1 Introduction
Formal argumentation [2] describes a family of approaches to modeling rational decisionmaking through the representation of arguments and their relationships. A particular important representative approach is that of abstract argumentation [7], which focuses on the representation of arguments and a conflict relation between arguments through modeling this setting as a directed graph. Here, arguments are identified by vertices and an attack from one argument to another is represented as a directed edge. This simple model already provides an interesting object of study, see [3] for an overview. Reasoning is usually performed in abstract argumentation by considering extensions, i. e., sets of arguments that are jointly acceptable given some formal account of “acceptability”. Therefore, this classical approach differentiates between “acceptable” arguments and “rejected” arguments. However, empirical cognitive studies such as the ones described in [16, 15] have shown that humans assess arguments in a more finegrained manner. For example, while [16] provides evidence that while humans adopt the “reinstatement principle” (which states that arguments defended by accepted arguments should also be accepted), they usually assign lower confidence to reinstated arguments than nonattacked ones. Similarly, the experiments described in [15] advocate that a probabilistic interpretation of arguments [10] is more suitable than the classical twovalued interpretation.
In order to formally address the observations described above, rankingbased semantics [1] provide a finegrained assessment of arguments. Here, we follow this line of work and make some first steps towards the use of conditional logic and the System Z inference mechanism [8] to define rankings between arguments. Conditional logic is a general nonmonotonic representation formalism that focuses on default rules of the form “if A then B” and there exist some interesting relationships between this formalism and that of formal argumentation [13, 9]. We make use of these relationships here for the purpose of defining a novel rankingbased semantics for abstract argumentation.
The rest of this work is organized as follows: In Section 2 all necessary preliminaries will be stated. Then we discuss our ranking idea in Section 3 and with Section 4 we conclude this paper.
2 Background
In the following, we want to briefly recall some general preliminaries on argumentation frameworks and conditional logics.
2.1 Abstract Argumentation Frameworks
In this work, we use argumentation frameworks first introduced in [7]. An argumentation framework is a pair , where is a finite set of arguments and is a set of attacks between arguments with . An argument is said to attack if . We call an argument acceptable with respect to a set if for each attacker of this argument with , there is an argument which attacks , i. e., ; we then say that is defended by . An argumentation framework can be illustrated by a directed graph with vertex set and edge set .
Example .
Let with and be an argumentation framework. The corresponding graph is shown in Figure 1. Argument is not acceptable with respect to any set of arguments, as is not defended against ’s attack. On the other hand, is acceptable with respect to , as defends against ’s attack and defends itself against ’s attack.
Up to this point the arguments can only have the two statuses of accepted or not accepted^{1}^{1}1However, using labelingbased semantics we can generate a threevalued model [18]., but we want to have a more finegraded comparison between arguments. For this we use the idea of rankingbased semantics [1, 6].
Definition (Rankingbased semantics).
A rankingbased semantics associates to any argumentation framework a preorder on . means that is at least as acceptable as . With we describe that and are equally acceptable, i. e., and . Finally we say is strictly more acceptable than , denoted by , if and not . We denote by the ranking on returned by .
2.2 Conditional Logics
We use a set of atoms and connectives (and), (or), and (negation) to generate the propositional language . is an interpretation (or possible world) for when . We denote the set of all interpretations as . An interpretation satisfies an atom (), if and only if . The relation is extended to arbitrary formulas in the usual way. We will abbreviate an interpretation with its complete conjunction, i. e., if are the atoms that are assigned true by and are the ones assigned with false, will be identified with . For we define if and only if for every . With we define the set of models for a set of formulas . A conditional is a structure of the form and represents a rule “If than (usually) ”.
We can consider conditionals as generalized indicator functions [5] for possible worlds as follows:
(1) 
where stands for unknown. Informally speaking, a world verifies a conditional iff it satisfies both antecedent and conclusion ; it falsifies iff is satisfies the antecedence but not the conclusion ; otherwise the conditional is not applicable . A conditional is satisfied by if it does not falsify it.
Semantics are given to sets of conditionals via ranking functions [8, 17]. With a ranking function, also called ordinal conditional function (OCF), we can express the degree of plausibility of possible worlds . With the help of OCFs we can express the acceptance of conditionals and nonmonotonic inferences, so is accepted by iff . With we denote the most plausible worlds.
As there are an infinite number of ranking functions that accept a given set of conditionals, we consider System Z [8] as an inference relation, which yields us a uniquely defined ranking function for reasoning.
Definition (System Z).
is tolerated by a finite set of conditionals if there is a possible world with and for all . The Zpartition of is defined as:


is the Zpartition of
For : iff and is the Zpartitioning of .
We define a ranking function as , with . Finally if and only if .
Example .
