Paper deep dive
Safety, Liveness, and Fairness in Quantitative Argumentation Dialogues
Arunavo Ganguly, Julian Alfredo Mendez, Timotheus Kampik
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
Last extracted: 7/8/2026, 7:32:56 PM
Summary
This paper introduces formal notions of safety, liveness, and fairness to quantitative bipolar argumentation dialogues. It defines strong and weak safety based on whether argument strengths remain above or eventually reach a justification threshold across a sequence of argumentation graphs. Liveness captures fluctuating argument strengths across this threshold, while fairness evaluates the distribution of safe arguments. The authors formally relate these properties, demonstrate that strong safety implies weak safety and is equivalent to the absence of liveness, and discuss analytical challenges in guaranteeing these properties dynamically.
Entities (7)
Relation Signals (5)
Strong Safety → implies → Weak Safety
confidence 98% · It is immediate from the definition that strong safety implies its weak counterpart.
Strong Safety → equivalentto → Not Liveness
confidence 96% · A topic T is strongly safe iff it is not live w.r.t. G, sigma and threshold t.
Liveness → requires → Fluctuation Across Threshold
confidence 95% · Liveness requires that arguments’ strengths fluctuate across the threshold of justification.
Fairness → assesses → Safe Arguments
confidence 94% · Fairness notions assess how safe arguments are spread within a sequence of argumentation graphs.
DFQuAD Semantics → isa → Modular Semantics
confidence 93% · Modular semantics are well-studied sub-class of gradual semantics... In our examples we make use of DFQuAD semantics
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:We introduce notions of safety, liveness, and fairness, as commonly used in temporal reasoning, to quantitative (bipolar) argumentation dialogues where repeated inferences are drawn from argumentation graphs with weighted nodes. Between inferences, these graphs undergo updates. Strong and weak safety capture that arguments' (final) strengths remain above a specific threshold of justification and always reach the threshold eventually, respectively. Liveness requires that arguments' strengths fluctuate across the threshold of justification. Fairness notions assess how safe arguments are spread within a sequence of argumentation graphs. We formally show how these notions are related, and discuss some analytical challenges with respect to providing general guarantees for our properties.
Tags
Links
- Source: https://arxiv.org/abs/2605.23578v1
- Canonical: https://arxiv.org/abs/2605.23578v1
PDF not stored locally. Use the link above to view on the source site.
Full Text
41,251 characters extracted from source content.
Expand or collapse full text
1] of Computing Science, å University, å, Safety, Liveness, and Fairness in Quantitative Argumentation Dialogues aganguly@cs.umu.se Alfredo julian.mendez@umu.se tkampik@cs.umu.se [ Abstract We introduce notions of safety, liveness, and fairness, as commonly used in temporal reasoning, to quantitative (bipolar) argumentation dialogues where repeated inferences are drawn from argumentation graphs with weighted nodes. Between inferences, these graphs undergo updates. Strong and weak safety capture that arguments’ (final) strengths remain above a specific threshold of justification and always reach the threshold eventually, respectively. Liveness requires that arguments’ strengths fluctuate across the threshold of justification. Fairness notions assess how safe arguments are spread within a sequence of argumentation graphs. We formally show how these notions are related, and discuss some analytical challenges with respect to providing general guarantees for our properties. keywords: Formal argumentation, , , . 