymemes [2]. Misunderstandings may arise due to interlocutors’ incorrect assumptions. Consider for example the dialogue below, which is annotated with the relevant locutions from the dialogue system proposed in this paper. Example 1. 1. Alice: I found a job so I can eat at a restaurant today. (assert 𝑏; 𝑏 ⇒ 𝑎; 𝑎) 2. Bob: You can’t afford to eat out, you owe money. (assert 𝑐; 𝑐 ⇒ ¬𝑎; ¬𝑎) 3. Alice: But I made a deal with my creditors. (assert 𝑞) 4. Bob: And so? What do you mean? (what do you intend) 5. Alice: So I don’t need to pay the bills today. (intend 𝑞; 𝑞 ⇒ ¬𝑒; ¬𝑒) 6. Bob: Why is that relevant? (what did you assume by 𝑐; 𝑐 ⇒ ¬𝑎; ¬𝑎) 7. Alice: I thought that you thought that I can’t afford to eat at a restaurant today because I owe money since I have bills to pay today. (assumed 𝑒; 𝑒 → 𝑐; 𝑐; 𝑐 ⇒ ¬𝑎; ¬𝑎) 8. Bob: No! I meant that you can’t afford to eat at a restaurant today because you owe money since you need to pay Kate back today. (meant 𝑝; 𝑝 → 𝑐; 𝑐; 𝑐 ⇒ ¬𝑎; ¬𝑎) SAFA’22: Fourth International Workshop on Systems and Algorithms for Formal Argumentation 2022, September 13, 2022, Cardiff, Wales, United Kingdom † Corresponding author E-mail: andreas.xydis@kcl.ac.uk © 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) 19 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 Alice first asserts an argument (1), to which Bob responds with a counterargument (2). Alice then moves an enthymeme (3) for an argument that she believes counters Bob’s argument. Note, the enthymeme Alice presents does not explicitly contradict anything that Bob has said, and so Bob asks for clarification (4) on what he is meant to infer from this enthymeme, which Alice provides (5)1 . However, Alice’s clarification still does not explicitly contradict anything Bob has said. Since Bob does not understand why Alice’s enthymeme is relevant to what he said, he asks Alice to explain what she assumed (6). Alice explains the assumption she had made (7), which Bob then corrects (8). Notice that Bob’s assertion in (2) is an enthymeme of his intended argument in (8). This simple example illustrates the need for a dialogue system that allows agents to ask for and provide clarifications when enthymemes are involved (4 and 5, respectively). It also shows that agents need to be able to ask what another agent has assumed was intended by an enthymeme (6), to answer such a question (7), and to correct any erroneous assumptions (8). Additionally, observe that before Bob resolves the misunderstanding regarding what he intended and what Alice assumed (6-8), although it appears that Alice wins the dialogue (since she moves her enthymeme in 3 against Bob’s argument in 2), there is no formal defeat on Bob’s argument and so Bob’s argument (2) is determined acceptable in the argument framework AF constructed by the contents of the enthymemes revealed by the agents. In other words, a mismatch can exist between the pragmatic and the logical conclusions implied by a dialogue in which enthymemes are used. Most works dealing with enthymemes focus on how an enthymeme can be constructed from the intended argument and how the intended argument can be reconstructed from a received enthymeme, based on assumptions that the sender and the receiver make about their shared knowledge and context, e.g. [4, 5, 6]. Notice that [7] allows for both backward and forward extending of enthymemes, as does the dialogue system in [8], which additionally enables resolution of misunderstandings that arise due to use of enthymemes. However, unlike our work, these works do not consider how the outcome of the dialogue relates to the AF that is instantiated based on contents of the enthymemes moved during a dialogue, meaning that there is no guarantee that the dialogue outcome respects the underlying argumentation theory. Notable exceptions are [9] and [10], but these works only address backward and forward extension of enthymemes, respectively, whereas our paper allows for the full extension of any kind of enthymeme to the intended argument it was instantiated from as well as for the resolution of misunderstandings that may occur because of enthymemes. Our primary contribution is the development of a dialogue system in which agents can move any kind of enthymeme and seek clarification to elicit the intended arguments of these enthymemes as well as resolve any misunderstanding that may occur between the participants regarding what an agent assumes and what its counterparts intends, such that under certain conditions, the dialogical status of the moves made during the