argumentation, Support, Structured argumentation CEUR Workshop Proceedings
Although abstract argumentation has provided a highly efective way to analyse and evaluate
sets of arguments, end users require a more intuitive interface (see [
We build on ideas raised in [
+ and BAF semantics. We take inspiration from the discussion of types
https://jamumford.github.io (J. Mumford); https://www.csc.liv.ac.uk/~katie/ (K. Atkinson);
nEvelop-O
LGOBE
(T. Bench-Capon)
https://www.csc.liv.ac.uk/~tbc/ (T. Bench-Capon)
of structured and abstract support from [
We take our definitions of the relevant theory from [ 11]. Four types of support are described as relevant in +. An argument is a proper subargument of some argument if consists solely of premises pertaining to and is not equal to . Arguments and conclusion support one another if and are independent and have the same conclusion. An argument premise-supports some argument if the conclusion of is a premise of . An argument intermediate-supports some argument if the conclusion of is not a premise of but is the conclusion of a proper subargument of . The last three represent defences to the standard types of attack: rebuttal, undermining and undercutting respectively.
BAFs present the support relation as distinct from the attack relation, such that the intersection
of the two is the empty set. Graphically, a BAF is depicted as a digraph in which any edge
between any two arguments in a specific direction may be an attack or support but not both.
There are four types of attack which are used to define support in BAFs [
There are three BAF semantics, which are derived from closure properties under the four
types of attack. General support [
The methodology represents support via the use of an attack relation and the explicit invocation of the complements of statements/arguments in accordance with the type of support intended. Whilst our methodology should apply to asymmetric frameworks, here we assume symmetric attacks by default, to represent the symmetry of conflict and that elements are defeasible by default. We regard asymmetric attacks to represent some abstract preference ranking, although we do not discuss the means by which such preferences are determined (such as via valuebased approaches, or argument schemes). We conjecture that the complement-based approach will maintain efective representation regardless of whether any given attack is symmetric or asymmetric, since the presence of conflict is what determines the relevant properties.
The objective is to produce semantics corresponding to the three BAF and the four interpretations of support using only an attack relation. In this paper, we do not present formal proofs of the relevant semantics, but we do present the foundations for this research in terms of appropriate representations and definitions. Firstly, we present in Definition 1 our specification of CoAFs used throughout this paper. Intuitively, one can see that Dung’s AFs present a special case of CoAF in which no complements are expressed in . Formal semantics for CoAFs are not presented in this paper. in
− must also attack and/or be attacked by .
Definition 1. (CoAFs) A complement-based argumentation framework is a pair = ⟨, ⟩ is a finite set of nodes representing arguments/statements, where for each node ∈ its complement may also be in . is a binary attack relation on such that ⊆ × for each node in the set of attackers of is denoted − and the set of nodes attacked by as +. Finally, to preserve non-contradiction, for any node that attacks another node , each node , where
We now move on to the BAF interpretations, Definitions 2 and 3 ofer formal expressions that we conjecture satisfy the closure properties for necessary and suficient support respectively. In essence, support is indicated by an conflict with complements, where is necessary support for if the complement of is in conflict with , and is suficient support for if is in conflict with the complement of .
−
Of course, as already stated, Figure 1 depicts necessary and suficient support with symmetric attacks. However, Definitions
2 and 3 generalise to asymmetric attack relations, in which contrary statements/arguments may have some preference ordering and modus tollens is abandoned. Figures 2 and 3 unpack Figure 1 to illustrate the two cases for each type of support in accordance with Definitions 2 and 3. In both figures, the left graph represents the graphical architecture where the second condition is trivially satisfied ( ∉ − for necessary support, and for suficient support), and the right graph represents where the second condition is + . tionship between and is leveraged, we have closure under secondary and extended attacks, but not under supported and mediated attacks as per the BAF semantics for necessary support. By duality, is suficient support for , and where if a defeat relationship between and is leveraged, we have closure under supported and mediated attacks, but not under secondary and extended attacks as per the BAF semantics for suficient support. meaningfully satisfied. In the right graphs, the thick line attack is analogous to the thick line attack that is added in Figure 1 in order to maintain rationality. Note that in Figures 2 and 3 the symmetry of attack of the thick line is not mandatory; as long as at least one node attacks the other then the definitions are satisfied and the closure properties upheld.
Definition 2. (Necessary support) For any , ∈ , for some argumentation framework (, ) we say that is necessary support for if 1. ∈ − ∪ +; and 2. ∀ ∶ ( ∈ −), ∈ −
⟹ ∈ − ∪ +. secondary and extended attacks. In the right graph, since ∈ − we must have ∈ − ∪ +. Definition 3.
