We present an approach to explaining inference in Quantitative Bipolar Argumentation Graphs (QBAGs): we propose the notion of a Quantitative Dispute Tree (QDT) that effectively collects the QBAG's directed paths between a topic argument and its children, labelled as proponent and opponent. A QDT is intended to be interpreted as a dispute that is sufficient to establish the direction of the change of strength of the topic argument. We propose to define pro and con arguments by using contribution functions that quantify a given argument's contribution to the strength of another argument. We advance some principles for contribution functions as well as gradual semantics to ensure some reasonable properties of QDTs. 1
In AI research focused on reasoning, formal argumentation has emerged as a promising facilitator of eXplainable AI (XAI) [1,2]. However, to properly foster explainability formal argumentation needs to be explainable in itself. One popular way to explain argumentation-based inference is the construction of Dispute Trees (DTs) [3,4]: given a topic argument in an argument graph, DTs collect arguments pro and con the topic argument, expanding along the edges. With an intuitive interpretation of the expansion and labelling of arguments as proponents and opponents, DTs facilitate explainability of (non-)acceptance of arguments in argument graphs (see [1] for an overview). Previously, DTs have been introduced to abstract, structured and, recently, bipolar argumentation approaches [3,4,5]. We here introduce DTs to quantitative bipolar argumentation which features both support and attack relations as well as weighted nodes (argument strengths) in argument graphs. Quantitative argumentation is of interest to the community as a family of argumentative reasoning methods that use numerical information [6]. It is also of interest as a potential bridge to connectionist AI approaches and explainability thereof, e.g. when interpreting feed-forward neural networks [7] or used in explainable recommender systems [8].
In this spirit of aiming at explainable reasoning approaches that can incorporate numerical (statistical or otherwise) information, we advance Quantitative Dispute Trees (QDTs) for explainability of argument strengths in quantitative (bipolar) argumentation. We specifically tackle the problem of explaining the change from initial to final argument strength in quantitative bipolar argumentation graphs (QBAGs) evaluated using (numerical) gradual semantics. We will define QDTs as formal graph-based structures that help to explain or justify why the (numerical) strength of a given topic argument increases or decreases when a QBAG is evaluated, and how other arguments contribute to this increase/decrease. The following example provides an intuition of this paper's main contribution.
Example 1.1. Consider the QBAG G depicted in Figure 1a (we give formal definitions of QBAGs and their semantics in Preliminaries). There are six arguments a, b, c, d, e, f with initial strengths given in brackets, and final strengths (given in boldface) that result after evaluating G using the Quadratic Energy (QE) gradual semantics. We consider a to be the topic argument whose strength increase (from 0.5 to 0.519) is to be explained. We aim to answer these questions:
• How much does each argument contribute to the final strength of a?
• Which arguments are pro and which con (a getting stronger)?
• Which arguments when disclosed are sufficient to guarantee that the final strength of a will not decrease below the initial strength even when further available arguments are disclosed? It is not immediate how much each argument contributes to a being stronger -we will propose some ways to quantify that. It is also not clear which arguments contribute positively or negatively, e.g. f, as an attacker of an attacker, can be said to be pro a getting stronger. But it would not be so straightforward if f also attacked d -for simplicity, this situation is not included in this example, but our approach will allow to qualify that as well. It is further not clear whether all the arguments are needed to establish that a gets stronger. Perhaps only some will suffice, e.g. the strong con argument c and the pro arguments d and f, so that whether b or e are considered would not change the fact that a only gets stronger. We will advance QDTs (some depicted in Figure 1) that capture these considerations.
In this paper we will propose a generic notion of argument contribution to another argument that is meant to both qualify whether one argument is pro or con another, and to quantify how much one arguments contributes to the final strength of another one. We will advance a few concrete instantiations of argument contribution functions applicable to quantitative argumentation and suggest a few reasonable principles that contribution functions should arguably satisfy. We will then propose a definition of QDTs that use proponent and opponent labels defined by a contribution function for an intepretation as a kind of dispute regarding the change in the strength of a given topic argument. Our QDTs will effectively amount to taking a connected sub-graph of a QBAG and collecting the directed paths between the topic argument and its children into a tree that is in the end sufficient to establish the topic argument getting stronger or weaker.