dialogue—determined by what we call the dialogue framework—corresponds to the acceptability of the arguments in the Dung AF instantiated from the declarative contents of the moves made at that stage in the dialogue. This correspondence—not considered previously for dialogue systems that support 1 This example highlights requirements for dialogical ‘reification’ of relations between locutions, as proposed in [3]; requirements that arise due to the use of enthymemes 20 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 resolution of misunderstandings in dialogues—is significant since it demonstrates that the dialogue system we propose respects the logic and semantics of the underlying argumentation theory. Essentially, this correspondence means that there is no disadvantage to the use of enthymemes in dialogues (even when misunderstandings occur); a common real-world feature of dialogues that supports efficient inter-agent communication. If we are to enable effective human-computer interaction and provide normative support for human-human dialogue, we need to account for the ubiquitous use of enthymemes in real-world dialogues and make sure that agents can still reach the “correct" conclusions based on the knowledge they have shared. 2. Preliminaries The 𝐴𝑆𝑃 𝐼𝐶 + framework [11] abstracts from the particularities of the underlying language, the nature of conflict, the defeasible and strict inference rules (which can encode a deductive logic of one’s choosing) that are used to chain inferences from premises to an argument’s conclusion. 𝐴𝑆𝑃 𝐼𝐶 + arguments are evaluated in a Dung AF [12], and 𝐴𝑆𝑃 𝐼𝐶 + provides guidelines for ensuring that the outcome of evaluation yields rational outcomes. An 𝐴𝑆𝑃 𝐼𝐶 + argumentation theory AT is a tuple ⟨𝐴𝑆, 𝐾⟩ consisting of an argumentation system 𝐴𝑆 and a knowledge base 𝐾. 𝐴𝑆 is a tuple ⟨𝐿, ( · ), 𝑅, nom⟩ where 𝐿 is a logical language and ( · ) : 𝐿 → (2𝐿 − {∅}) is a function that generalises the notion of negation, so as to declare that two formulae are in conflict (e.g., married = {single, unmarried}). Additionally, 𝑅 = 𝑅𝑠 ∪ 𝑅def is a set of strict (𝑅𝑠 ) and defeasible (𝑅def ) inference rules and nom : 𝑅def* →𝐿 * (where 𝑅def is the class of all defeasible rules) is a naming function which assigns a name (or nominal) to each defeasible rule (so that rules can be referenced in the object language). Lastly, 𝑅 = 𝑅𝑠 ∪ 𝑅def is the (disjoint) union of a set of strict rules 𝑅𝑠 of the form 𝑝1 , . . . , 𝑝𝑛 → 𝑝, and a set of defeasible rules 𝑅def of the form 𝑝1 , . . . , 𝑝𝑛 ⇒ 𝑝, for 𝑝, 𝑝1 , . . . , 𝑝𝑛 ∈ 𝐿. For each rule 𝑟 ∈ 𝑅 let antecedents(𝑟) = {𝑝1 , . . . , 𝑝𝑛 } denote the set of antecedents of 𝑟 and consequent(𝑟) = 𝑝 denote the consequent of 𝑟. Finally, a knowledge base 𝐾 ⊆ 𝐿 is a set of premises. In our representation of an argument, a strict/defeasible inference rule is incorporated as a node, intermediating between the parent node (the rule’s conclusion) and the child node(s) (the rule’s antecedent(s)). This contrasts with the standard 𝐴𝑆𝑃 𝐼𝐶 + notion of an argument in which the rules are represented by undirected edges (where solid lines represent strict rules and dotted lines represent defeasible rules) linking the conclusion to the rule’s antecedents (see Figures 1.(a) and 1.(b) for an example of an 𝐴𝑆𝑃 𝐼𝐶 + argument and an argument as depicted in this paper, respectively). Definition 1. Given an argumentation system AS = ⟨𝐿, ( · ), 𝑅, nom⟩ and an argumen- tation theory AT = ⟨AS , 𝐾⟩, an argument is a labelled (downward directed) tree 𝐴 = ⟨Nodes(𝐴), Edges(𝐴), lab𝐴 ⟩ such that: 1. lab 𝐴 : Nodes(𝐴) → 𝐿 ∪ 𝑅 is a node labelling; 2. Edges(𝐴) ⊆ Nodes(𝐴) × Nodes(𝐴) such that if (𝑛𝑖 , 𝑛𝑗 ) ∈ Edges(𝐴), then either: (a) lab 𝐴 (𝑛𝑖 ) ∈ 𝐿, lab 𝐴 (𝑛𝑗 ) ∈ 𝑅 and consequent(lab 𝐴 (𝑛𝑗 )) = lab 𝐴 (𝑛𝑖 ), or (b) lab 𝐴 (𝑛𝑖 ) ∈ 𝑅 and antecedents(lab 𝐴 (𝑛𝑖 )) = {lab 𝐴 (𝑛𝑗 ) | (𝑛𝑖 , 𝑛𝑗 ) ∈ Edges(𝐴)}, 21 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 and if (𝑛𝑖 , 𝑛𝑗 ), (𝑛𝑖 , 𝑛𝑘 ) ∈ Edges(𝐴) and 𝑛𝑗 ̸= 𝑛𝑘 then lab 𝐴 (𝑛𝑗 ) ̸= lab 𝐴 (𝑛𝑘 ); 3. for every node 𝑛 ∈ Leaves(𝐴), we have lab 𝐴 (𝑛) ∈ 𝐾; 4. |Roots(𝐴)| = 1; 5. lab 𝐴 (Conc(𝐴)) ∈ 𝐿, where Conc(𝐴) is the unique element in Roots(𝐴); where Leaves(𝐴) = {𝑛𝑖 ∈ Nodes(𝐴) | ∄(𝑛𝑖 , 𝑛𝑗 ) ∈ Edges(𝐴)}, Roots(𝐴) = {𝑛𝑖 ∈ Nodes(𝐴) | ∄(𝑛𝑗 , 𝑛𝑖 ) ∈ Edges(𝐴)}. Let A𝐴𝑇 denote the set of arguments instantiated by 𝐴𝑇 , A* the class of all arguments, and Rules(𝐴) = {𝑛 ∈ Nodes(𝐴) | lab 𝐴 (𝑛) ∈ 𝑅}, for all 𝐴 ∈ A* . The attack and defeat relations over arguments are defined as for 𝐴𝑆𝑃 𝐼𝐶 + arguments. Argu- ment 𝑋 attacks argument 𝑌 if 𝑋’s claim conflicts with an