(Suficient support) For any , ∈ , for some argumentation framework (, ) we say that is suficient support for if −
+ 1. ∈ ∪ ; and 2. ∀ ∶ ( ∈ −), ∈
⟹ ∈ − ∪ +.
−
For some argument/statement to be necessary for , then for every labelling in which () = in ⟹ () = in. For some argument/statement to be suficient for , then for every − supported and mediated attacks. In the right graph, since ∈ we must have ∈ − ∪ +. labelling in which () = in ⟹ () =
in. Yet, these specifications are not aligned with an intuitive notion of support as used independently of existing attacks. That is, necessary and suficient support should be expressible in an attack relation, but not be a context-dependent artefact of argumentation dynamics. Let us consider an illustrative example: Example 1. (BAF support) Given statements = ℎ , = it is a plane, = it is not mechanical and = it cannot fly , we can derive a CoAF as in Figure 1. Intuitively is necessary support for , and is suficient support for . Indeed every labelling in which () = in, we have () = have () = in. On the other hand, we can see that in every labelling where () = in, we in. This might imply that is suficient support for , and that is necessary support for . But intuitively we can think of counter examples to this relationship. If we were to add a statement =
it is a bird, then the attack relation would be adjusted with conflict between and , and , leaving a possible labelling in which () = in, () = in and () = out.
Conversely, no additional statements or attacks can be added in a manner coherent with CoAF ̸ semantics such that the necessary and suficient relationships between and are removed such that () = in
in. Hence we distinguish between artefacts of the labellings resulting from incomplete knowledge representations, and genuine necessary and suficient support relations which are evoked by specific interactions in the attack relation, which will hold regardless of any growth of the statement/argument set and the accompanying attack relation.
We suggest that necessary support and suficient support are readily compatible with expression under modified abstract argumentation semantics and will adhere to the rationality postulates from [17]. However, we suggest that the modifications required to express general support are rationally incoherent, indicating problems with the use of general support in practical reasoning. Whilst not formally proven here, one can intuitively see in Figure 1 how closure under secondary and extended attacks are connected graphically, and how closure under supported and mediated attacks are connected. Trying to separate closure under secondary attacks from closure under extended attacks, and closure under supported attacks from closure under mediated attacks, appears to be highly problematic. Formal proofs will need to be forthcoming; it was suggested in [18] that the attack relation was incapable of expressing this notion of support, but this would require confirmation with explicit use of complements. Nonetheless, the trouble with representing general support lends weight to the criticism of the rationality of this type of support that was raised in [11].
The descriptions of necessary and suficient support have been framed as applying to frameworks in which the nodes can be either statements or arguments. However, one might find more application when the nodes are statements and the attack relation can be used to express support in the form of argument structure, as in Example 1. If a CoAF consists of statements as nodes, one can express argument structure and support as defined for + in accordance with Definitions 1, 2 and 3 and their graphical representations in Figures 2 and 3.
Recall that there are four types of support available in an + framework, which are illustrated in Figure 4 as a CoAF. Arguments can be extracted from a CoAF by selecting a starting node to act as the claim, and establishing the remaining structure in accordance with nodes providing suficient support in an iterative manner. In order to represent the structure appropriately, we allow for premises, claims, and collectors to be expressed as nodes. Collectors are nodes that are used to represent rules from premise/s to claim by presenting suficient support for the claim and receiving necessary support from the premise/s. Collectors can represent defeasible rules from a conjunction or single premise, as well as strict rules from a suficient conjunction of premises (strict rules from a single premise do not need a collector node). Strict rules from a conjunction of necessary premises require a strict collector node, and require that every necessary supporter not in the conjunction is suficiently supported by a node in the conjunction (see Examples 2, 3, 4 and 5). Figure 4 is restricted to strict rules in order to more concisely represent the four types of + support, and is illustrated further in Example 2. We will demonstrate use with defeasible rules in Examples 3, 4 and 5 when indicating how the attack relation incorporated the three types of + attack: rebuttal, undercut, and undermine.