Let . For this set of conditionals, with and therefore we have the values from Table 1.
u  u  u  1  0  
u  u  u  0  u  
u  u  u  1  0  
u  u  u  0  u  
1  u  u  1  0  
1  u  u  0  u  
1  u  u  1  0  
1  u  u  0  u  
u  1  u  1  0  
u  1  u  0  u  
u  1  u  1  0  
u  1  u  0  u  
0  0  u  1  1  
0  0  1  0  u  
0  0  u  1  1  
0  0  0  0  u 
So we can derive and .
3 Rankingbased Semantics with Conditional Logic Semantics
In this work we want to extend previous works [9, 13] to not only combine abstract argumentation and conditional logics, but also present ideas to rank arguments using this combination.
The general idea is to represent an abstract argumentation framework as a set of conditionals, using System Z in order to determine a ranking function that accepts these conditionals, and then extract rankings on arguments from this ranking function. First, we need a translation from an argumentation framework to a set of conditionals. It is clear, that for an argument to be acceptable every attacker has to be not acceptable. With this idea we can construct the conditional logic knowledge base. Let be an argumentation framework and , where is the set of conditional knowledge bases over the propositional language generated by .
(2) 
In other words, models that an argument is accepted if all its attackers are not accepted.
We can use inference systems like System Z on these conditional knowledge bases to generate a ranking over the possible worlds. Based on this ranking we want to rank the arguments. Our first idea is to count the number of occurrences of a positive literal in the set of worlds and then rank the corresponding arguments based on this number. So if an argument has a higher count then an argument , we say . This simple idea yields a clear and uniquely defined ranking, while not needing a complex algorithm to be computed.
Definition .
Let be an argumentation framework translated with help of and an inference system to the set of worlds . Define
(3) 
We can then use this counting function for our rankingbased semantics.
Definition (Conditionalcountingbased semantics).
The Conditionalcountingbased semantics (Ccbs) associates to any argumentation framework a ranking on such that with respect to a translation and a ranking function .
Example .
Let with and be an argumentation framework. The corresponding graph can be found in Figure 2. Using Equation 2 we obtain . With we have . Now we can count the number of occurrences of each argument. So , , and . This results in .
Looking at the graph we see, that argument is unattacked, so it is intuitive that this argument is ranked at the highest position. Also the arguments and are attacking each other and are not attacked by any other argument. These two arguments are there indistinguishable and should be ranked on the same level, but both arguments have at least one attacker so it should be ranking lower then . Argument is attacked by three other arguments and defended by none, hence this argument should be ranked lower then its attackers.
For some further ideas of other translations we recommend [9]. Instead of System Z we could also use crepresentations [11].
Rankingbased semantics are usually evaluated wrt. a series of rationality postulates [1, 6]. In this work, we provide some first steps in this direction and look at four simple ones, namely Abstraction [1], Independence [14], Void Precedence [14, 1] and SelfContradiction [14]. With the property of Abstraction we can ensure, that a ranking over arguments only depends on the attacks between arguments and not on the identity of the arguments.
Definition .
An isomorphism between two argumentation framework and is a bijective function such that if and only if . With a slight abuse of notation, we will note .
Definition (Abstraction).
A rankingbased semantics satisfies Abstraction iff for any AF, AF’, for every isomorphism such that , we have iff .
It is natural, that arguments from two different disconnected subgraphs should have no influence on each other for a ranking. A ranking, which satisfies Independence, ensures this idea.
Definition .
The connected components of an argumentation framework are the set of largest subgraphs of , denoted by , where two arguments are in the same component of iff there exists some path between them.
Definition (Independence).
A rankingbased semantics satisfies Independence iff for any argumentation framework such that , , iff .
Proposition .
Ccbs satisfies Abstraction and Independence.
Proof.
For Abstraction we can see, that using an isomorphism does not change the structure of an argumentation framework nor does it change relationships between arguments. Ccbs does not change if an argument is renamed to as long as incoming and outgoing attacks are still the same.
Adding independent arguments to an argumentation framework does not change the ranking between two arguments. Given an argumentation framework with two arguments and . If we add an argument to this to create , at most we would change the score of and by two, but this change takes place for both. So if , then it holds that . ❑
The idea of Void Precedence states that a nonattacked argument should be strictly more acceptable than an attacked argument.
Definition (Void Precedence).
A rankingbased semantics satisfies Void Precedence if and only if for any and , if and with , then .
On the contrary, a selfattacking argument should always be ranked worse than any other argument, because these arguments are contradicting themselves. This is handled with the property SelfContradiction.
Definition (SelfContradiction).
A rankingbased semantics satisfies SelfContradiction if and only if for any and , if and then .
Proposition .
Ccbs does not satisfy Void Precedence nor SelfContradiction.
Proof.
To prove this we look at the following example. Let with and be an argumentation framework. Using Equation 2 we obtain . With we have and , . This results in . Therefore is not strictly less acceptable then . So Ccbs does not satisfy Void Precedence nor SelfContradiction. ❑
Hence this semantics has a few shortcomings, we propose an extension. Before we count the occurrences we rank every argument with an selfattack at the lowest possible position.
Definition .
Ccbs’ associates to any argumentation framework a ranking on such that with respect to a translation and a ranking function .