ordinary premise or the consequent or name of a defeasible rule in 𝑌 , and may succeed as a defeat if the targeted sub-argument 𝑌 ′ of 𝑌 (𝑌 ′ ∈ Sub(𝑋)) is not strictly preferred to 𝑋 (for details see [11]). An argument framework AF instantiated by an argumentation theory 𝐴𝑇 is a tuple ⟨A𝐴𝑇 , Dfs⟩ where Dfs ⊆ A𝐴𝑇 × A𝐴𝑇 is the 𝐴𝑆𝑃 𝐼𝐶 + defeat relation. According to a complete labelling function L on AF (as defined in [13]) ∀𝐴 ∈ A𝐴𝑇 : 𝐴 is in iff every 𝐵 that defeats 𝐴 is out ((𝐵, 𝐴) ∈ Dfs → 𝐵 is out); 𝐴 is out iff ∃(𝐵, 𝐴) ∈ Dfs, 𝐵 is in; undec iff 𝐴 is neither in nor out. in(L), out(L) and undec(L) respectively denote the set of arguments that are in, out and undec. A preferred labelling L𝑝𝑟 is a complete labelling where in(L𝑝𝑟 ) is maximal, the grounded labelling L𝑔𝑟 is a complete labelling where in(L𝑔𝑟 ) is minimal and a stable labelling L𝑠𝑡 is a complete labelling where undec(L𝑝𝑟 ) is empty. Enthymemes are incomplete arguments. Contrary to other approaches that handle en- thymemes [14, 15], we allow omission of an argument’s claim, as well as its premises, and so may obtain a disjointed graph (as the claim is the root of the tree that is the intended argument). Hence we represent enthymemes as a forest of trees (see Figure 1.(c) for an example). Since an enthymeme 𝐸 is constructed from an intended argument 𝐴, one may choose to remove the conclusion of an inference rule while retaining the inference rule, or remove a sub-argument whose conclusion is the antecedent of a strict/defeasible rule. So, an enthymeme 𝐸 of an argument 𝐴 is a forest of trees whose nodes and edges are a subset of the nodes and edges of 𝐴, and the label of each node in 𝐸 is the same label of the corresponding node in 𝐴 (for example see 𝑋 ′ in Figure 1.(b) and Figure 1.(c)). Definition 2. Let AS = ⟨𝐿, ( · ), 𝑅, nom⟩. An enthymeme 𝐸 is a forest of labelled (downward directed) trees 𝐸 = ⟨Nodes(𝐸), Edges(𝐸), lab 𝐸 ⟩ such that Nodes(𝐸) ̸= ∅ and conditions 1 and 2 from Definition 1 hold. We say that 𝐸 is an enthymeme of an argument 𝐴 (written 𝐸 ≤ 𝐴) if Nodes(𝐸) ⊆ Nodes(𝐴), Edges(𝐸) ⊆ Edges(𝐴) and lab 𝐸 (𝑛) = lab 𝐴 (𝑛), for all 𝑛 ∈ Nodes(𝐸). Then Roots(𝐸) = {𝑛𝑖 ∈ Nodes(𝐸) | ∄(𝑛𝑗 , 𝑛𝑖 ) ∈ Edges(𝐸)} and Leaves(𝐸) = {𝑛𝑖 ∈ Nodes(𝐸) | ∄(𝑛𝑖 , 𝑛𝑗 ) ∈ Edges(𝐸)}. Note that an argument 𝐴 is an enthymeme of itself, where Roots(𝐴) = {Conc(𝐴)} and {lab(𝑛) : 𝑛 ∈ Leaves(𝐴)} ⊆ 𝐾. Thus, we also write that 𝐸 ′ ≤ 𝐸 (where 𝐸 ′ and 𝐸 are enthymemes) to denote that 𝐸 ′ is an enthymeme of 𝐸 (or else 𝐸 extends 𝐸 ′ , e.g. 𝐵 ′ ≤ 𝐵 and 𝐵 extends 𝐵 ′ in Figure 1.(b)). The class of all enthymemes is denoted E * and so includes the class of all arguments (A* ⊆ E * ). 22 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 X X′ ¬a X c ⇒ ¬a ¬a c E ¬a c p→c c ⇒ ¬a p p p c (a) (b) (c) Figure 1: (a) Bob’s argument 𝑋 (enclosed by a dashed line) in line 8 in Example 1 as represented in 𝐴𝑆𝑃 𝐼𝐶 + . ¬𝑎 is the claim of 𝑋 and 𝑝 is a premise that strictly (since connected with a solid line) infers the intermediate claim 𝑐, and 𝑐 defeasibly (since connected with a dashed line) infers ¬𝑎; (b) 𝑋 as represented in this paper, where the inference rules applied (⇒ denotes defeasible inference, → strict inference) are explicitly represented and a claim is connected to its premises via the inference rules applied. Notice that 𝑋 ′ (enclosed by a dotted line) is the enthymeme moved in line 2 in Example 1. (c) An enthymeme 𝐸 (of 𝑋). 3. Dialogue system An enthymeme dialogue 𝑑 is a sequence of moves. Each move is a 5-tuple comprising the move’s sender, locution, content, reply and target. The sender is either Prop or Op (the participants of the dialogue). Table 1 shows the locutions permitted in our system. A move’s content depends on its locution: if a move’s locution is assert or w.d.y.a.b. its content is an enthymeme. If a move’s locution is intend, assumed, meant or agree its content is an argument, otherwise the move’s content is ∅. The w.d.y.i., intend, assumed, meant and agree moves explicit responses to a previous move. For any move with one of the aforementioned locutions, its reply is a natural number indexing the move to which it replies, otherwise its reply is ∅. An asserted enthymeme is moved as a defeat against a previously moved enthymeme — the target of the assertion — whereas a w.d.y.a.b. move questions the defeat relation between the contents of two moves. So, if a move’s locution is assert, its target is the natural number indexing the move whose content is the enthymeme being defeated, whereas if a move’s locution is w.d.y.a.b., its target is the pair of natural