=Paper=
{{Paper
|id=Vol-3209/0872
|storemode=property
|title=Dispute Trees as Explanations in Quantitative (Bipolar) Argumentation
|pdfUrl=https://ceur-ws.org/Vol-3209/0872.pdf
|volume=Vol-3209
|authors=Kristijonas Čyras, Timotheus Kampik, Qingtao Weng
|dblpUrl=https://dblp.org/rec/conf/comma/CyrasKW22
}}
==Dispute Trees as Explanations in Quantitative (Bipolar) Argumentation==
https://ceur-ws.org/Vol-3209/0872.pdf
Dispute Trees as Explanations in Quantitative
(Bipolar) Argumentation
Kristijonas Čyras1 , Timotheus Kampik2 and Qingtao Weng1
1 Ericsson Research, Sweden and USA
2 Umeå University, Sweden
Abstract
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
Keywords
Explainable AI, Quantitative argumentation, Dispute trees
1. Introduction
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].
1At 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
$ kcyras@gmail.com (K. Čyras); tkampik@cs.umu.se (T. Kampik); qingtao.weng@ericsson.com (Q. Weng)
0000-0002-4353-8121 (K. Čyras); 0000-0002-6458-2252 (T. Kampik); 0000-0001-7574-8951 (Q. Weng)
© 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)
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Kristijonas Čyras et al. CEUR Workshop Proceedings 1–12
In this spirit of aiming at explainable reasoning approaches that can incorporate numerical
(statistical or otherwise) information, we advance Quantitative Dispute Trees (QDTs) for explain-
ability 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?
a : (0.5)
0.519
[P : a] [P : a] [P : a]
+ - +
b : (0.2) c : (0.85) d : (0.65)
0.172 0.625 0.65
[O : c] [P : d] [O : c] [P : d] [P : b] [O : c] [P : d]
- + -
e : (0.4) f : (1)
0.4 1
[P : f] [O : e] [P : f] [O : e] [O : e] [P : f]
(a) G (b) T (c) T′ (d) T◦
Figure 1: Subfigure 1a shows a QBAG G. Here, a node labelled x :f(i) carries argument x with initial
strength τ(x) = i and final strength σ (x) = f. Edges labelled + and − respectively represent attack and
support. Subfigures 1b, 1c and 1d show quantitative dispute trees (QDTs) constructed from G, to be
defined and discussed in Section 4.
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.
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Kristijonas Čyras et al. CEUR Workshop Proceedings 1–12
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.
2. Preliminaries
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 AttG (x) := {z | z ∈ Args, (z, x) ∈ Att} is the set of attackers of x and
each z ∈ AttG (x) is an attacker of x; the set SuppG (x) := {y | y ∈ Args, (y, x) ∈ Supp} is the set
of supporters of x and each y ∈ SuppG (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 σG : Args → I ∪ {⊥} that assigns
the final strength σG (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.
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Kristijonas Čyras et al. CEUR Workshop Proceedings 1–12
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.
Aggregation Functions
Sum αvΣ : [0, 1]n → R αvΣ (s) = ∑ni=1 vi × si
Product αvΠ : [0, 1]n → [−1, 1] αvΠ (s) = ∏i:vi =−1 (1 − si ) − ∏i:vi =1 (1 − si )
Top αvmax : [0, 1]n → [−1, 1] amax
v (s) = Mv (s) − M−v (s),
where Mv (s) = max{0, v1 × s1 , . . . , vn × sn }
Influence Functions
Linear(k) lwl : [−k, k] → [0, 1] lwl (s) = w − wk × max{0, −s} + 1−wk × max{0, s}
1−w2
Euler-based lwe : R → [w2 , 1] lwe (s) = 1 − 1+w×e s
p p
p-Max(k) lw : R → [0, 1] lw (s) = w − w × h(− ks ) + (1 − w) × h( ks ),
max{0,x} p
for p ∈ N where h(x) = 1+max{0,x} p
Table 1
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 si and
vi ∈ {−1, 1} the strengths and relationships, respectively, of the argument’s attackers/supporters.
Semantics Aggregation Influence
QuadraticEnergyModel Sum 2-Max(1)
SquaredDFQuADModel Product 1-Max(1)
EulerBasedTopModel Top EulerBased
EulerBasedModel Sum EulerBased
DFQuADModel Product Linear(1)
Table 2
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 σG assigning the final strengths to all arguments
1 Here, leaves are nodes without incoming edges.
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Kristijonas Čyras et al. CEUR Workshop Proceedings 1–12
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 :
max{E(x), 0}2 max{−E(x), 0}2
σ (x) = τ(x) + (1 − τ(x)) · − τ(x) ·
1 + max{E(x), 0}2 1 + max{−E(x), 0}2
where
E(x) = ∑ σ (y) − ∑ σ (z).
