Introduction

Abstract argumentation Frameworks (AFs) [1] are promising tools in the Explainable AI (XAI) field [2] due to their transparency and interpretability, as well as their ability to support reasoning about conflicting information [3,4,5]. Quantitative Bipolar AFs (QBAFs) [6] are an extension of traditional AFs, which consider the (dialectical) strength of arguments and the support relation between arguments. In QBAFs, each argument has a base score, and its dialectical strength is computed by gradual semantics based on its base score and the strength of its attackers and supporters [7]. QBAFs can be deployed to support several applications like product recommendation [8], review aggregation [9] or stance aggregation [10].

Another interesting application that has been considered recently are truth discovery networks [11,12,13]. Figure 1 shows an example of a Truth-Discovery QBAFs (TD-QBAF) to evaluate the trustworthiness of sources and the reliability of claims made about an exhibition. We have 11 sources and 6 claims, each represented as an abstract argument. The nodes on the left represent the 11 source arguments (𝑠0 to 𝑠10), while the ones on the right represents the 6 claim arguments. The claim arguments are categorized into three types -year, place, and theme

ArgXAI-24: 2nd International Workshop on Argumentation for eXplainable AI

x.yin20@imperial.ac.uk (X. Yin); potykan@cardiff.ac.uk (N. Potyka); ft@imperial.ac.uk (F. Toni) 000-0002-6096-9943 (X. Yin); 0000-0003-1749-5233 (N. Potyka); 0000-0001-8194-1459 (F. Toni) of the exhibition -each distinguished by different colors. For pairs of contradictory claims, where different values are asserted for the same object, a bi-directional attack relationship is introduced between the claims. For each report (one for each pair of source and claim), a bi-directional support relationship is established between the source and the claim. Following [13], we use a base score of 0.5 for source argument (we are initially indifferent about the trustworthiness of a source), and a base score of 0 for claims (we do not believe claims without evidence). We compute the dialectical strength of arguments using the Quadratic Energy (QE) gradual semantics [14], and the final strengths of arguments are displayed on their side in Figure 1. While the strength values seem plausible, it can be challenging to understand why certain claims and sources receive higher or lower trust scores.

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To address this problem, attribution explanations (AEs) have been proposed. Specifically, given an argument of interest (topic argument) in a QBAF, AEs can explain the impact of arguments on the topic argument. AEs can be broadly categorized into Argument Attribution Explanations (AAEs) (e.g., [15,16,17]) and Relation Attribution Explanations (RAEs) (e.g., [18,19]). AAEs explain the strength of the topic argument by assigning attribution scores to arguments: the greater the attribution score, the greater the argument's contribution to the topic argument. Similarly, RAEs assign the attribution scores to edges to measure their contribution. Removalbased and Shapley-based techniques are commonly used for computing the attribution scores.

However, most existing studies focus on explaining acyclic QBAFs rather than cyclic ones, leaving a gap in understanding the complexities of the latter. In addition, current research typically examines only one type of attribution -either AAEs or RAEs -without providing a comprehensive comparison of both methods. In this paper, we aim to address these gaps by investigating the applicability of removal and Shapley-based AAEs and RAEs in the context of cyclic TD-QBAFs. Furthermore, we offer a comprehensive comparison between them to better understand the applicability of these AEs.

Preliminaries

QBAFs and the QE Gradual Semantics

We briefly recall the definition of QBAFs and the QE gradual semantics [14]. The base score function in QBAFs assigns an apriori belief to arguments. QBAFs can be represented graphically (as in Figure 1) using nodes to represent arguments and edges to show the relations between them. Then QBAFs are said to be (a)cyclic if the graphs representing them are (a)cyclic.