numbers (𝑖, 𝑗) where 𝑚𝑖 targets 𝑚𝑗 (𝑚𝑖 ’s enthymeme is moved as a defeat on 𝑚𝑗 ’s enthymeme), otherwise its target is ∅. When an agent asserts an enthymeme, reveals its intended argument, corrects or confirms its counterpart’s interpretation of an enthymeme it has in mind a complete argument and this is the intended argument of the move. Note that since we accommodate nefarious agents, we do not insist that a move’s intended argument necessarily extends the move’s enthymeme. Similarly, when an agent asserts an enthymeme 𝐸 so as to defeat some targeted enthymeme 𝐸 ′ , the agent has in mind an argument 𝐴 that it believes was intended as the complete argument extending 23 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 Locution Meaning assert Assert an enthymeme. w.d.y.i. Request the intended argument of an asserted enthymeme by asking “what do you intend”. intend Provide the intended argument of an asserted enthymeme. w.d.y.a.b. Check the other participant’s understanding of an enthymeme by asking “what did you assume by . . . ”. assumed Provide their own interpretation of an enthymeme. meant Correct the other participant’s interpretation of an enthymeme. agree Confirm the other participant’s interpretation of an enthymeme. stop Indicate the participant’s intention to stop the dialogue. Table 1 Table of locutions permitted in our dialogue system and their meanings. the targeted 𝐸 ′ . 𝐴 is said to be the assert move’s intended target argument. See Table 2 for an example of an enthymeme dialogue. Definition 3. An enthymeme dialogue between two participants Prop and Op is a sequence of moves 𝑑 = [𝑚0 , 𝑚1 , . . . , 𝑚ℓ ], where each move 𝑚𝑖 = ⟨s(𝑚𝑖 ), l(𝑚𝑖 ), c(𝑚𝑖 ), re(𝑚𝑖 ), t(𝑚𝑖 )⟩ is a 5-tuple comprising the following: 1. Sender: s(𝑚𝑖 ) ∈ {Prop, Op}; 2. Locution: l(𝑚𝑖 ) ∈ {assert, w.d.y.i., intend, w.d.y.a.b., assumed, meant, agree, stop}; 3. Content: if l(𝑚𝑖 ) ∈ {assert, w.d.y.a.b.} then c(𝑚𝑖 ) ∈ E * , else if l(𝑚𝑖 ) ∈ {intend, assumed, meant, agree} then c(𝑚𝑖 ) ∈ A* , otherwise c(𝑚𝑖 ) = ∅; 4. Reply: if l(𝑚𝑖 ) ∈ {w.d.y.i., intend, assumed, meant, agree}, re(𝑚𝑖 ) ∈ {0, . . . , (𝑖 − 1)}, otherwise re(𝑚𝑖 ) = ∅; 5. Target: if l(𝑚𝑖 ) ∈ {assert}, t(𝑚𝑖 ) ∈ {0, . . . , (𝑖 − 1)} ∪ {∅}, otherwise if l(𝑚𝑖 ) = w.d.y.a.b., t(𝑚𝑖 ) ∈ ({0, . . . , (𝑖 − 1)} × {0, . . . , (𝑖 − 1)}), otherwise t(𝑚𝑖 ) = ∅. We denote the class of all moves as M * . If l(𝑚𝑖 ) ∈ {assert, intend, meant, agree}, IntArg(𝑚𝑖 ) ∈ (A* ∪ {∅}) is the intended argument of 𝑚𝑖 and if l(𝑚𝑖 ) = assert, IntTarArg(𝑚𝑖 ) ∈ (A* ∪ {∅}) is the intended target argument, else IntArg(𝑚𝑖 ) = ∅ and IntTarArg(𝑚𝑖 ) = ∅, respectively. We let 𝑑𝑖 = [𝑚0 , . . . , 𝑚𝑖 ] be the enthymeme dialogue whose last move is 𝑚𝑖 and say that 𝑑𝑘 = [𝑚0 , . . . , 𝑚𝑖 , . . . , 𝑚𝑘 ] expands 𝑑𝑖 where 𝑖 < 𝑘 < ℓ. Abusing notation, we identify re(𝑚𝑖 ) with the move 𝑚re(𝑚𝑖 ) to which 𝑚𝑖 replies and t(𝑚𝑖 ) with the move 𝑚t(𝑚𝑖 ) (or the pair of moves (𝑚𝑗 , 𝑚𝑘 )) that 𝑚𝑖 targets, instead of the indices re(𝑚𝑖 ) and t(𝑚𝑖 ) (or (𝑗, 𝑘)) of 𝑚re(𝑚𝑖 ) and 𝑚t(𝑚𝑖 ) (or (𝑚𝑗 , 𝑚𝑘 )), respectively. We now define a well-formed enthymeme dialogue 𝑑 with two participants, Prop and Op. We assume that the agents share the same underlying argumentation system, ensuring they can understand each other, except that the defeasible rules they are aware of may differ. They may of course have different knowledge (belief) bases. The participants alternate turns and cannot repeat moves. Prop must start by asserting an argument (the conclusion of which we call the topic of the dialogue). Apart from this first assert move, which has no target, an assert move’s enthymeme targets (i.e., is moved as a defeat against) a previously asserted enthymeme. 24 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 Note that while 𝑚𝑖 targets 𝑚𝑗 , this does not necessarily imply that the content 𝐸 of 𝑚𝑖 is a valid defeat on the content 𝐸 ′ of 𝑚𝑗 , according to the 𝐴𝑆𝑃 𝐼𝐶 + definition of defeat. Since enthymemes may only partially reveal the content of an argument, one may not be able to validate that a target relation corresponds to an 𝐴𝑆𝑃 𝐼𝐶 + defeat; hence our definition of well- formed enthymeme dialogues does not enforce such a correspondence. This of course means that participants may be mistaken in the assumptions they have made regarding the other’s intended arguments (i.e., 𝐸 may not legitimately defeat the intended complete argument 𝐴 that extends 𝐸 ′ ), or, for strategic purposes, an agent may dishonestly target a move’s enthymeme 𝐸 ′ , even though they know that their enthymeme 𝐸 does not defeat the intended target argument 