1 Introduction Quantitative (bipolar) argumentation is a formal argumentation variant in which inferences are drawn from graphs whose nodes (arguments) with weights (initial strengths) are connected by support and attack relations. Argumentation semantics then draw inferences from a quantitative bipolar argumentation graph (QBAG) by updating arguments’ initial strengths to final strengths, considering initial strengths and the QBAG’s topology. QBAGs are of substantial interest to the formal argumentation community, notably because of their potential to complement machine learning approaches and help make them explainable [Potyka0T23, DBLP:conf/ecai/0007PT23] and contestable [yin2025contestabilityquantitativeargumentation]. Currently, QBAG applications are emerging that facilitate explainability and reliability of large language model-based inference [jin2026argoraorchestratedargumentationcausally, chen2026latentdebatesurrogateframework]. In such applications, argumentation can be viewed as a dynamic process during which QBAGs are manipulated, and repeated inferences are drawn. Accordingly, it may be interesting to inquire about the dynamic properties of QBAG-based inference. This paper sets out to systematise such inquiries by drawing inspiration from well-known notions of safety, liveness, and fairness (SLF), which are prominent in various temporal reasoning approaches, such as linear temporal logic [Alpern-Schneider, DBLP:conf/popl/GabbayPSS80] and Petri nets [DBLP:journals/ipl/KindlerA99]111However, we do not formally relate to these notions of “tracial control”, as we are interested in conceptually somewhat different questions, namely the dynamics of quantitative argument strengths.. Specifically, given a sequence of QBAGs (which we call a chain or dialogue), a set of topic arguments occurring in all of the sequence’s graphs, and a specified strength threshold, these notions allow us to ask the following questions: Safety: Do all of the topic arguments always stay above the threshold, or do they at least cross it eventually? The former property, which we call strong safety can, e.g., help us check whether obviously credible arguments always retain reasonable strengths in the dialogue, or if the dialogue, colloquially speaking, at times reaches “cuckoo-land” territory. The latter property (weak safety) allows us to check whether an agent reaches an objective represented by the topic arguments’ final strength at the dialogue’s end. Liveness: Do all of the topic arguments “cross the threshold” in the course of the dialogue, i.e,, for each argument, is there a QBAG where its final strength is below the threshold, as well as a QBAG where its final strength is above the threshold? Liveness helps us understand whether an intuitively “interesting” deliberation takes place with respect to the arguments: if the dialogue is not live with respect to the arguments, we do not observe any notable situations in which an eventually credible argument is substantially challenged, or where an eventually not credible argument appears to be credible. Fairness: Do all topic arguments get to have an equal shot at credibility? We present several drastic and gradual fairness notions, which are based on safety. The following example helps illustrates the intuitions provided above. Example 1. (0.5):0.6 a~(0.5)\!:\!0.6(0.7):0.7 b~(0.7)\!:\!0.7(0.2):0.2 c~(0.2)\!:\!0.2e+ (a) Initial QBAG G. (0.5):0.1 a~(0.5)\!:\!0.1(0.7):0.0 b~(0.7)\!:\!0.0(0.2):0.2 c~(0.2)\!:\!0.2(1.0):1.0 d~(1.0)\!:\!1.0e+-- (b) Normal expansion G′G . (0.5):0.5 a~(0.5)\!:\!0.5(0.7):0.56 b~(0.7)\!:\!0.56(0.2):0.2 c~(0.2)\!:\!0.2(1.0):0.2 d~(1.0)\!:\!0.2(0.8):0.8 e~(0.8)\!:\!0.8e+--- (c) Normal expansion G′G . Figure 1: From left to right, the initial QBAG expands by successively adding d (normal expansion G′G ) and e (normal expansion G′G ), respectively. Such scenarios capture the essence of dynamic argumentation. Consider the QBAGs in Figure 1. Each argument is labelled (): x( y): z, denoting the name, initial strength and final strength, respectively. Intuitively, initial strengths represent the credibility of arguments per se, whereas final strengths reflect the justification reached through interactions with others. Interactions are represented via attack and support relations, labelled −- and ++, respectively. The QBAGs in Figure 1 portray a normal expansion chain: new arguments and relationships are added, but relationships between existing arguments (as well as arguments’ initial strengths) remain the same. Our SLF notions are introduced with a “threshold of justification"; justified arguments are assumed to exceed the aforementioned threshold. To that end, we understand safe argument sets to be justified, either throughout the process (strong safety) or at the end (weak safety). For instance, \ c\ and \ b\ are strongly and weakly safe, respectively, against the threshold t=0.1t=0.1 in Figure 1. Liveness accounts for arguments that fluctuate across the threshold. In Figure 1, \ b\ is live as as