1 4 1 1 4 1 1 ∧1 1 2 2 3 2 2 3 4 1 1 2 3 s1 s2 s5 1 1 2 2 a b s3 s4 e 1 1 c d Example 2. ( + support) Given statements 1 = Josh has four oranges, 2 = Josh has six apples, 3 = Josh has ten nectarines, 4 = Josh has two limes, 1 = Josh has citrus fruit, 2 = Josh has at least ten stone fruit and 1 = Josh has at least ten fruit, we can derive a CoAF and extract arguments as in Figure 4. We can provide two examples of argument extraction, for arguments d and c. When a collector node is not involved then the process is simple: for argument d we begin with 1 as the claim, and since 2 is a suficient supporter by itself then we have a strict inference rule from 2 to 1. For argument c, we begin with 1 as the claim, but the suficient supporter ∧1 is a collector, which means we use necessary supporters of ∧1, the premises 1 and 2, which provide a strict inference rule for 1, since 1 is suficiently supported by 1.
We will now showcase how the CoAF approach can express the three types of attack defined for +: rebutting, undercutting, and undermining. Figure 5 provides the arguments used in Figures 6, 7 and 8 and in Examples 3, 4 and 5.
1 2 a 1 s1 1 1 1 2 1
2
Example 3. (Rebutting) Given statements 1 = Murphy is devilishly handsome, 2 = Murphy has missed the date, and 1 = Murphy will have a successful date, we can derive a CoAF and extract arguments in accordance with the top right graph in Figure 6. We regard the rule 2 = 2 ⟹ 1 as strict and so argument b is straightforward to extract. However the rule 1 = 1 ⟹ 1 is regarded as defeasible, which means it must be made explicit in the graph and the collector node ∧1 is added to collect the rule. Argument a’ is extracted by beginning with 1, moving to ∧1 as a suficient b’ supporter, and selecting 1 as a necessary supporter. Since ∧1 has been marked as defeasible, the inference rule 1 = 1 ⟹ 1 must be defeasible. Thus we have extracted arguments a’ and b which attack one another via rebuttal.
Example 4. (Undercutting) Given statements 1 = The weather forecaster says it will rain tomorrow, 3 = Weather forecasters are wrong sometimes, and 1 = It will rain tomorrow, we can derive a CoAF and extract arguments in accordance with the left graph in Figure 7. We regard the rule 1 ∧ 3 ⟹ 1 as strict and so ∧1 is a strict collector. Nonetheless we extract argument a’ by beginning with 1, moving to ∧1 as a suficient supporter, and selecting 1 as a necessary supporter. Since 1 is not the sole necessary supporter of ∧1 and 3 is not suficiently supported by 1, the inference rule 1 = 1 ⟹ 1 must be defeasible. Argument c is extracted by beginning with ∧1 (∧1 is not represented graphically since it has no efect other than symbolic) and moving to the suficient supporter 3 to derive the strict rule 3 = 3 ⟹ 1, since 3 attacks all rules that rely on ∧1 (which is only 1 in this case). Thus we have extracted arguments a’ and c, where c undercuts a’.
1 4 1 4
1 4 1 1 ∧1 1 4 ∧2 1 1 ∧1 1
Example 5. (Undermining) Given statements 1 = A Bordeaux is a vastly superior wine to a Claret, 4 = Bordeaux and Claret are the same, and 1 = I shall order a Bordeaux wine, we can derive a CoAF and extract arguments in accordance with the top right graph in Figure 8. We regard the rule 4 = 4 ⟹ 1 as strict and so argument d is straightforward to extract. Argument a’ is extracted as in Example 3. Thus we have extracted arguments a’ and d, where d undermines a’.
We have presented a means of representing support solely via the use of the attack relation. The various types of support for abstract BAF semantics, and + frameworks, have been examined and we proposed definitions for necessary and suficient support that we conjecture are capable of representing these types of support. We do not, however, capture general support for BAF semantics: we believe that general support as a practical and natural notion of support is problematic and intend to explore this unease further. Several examples were suggested in order to illustrate how one may extract arguments from our CoAFs. This would enable supporting arguments to be used as part of the explanations ofered to users. Next steps will be to formally prove that our definitions of necessary and suficient support fulfill the BAF definitions via closure under the various types of attack. Extending the formal analysis to general support and evidentiary support as CoAFs, would be fruitful research directions.
Being able to incorporate support into the attack relation allows for the calculation of acceptability via an AF and some labelling semantics, which has potential benefits in terms of ease in comparison with, for instance, + which frequently duplicates statements that are expressed in more than one argument, complicating calculation. We also consider that an AF may be easier to integrate with machine learning (ML) models that are commonly graph-based, which would be advantageous for building hybrid ML/argumentation systems. [9] L. Yu, R. Markovich, L. van der Torre, Interpretation of support among arguments, in:
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