In this paper, we use the QE gradual semantics [14] to evaluate the strength of arguments in QBAFs. Like most QBAF semantics, it computes strength values iteratively by initializing the strength value of each argument with its base score and repeatedly applying an update function. Let us represent the strength of arguments in the 𝑖-th iteration by a function

𝜎 𝑖 : 𝒜 → [0, 1],

where 𝜎 0 (𝛼) = 𝜏 (𝛼) for all 𝛼 ∈ 𝒜. In order to compute 𝜎 𝑖+1 from 𝜎 𝑖 , the update function first computes the energy 𝐸 𝑖 𝛼 of attackers and supporters of each argument 𝛼 defined by

𝐸 𝑖 𝛼 = ∑︁ {𝛽∈𝒜|(𝛽,𝛼)∈ℛ + } 𝜎 𝑖 (𝛽) − ∑︁ {𝛽∈𝒜|(𝛽,𝛼)∈ℛ − } 𝜎 𝑖 (𝛽).

It then computes the strength in the next iteration via

𝜎 𝑖+1 (𝛼) = ⎧ ⎨ ⎩ 𝜏 (𝛼) − 𝜏 (𝛼) • (𝐸 𝑖 𝛼 ) 2 1+(𝐸 𝑖 𝛼 ) 2 𝑖𝑓 𝐸 𝑖 𝛼 ≤ 0; 𝜏 (𝛼) + (1 − 𝜏 (𝛼)) • (𝐸 𝑖 𝛼 ) 2 1+(𝐸 𝑖 𝛼 ) 2 𝑖𝑓 𝐸 𝑖 𝛼 > 0.

The final dialectical strength of each argument 𝛼 is then defined as the limit lim 𝑡→∞ 𝜎 𝑡 (𝛼). In cyclic graphs, the strength values may start oscillating and the limit may not exist [20]. In all known cases, the problem can be solved by continuizing the semantics [14,13]. However, we do not have space to discuss these issues in more detail here and will just restrict to examples where the strength values converge.

To better understand the QE gradual semantics, let us look at an example. Example 1. Consider the QBAF in Figure 2, where the base scores are given as 𝜏 (𝛼) = 0.8, 𝜏 (𝛽) = 0.6, 𝜏 (𝛾) = 0.9, and 𝜏 (𝛿) = 0.7. Since 𝛽 and 𝛾 have no parents, we have 𝐸 𝑖 𝛽 = 𝐸 𝑖 𝛾 = 0 for all 𝑖 and thus 𝜎(𝛽) = 𝜏 (𝛽) = 0.6 and 𝜎(𝛾) = 𝜏 (𝛾) = 0.9. For 𝛿, we have

𝐸 𝑖 𝛿 = 𝜎 𝑖 (𝛾) − 𝜎 𝑖 (𝛽) = 0.3 for all 𝑖, hence 𝜎(𝛿) = 𝜏 (𝛿) + (1 − 𝜏 (𝛿)) • 0.3 2 /(1 + 0.3 2 ) = 0.72. For 𝛼, we have 𝐸 𝑖 𝛼 = 𝜎 𝑖 (𝛾) + 𝜎 𝑖 (𝛿) − 𝜎 𝑖 (𝛽) = 1.02 for all 𝑖 ≥ 1. Hence, 𝜎(𝛼) = 𝜏 (𝛼) + (1 − 𝜏 (𝛼)) • 1.02 2 /(1 + 1.02 2 ) = 0.90.

In the remainder, unless specified otherwise, we assume as given a generic QBAF 𝒬 = ⟨𝒜, ℛ − , ℛ + , 𝜏 ⟩ and we let ℛ = ℛ − ∪ ℛ + We will often need to restrict QBAFs to a subset of the arguments or edges, or change the base score function, as follows.

Notation 1. For 𝒰 ⊆ 𝒜, let 𝒬 | 𝒰 = ⟨𝒜 ∩ 𝒰, ℛ − , ℛ + , 𝜏 ⟩. Then, for any 𝛼 ∈ 𝒜, we let 𝜎 𝒰 (𝛼) denote the strength of 𝛼 in 𝒬 | 𝒰 . Notation 2. For 𝒮 ⊆ ℛ, let 𝒬 | 𝒮 = ⟨𝒜, ℛ − ∩ 𝒮, ℛ + ∩ 𝒮, 𝜏 ⟩.