it drops below t=0.1t=0.1 in G′G , and exceeds it again in G′G . Our notions of fairness then cover two dimensions: several binary classifications based on safety, and gradual notions based on the established measures of the Gini index and Shannon entropy. In the example, ,\ a, b\ is cautiously fair because both \ a\ and \ b\ are weakly safe. However, our gradual fairness notions provide more nuance: ,,\ a, b, c\ yields fairness scores of 0.554490.55449 and 0.462120.46212 given the Gini-based and Shannon-based notions, respectively. Intuitively, this occurs because \ c\ has a better shot at justification than ,\ a, b\. Next, we introduce relevant preliminaries (Section 2), to then formally present our SLF notions, observing some of their properties, as well as how the notions are interrelated (Section 3). We also discuss why giving guarantees with respect to the SLF notions may be challenging by an empirical demonstration of semantics behaviour and highlight future research directions in the context of related work (Section 4). 2 Preliminaries Let ⟨,⪯⟩ , be a partial ordering. Typically, I will be a real interval along with the canonical ordering. A quantitative bipolar argumentation graph (QBAG) is a quadruple G=⟨,τ,,⟩G= ,τ,Att,Supp , where Args is a set of arguments, Att and Supp are disjoint binary relations on Args, called attacks and supports, respectively, and τ:→τ:Args assigns an initial strength in I to each argument. Intuitively, an initial strength may represent the initial confidence in the argument without any influence: greater initial strengths represent higher confidence. For some examples of QBAGs, see Figure 1 (Example 1). Unless stated otherwise, let G=⟨,τ,,⟩G= ,τ,Att,Supp be an arbitrary but fixed QBAG, with finite Args. Given an argument ∈ x , the attackers and supporters of x are denoted by G():=∈:(,)∈Att_G( x):=\ a :( a, x) \ and G():=∈:(,)∈Supp_G( x):=\ a :( a, x) \, respectively. We drop the subscript G whenever it is clear from context. Given two arguments ,∈ x, y , we say that x reaches y iff there is a path from x to y in the graph ⟨,∪⟩ ,Att . An acyclic QBAG does not contain any loops, i.e. no argument ∈ x can reach itself. For any ′⊆Args , we define the restriction of G to Args as G↓′:=⟨′,τ∩(′×),∩(′×′),∩(′×′)⟩G _Args := ,τ∩(Args ×I),Att∩(Args ×Args ),Supp∩(Args ×Args ) . Finally, we say G is a sub-QBAG of G′=⟨′,τ′,′,′⟩G = ,τ ,Att ,Supp , denoted by G′⊑G G, iff ′⊆Args , ′⊆Att , ′⊆Supp and τ′()=τ()τ ( x)=τ( x), for every ∈′ x . If G⊑G′G G but G≠G′G =G , we say G⊏G′G G . Our idea of dynamic argumentation is captured via QBAG chains. Definition 1 (QBAG Chains). A QBAG chain (short: chain) is a sequence (totally ordered set) of QBAGs =(G1,…,Gn) G=(G_1,…,G_n). Intuitively, chains are evolutions of a given QBAG along arbitrary dimensions, such as expansions, initial strength updates etc. If a chain only encompasses expansions we call it a QBAG expansion chain; i.e., an QBAG expansion chain (short: expansion chain) is a chain (G1,…,Gn)(G_1,…,G_n) with Gi⊏Gi+1G_i G_i+1 for each i. Given a chain =(G1,…Gn) G=(G_1,… G_n), we use G≺G′G _ GG to denote that for some 1≤i<n1≤ i<n it holds that G=GiG=G_i and G′=Gi+1G =G_i+1. When we write G∈G∈ G, we mean that for some GiG_i, 1≤i≤n1≤ i≤ n it holds that Gi=G_i=G. An expansion chain G is: (i) normal iff for every G,G′∈G,G ∈ G it holds that ∀⟨,⟩∈(′∪′)∖(∪)∀ x, y ∈(Att ) (Att ), either x or y is in ′∖Args ; (i) weak iff for every G,G′∈G,G ∈ G it holds that none of the arguments ∈′∖ x can reach the arguments in Args. Sometimes, an expansion chain G is represented as G1≺G1⋯≺GnG_1 G_1… G_n, where G1G_1 and GnG_n are start and end points of G, respectively. Consider the next example. Example 2 (Expansions and Chains). Referring to Example 1, G′G and G′G , are (normal) expansions of G and G′G , respectively. Furthermore, G≺G′≺G′G G G constitute a chain with G and G′G being the start and end points, respectively. Note that while the examples provided in this paper are catered towards expansion chains, our SLF notions are applicable to arbitrary chains. A gradual semantics is a function σG:→∪⟂ _G:Args ∪\ \ mapping the arguments in a QBAG G, to the partial ordering I, along with the symbol ⟂ for undefined [baroni-toni-rago]. Modular semantics are well-studied sub-class of gradual semantics, where the final strength is assigned by aggregating the final strengths of its attackers and supporters, and then