This work is a very preliminary step towards defining QDTs for QBAGs. We will see that there are many ways of going about defining argument contributions and subsequent explanations of argument strength changes, and rather than proposing a definite approach, we would like to initiate a discussion around the possible merits and drawbacks of different formalisations.
This section introduces the formal preliminaries of our work. Let I be a real interval. Typically, I = [0, 1] is the unit interval. A quantitative bipolar argumentation graph contains a set of arguments related by binary attack and support relations, and assigns an initial strength in I to the arguments. The (initial) strength can be thought of as initial credence in, or importance of, arguments. Typically, the greater the strength in I, the more credible or important the argument is. Definition 2.1 (Quantitative Bipolar Argumentation Graph (QBAG) [9,6]). A Quantitative Bipolar Argumentation Graph (QBAG) is a quadruple (Args, τ, Att, Supp) consisting of a set of arguments Args, an attack relation Att ⊆ Args × Args, a support relation Supp ⊆ Args × Args and a total function τ : Args → I that assigns the initial strength τ(x) to every x ∈ Args.
Henceforth, we assume as given a fixed but otherwise arbitrary QBAG G = (Args, τ, Att, Supp), unless specified otherwise. We also assume that Args is finite.
Given a ∈ Args, the set Att G (x) := {z | z ∈ Args, (z, x) ∈ Att} is the set of attackers of x and each z ∈ Att G (x) is an attacker of x; the set Supp G (x) := {y | y ∈ Args, (y, x) ∈ Supp} is the set of supporters of x and each y ∈ Supp G (x) is a supporter of x. We may drop the subscript G when the context is clear.
Reasoning in QBAGs amounts to updating the initial strengths of arguments to their final strengths, taking into account the strengths of attackers and supporters. Specifically, given a QBAG, a strength function assigns final strengths to arguments in the QBAG. Different ways of defining a strength function are called gradual semantics. Definition 2.2 (QBAG Semantics and Strength Functions [6,9]). A gradual semantics σ defines for G = (Args, τ, Att, Supp) a (possibly partial) strength function σ G : Args → I ∪ {⊥} that assigns the final strength σ G (x) to each x ∈ Args, where ⊥ is a reserved symbol meaning 'undefined'.
Note that we restrict ourselves to numeric strength functions by stipulating I to be an interval, instead of a generic set of elements with a preorder. This is typical of many gradual semantics and will simplify our exposition by allowing arithmetic operations with strengths.
We may abuse the notation and drop the subscript G so that σ denotes the strength function. A gradual semantics can define a strength function as a composition of multivariate real-valued functions that determines the strength of a given argument by aggregating the strengths of its attackers and supporters, taking into account the initial strengths [9]. A strength function so defined is recursive and generally takes iterated updates to produce a sequence of strength vectors, whence the final strengths are defined as the limits (or fixed points) if they exist. However, for acyclic QBAGs -namely, QBAGs without directed cycles -defining a semantics and computing the final strengths can be more straightforward: in the topological order of an acyclic QBAG as a graph, start with the leaves,1 set their final strengths to equal their initial strengths, and then iteratively update the strengths of parents whose all children already have final strengths defined.
Table 1 gives a list of common influence and aggregation functions. Table 2 shows some examples of gradual semantics.
Common aggregation α and influence ι functions [9, pp. 1724 Table 1; with a fixed typo for p-Max(k)]. Parameter s represents the strength of an argument at that state, w the initial strength, and s i and v i ∈ {−1, 1} the strengths and relationships, respectively, of the argument's attackers/supporters.