Supp(y,x) Att(z,x)
Concretely, σ (f) = τ(f) = 1, σ (e) = τ(e) = 0.4 and σ (d) = τ(d) = 0.65. For σ (b), note
max{−τ(e),0}2 max{τ(e),0}2
that E(b) = −σ (e), so σ (b) = τ(b) + (1 − τ(b)) · 1+max{−τ(e),0} 2 − τ(b) · 1+max{τ(e),0}2 = τ(b) −
2
τ(e)
τ(b) · 1+τ(e) rd decimal in this example).2 Similarly,
2 = 0.172 (all numbers are rounded to the 3
(τ(f)−τ(e)) 2
E(c) = σ (e) − σ (f), so that σ (c) = τ(c) − τ(c) · 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.
3. Argument Contributions
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 relationships –
in 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 ↓A :=
(A, τ ∩ (A × I), Att ∩ (A × A), Supp ∩ (A × A)).
We next summarise some ways to attribute some argument’s x ∈ Args contribution to the
final strength σG (a) of a given argument a ∈ Args in G. We will denote such contribution by
CtrbG,a (x).
• Compute σG (a) − σG′ (a), where G′ is obtained from G by removing x:
CtrbG,a (x) = σG (a) − σG↓Args\{x} (a) (1)
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
2 These computations can be done using the code at github.com/kcyras/QDT and found on Jupyter notebook
github.com/kcyras/QDT/blob/main/exampleQDT.ipynb.
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Kristijonas Čyras et al. CEUR Workshop Proceedings 1–12
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 σG− (a) − σG′ (a) where G− is obtained from G by
removing direct relations to x and G′ is obtained from G by removing x entirely:
CtrbG,a (x) = σ(Args,τ,Att\{(y,x)|(y,x)∈Att}),Supp\{(y,x)|(y,x)∈Supp}) (a) − σG↓Args\{x} (a) (2)
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 inconsis-
tency [12] or of inputs on outputs in decision making systems [13, 14, 15, 16, 17], we can for
instance use Shapley(-like) values:
|X|! · (|Args|! − |X| − 1)! (︂ )︂
CtrbG,a (x) = ∑ σ G↓ (a) − σ G↓ (a) (3)
X⊆Args\{x}
|Args|! Args\(X∪{x}) Args\X
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(b1 ), . . . , τ(bK )), where {b1 , . . . , bK } are all the
arguments from which there is a path3 to a in G and f is a composition of aggregation α and
f
influence ι functions. Then, assuming f is differentiable, compute the partial derivative ∂ ∂τ(x) of
f with respect to the initial strength of x and evaluate it at the point of all the initial strengths.
⃓
∂ f ⃓⃓
CtrbG,a (x) = (4)
∂ τ(x) ⃓(τ(a),τ(b1 ),...,τ(x),...,τ(bK ))
Example 3.1. In Example 1.1, G is acyclic and we have spelled out in Example 2.1 e.g. σ (b) in
terms of all the other arguments connected to b via a direct path, namely e. So we can use the
gradient saliency method given in Equation 4 to determine the contributions. For instance,
τ(e)2 2τ(b)τ(e)3
(︃ )︃
∂ σ (b) ∂ 2τ(b)τ(e)
= τ(b) − τ(b) · = 2
− .
∂ τ(e) ∂ τ(e) 1 + τ(e)2 2
(1 + τ(e) ) 1 + τ(e)2
3
⃓
So CtrbG,b (e) = 2τ(b)τ(e)
2 2
− 2τ(b)τ(e) ⃓
1+τ(e)2 ⃓
≈ −0.119.
(1+τ(e) ) τ(b)=0.2,τ(e)=0.4
We will be interested in explaining the strength change of argument a in Example 1.1, so we
compute the following contributions: Ctrba (b) = 0.158, Ctrba (c) = −0.134, Ctrba (d) = 0.183,
Ctrba (e) = −0.123, Ctrba (f) = 0.101.
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.
3 There is a directed path in G from x to a iff ∃ (xR b . . . b R a) with R ∈ {Att, Supp} and b ∈ Args ∀k ∈
1 1 n n k k
{1, . . . , n}, n ⩾ 1.
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Kristijonas Čyras et al. CEUR Workshop Proceedings 1–12
Definition 3.2 (Contribution). Given a QBAG G, a topic argument a ∈ Args and any x ∈ Args,
we let CtrbG,a (x) ∈ R (or simply Ctrba (x) where the context is clear) denote the contribution of
x to the topic argument’s a final strength σG (a). We call the function Ctrb : Args × Args → R
defined by Ctrb(a, x) = Ctrba (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 Ctrba (a), because σG′ (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 de-
fined 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.