𝐴. A w.d.y.i. move is used to request the intended argument of a previously asserted enthymeme 𝐸, while an intend move replies to a w.d.y.i. move and supplies the complete argument extending 𝐸. Recall that the receiver of an asserted enthymeme 𝐸 ′ might not be able to interpret correctly the intended argument 𝐴′ extending 𝐸 ′ . Thus, if 𝑚𝑗 targets 𝑚𝑖 , the agent Ag who moved 𝑚𝑖 might not be able to understand why the content 𝐸 ′ of 𝑚𝑗 has been moved against the content 𝐸 of 𝑚𝑖 (i.e., Ag is unable to reconstruct an argument 𝐴′ from 𝐸 ′ , such that 𝐴′ represents a valid defeat on Ag’s intended argument 𝐴 extending 𝐸). Moreover, the sender Ag′ of 𝑚𝑗 might be mistaken in her interpretation of the intended target argument 𝐴 extending 𝐸 in 𝑚𝑖 . So, Ag can check whether Ag′ correctly assumes that 𝐴 extends 𝐸, by making a w.d.y.a.b. move 𝑚𝑘 (whose content repeats the content 𝐸 of 𝑚𝑖 ) and which essentially amounts to questioning the validity of the targeting relationship (𝑚𝑗 , 𝑚𝑖 ) (i.e., whether it corresponds to a valid defeat); thus, the target of 𝑚𝑘 is (𝑚𝑗 , 𝑚𝑖 ). A participant of a well-formed enthymeme dialogue 𝑑 replies to a w.d.y.a.b. move 𝑚𝑘 only with an assumed move 𝑚ℎ . We also impose that the content of 𝑚ℎ is an argument that extends the content of 𝑚𝑘 since by using 𝑚ℎ its sender (Ag′ ) reveals its understanding of the argument extending Ag’s enthymeme (𝐸) in 𝑚𝑘 . A reply to the assumed move 𝑚ℎ is a move 𝑚𝑔 with locution meant or agree, in which Ag reveals the intended argument 𝐴 extending the enthymeme 𝐸 in 𝑚𝑖 (and which is repeated in 𝑚𝑘 ). Now, if Ag′ (in the move 𝑚ℎ ) was mistaken in its assumption as to what was the intended argument, then 𝑚𝑔 ’s locution is meant and its content 𝐴 must not be the same as the mistaken content 𝐵 of 𝑚ℎ . Otherwise, if Ag′ ’s interpretation was correct (i.e., Ag′ moved 𝐴 in 𝑚ℎ ), then 𝑚𝑔 ’s locution is agree and its content is 𝐴; that is, the argument in 𝑚ℎ and 𝑚𝑔 is the intended argument 𝐴 that extends the enthymeme 𝐸 in Ag’s earlier move 𝑚𝑖 . Finally, 𝑑 is terminated when two stop moves are made consecutively (indicating that neither agent wishes to continue the dialogue). For an example of a well-formed enthymeme dialogue see Table 2. Definition 4. Let AS Ag = ⟨𝐿, ( · ), 𝑅Ag , nom⟩ be an argumentation system for Ag ∈ Ag Ag {Prop, Op} such that 𝑅Ag = 𝑅𝑠 ∪ 𝑅def , where 𝑅def is Ag’s defeasible rules. An enthymeme dialogue 𝑑 = [𝑚0 , . . . , 𝑚ℓ ] between Prop and Op is said to be well-formed if 𝑚𝑖 ̸= 𝑚𝑗 for all 𝑖 ̸= 𝑗, and for all 𝑖 ≤ ℓ: 1. s(𝑚𝑖 ) = Prop if 𝑖 is even, otherwise s(𝑚𝑖 ) = Op; 2. if 𝑖 = 0, then l(𝑚𝑖 ) = assert, c(𝑚𝑖 ) ∈ A* , and t(𝑚𝑖 ) = ∅; 25 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 3. if 𝑖 > 0 and l(𝑚𝑖 ) = assert, then c(𝑚𝑖 ) ∈ E * and l(t(𝑚𝑖 )) = assert; 4. if l(𝑚𝑖 ) = w.d.y.i., then l(re(𝑚𝑖 )) = assert; 5. if l(𝑚𝑖 ) = intend, then l(re(𝑚𝑖 )) = w.d.y.i. and c(re(𝑚𝑗 )) ≤ c(𝑚𝑖 ); 6. if l(𝑚𝑖 ) = w.d.y.a.b., then t(𝑚𝑖 ) = (𝑚𝑗 , t(𝑚𝑗 )) such that l(𝑚𝑗 ) = assert, and c(𝑚𝑖 ) = c(t(𝑚𝑗 )); 7. if l(𝑚𝑖 ) = assumed, then l(re(𝑚𝑖 )) = w.d.y.a.b. and c(𝑚𝑖 ) ∈ A* such that c(re(𝑚𝑖 )) ≤ c(𝑚𝑖 ); 8. if l(𝑚𝑖 ) = meant, then l(re(𝑚𝑖 )) = assumed, c(𝑚𝑖 ) ∈ A* and c(𝑚𝑖 ) ̸= c(re(𝑚𝑖 )); 9. if l(𝑚𝑖 ) = agree, then l(re(𝑚𝑖 )) = assumed, c(𝑚𝑖 ) ∈ A* , and c(𝑚𝑖 ) = c(re(𝑚𝑖 )). The topic of 𝑑 (denoted Topic(𝑑)) is the label of the conclusion of the argument moved in 𝑚0 . If ∃𝑚𝑖 , 𝑚𝑗 (0 ≤ 𝑖 < 𝑗 ≤ ℓ) such that 𝑗 = 𝑖 + 1 and l(𝑚𝑖 ) = l(𝑚𝑗 ) = stop, then 𝑗 = 𝑙. We say that 𝑑 is terminated iff l(𝑚ℓ ) = l(𝑚ℓ−1 ) = stop. Step Dialogue 𝑑 IntArg(𝑚𝑖 ) IntTarArg(𝑚𝑖 ) 1 𝑚0 = (Prop, assert, 𝐴, ∅, ∅) 𝐴 ∅ 2 𝑚1 = (Op, assert, 𝑋 ′ , ∅, 0) 𝑋 𝐴 3 𝑚2 = (Prop, assert, 𝐵 ′ , ∅, 1) 𝐵 𝐶 4 𝑚3 = (Op, w.d.y.i., ∅, 2, ∅) – – 5 𝑚4 = (Prop, intend, 𝐵, 3, ∅) 𝐵 – 6 𝑚5 = (Op, w.d.y.a.b., 𝑋 ′ , ∅, (2, 1)) – – 7 𝑚6 = (Prop, assumed, 𝐶, 5, ∅) – – 8 𝑚7 = (Op, meant, 𝑋, 6, ∅) 𝑋 – Table 2 The enthymeme dialogue 𝑑 from Example 1. The internal structure of arguments and enthymemes moved in 𝑑 is shown in Figure 2. C X ¬a X′ ¬a A B c ⇒ ¬a c ⇒ ¬a a ¬e c c b⇒a q ⇒ ¬e e→c p→c ′ b q B e p (a) (b) (c) (d) Figure 2: The internal structure of arguments (enclosed by a dashed line) and enthymemes (enclosed by a dotted line) moved in 𝑑 shown in Table 2. Henceforth, we use ‘dialogue’ as shorthand for a ⟨︀‘well-formed enthymeme dialogue’. A dialogue 𝑑’s dialogue framework is a 4-tuple DF 𝑑 = M 𝑌 𝑁 , T , Re, S where M 𝑌 𝑁 = M ∪ ⟩︀ 26 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 {Yes, No}, M is the set of moves in 𝑑 (excluding stop moves), Yes and No are auxiliary elements used to respectively