Examples of gradual semantics. In this paper we use QuadraticEnergyModel for illustrations.
While many gradual semantics can be defined for QBAGs in general, their convergence is not always guaranteed in a particular QBAG. For several semantics, convergence is however guaranteed in acyclic QBAGs. (See e.g. [9] for a neat exposition of convergence results under various semantics.) We restrict our attention to QBAGs for which a fixed but otherwise arbitrary gradual semantics is well-defined. In other words, our study applies to the setting where a gradual semantics σ defines a total strength function σ G assigning the final strengths to all arguments of a given G. We specifically use acyclic QBAGs and the Quadratic Energy (QE; see Table 2) semantics [10] (computable in accordance with a topological ordering of an acyclic QBAG). We however note that both the formal definitions and theoretical analysis given in this paper apply to the more general setting with well-defined gradual semantics giving total strength functions.
Example 2.1. In Example 1.1, QE semantics defines the following strength function σ : 2where
1+τ(e) 2 = 0.172 (all numbers are rounded to the 3 rd decimal in this example). 2 Similarly,
1+(τ(f)−τ(e)) 2 = 0.625. Finally, σ (a) = 0.519. The final strengths thus obtained are given in bold within nodes in Figure 1a.
To begin with, in order to determine if one argument positively or negatively contributes to the strength of another argument -akin to being a proponent or an opponent in a fictitious dispute regarding the final strength of arguments -we may consider different ways to define such contributions. For instance, we could ask how different the final strengths of a given topic argument would be in the absence of the contributing argument and/or relevant relationshipsin other words, in various sub-graphs of a QBAG. A useful notion for talking about QBAGs as sub-graphs of one another is that of a restriction of a QBAG to a set of arguments, as follows.
Definition 3.1 (QBAG Restriction). Given a QBAG G = (Args, τ, Att, Supp) and a set of arguments A ⊆ Args, we define the restriction of G to A as the QBAG G ↓
We next summarise some ways to attribute some argument's x ∈ Args contribution to the final strength σ G (a) of a given argument a ∈ Args in G. We will denote such contribution by Ctrb G,a (x).
, where G ′ is obtained from G by removing x:
This is the basic idea found in [11], but the authors therein stipulate that it ignores the fact that the children of x contribute to the strength of x and hence to the strength of a, and that this
definition is thus, arguably, not the right one. We could argue against that: after all, x "absorbs" the contributions from its children, and so its mere existence enables those to propagate further to a. We, however, do not intend to settle whether this is a reasonable definition of attributing contributions, but merely give it as an option, following [11]. • Similarly following [11], we can compute σ G − (a) − σ G ′ (a) where G − is obtained from G by removing direct relations to x and G ′ is obtained from G by removing x entirely:
In other words, we determine the strength of a with attackers and supporters of x removed (thus considering only the "direct contribution" of x) and take away the strength of a with x removed. • Following the game-theoretic approaches to capture the degree of influence on inconsistency [12] or of inputs on outputs in decision making systems [13,14,15,16,17], we can for instance use Shapley(-like) values:
This measure is in general computationally intractable, but can nevertheless serve as an option. • Many attribution techniques in numerical analysis aim to quantify which variable a given function f is sensitive to: gradient saliency, for instance, calculates the derivative of f 's outputs with respect to its inputs [18]. In our setting, assuming G is acyclic, we can traverse G in its topological order and write σ (a) = f (τ(a), τ(b 1 ), . . . , τ(b K )), where {b 1 , . . . , b K } are all the arguments from which there is a path 3 to a in G and f is a composition of aggregation α and influence ι functions. Then, assuming f is differentiable, compute the partial derivative ∂ f ∂ τ(x) of f with respect to the initial strength of x and evaluate it at the point of all the initial strengths. Whatever the specific method for determining an argument's contribution to the final strength of another argument, we will rely on a generic notion of contribution that we stipulate as follows.