computing the influence on the specific argument given the aggregation result and the argument’s initial strength. More formally, an aggregation function has the form f(A,S)f(A,S), given multi-sets of attack values A and support values S; the application of an influence function gs(f(A,S))g_s(f(A,S)) then yields an update of an argument’s strength value s∈s . Modular semantics first initialize all arguments by assigning the their initial strengths as the temporary strength value, and then update these values by applying aggregation and influence function to these values following the topological order of the QBAG (in the direction of the binary relations). In our examples we make use of DFQuAD semantics, given =[0,1]I=[0,1]. Fix A,SA,S to be (),()Att( x),Supp( x), respectively, for ∈ x . For the aggregation function f((),())=∏∈()(1−σ())−∏∈()(1−σ())f(Att( x),Supp( x))= _ b ( x)(1-σ( b))- _ c ( x)(1-σ( c)) and initial strength τ(),τ( a), DFQuAD is defined over acyclic graphs with a topological ordering as per the equation given below. g(τ(),f(()),()):= g(τ( a),f(Att( x)),Supp( x)):=\ τ()−τ()×max0,−f((),())+ τ( a)-τ( a)× \0,-f(Att( x),Supp( x))\ 3.99994pt+ (1−τ())×max0,f((),()). (1-τ( a))× \0,f(Att( x),Supp( x))\. Assumption. Notice that given cyclic QBAGs, modular semantics may not converge and thus yield undefined final strengths in some cases [potyka2024empirical]. This is a problem for us, as we assume that final strengths are comparable. However, for the sake of simplicity, we do not handle this edge case and the reader may assume that we focus on acyclic QBAGs, which is a common assumption in the literature [kotonya2019gradual, cocarascu2019extracting, lidecision, chi2021optimized]. 3 Safety, Liveness, and Fairness Properties In this section, we formally introduce the SLF notions for QBAGs. The notions are validated w.r.t. an initial QBAG G, a chain of updates G, a semantics σ, and a threshold t∈t . We call t the threshold of justification. Throughout this section, we fix a partial ordering I, arbitrary QBAGs G=⟨,τ,,⟩G= ,τ,Att,Supp , G′=⟨′,τ′,′,′⟩G = ,τ ,Att ,Supp , G′=⟨′,τ′,′,′⟩G = ,τ ,Att ,Supp , a QBAG chain =(G1,…Gn) G=(G_1,… G_n). We also fix a set of topic arguments T⊆⋂1≤i≤nGiT _1≤ i≤ nG_i w.r.t. which we analyse the SLF notions. Safety Strongly safe arguments exceed the threshold of justification against the repeated drawing of inferences during the expansion. Definition 2 (Strong safety). We say that a topic set T is safe w.r.t. a chain , G, a semantics σ, and t∈t iff for every G′∈G ∈ G, and ∈T x∈ T, σG′()≥t _G ( x)≥ t. In contrast, weak safety stipulates that arguments eventually exceed the threshold, thereby becoming justifiable at the end. Definition 3 (Weak safety). We say that a topic set T is weakly safe w.r.t. a chain G, a semantics σ and t∈t iff for every ∈T x∈ T, σG′()≥t _G ( x)≥ t, where G′G is the end-point of G. We revisit our initial example to illustrate the safety notions. Example 3 (Strong and weak safety). In Figure 1, the arguments c and b are strongly and weakly safe, respectively, w.r.t. (G,G′,G′)(G,G ,G ) and the threshold t=0.1t=0.1. However, if the threshold is changed to t=0.2t=0.2, c is neither strongly safe nor weakly safe. On the flip side, d continues to be weakly safe. It is immediate from the definition that strong safety implies its weak counterpart. Theorem 1. If an argument ⊆E\ x\ E of a topic E is safe, then it is weakly safe. Proof. Let T⊆T be safe w.r.t. a chain (G1,…,Gn)(G_1,…,G_n), a semantics σ and t∈t . This means σG′()≥t _G ( x)≥ t for each ∈T x∈ T, where G′∈G ∈ G. In particular, σGn()≥t _G_n( x)≥ t. The previous statement is the condition for weak safety, as GnG_n is the end-point of the chain. ∎ In general, the converse of Theorem 1 does not hold, as seen in Example 3. However, under certain restrictions upon the expansion sets and the choice of semantics a partial converse can be achieved. In the subsequent theorem, we propose a few of such cases. Theorem 2. Suppose either of the following cases hold: • The chain G has length 11, or • G is a weak chain and σ is a modular semantics, or • Only arguments that cannot reach Args are added to the chain G. Then strong and weakly safe arguments coincide w.r.t. G,,σG, G,σ. Proof. Assume T is weakly safe w.r.t. ,σ G,σ and t∈t , where the chain G is denoted by (G1,…,Gn)(G_1,…,G_n). Now, we prove our claim by cases. • If G has length 11, then =(G1) G=(G_1), hence