confirm and reject the other participant’s interpretation of an enthymeme, T is a defeat relation, Re a reply relation, and S a support relation. Recall that a dialogue framework is used to determine the dialogical status of the dialogue moves, similarly to a Dung AF which is used to determine the acceptability status of arguments. When a move is made in 𝑑, it is added to M 𝑌 𝑁 (unless it is a stop). An agent Ag’s assert move 𝑚𝑖 moves the content of 𝑚𝑖 as a defeat on the content of 𝑚𝑖 ’s target, whereas Ag makes a w.d.y.a.b. move to question the defeat relation between two moves 𝑚𝑗 and 𝑚𝑘 ((𝑚𝑗 , 𝑚𝑘 ) is the target of 𝑚𝑖 ). So, in both cases, we add a defeat relation between 𝑚𝑖 and its target; in all other cases T remains the same. If 𝑚𝑖 is a w.d.y.i., or intend, or assumed that replies to 𝑚𝑗 , then this reply relation is added to Re, since these moves respectively request the intended argument of a move, provide the intended argument of a move and provide an interpretation of an enthymeme. If 𝑚𝑖 is a meant move, a reply relation between No and 𝑚𝑗 is added to Re. Essentially, 𝑚𝑖 has two effects on DF 𝑑 : firstly, the sender Ag of 𝑚𝑖 rejects its interlocutor’s Ag′ ’s understanding of Ag’s enthymeme 𝐸, thus, we interpret 𝑚𝑖 as a negative reply to 𝑚𝑗 ; secondly, using 𝑚𝑖 , Ag reveals the intended argument from which 𝐸 was constructed. If we are to draw accurate conclusions regarding the acceptability of moves in DF 𝑑 , then 𝑚𝑖 is not enough to express both effects. Hence, we use No to validate that Ag′ ’s interpretation of 𝐸 is not acceptable, and (as we will see below) we connect 𝑚𝑖 (through a support relation) to the move 𝑚𝑘 , which contains 𝐸, in order to show that the acceptability of 𝑚𝑘 (i.e., 𝐸) influences the acceptability of 𝑚𝑖 (i.e., the intended argument from which 𝐸 was constructed). In this way, changes in the status of 𝑚𝑖 do not influence the status of 𝑚𝑗 which remains unacceptable in DF 𝑑 since Ag′ ’s understanding of 𝐸 was mistaken. In all other cases Re does not change. If 𝑚𝑖 is an intend move with content 𝐴, then a support relation between 𝑚𝑖 and 𝑚𝑗 is added to S . Intuitively, prior to 𝑚𝑖 , 𝑚𝑘 replies to (and so challenges) 𝑚𝑗 , by requesting the sender of 𝑚𝑗 to provide the intended argument from which the enthymeme 𝐸 of 𝑚𝑗 was constructed (i.e., 𝐴 extends 𝐸). 𝑚𝑖 satisfies this request (and so replies to 𝑚𝑘 ), and thus 𝑚𝑖 is moved to support 𝑚𝑗 against 𝑚𝑘 ’s challenge (essentially 𝑚𝑘 questions why 𝑚𝑗 was moved, whereas 𝑚𝑖 justifies why 𝑚𝑗 was moved). A meant or agree move 𝑚𝑖 also provides the intended argument of a move 𝑚𝑗 (where 𝑚𝑗 is targeted by a move 𝑚𝑘 , the defeat relation (𝑚𝑘 , 𝑚𝑗 ) is challenged by a move 𝑚ℎ requesting the interpretation of 𝑚𝑗 ’s content by the sender Ag of 𝑚𝑘 , a move 𝑚𝑔 replies to 𝑚ℎ providing Ag’s interpretation of 𝑚𝑗 ’s content and 𝑚𝑖 replies to 𝑚𝑔 ) and so we add a support relation between 𝑚𝑖 and 𝑚𝑗 . Finally, similarly to a meant move, an agree move has two effects on DF 𝑑 : by moving 𝑚𝑖 the sender Ag of 𝑚𝑖 confirms Ag′ ’s (i.e., the other participant’s) understanding of Ag’s enthymeme 𝐸 and, thus, we interpret 𝑚𝑖 as a positive reply to 𝑚𝑔 , thus supporting it; secondly, using 𝑚𝑖 , Ag reveals the intended argument from which 𝐸 was constructed. Hence, we use Yes to validate that Ag′ ’s interpretation of 𝐸 is acceptable, and we connect 𝑚𝑖 (through a support relation) to the move 𝑚𝑗 , which contains 𝐸, in order to show that the acceptability of 𝑚𝑗 (i.e., 𝐸) influences the acceptability of 𝑚𝑖 (i.e., the intended argument from which 𝐸 was constructed). In this way, changes in the status of 𝑚𝑖 do not influence the status of 𝑚𝑔 which remains acceptable in DF 𝑑 since Ag′ ’s understanding of 𝐸 was correct. In all other cases S remains the same. For an example of a DF see Figure 3. 27 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 Re Re T T m4 m3 m2 m1 m0 T S S Re Re Yes No m6 m5 m7 Figure 3: The DF 𝑑 instantiated by 𝑑 in Table 2. Each arrow represents a relation between moves in the DF 𝑑 . There is a single complete (and hence preferred, grounded and stable) labelling L of the DF 𝑑 . in and out moves are enclosed by a green, respectively brown, circle. Definition 5. Let DF 𝑑0 = M0𝑌 𝑁 , T0 , Re 0 , S0 be the dialogue framework of a dialogue 𝑑0 = ⟨︀ ⟩︀ [𝑚 ⟨︀ 0𝑌] 𝑁 where DF 𝑑0 = ⟩︀ ⟨𝑚0 , ∅, ∅, ∅⟩. Let 𝑑ℓ = [𝑚0 , . . . , 𝑚ℓ ] expand 𝑑0 . Then, for each DF 𝑑𝑖 = M𝑖 , T𝑖 , Re 𝑖 , S𝑖 , where 0 ≤ 𝑖 ≤ ℓ we have that: 1. M𝑖𝑌 𝑁 = M𝑖−1 𝑌 𝑁 ∪ {𝑚 } if l(𝑚 ) ̸= stop, otherwise M 𝑌 𝑁 = M 𝑌 𝑁 ; 𝑖 𝑖 𝑖 𝑖−1 2. T𝑖 = T𝑖−1 ∪ {(𝑚𝑖 , t(𝑚𝑖 ))} if l(𝑚𝑖 ) ∈ {assert, w.d.y.a.b.