Definition 3.2 (Contribution). Given a QBAG G, a topic argument a ∈ Args and any x ∈ Args, we let Ctrb G,a (x) ∈ R (or simply Ctrb a (x) where the context is clear) denote the contribution of x to the topic argument's a final strength σ G (a). We call the function Ctrb : Args × Args → R defined by Ctrb(a, x) = Ctrb a (x) a contribution function.
Note that the first three contribution function examples given above are partial functions; in particular, they do not define the topic argument's contribution to itself Ctrb a (a), because σ G ′ (a) is undefined when a is not in (the set of arguments of) G ′ . The gradient saliency contribution function (Equation 4), however, defines contributions for all arguments to all others.
We expect some intuitive behaviours from a contribution function and can formalise these expectations as contribution function principles, analogously to principles that have been defined for gradual semantics [6]. In the case of contribution functions, principle satisfaction obviously depends on the properties of the semantics used. One intuitive requirement is that some contributions exist whenever the final strength of the topic argument changes. On the other hand, the gradient saliency contribution function together with a differentiable strength function such as given by the QE semantics does satisfy Principle 3.1 (basic calculus).
Another intuitive principle, which we call directionality, stipulates that an argument can only contribute to another argument if there is a directed path from the former to the latter. As a shorthand, we denote the existence of a directed path from x to a (in G) by r G (x, a), and non-existence of any such path by ¬r G (x, a).
Ctrb satisfies the directionality principle w.r.t. a gradual semantics σ iff for every QBAG G = (Args, τ, Att, Supp), for each a, x ∈ Args it holds that whenever ¬r G (x, a), then Ctrb a (x) = 0.
We can image a gradual semantics that makes an argument's final strength dependent on the arguments it attacks or supports (somewhat reflecting the idea of range-based semantics in abstract argumentation [20]). For such a semantics, all of the contribution functions introduced above would violate the directionality principle. For instance, consider the strength function σ (x) = τ(x) − ∑ {y|x∈Att(y)} τ(y), the QBAG ({a, b}, {(a, 1), (b, 1)}, {(b, a)}, {}) with b attacking a, and Ctrb defined using Equation 1. There is no directed path from a to b, but removing a changes the final strength of b from 0 to 1, whence the contribution of a to b is 0 − 1 ̸ = 0.
We can also observe that for instance the gradient saliency contribution function with QE semantics satisfies Principle 3.2 (as the derivative on a variable not in the function equals 0).
We now want to propose a use of argument contributions towards explaining argument strength changes in QBAGs by means of labelling quantitative dispute trees that we define next.
Given a topic argument a in G, a quantitative dispute tree (QDT) can be defined as a tree T consisting of alternating proponent and opponent arguments (according to their contributions Ctrb a (•)) and undirected links among them following the relationships from G. When the final strength of a increases (or at least does not decrease) compared to its initial strength τ(a), we will be interested in QDTs that correspond to sub-graphs of G in which the final strength of a increases (or at least does not decrease) and does not oscillate around σ (a) upon addition of further arguments. Analogously for the case when the final strength of a decreases in G compared to τ(a). We first define QDTs and then formalise the desired condition. Definition 4.1 (Quantitative Dispute Tree (QDT)). A quantitative dispute tree (QBDT) for the topic argument a ∈ Args in an acyclic QBAG G = (Args, τ, Att, Supp) is an in-tree 4 T, such that:
1. every node of T is of the form [L : x], with L ∈ {P, O} and x ∈ Args: the node holds argument named x and is • either a proponent (P) node, denoted
or c ∈ Supp G (x).
Note that the above P and O are "graph-global" notions of proponent and opponent, given topic a, rather than relative to attackers and/or supporters of a given argument, as is common in dispute trees defined for non-quantitative argumentation [3,4,5].
For the sake of representational simplicity, we avoid explicit naming of nodes in QDTs and implicitly force nodes holding the same argument to appear as multiple distinct nodes. In particular, if some c ∈ Args relates to (attacks or supports) distinct x, y ∈ Args, then [L : c] can appear as children of both nodes holding x and y in T only as implicitly distinctly named copies. We likewise keep the argument relationships implicit without labelling edges of QDTs.