Truth Discovery QBAFs (TD-QBAFs)

TD-QBAFs allow reasoning about truth discovery problems using quantitative argumentation. Truth discovery problems can be described concisely as truth discovery networks (TDNs) [11]. Formally, a TDN is a tuple N = (𝒮, 𝒪, 𝒟, 𝒫) consisting of a finite set of sources 𝒮, a finite set of objects 𝒪, a set 𝒟 = {𝐷 𝑜 } 𝑜∈𝑂 of domains of the objects, and a set of reports 𝒫 ⊆ 𝒮 × 𝒪 × 𝑉 , where 𝑉 = ⋃︀ 𝑜∈𝑂 𝐷 𝑜 , and for all (𝑠, 𝑜, 𝑣) ∈ 𝒫, we have 𝑣 ∈ 𝐷 𝑜 , and there is no (𝑠, 𝑜, 𝑣 ′ ) ∈ 𝒫 with 𝑣 ̸ = 𝑣 ′ . Given a TDN N , we are interested in a truth discovery operator that assigns a trust score to each source and each claim [11].

Singleton suggested to reason about TDNs using bipolar argumentation frameworks, where we have bi-directional support edges between sources and their claims (trustworthy sources make claims more believable, and, conversely, believable claims make sources more trustworthy) and contradictory claims attack each other [12]. TD-QBAFs implement this idea with QBAFs, where sources have a base score of 0.5 (we are initially indifferent about the trustworthiness of sources) and claims have a base score of 0 (we do not believe anything without evidence).

Definition 2 (TD-QBAF induced from a TDN). The TD-QBAF induced from the TDNN = (𝒮, 𝒪, 𝒟, 𝒫) is defined as 𝑄 = (𝒜, ℛ − , ℛ + , 𝜏 ), where 𝒜 = 𝒮 ∪{(𝑜, 𝑣) | ∃𝑠 ∈ 𝒮 : (𝑠, 𝑜, 𝑣) ∈ 𝒫}, ℛ − = {(𝑐, 𝑐 ′ ) ∈ 𝒜 2 ∩ 𝐶 2 | obj(𝑐) = obj(𝑐 ′ ), val(𝑐) ̸ = val(𝑐 ′ )}, ℛ + = {(

𝑠, (𝑜, 𝑣)), ((𝑜, 𝑣), 𝑠) | (𝑠, 𝑜, 𝑣) ∈ 𝒫}. 𝜏 (𝑠) = 0.5 for all 𝑠 ∈ 𝒮 and 𝜏 (𝑐) = 0 for all 𝑐 ∈ 𝐶.

Every QBAF semantics gives rise to a truth discovery operator that is defined by associating each source and claim with its final strength under the semantics. The semantical properties of QBAF semantics like balance and monotonicity directly translate to meaningful guarantees for the derived trust scores.

Argument Attribution Explanations

In order to explain trust scores in TD-QBAFs, we recall the removal-based and Shapley-based AAEs. AAEs aim at evaluating the impact of an argument on a given topic argument. The removal-based AAEs proposed by [15] measure how the strength of the topic argument changes if an argument is removed.

Definition 3 (Removal-based AAEs).

Let 𝛼, 𝛽 ∈ 𝒜. The removal-based AAE from 𝛽 to 𝛼 under 𝜎 is:

𝜙 𝛼 𝜎 (𝛽) = 𝜎(𝛼) − 𝜎 𝒜∖{𝛽} (𝛼). The Shapley-based AAEs [16,21] use the Shapley value from coalitional game theory [22] to assign attributions. Each argument in a QBAF is seen as a player that can contribute to the strength of the topic argument. Intuitively, Shapley-based AAEs look at all possible ways how the argument could be added to the QBAF and average its impact on the topic argument.

Relation Attribution Explanations

RAEs are similar to AAEs, but measure the impact of edges rather than the impact of arguments. Analogous to the idea of removal-based AAEs [15], we consider the removal-based RAEs.

Definition 5 (Removal-based RAEs).

Let 𝛼 ∈ 𝒜 and 𝑟 ∈ ℛ. The removal-based RAE from 𝑟 to 𝛼 under 𝜎 is: [18,19] share the same idea with Shapley-based AAEs, but the attribution objects are changed from arguments to edges.