G1=GnG_1=G_n. Since the start and end-point of G coincides, as per Definition 3, σG1()≥t _G_1( x)≥ t, for every ∈T x∈ T. Since G is singleton, ∀G′∈∀ G ∈ G, σG′()≥t _G ( x)≥ t, which is the condition for strong safety. • Weak expansions under any modular semantics σ do not alter the final strengths of G. This is due to the strong directionality argument in [kampik-potyka] (Principle 4, Lemma 6.1). So ∀G′∈∀ G ∈ G, and ∈ x , σG()=σG′() _G( x)= _G ( x). In particular, if any topic argument y has σGn()≥t _G_n( y)≥ t iff ∀G′∈,σG′()≥t∀ G ∈ G, _G ( y)≥ t. • Again as per [kampik-potyka], any modular semantics σ satisfy strong directionality, i.e., arguments that cannot reach T, added during the updates, do not change the final strength of the topics. Hence, ∀G′∈∀ G ∈ G, and ∈ x , σG()=σG′() _G( x)= _G ( x). Now, we follow the argument as given in the previous case. ∎ Liveness Safe arguments might appear monotone or unalive due to their consistent levels of high credibility, as seen in Definitions 2 and 3. Our notion of liveness aims to capture the contrasting case of such observations. The idea is formally introduced through fluctuations across the threshold-of-justification. Intuitively, fluctuations quantify “flip-flopping” behaviour exhibited by an argument. Definition 4. We say x shows k-fluctuations w.r.t. a chain =(G1,…,Gn),σ G=(G_1,…,G_n),σ and t∈t iff there are 1≤i0≤⋯≤ik≤n1≤ i_0≤…≤ i_k≤ n s.t. for each 1≤j<n1≤ j<n, either σGij()≤t≤σGij+1() _G_i_j( x)≤ t≤ _G_i_j+1( x), if possible or σGij+1()≤t≤σGij() _G_i_j+1( x)≤ t≤ _G_i_j( x), otherwise. We say x shows exactly n-fluctuations over a chain G iff x cannot show n+1n+1-fluctuations. Based upon this idea, we introduce the notion of liveness. Definition 5 (Liveness). We say topic set T is live w.r.t. a chain G a semantics σ, and a∈a iff each ∈T x∈ T shows at-least 11-fluctuation w.r.t. ,σ G,σ and threshold t. Again, we go back to our running example. Example 4 (Liveness Example). The chain =(G,G′,G′) G=(G,G ,G ) has the topic set ,\ a, b\ as live w.r.t. the threshold 0.20.2. On the other hand, \ c\ shows no fluctuation w.r.t. the threshold t=0.2t=0.2, thereby being strongly safe but unalive. The latter part of the above observation can be generalised, i.e. a topic is not live iff it is strongly safe. This observation provides an equivalent formulation of strong safety. Theorem 3. A topic T is strongly safe iff it is not live w.r.t. ,σ G,σ and threshold t∈t . Proof. Suppose T is strongly safe, then as per definition σG()≥t _G( x)≥ t, ∀G∈,∀∈T∀ G∈ G,∀ x∈ T. Now, the above condition is equivalent to saying G never goes below t in the chain,i.e. there are 0 fluctuations w.r.t. ,σ G,σ and t. ∎ We can guarantee that in certain cases no fluctuations are observed in the topic set. These include cases of weak expansions, and evolutions that only append downstream arguments. We formally observe it below as a corollary; the proof is a straight-forward application of Theorem 2 and Theorem 3. Corollary 1. Suppose either of the following cases hold: • The chain G has length 11, or • G is a weak chain and σ is a modular semantics, or • Only down-stream arguments are added to Args in the chain G. Then ∈ x shows no fluctuations w.r.t. ,σ G,σ and t∈t . □ Fairness We provide three notions towards binary classification of strength-fairness. Intuitively, these notions condition the safety of all topic arguments on the existence of safety in one topic argument in terms of a necessary condition. Here, the existential safety notion (“one topic argument”) must be at least as strong as the universal safety notion (“all topic arguments”). Notice that given the argumentation-based setting, our fairness notions purposefully diverge conceptually from fairness notions in temporal reasoning, which tend to require executions infinitely often; in contrast, we are interested in notions reflecting fairness in finite resource distribution. Definition 6 (Fairness). We say a topic set T, w.r.t. ,σ,t∈ G,σ,t is • Ideally Fair iff some ⊆T\ x\ T is strongly safe implies T is strongly safe. • Lively Fair iff some ⊆T\ x\ T is weakly safe implies T is weakly safe. • Cautiously Fair iff some ⊆T\ x\ T is strongly safe implies T is weakly safe. In our running example, sets of arguments satisfy different fairness notions. Example 5 (Strength Fairness Example). Refer again to Figure 1, and consider ,,\ a, b, c\ to be the topic set. Example 3 states c is