} and t(𝑚𝑖 ) ̸= ∅, otherwise T𝑖 = T𝑖−1 ; 3. Re 𝑖 = Re 𝑖−1 ∪ {(𝑚𝑖 , re(𝑚𝑖 ))} if l(𝑚𝑖 ) ∈ {w.d.y.i., intend, assumed}, Re 𝑖 = Re 𝑖−1 ∪ {(No, re(𝑚𝑖 ))} if l(𝑚𝑖 ) = meant, otherwise Re 𝑖 = Re 𝑖−1 ; 4. S𝑖 = S𝑖−1 ∪ {(𝑚𝑖 , re(re(𝑚𝑖 )))} if l(𝑚𝑖 ) = intend, else S𝑖 = S𝑖−1 ∪ {(𝑚𝑖 , 𝑚𝑗 )} if l(𝑚𝑖 ) = meant and t(re(re(𝑚𝑖 ))) = (𝑚𝑘 , 𝑚𝑗 ) , else S𝑖 = S𝑖−1 ∪ {(Yes, re(𝑚𝑖 )), (𝑚𝑖 , 𝑚𝑗 )} if l(𝑚𝑖 ) = agree and t(re(re(𝑚𝑖 ))) = (𝑚𝑘 , 𝑚𝑗 ), otherwise S𝑖 = S𝑖−1 , where 0 ≤ 𝑗 < 𝑘 < 𝑖. We define a complete labelling on DF 𝑑 of a dialogue 𝑑 as a function assigning in to a move 𝑚𝑖 in 𝑑 iff: 1) every move replying to 𝑚𝑖 is out; 2) every move 𝑚𝑗 targeting 𝑚𝑖 , and such that the defeat relation from 𝑚𝑗 ’s enthymeme to 𝑚𝑖 ’s enthymeme is not targeted by an in move, is out, and; 3) every move that is supported by 𝑚𝑖 is in. A move 𝑚𝑖 is assigned out iff either a move replying to 𝑚𝑖 is in, or a move targeting 𝑚𝑖 (such that the defeat relation is not targeted by an in move) is in, or a move that is supported by 𝑚𝑖 is out (intuitively, the content 𝐸 of a move 𝑚𝑗 supported by 𝑚𝑖 is an enthymeme of 𝑚𝑖 ’s content and so 𝑚𝑗 ’s acceptability influences 𝑚𝑖 ’s acceptability); a move is undec iff it is neither in nor out. Definition 6. Let DF 𝑑 = M 𝑌 𝑁 , T , Re, S . We define a complete labelling on DF 𝑑 to be a ⟨︀ ⟩︀ function L : M 𝑌 𝑁 → {in, out, undec} such that, for every 𝑚𝑖 ∈ M 𝑌 𝑁 : 1. L(𝑚𝑖 ) = in iff for all 𝑚𝑗 , 𝑚𝑘 ∈ M 𝑌 𝑁 : (a) if (𝑚𝑗 , 𝑚𝑖 ) ∈ Re then L(𝑚𝑗 ) = out, and (b) if (𝑚𝑗 , 𝑚𝑖 ) ∈ T , and ∄𝑚𝑘 ∈ M 𝑌 𝑁 such that t(𝑚𝑘 ) = (𝑚𝑗 , 𝑚𝑖 ) and L(𝑚𝑘 ) = in, then L(𝑚𝑗 ) = out, and (c) if (𝑚𝑖 , 𝑚𝑗 ) ∈ S , then L(𝑚𝑗 ) = in; 2. L(𝑚𝑖 ) = out iff there is some 𝑚𝑗 , 𝑚𝑘 ∈ M 𝑌 𝑁 such that: (a) (𝑚𝑗 , 𝑚𝑖 ) ∈ Re and L(𝑚𝑗 ) = in, or (b) (𝑚𝑗 , 𝑚𝑖 ) ∈ T , and ∄𝑚𝑘 ∈ M 𝑌 𝑁 such that t(𝑚𝑘 ) = (𝑚𝑗 , 𝑚𝑖 ) and L(𝑚𝑘 ) = in, and L(𝑚𝑗 ) = in, or (c) (𝑚𝑖 , 𝑚𝑗 ) ∈ S and L(𝑚𝑗 ) = out. 28 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 If L is a complete labelling on a DF 𝑑 , then in𝑑 (L), out𝑑 (L) and undec𝑑 (L) respectively denote the set of all moves labelled in, out and undec. Below we also define preferred, preferred and stable labellings on DF 𝑑 (for an example see Figure 3). Definition 7. Let DF 𝑑 be the dialogue framework of a dialogue 𝑑 and let L be a complete labelling function on DF 𝑑 . Then: 1. L is a preferred labelling function on DF 𝑑 iff there does not exist a complete labelling function L′ on DF 𝑑 such that in𝑑 (L) ⊂ in𝑑 (L′ ); 2. L is the (unique) grounded labelling function on DF 𝑑 iff there does not exist a complete labelling function L′ on DF 𝑑 such that in𝑑 (L′ ) ⊂ in𝑑 (L); 3. L is a stable labelling function on DF 𝑑 iff undec𝑑 (L) = ∅. We define an argumentation theory instantiated by a dialogue 𝑑 as a tuple AT 𝑑 = ⟨AS 𝑑 , 𝐾𝑑 ⟩, where the logical language 𝐿, the contrariness function ( · ), the naming function nom, and the strict rules 𝑅𝑠 of AS 𝑑 are those shared by the participants of 𝑑. The defeasible rules of AS 𝑑 is the set of the defeasible rules revealed during 𝑑, excluding the inference rules of enthymemes in assumed moves as these are assumptions which may be mistaken (in case they are correct, they are repeated within the content of an agree move following an assumed move). A premise 𝜑 of an enthymeme 𝐸 belongs to the set of premises in AT 𝑑 (i.e., 𝐾𝑑 ) iff 𝜑 is a leaf in 𝐸 and if the intended argument 𝐴 from which 𝐸 was constructed has been revealed by the sender of 𝐸, then 𝜑 is also a leaf in 𝐴. Definition 8. Let 𝑑 = [𝑚0 , . . . , 𝑚ℓ ] be a dialogue between Prop and Op, where AS Ag = ⟨𝐿, ( · ), 𝑅Ag , nom⟩ is the argumentation system for Ag ∈ {Prop, Op}, 𝐾Ag is the knowledge Ag . Let DF 𝑑 = M 𝑌 𝑁 , T , Re, S be the dialogue framework of ⟨︀ ⟩︀ base of Ag and 𝑅Ag = 𝑅𝑠 ∪ 𝑅def 𝑑. The set of defeasible rules in 𝑑 is DefDRules(𝑑) = ℓ ⋃︁ * | l(𝑚𝑖 ) ̸= assumed, c(𝑚𝑖 ) ∈ E * and 𝑛 ∈ Nodes(c(𝑚𝑖 )) . {︀ }︀ lab c(𝑚𝑖 ) (𝑛) ∈ 𝑅def 𝑖=0 The set of premises of Ag in 𝑑 is denoted DPremAg (𝑑) = ⎧ ⃒ ⎫ ⎪ ⃒ s(m𝑖 ) = Ag, l(𝑚𝑖 ) ̸∈ {w.d.y.i., w.d.y.a.b., assumed}, ⎪ 𝑙 ⎨ ⎪ ⃒ ⎪ n∈ ⋃︁ ⃒ ⎬ lab (𝑛) . ⎪ c(𝑚𝑖 ) ⃒⃒ Leaves(c(𝑚𝑖 )), and ∄𝑚𝑗 (𝑗 ≤ ℓ) such that (𝑚𝑗 , 𝑚𝑖 ) ∈ S ⎪ ⃒ 𝑖=0 ⎪ ⎪ and n ̸∈ Leaves(c(𝑚𝑗 )) ⎩ ⃒ ⎭ The set of all premises in 𝑑 is denoted DPrem(𝑑) = DPremProp (𝑑) ∪ DPremOp (𝑑). The argumentation theory instantiated by a dialogue 𝑑 is AT 𝑑 = ⟨AS 𝑑 , 𝐾𝑑 ⟩, where AS 𝑑 = ⟨𝐿, ( · ), 𝑅𝑠 ∪ DefDRules(𝑑), nom⟩ and 𝐾𝑑 = DPrem(𝑑). The argumentation theory of Ag instantiated by a dialogue 𝑑 is AT Ag Ag Ag 𝑑 = ⟨AS 𝑑 , 𝐾𝑑 ⟩, where AS Ag Ag 𝑑 = ⟨𝐿, ( · ), 𝑅Ag ∪ DefDRules(𝑑), nom⟩ and 𝐾𝑑 = DPrem(𝑑) ∪ 𝐾Ag . 