𝜆 𝛼 𝜎 (𝑟) = 𝜎(𝛼) − 𝜎 ℛ∖{𝑟} (𝛼). Shapley-based RAEs

Definition 6 (Shapley-based RAEs).

Let 𝛼 ∈ 𝒜 and 𝑟 ∈ ℛ. The Shapley-based RAE from 𝑟 to 𝛼 under 𝜎 is:

𝜑 𝛼 𝜎 (𝑟) = ∑︁ 𝒮⊆ℛ∖{𝑟} (|ℛ| − |𝒮| − 1)! |𝒮|! |ℛ|! [︀ 𝜎 𝒮∪{𝑟} (𝛼) − 𝜎 𝒮 (𝛼) ]︀ .

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Explaining TD-QBAFs with AAEs and RAEs

Settings

To compare the different AEs, we explain the strength of argument 𝑐5 in Figure 1. Since there are 17 arguments and 32 edges in Figure 1, computing Shapley-based AAEs and RAEs exactly is prohibitively expensive. We therefore apply the approximation algorithm from [19] that approximates the Shapley values using sampling (we set the sample size to 1000).

We report the removal and Shapley-based AAEs and RAEs in Figure 3 and 4 1 . In addition, to provide intuitive explanations for argument 𝑐5, we visualize the removal and Shapleybased AAEs and RAEs as shown in Figure 3 and 4, where blue/red arguments or edges denote positive/negative AAEs or RAEs. The darkness of the color of arguments and the thickness of the edges denote the magnitude of the their AAEs and RAEs, respectively 2 . For the removal-based AAEs, we observe that 𝑠7, 𝑠8, 𝑠9, and 𝑠10 have noticeably positive influences on 𝑐5, followed by minor positive influences from 𝑠3, 𝑐1, and 𝑐2. This is because 𝑠7 to 𝑠10 are direct supporters for 𝑐5, whereas 𝑠3, 𝑐1, and 𝑐2 indirectly support 𝑐5. Specifically, 𝑐2 supports 𝑠3, 𝑠3 supports 𝑐1, 𝑐1 supports 𝑠7, and then 𝑠7 supports 𝑐5, meaning 𝑠3, 𝑐1, and 𝑐2 all indirectly support 𝑐5. These indirect influences also explain why the AAEs of 𝑠3, 𝑐1, and 𝑐2 are much smaller than those of 𝑠7 to 𝑠10. Besides, since 𝑠7 is supported by 𝑐1, its AAE is slightly larger than those of 𝑠8 to 𝑠10, which have consistent AAEs due to their symmetrical structure to 𝑐5. In contrast, 𝑠0, 𝑠1, 𝑠2, and 𝑐0 have minor negative influences on 𝑐5 because 𝑐0 attacks 𝑐1, an indirect supporter for 𝑐5. Furthermore, 𝑠0 to 𝑠2 support 𝑐0, and thus they have negative influences on 𝑐5 as well. However, their negative influences are not obvious due to the indirect influences. Finally, the remaining arguments have AAEs close to 0, indicating their negligible influences on 𝑐5.

Results and Analysis for AAEs

When considering the Shapley-based AAEs, the results are similar to those of removal-based AAEs, where 𝑠7 to 𝑠10 still have significant influences on 𝑐5. Unlike removal-based AAEs, however, we notice that both 𝑐4 and 𝑠6 have minor negative influences on 𝑐5. This is because 𝑐4 directly attacks 𝑐5, while 𝑠6 indirectly attacks 𝑐5 by supporting 𝑐4, although the QE strength of 𝑐4 is very small (close to 0). Also, the negative influences of 𝑠0 to 𝑠2 and 𝑐0 and positive influence of 𝑐2 are relatively negligible compared with those of in removal-based AAEs due to their indirect connection to 𝑐5.

In this case study, both removal and Shapley-based AAEs can effectively capture the main influential arguments despite having some tiny differences in those low contributing arguments. This is mainly because of their different mechanisms of computing the AAEs. Another important reason is probably due to the approximation algorithm used for Shapley-based AAEs, leading to different AAEs even with the same sample size for the coalitions. We also noticed that the qualitative influence (the sign) of those Shapley-based AAEs close to 0 is sensitive when 1 The numerical AAEs and RAEs can be found in the Appendix 2 The code of all experiments is available at https://github.com/XiangYin2021/TD-QBAF-AAE-RAE. applying the approximation algorithm, thus we do not visualize those close to 0. However, this should not be a concern since their influence is negligible.