We first note that QDTs always exist:
Proposition 4.1 (Existence of QDTs). For every acyclic QBAG G = (Args, τ, Att, Supp), for every a ∈ Args, there exists a QDT T for a.
Proof. By definition of a QDT, using only condition 2, one can always construct the trivial QDT with exactly one node [P : a]. Still, observe that non-trivial QDTs exist as well: take the unique connected component of G containing a and map each of its directed paths from every x ∈ Args \ {a} into a branch of QDT T by traversing the path from a to x in the opposite direction of attacks and supports. This procedure is well-defined due to conditions 1 and 3.
Note that the P and O labels of nodes do not determine the construction of a QDT, but are intended to help to "read" the QDT, by providing interpretation as to which nodes are pro and con, as determined by argument contributions.
In Example 1.1, Figure 1d depicts QDT T • that is constructed from the whole of G, using contributions from Example 3.1 to label nodes of T • as proponent or opponent. Now, observe that there is a natural correspondence between a QDT T constructed for some topic argument in G and a unique sub-graph of G that has the same arguments as those held by the nodes of T (with the same argument held by multiple nodes in T, if any, mapping to the same node in G) and the relations recovered using condition 3 of Definition 4.1. So with an abuse of notation we will denote by σ T (x) the final strength of x in the sub-graph of G to which the QDT T thus corresponds. With this, we are ready to define the desired property of QDT sufficiency. Definition 4.2 (Sufficiency). A QDT T is sufficient for a if for any (non-necessarily proper) super-tree T ′ of T satisfying the QDT conditions 1, 2 and 3 from Definition 4.1, it holds that:
Note that the sufficiency condition entails that T itself must satisfy σ T (a) ⩾ τ(a), in case σ G (a) ⩾ τ(a) (respectively, σ T (a) < τ(a), in case σ G (a) < τ(a)). Intuitively, a QDT witnesses the (non-)decrease of the topic argument's strength when compared to the initial one.
In Example 1.1, T • given in Figure 1d (as before, the proponent and opponent labels are determined using contributions from Example 3.1) is sufficient for a: it is a QDT as observed after Proposition 4.1; it satisfies σ T • (a) = σ G (a) ⩾ τ(a); and it has no proper super-trees.
While QDTs for a exist in general (Proposition 4.1), existence of sufficient QDTs may need some reasonable restrictions of the strength function. We identify one such property of gradual semantics next, which says that the final strength of any argument depends only on the arguments from which there is a directed path to the argument in question.
Principle 4.1 (Directional Connectedness). A gradual semantics σ satisfies the directional connectedness principle iff for every QBAG G = (Args, τ, Att, Supp), for any x ∈ Args it holds that σ G (x) = σ G↓ {x}∪{y∈Args | r G (y,x)} (x).
We claim that our directional connectedness principle is different from the existing gradual argumentation principles such as General Principles (GP) 1 and 6 in [6]. Roughly, GP1 insists that the final strength of an argument differs from its initial strength only if it has direct attackers or supporters; however, it does not exclude contributions from "downstream" (i.e. children of attackers/supporters) or unconnected arguments. GP6 insists that arguments of equal initial strength, with equally strong attackers and supporters, are of equal final strength; however, this does not preclude equal contribution of all arguments to all other arguments, roughly speaking.
Assuming directional connectedness, sufficient QDTs are guaranteed to exist: In practice, we can construct a sufficient QDT by inspecting how adding various arguments and relationships, starting with the topic a, affects the final strength of a in the (QBAG corresponding to the) QDT being thus constructed. Once can also check that in this example T is actually a unique minimal sufficient QDT for a. For instance, the candidate tree with nodes {[P : a], All in all, super-trees of T are all and only QDTs sufficient for a, with T being the unique minimal such. T can thus be seen as a minimum dispute that suffices to establish that the final strength of a will be above τ(a), increasing towards σ G (a).