When we evaluate this ranking semantics it is easy to see that Ccbs’ satisfies SelfContradiction (we omit the proof).
Proposition .
Ccbs’ satisfies Void Precedence.
So with this small extension we now satisfy an additional two very intuitive properties for rankingbased semantics. We leave an investigation of further properties for future work.
4 Conclusion
In this work we have presented a first idea to rank arguments with conditional logics. For this we first looked at a simple translation from an argumentation framework to conditional logic and applied an inference relation. Using a simple counting idea results in a ranking over arguments.
Although this semantics does not satisfy two desired properties, with a small extension we have shown that these two properties are satisfied. Also we have established a simple connection between ranking arguments and conditional logic. In the future we can improve this idea and hopefully present a rankingbased semantics, which satisfies a good number of properties presented in [6].
Another future work approach is to look at other frameworks like ADFs presented in [4], which uses an acceptance function for every argument. This could prove to be helpful in finding a ranking with conditional logic.
[12] used a similar idea to rank arguments from a Defeasible Logic Programming
(DeLP), a system, which combines logics programming with defeasible argumentation. They used System Z to identify “good” arguments.
Acknowledgements
The research reported here was supported by the Deutsche Forschungsgemeinschaft under grant KE 1413/111.
References
 [1] L. Amgoud and J. BenNaim, ‘Rankingbased semantics for argumentation frameworks’, in Proceedings of the 7th International Conference on Scalable Uncertainty Management (SUM’13), pp. 134–147, (2013).
 [2] K. Atkinson, P. Baroni, M. Giacomin, A. Hunter, H. Prakken, C. Reed, G.R. Simari, M. Thimm, and S. Villata, ‘Toward artificial argumentation’, AI Magazine, 38(3), 25–36, (October 2017).
 [3] Handbook of Formal Argumentation, eds., P. Baroni, D. Gabbay, M. Giacomin, and L. van der Torre, College Publications, 2018.

[4]
G. Brewka, S. Ellmauthaler, H. Strass, J. Wallner, and S. Woltran, ‘Abstract
dialectical frameworks revisited’, in
Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI’13)
, pp. 803–809, (2013). 
[5]
B. De Finetti,
Theory of probability: A critical introductory treatment
, volume 6, John Wiley & Sons, 2017.  [6] Jérôme Delobelle, Rankingbased Semantics for Abstract Argumentation, Ph.D. dissertation, Artois University, 2017.
 [7] P. Dung, ‘On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and person games’, Artificial Intelligence, 77(2), 321–357, (1995).
 [8] M. Goldszmidt and J. Pearl, ‘Qualitative probabilities for default reasoning, belief revision, and causal modeling’, Artificial Intelligence, 84(12), 57–112, (1996).
 [9] J. Heyninck, G. KernIsberner, and M. Thimm, ‘On the correspondence between abstract dialectical frameworks and nonmonotonic conditional logics’, in Proceedings of the 33rd International Florida Artificial Intelligence Research Society Conference (FLAIRS33), (2020).
 [10] A. Hunter, S. Polberg, and M. Thimm, ‘Epistemic graphs for representing and reasoning with positive and negative influences of arguments’, Artificial Intelligence, 281, 103236, (April 2020).
 [11] G. KernIsberner, Conditionals in nonmonotonic reasoning and belief revision: considering conditionals as agents, Springer, 2001.
 [12] G. KernIsberner and G.R. Simari, ‘A default logical semantics for defeasible argumentation’, in TwentyFourth International FLAIRS Conference, (2011).
 [13] G. KernIsberner and M. Thimm, ‘Towards conditional logic semantics for abstract dialectical frameworks’, in Argumentationbased Proofs of Endearment  Essays in Honor of Guillermo R. Simari on the Occasion of his 70th Birthday, College Publications, (2018).
 [14] P. Matt and F. Toni, ‘A gametheoretic measure of argument strength for abstract argumentation’, in Proceedings of the 11th European Conference on Logics in Artificial Intelligence, pp. 285–297. Springer, (September 2008).
 [15] S. Polberg and A. Hunter, ‘Empirical evaluation of abstract argumentation: Supporting the need for bipolar and probabilistic approaches’, International Journal of Approximate Reasoning, 93, 487–543, (2018).
 [16] I. Rahwan, M. I. Madakkatel, J. F. Bonnefon, R. N. Awan, and S. Abdallah, ‘Behavioural experiments for assessing the abstract argumentation semantics of reinstatement’, Cognitive Science, 34(8), 1483–1502, (2010).
 [17] W. Spohn, ‘Ordinal conditional functions: a dynamic theory of epistemic states’, in Causation in Decision, Belief Change, and Statistics, 105–134, Kluwer, (1988).
 [18] Y. Wu, M. Caminada, and M. Podlaszewski, ‘A labellingbased justification status of arguments’, Studies in Logic, 3(4), 12–29, (2010).
Comments
There are no comments yet.