Results and Analysis for RAEs

Figure 4 shows the results of removal and Shapley-based RAEs. Let us first discuss the removal-based RAEs. We see that (𝑠7, 𝑐5) has the largest positive impact on 𝑐5. Following closely are (𝑠8, 𝑐5), (𝑠9, 𝑐5), and (𝑠10, 𝑐5), which also have notably positive influences on 𝑐5 because they are direct incoming supports to 𝑐5. There are also four outgoing supports from 𝑐5, namely (𝑐5, 𝑠7), (𝑐5, 𝑠8), (𝑐5, 𝑠9), and (𝑐5, 𝑠10), with positive influences but their RAEs are greatly smaller than that of the previous four as they are indirect supports. For instance, 𝑐5 first supports 𝑠7, and then 𝑠7 supports 𝑐5, indicating the indirect positive influence of (𝑐5, 𝑠7). Additionally, (𝑐5, 𝑐4) also contributes positively to 𝑐5 because 𝑐5 attacks its attacker 𝑐4, thus the attack from 𝑐4 to 𝑐5 is weakened. We can also observe In this case study, both removal and Shapley-based RAEs have a consistent ranking for the main influential edges despite having some tiny differences in those low contributing edges. The reasons are the same we discussed above.

Let us further compare the results of AAEs and RAEs. In this case study, we observe some connections between AAEs and RAEs. For example, in both AAEs, the top-4 influential arguments are 𝑠7 to 𝑠10, while in both RAEs, the outgoing edges from these arguments ((𝑠7, 𝑐5), (𝑠8, 𝑐5), (𝑠9, 𝑐5), and (𝑠10, 𝑐5)) also rank in the top-4. In addition, 𝑠0 to 𝑠4 and 𝑐0 to 𝑐2 have minor influences in the removal-based AAEs, while their incoming or outgoing edges also have minor influences in the removal-based RAEs. A similar phenomenon can be found in the Shapley-based AAEs and RAEs. While it is expected that the RAEs for outgoing edges of important arguments are relatively high, the consistency observed across different sets of arguments and edges is noteworthy. Besides, we found that removal or Shapley-based AAE of an argument does not necessarily equate to the sum of RAEs of all its incoming and outgoing edges, which goes against a reasonable expectation. We will leave the investigation of their formal relationships for future work.

Conclusion

Since most existing applications of AAEs and RAEs focus on acyclic QBAFs, this paper investigated their applicability in cyclic QBAFs. First, we found that AAEs and RAEs can provide intuitive explanations. By displaying the ranking of arguments or edges, it is easy to identify the most influential arguments or edges in the QBAF without delving into the complex (cyclic) structure of the QBAFs, particularly in TD-QBAFs where the number of arguments is typically large and the connections between source arguments and claim arguments are bi-directional. Second, AAEs and RAEs can provide interesting or even surprising explanations. For example, in the case study provided earlier, one might overlook the influence between claim arguments 𝑐1 and 𝑐5 because they are in different topics (Year=1962 and Theme=Art), but AEs demonstrate that 𝑐1 can contribute to 𝑐5 through 𝑠7. Third, RAEs provide more fine-grained explanations than AAEs. This is because when computing AAEs, such as removal-based AAEs, removing an argument means removing all the incoming and outgoing edges associated with that argument, whereas RAEs offer a more detailed insight by processing every incoming and outgoing edge individually. One can choose between them depending on the granularity for practical use.

For future work, it would be worthwhile to investigate how different gradual semantics influence AAEs and RAEs, because the property satisfaction of semantics have an influence on the property satisfaction of explanations. Additionally, the formal relationship between AAEs and RAEs requires further exploration. However, we believe AAEs and RAEs can complement each other, providing a deeper and more comprehensive understanding of the internal mechanisms of QBAFs, particularly the interactions between arguments and edges in complex QBAFs.

Additional Results for AAEs and RAEs