Our running example above is with respect to the gradient saliency contribution function. Using different contribution functions, including those offered in Section 3, may result in different QDTs. This would mean that using QDTs as a form of justification or explanation of argument strength changes in QBAGs depends and requires agreement on the notion of argument contribution to begin with. We think investigations of QDT dependence on argument contribution quantification is an interesting line of research that we leave for the future. We can nonetheless observe that assuming all the above proposed principles for reasonable strength and contribution functions, there is at least one sufficient QDT with a non-trivially contributing argument: Proposition 4.3. If a gradual semantics σ satisfies the directional connectedness principle and a contribution function Ctrb satisfies both the contribution existence and directionality principles w.r.t. σ , then for every acyclic QBAG G = (Args, τ, Att, Supp), for every a ∈ Args with σ (a) ̸ = τ(a), there exists a QDT T sufficient for a with a node [L : x] such that Ctrb a (x) ̸ = 0.
Proof. Using Principle 4.1, we can construct QDT T sufficient for a from the sub-graph of G containing all and only the children of a, as in the proof of Proposition 4.2. Using Principle 3.1, there is x ∈ Args with Ctrb a (x) ̸ = 0. Using Principle 3.2, by contraposition, there is a directed path from x to a in G. Thus, using Principle 4.1 again, T has a node [L : x], as required.
Our work adds to the growing body or research on argumentative explainability [1,2]. Our focus lies in formally explaining argumentative inference using dispute trees in gradual argumentation. DTs are a well-established concept in formal argumentation and play a crucial role in research on formal -and in particular, abstract -argumentation semantics [4]. More recently, DTs have been positioned as facilitators of argumentative explainability, supporting both abstract and structured argumentation [21]. However, DTs have so far not been introduced to quantitative argumentation. Generally, with the exception of the recent work (by the first two authors of this paper) on change explainability in QBAGs [22], no research appears to exist that formally studies the explainability of gradual argumentation-based inference. Instead, most works on quantitative argumentation and explainability focus on the application of the former to facilitate the latter, in the context of recommender systems [8,23]. We speculate that a better formal understanding of gradual argumentation explainability can serve as a facilitator of its application potential.
The formal work on quantitative dispute trees (QDTs) we present in this paper has plenty of limitations and thus space for open discussion. For instance, different contribution functions may adhere to different desirable principles: the contribution function given by Equation 1 may satisfy that given two attackers/supporters of an argument, the one with the higher absolute final strength has the higher contribution; whereas the gradient saliency contribution function may satisfy that given two attackers/supporters of an argument, the one whose final strength is higher proportionally to its initial strength has the higher contribution; this different behaviour can be inspected in Example 3.1. Contribution function dependency on gradual semantics is also an important issue: for instance, if an attacker of the topic argument is "unreasonably" assigned positive contribution, then the attacker may "unintuitively" appear as a proponent node in a QDT. Delineating further desirable properties of contribution functions and characterising their satisfaction with respect to various gradual semantics would be an interesting research direction.
The current work can also potentially be extended in various other directions. One is to explain why an argument's final strength exceeds or falls below some threshold value or to explain the differences in final strengths of two or more arguments (we are studying definitions of multi-topic quantitative dispute graphs). Another is a comparison of quantitative and abstract dispute trees. And as with most explainability research, the usefulness of QDTs as explanations in realistic applications and with user studies should be investigated. We hope to pursue this in the future.
At the time of writing this paper, Kristijonas Čyras was partially unaffiliated, but the paper is based on the research carried out with Ericsson Research in Sweden. Qingtao Weng is affiliated with Ericsson in Sweden. 1st International Workshop on Argumentation for eXplainable AI (ArgXAI, co-located with COMMA '22), September 12, 2022, Cardiff, UK