strongly safe w.r.t. (G,G′,G′)(G,G ,G ) and threshold 0.20.2. All the topic-arguments are above 0.20.2 at the end-point G′G , thereby making the set ,,\ a, b, c\ cautiously and lively fair. However, ideal fairness is violated by ,,\ a, b, c\ as \ c\ is strongly safe, whereas ,\ b, c\ are weakly so. The following connections are present between the notions of fairness. Theorem 4. For any topic set T, the following holds w.r.t. ,σ,t∈ G,σ,t . • Lively fairness implies cautious fairness. • Ideal fairness implies lively fairness, in the presence of a strongly safe argument. Proof. For the first case, assume T is lively fair w.r.t. G, σ and t∈t . If there is a strongly safe ∈T x∈ T then by Theorem 1, x is weakly safe, and consequently T is weakly safe due to live fairness. This satisfies the constraint for being cautiously fair. For the second case, suppose T is ideally fair with a strongly safe topic argument x, which implies T is strongly safe, and by Theorem 1, T is weakly safe. Now trivially, lively fairness is satisfied. ∎ Additionally, ideal fairness prohibits the liveness of topic arguments, thereby implying a “dull” behaviour of the dialogue with respect to the concerned arguments. Theorem 5. If T is ideally fair then it is not live whenever there is a strongly safe ⊆T\ x\ T. Proof. Suppose T is ideally fair w.r.t. ,σ,t G,σ,t with a strongly safe ⊆T\ x\ T. This implies every ∈T x∈ T satisfies σ()≥tσ( x)≥ t through the update chain G, showing no fluctuations. Hence, T is not live as per definition. ∎ Towards the gradual notions of fairness, we firstly draw on the Gini index [gini1912], which traditionally measures “inequalities” in the distribution of wealth, income, or consumption. The idea is to analogously measure differences in the safety of topic arguments. Consider the example below. Example 6. In Figure 1, a and b exceed the threshold 0.20.2 for the QBAGs G,G′\G,G \, whereas c exceeds the threshold for the whole chain G,G′,G′\G,G ,G \. Thus the set ,,\ a, b, c\ has the ordering (,,)( a, b, c) when arranged by increasing number of threshold-exceeding QBAGs in the chain (G,G′,G′)(G,G ,G ). Figure 2 plots the safety curve (solid line) using Equation 2 , where T=T=Args is topic set, t=0.2t=0.2, =(G,G′,G) G=(G,G ,G), and S()S( x) returns the number of QBAGs within G, for which x exceeds the threshold t=0.2t=0.2. The fairness line (dotted line) for the topic set ,,\ a, b, c\ is a line joining (0,0)(0,0) and (3,7)(3,7), as per Equation 1. Intuitively, the fairness line shows that the cumulative sum of threshold-exceeding arguments increase linearly w.r.t. the aforementioned ordering of the topics. Fairness is now understood in terms of deviation of the safety curve from the fairness line. The deviation, in turn, is measured in terms of area enclosed within their graphs, and normalised using a modified sigmoid function. For this case, the Gini-based fairness score is ≈ 0.46212. Figure 2: Plot of Fairness Line v/s ζ. The solid line is due to the safety-curve ζ, and the fairness line is represented using dashes. The area enclosed between the two curves measures the inequality in fluctuation. Formally, given a chain G, any argument ∈T x∈ T, and a threshold t∈t , we define S,σ,t()S_ G,σ,t( x) to return the cardinality of the set |G∈:σ()≥t|,|\G∈ G:σ( x)≥ t\|, i.e. the number of QBAGs within G for which x exceeds t. We drop the subscripts whenever it is clear from context. The fairness line f(x)f(x) is the line joining (0,0)(0,0) to (|T|,∑∈TS())(|T|, _ y∈ TS( y)), which is defined piecewise f(x)=(∑∈TS()|T|)×x;0≤x≤|T|,0;x>|T|,x<0.f(x)= cases ( _ y∈ TS( y)|T| )× x;&0≤ x≤|T|,\\ 0;&$x>|T|,\,x<0$. cases (1) The safety curve can be formalised as follows: suppose 1,…,n x_1,…, x_n are the arguments in T arranged in an increasing order, then the fluctuation-curve is defined piece-wise: ζ,T,σ(x)=(∑0≤j≤iS,σ,t(j))+(S,σ,t(i)i+1−i)×x;i≤x≤i+1,0;x>|T|,x<0. _ G,T,σ(x)= cases ( _0≤ j≤ iS_ G,σ,t( x_j) )+ ( S_ G,σ,t( x_i) x_i+1- x_i )× x;&i≤ x≤ i+1,\\ 0;&$x>|T|,\,x<0$. cases (2) Based on these notions, we introduce Gini-based fairness. Definition 7 (Gini-based fairness). For the topic set T, define the un-normalised Gini-based fairness score using Equation 2 w.r.t. ,σ,t∈ G,σ,t .: ℱginiu(,T,σ):=∫0|T||ζ,σ,t(x)−f(x)|x. F^u_gini( G,T,σ):= _0^|T|| _ G,σ,t(x)-f(x)|dx. (3) Now, Gini-based fluctuation fairness as ℱgini(,T,σ):=(21+e−ℱginiu(,T,σ)−1). F_gini( G,T,σ):= ( 21+e^- F^u_gini( G,T,σ)-1 ). Intuitively, the integral in Equation 3 measures the deviation of the safety curve from the fairness line in terms of area enclosed between them. Hence, as inequality grows, ℱgini F_gini approaches 1,whereas the case of perfect equality encloses a non-positive area. This is formally observed in the theorem below. Theorem 6. Fairness of fluctuations is a positive function having its range in the unit interval [0,1)[0,1). The value 0 represents perfect equality, i.e. when the safety curve coincides with the fairness line. Proof. Notice that, as ℱginiu(,T,σ)≥0 F^u_gini( G,T,σ)≥ 0, it implies 0≤ℱgini(,T,σ)<10≤ F_gini( G,T,σ)<1. In the case of ℱginiu(T,,σ)=0 F^u_gini(T, G,σ)=0, i.e. when the fairness line and ζ coincides, (21+e−ℱginiu(,T,σ))=1 ( 21+e^- F^u_gini( G,T,σ) )=1 and by definition, ℱgini(,T,σ)=0 F_gini( G,T,σ)=0. ∎ We propose another gradual notion, inspired by Shannon entropy [shannon_entropy]. To that end, we define a threshold-exceeding probability on the topic set T w.r.t. ,σ,t∈ G,σ,t : p,T,σ():=Total threshold exceeding QBAGs of Total fluctuations shown by the topic set T=S,T,σ()∑∈TS,T,σ().p_ G,T,σ( x):= Total threshold exceeding QBAGs of xTotal fluctuations shown by the topic set T= S_ G,T,σ( x) _ y∈ TS_ G,T,σ( y). Again, we drop the subscript whenever clear from context. Notice that p is undefined in the absence of threshold exceeding behaviour. Given a random threshold exceeding behaviour, we want to measure how surprising it is. If the behaviour is from a topic with low number of fluctuation then it is highly surprising, and vice-versa. Formally, we denote it as I(p):=logb(1p())=−logb(p()),I(p):= _b ( 1p( x) )=- _b(p( x)), where b is the least common multiple of the denominators of p()p( x). Now, Shannon-based fluctuation fairness is defined as the expectation of the surprise factor. Definition 8 (Shannon-based fairness). For the topic arguments E w.r.t. the chain G, semantics σ and threshold a∈a , Shannon-based fluctuation fairness is defined as: ℱshannon(,T,σ):=−∑∈Tp()logb(p());if p is defined,1;if p is undefined.F_shannon( G,T,σ):= cases- _ x∈ Tp( x) _b(p( x));&if $p$ is defined,\\ 1;&if $p$ is undefined. cases (4) where b is the least common multiple of the denominators of p()p( x). Example 7. Let us consider the Shannon-based fairness for the topic set T=,,T=\ a, b, c\ in Figure 1. The probability of fluctuation is given by p()=27,p()=27p( a)= 27,p( b)= 27 and p()=37p( c)= 37. It is easy to see that b=7b=7 and by definition, ℱshannon≈0.55449. F_shannon≈ 0.55449. The next result provides a case where Shannon-based fairness is maximised. Theorem 7. If ∀,∈T,S()=S()∀ x, y∈ T,S( x)=S( y) w.r.t. ,σ G,σ and t∈t then ℱshannon(E,,σ)F_shannon(E, G,σ) is 11. Proof. Suppose that ∀,∈T,S()=S()∀ x, y∈ T,S( x)=S( y), it immediately implies that ∀,,p()=p()∀ x, y,p( x)=p( y). The later now implies p()=1mp( x)= 1m, where m=|T|m=|T|, and consequently b=mb=m. Now the result is the given by the following chain of equality: ℱshannon=−∑∈Tp()⋅logb(p())=−m⋅(1m⋅logm(1m))=1. F_shannon=- _ x∈ Tp( x)· log_b(p( x))=-m· ( 1m· log_m ( 1m ) )=1. ∎ Theorem 7 shows the interplay between binary and gradual notions of fairness. If a topic T is ideally fair in the presence of some strongly safe ⊆T\ x\ T, then ℱshannon(,T,σ)=1 F_shannon( G,T,σ)=1. Similarly, given the previous scenario, ℱgini(,T,σ)=0 F_gini( G,T,σ)=0. Theorem 8. If T is ideally fair w.r.t. ,σ,t G,σ,t, and some ⊆T\ x\ T is strongly safe, then ℱgini(,T,σ)=0 F_gini( G,T,σ)=0, and ℱshannon(,T,σ)=1 F_shannon( G,T,σ)=1. Proof. Notice that if the theorem’s antecedent holds, then ∀,∈T∀ x, y∈ T, we have S()=S()S( x)=S( y). This means p is an uniform distribution, and by Theorem 7, ℱshannon(,T,σ)=1 F_shannon( G,T,σ)=1. On the other hand, the previous observation implies that ζ is a linear function satisfying ζ(0)=0ζ(0)=0, and ζ(|T|)=∑∈TS(x)ζ(|T|)= _ x∈ TS(x). This in turn implies that ζ coincides with the fairness line. Finally by Theorem 6, ℱgini(,T,σ)=0 F_gini( G,T,σ)=0. ∎ 4 Discussion and Conclusion Our work contributes to the line of research on argumentation dynamics that, broadly speaking, studies argumentative inference where the representation of the exchange of arguments undergoes changes. This naturally reflects the dynamic nature of dialectical reasoning. Argumentation dynamics are frequently studied for different variants of computational argumentation—see [doutre-argument] for a survey of works