29 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 4. Discussion If a dialogue 𝑑 is exhaustive and its participants honest, the dialectical status of the moves in the DF 𝑑 (determined by a complete, preferred, grounded and stable labelling) is sound and complete with respect to the dialectical status of the arguments in the AF instantiated by the contents of the moves made in the dialogue (determined by a complete, preferred, grounded, and stable labelling, resp.). Essentially, if 𝑑’s participants are honest, they only assert an enthymeme constructed from the intended argument of their move and such that it does defeat the intended targeted argument (i.e., what the sender assumes to be the target move’s intended argument) according to the 𝐴𝑆𝑃 𝐼𝐶 + definition of defeat. It also means that whenever they reveal their understanding of the argument from which their counterpart’s enthymeme 𝐸 is constructed, the argument does indeed extends 𝐸, and whenever they reveal the intended argument of their own move, this is indeed the argument they intended. If 𝑑 is exhaustive, then any available move that can be made, is indeed made. In other words, if a participant can move an argument (constructed by the contents of the moves made in the dialogue) as a defeat against the content of another move made in the dialogue, or question a defeat relation or reply to a move, they indeed do so. Due to lack of space we cannot present here the technical details for our claim, but for a better understanding of our claim the reader can look at Example 2. Example 2. Figures 4.(a) and 4.(b) respectively show the DF 𝑑 of Table 2’s exhaustive and honest dialogue 𝑑 2 , and the AF AT 𝑑 instantiated by AT 𝑑 . We also show a valid complete (resp. preferred, grounded and stable) labelling of DF 𝑑 ’s moves given by the labelling function L on DF 𝑑 , and a valid complete (resp. preferred, grounded and stable) labelling of arguments in AF AT 𝑑 given by the labelling function L′ on AF AT 𝑑 . In both frameworks the topic 𝑎 of 𝑑 is labelled out, (since 𝑚0 is labelled out in DF 𝑑 and IntArg(𝑚0 ) is labelled out in AF AT 𝑑 ) which means that 𝑎 does not belong to the complete (resp. preferred, grounded and stable) extension [13]. Moreover, the acceptability of moves in 𝑑 coincides with the acceptability of their intended arguments in the AF AT 𝑑 instantiated by AT 𝑑 . This paper introduces a novel dialogue system that allows participants to seek and provide clarifications regarding possible misinterpretations that arise from the use of enthymemes, and instantiates a dialogue framework that is used to determine the dialogical status of the dialogue moves. As far as we are aware, there are no other works that handle misunderstandings that may occur between the participants due to the use of enthymemes and also propose a mechanism for determining the status of the dialogue moves. Our work paves the way for a sound and complete dialogue system that accommodates use of any kind of enthymeme, and where any kind of uncertainty that arises from their use can be resolved. This is important since it ensures that the dialogue can be played out such that an enthymeme moved in the dialogue is only justified in the case that its intended argument is justified by the contents of the moves made in the dialogue. In future work we will explore how enthymemes may be used to give a strategic advantage to a participant. 2 We assume that 𝐶 and 𝑋 are preferred to 𝐴 since owing money takes precedence over eating at a restaurant, and 𝐵 is preferred to 𝐶 ′ since 𝐶 ′ is just a statement. 30 Andreas Xydis et al. CEUR Workshop Proceedings 19–32 Re Re T T A′ in A out B in B′ in m4 m3 m2 m1 m0 T S S Re Re Yes No m6 m5 m7 X′ in X in X ′′ in (a) (b) X C ¬a ¬a A B c ⇒ ¬a c ⇒ ¬a a ¬e c C ′′ c X ′′ b⇒a q ⇒ ¬e e→c p→c b A′ q B′ e C′ p X′ (c) (d) (e) (f) Figure 4: (a) The unique complete (preferred, grounded and stable) labelling L on the DF 𝑑 instantiated by 𝑑 in Table 2 (arrows represent relations between moves in DF 𝑑 ). in and out moves are respectively enclosed by green and brown circles. (b) The AF AT 𝑑 instantiated by AT 𝑑 . A complete (resp. preferred, grounded, stable) labelling function L′ on AF AT 𝑑 is shown. in and out arguments are enclosed by a green and a brown circle, respectively. Note, this is the only possible complete (resp. preferred, grounded, stable) labelling on AF AT 𝑑 for this example. (c)-(f) The arguments instantiated by AT 𝑑 enclosed by a dashed line. References [1] P. Besnard, A. Garcia, A. Hunter, S. Modgil, H. Prakken, G. Simari, F. 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