Principle 3.1 (Contribution Existence). A contribution function Ctrb satisfies the contribution
existence principle w.r.t. a gradual semantics σ iff for every QBAG G = (Args, τ, Att, Supp), for
every a ∈ Args it holds that whenever σG (a) ̸= τ(a), then ∃x ∈ Args s.t. Ctrba (x) ̸= 0.
Not all contribution functions given above always satisfy the contribution existence principle:
Example 3.2 (Violation of Contribution Existence). Assume a gradual semantics for acyclic
graphs where the final strength of an argument depends on the maximal absolute final strength of
its attackers and supporters: σ (x) = τ(x) − max{τ(y) | y ∈ Att(x)} (behaviour that reflects one of
the basic ideas in abstract argumentation [19]). Consider the following QBAG with a topic argu-
ment a having exactly two attackers b and c: ({a, b, c}, {(a, 0), (b, 1), (c, 1)}, {(b, a), (c, a)}, {}).
We have σ (a) = −1 ̸= τ(a). Using Equation 1 to define Ctrb, since removing only one attacker
still leaves the same final strength of a, we get Ctrba (b) = Ctrba (c) = 0. Ctrb thus violates the
contribution existence principle w.r.t. semantics σ .
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 short-
hand, we denote the existence of a directed path from x to a (in G) by rG (x, a), and non-existence
of any such path by ¬rG (x, a).
Principle 3.2 (Directionality). A contribution function 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 ¬rG (x, a), then Ctrba (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
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Kristijonas Čyras et al. CEUR Workshop Proceedings 1–12
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.
4. Quantitative Dispute Trees
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
Ctrba (·)) 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-tree4 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 [P : x], if Ctrba (x) ⩾ 0,
• or an opponent (O) node, denoted [O : x], if Ctrba (x) < 0;
2. the root of T is [P : a] – a P node holding a;
3. every non-root node [C : c] is a child of exactly one [L : x] in T such that either c ∈ AttG (x)
or c ∈ SuppG (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
4A directed rooted tree with edges oriented towards the root.
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Kristijonas Čyras et al. CEUR Workshop Proceedings 1–12
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 σT (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:
• σT′ (a) ⩾ τ(a), in case σG (a) ⩾ τ(a);
• σT′ (a) < τ(a), in case σG (a) < τ(a).
Note that the sufficiency condition entails that T itself must satisfy σT (a) ⩾ τ(a), in case
σG (a) ⩾ τ(a) (respectively, σT (a) < τ(a), in case σG (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 σT◦ (a) = σG (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 σG (x) = σG↓{x}∪{y∈Args | r (y,x)} (x).
G
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:
Proposition 4.2 (Sufficient QDTs). If σ satisfies the directional connectedness principle, then
for every QBAG G = (Args, τ, Att, Supp), for every a ∈ Args, there exists a suffcient QDT T for a.
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Proof. Let G∗ be the sub-graph of G containing a and all and only the children of a, i.e. G∗ =
G ↓{a}∪{y∈Args | rG (y,a)} . Using Principle 4.1, σG (a) = σG∗ (a). As per the proof of Proposition 4.1,
construct a QDT T using the whole of G∗ . Then G∗ trivially corresponds to T, so that σG∗ (a) =
σT (a). Since T has no proper super-trees, T is sufficient for a.
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.
Example 4.1. We claim that QDT T given in Figure 1b from Example 1.1 is sufficient for a.
First, we have σT (a) = 0.524 > τ(a). Extending T with a proponent b would only increase
the final strength of a. But even extending T with only the opponent e yields a super-tree T′
with σT′ (a) = 0.5003 > τ(a); see Figure 1c. So all super-trees of T (including itself) satisfy the
sufficiency condition given in Definition 4.2.
Note that T is actually a minimal sufficient QDT for a. Indeed, without [P : f], i.e. in the sub-tree
with nodes {[P : a], [O : c], [P : d]}, the final strength of a would be 0.481. This rules out both such a
sub-tree and the sub-tree with nodes {[P : a], [P : d]} from satisfying sufficiency. Similarly, without
[P : d], the final strength of a in the sub-tree of T with arguments {[P : a], [O : c], [P : f]} would be
0.424, which with the above rules out any proper sub-tree of T from satisfying sufficiency.
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], [P : b], [O : c], [P : d]} (and any of its
sub-trees) are ruled-out by the fact that in the tree T◦ without [P : f] the final strength of a is 0.499.
Similarly, without d and e, the final strength of a is 0.476, so that trees with subsets of nodes
{[P : a], [P : b], [O : c], [O : e], [P : f]} are ruled out from satisfying sufficiency as well.
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 σG (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 Ctrba (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 Ctrba (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.
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5. Discussion
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, e.g. 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.
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