on abstract argumentation dynamics. For the specific case of QBAGs, dynamics are often applied in order to facilitate some notion of explainability, notably by assessing the impact (a change to) one argument has on the final strength of another one [DBLP:conf/ecai/0007PT23, DBLP:journals/ijar/KampikPYCT24]. In these works, dynamics are thus essentially used as an analytical tool. Conversely, our work introduces analytical tools for studying dynamics that occur in sequences of QBAGs, which we call chains or, intuitively, dialogues. It may thus be most closely related to [KAMPIK-strength-change], which studies which changes—also called explanations—made to a QBAG affect the relative strength of two topic arguments. The current work could be extended to define explanations amounting to changes affecting safety, liveness, and fairness notions. Beyond the above, our study of SLF notions for quantitative (bipolar) argumentation dialogues leaves the following two key questions for future work. Providing SLF guarantees depending on argumentation semantics. Our work analyses only very simple cases where some of our SLF notions can be guaranteed, notably cases based on weak expansions, where changes do not affect the topic arguments. Indeed, we view the SLF notions as analysis tools that can be applied to specific instances of QBAG chains. Let us argue that safety, liveness, and fairness is challenging for more general cases. Figure 3 illustrates that even relatively simple scenarios may render it difficult to analytically show for general cases that SLF notions are satisfied or violated. Consider the QBAG in Figure 3a and topic arguments a and b. The strength of f is variable; consider a scenario where we have a chain of QBAGs that are all based on the one in the figure; however, the strength of f increases (either due to an initial strength change or influence of additional supporters), e.g., from 0.10.1 to 0.50.5 and finally to 0.90.9. The effects of these changes on the final strengths of a and b can be seen in Figure 3b. Notice that intuitively, the effect of strength changes to f on a and b is non-monotonic: it “switches” from positive to negative for a and inversely for b—this phenomenon is, e.g., discussed in [DBLP:journals/ijar/KampikPYCT24]. This makes it difficult to assess whether a topic set, even when consisting of a single argument is safe or live, and analogously, whether a topic set of several arguments is fair. In our example, \ a\ is strongly and weakly safe w.r.t. the thresholds 0.10.1 and 0.1750.175, respectively; however \ b\ fails to satisfy any of the safety notions w.r.t. the above-mentioned thresholds. Both a and b are live for the threshold 0.1750.175, however only b is live w.r.t. the other threshold of 0.10.1. \ b\’s weak safety failure prevents ,\ a, b\ from satisfying the binary notions of strength-fairness w.r.t. the thresholds 0.10.1 and 0.1750.175. Finally, the topic set ,\ a, b\ yields ≈0.76157≈ 0.76157 for Gini-based fairness, and ≈0.32945≈ 0.32945 for Shannon-based fairness. (0.2):? a~(0.2)\!:\! 0? 0(0.0):? b~(0.0)\!:\! 0? 0(0.0):? e~(0.0)\!:\! 0? 0(0.0):? c~(0.0)\!:\! 0? 0(0.0):? d~(0.0)\!:\! 0? 0(?):? f~(?)\!:\! 0? 0+---+++++ (a) QBAG with variable initial stength of f. (b) σ()σ( a) (solid) and σ()σ( b) (dashed), given τ()τ( f). Figure 3: SLF properties are difficult to guarantee, even in seemingly simple scenarios where we change the initial strength of a single argument that reaches two topic arguments. The horizontal lines in Figure 3.2. indicate different threshold values (of 0.10.1 and 0.1750.175, respectively). Defining SLF notions for partially ordered final strengths. In this work, we have made the crucial assumption that our assessments of safety, liveness, and fairness pertain to topic arguments that are comparable. However, this may not always be the case in QBAGs, as arguments may have undefined final strengths (practically, because a modular semantics may not converge for cyclic graphs). Pragmatically, one may claim that undefined final strengths are an edge case whose consideration introduces overhead and thus confusion. However, in some cases dealing with arguments of incomparable strengths can be useful, e.g., when moving from quantitative to labelling-based abstract argumentation where labels may form a lattice of argument strength (cf. [wu2010labelling]), as sketched in [KAMPIK-strength-change]. References