1 Introduction
In this paper we investigate conditions under which the nonmonotonic consequence relation of a given structured argumentation system is robust when irrelevant information is added or removed. Relevance can hereby be understood in two ways. First, syntactically as information that shares propositional variables with the information at hand. Second, semantically, as information that for some reason should not be considered to have defeating power over previously accepted arguments.
Structured argumentation has been studied in various settings such as ASPIC [19, 20], ABA [7, 21], and logicbased argumentation [2, 5, 6]. These frameworks share the underlying idea that arguments are to have a logical structure and attacks between them are at least partially determined by logical considerations. Although investigations into translations between these frameworks have been intensified recently [14], the frameworks are in various aspects difficult to compare and results obtained in one do not easily transfer to others. For this reason, we decided in this paper to study relevancerelated properties for structured argumentation on the basis of a simple framework for structured argumentation that allows us, on the one hand, to abstract away from particularities of the systems from the literature and, on the other hand, to translate these frameworks easily. The framework is simple in that arguments are premiseconclusion pairs obtained from a given consequence relation and it only allows for one type of attack (attacks in premises). The obtained simplicity makes studying metatheory technically straightforward and the availability of the translations makes results easily transferable.
The paper is structured as follows. In Section 2 we introduce our general setting for structured argumentation. In Section 3 we define the basic relevancerelated properties that we will investigate in this paper. In Section 4 we show how many of the most common systems of structured argumentation can be represented in our setting. In Section 5 we prove our main results. We conclude in Section 6.
2 General Setting
In the following we work with a simple setting for structured argumentation. It is abstract in the sense that it allows for instantiations that are adequate representations of many of the available systems of structured argumentation such as logicbased argumentation, ASPIC, ABA, etc. (see Sec. 4). In this contribution we restrict ourselves to nonprioritized settings.
We suppose to have available a formal language (we denote the set of wellformed formulas over also by ) and a relation (where denotes the set of finite subsets) which we will refer to as the deducability relation. We do not suppose any of the usual Tarskian properties in what follows (reflexivity, transitivity, and monotonicity).
[] Given a set of formulas we denote by the set of based arguments: iff for . Given , and .
To accommodate argumentative attacks we suppose to have two functions: a contrariness function that associates each formula with a set of conflicting formulas and a function that associates support sets with sets of formulas in which they can be attacked.
Often will simply be the identity function, although another option is, e.g., .
[] An (argumentation) setting is a triple . A setting based on is given by the quadruple .
A simple example of a setting is where is the deducability relation of classical propositional logic and .
Another example is the setting where and .
[Attacks] Given a setting , where and , attacks (in ) iff there is a for which . Our attack form is sometimes called premiseattack [20] or directed undercut [6]. In Section 4 we will show that by adjusting and adequately we are able to accommodate many other attack forms defined in the literature.
[Attack Diagram] Given a setting , its attack diagram is the directed graph with the set of nodes and edges between and iff attacks .
[Dung Semantics, [12]] Where is a setting and we define: is conflictfree iff there are no such that attacks . defends iff for each attacker of there is a that attacks . is admissible iff it is conflictfree and it defends every . is complete iff it is admissible and it contains every it defends. is preferred iff it is maximal complete. is grounded iff it is minimal complete. is stable iff it is admissible and for all there is a that attacks .
We denote the set of all admissible [complete, preferred, stable] sets (also called “extensions”) by [] and the grounded set by .
[Consequence Relations] Where , and given a setting we define: iff for all there is an with . Where the setting is clear from the context we will simply write to avoid clutter.
For reasons of space we restrict our focus in this paper on skeptical consequence as defined in Definition 2. Note that coincides with .
3 The Relevance Properties
3.1 Syntactic Relevance
A syntactical relevance property that has been proposed in the context of structured argumentation is noninterference [9]. Let us call two sets of formulas syntactically disjoint if no atom that occurs in a formula in also occurs in a formula in and vice versa: so where is the set of atoms occurring in formulas in . In such cases we write: .
[NonInterference, [9]] satisfies NonInterference iff for all for which we have:^{1}^{1}1A similar property is Basic Relevance [3, Definition 3.1].
[Contamination, [9]] Let be a consequence relation. A set , such that , is called contaminating (with respect to ), if for any set of formulas such that and for every , it holds that if and only if .
Consequence relations that are nontrivial and satisfy NonInterference also satisfy CrashResistance:^{2}^{2}2 is nontrivial if there are always two sets of formulas with the same atoms but different conclusions (see [9]).
[CrashResistance, [9]] A consequence relation satisfies CrashResistance iff there is no set that is contaminating with respect to .
Given a setting , a natural question is whether NonInterference is a property that gets inherited on the level of nonmonotonic inference from : we will show below that in case satisfies NonInterference so does . In fact, the following less requiring criterion is sufficient:
[PreRelevance] satisfies PreRelevance iff for all for which : if then there is a such that .
When considering attacks we need to extend the notion of PreRelevance by taking into account and . We first define:
[Prime settings]
A setting is prime iff for all sets of atoms and in for which , for all for which and , and for all and for which and , we have:
if then there are , , and for which .
[PreRelevant Settings] A setting is PreRelevant iff (i) is PreRelevant, (ii) is prime, and (iii) is monotonic (i.e., for all ).
Where (see Ex. 2) and , the PreRelevance of follows from the PreRelevance of .
Proof.
Items (i) and (iii) are trivial. For Item (ii) suppose that , where and and where are as in Def. 3.1. Thus, there is an s.t. . Thus, . By the PreRelevance of , there is an for which . ∎
Where (see Ex. 2), and is contrapositable (i.e., implies ), the PreRelevance of follows from the PreRelevance of .
In Section 5.1 we will show that: If satisfies PreRelevance then satisfies NonInterference for each .
We take the setting , where is the consequence relation of the semirelevance logic and . satisfies PreRelevance (see [4, Prop. 6.5]) and thus satisfies NonInference and CrashResistance. Similar for other relevance logics.
Although does not satisfy PreRelevance, does, where is the restriction of to pairs for which . Hence, where satisfies NonInterference. In [22] such a restriction is applied in the context of ASPIC.
Recently paraconsistent logics based on maximal consistent subsets [13] have been used in the context of structured argumentation. Let [] iff for all [some] maximal consistent subsets of , . ( is a maximal consistent subset of if it is consistent and there are no consistent such that .) Such consequence relations satisfy PreRelevance and thus, argumentative settings based on them satisfy NonInterference.
Given a setting let be the restriction of to pairs for which there is no such that for some .
Since arguments with empty supports have no attackers we have: Where and ,
If satisfies PreRelevance then satisfies NonInterference for each .
We illustrate the latter point with an example.
Also the setting in Ex. 2 satisfies NonInterference. Note for this that (where the latter is defined as in Ex. 3.1) in the context of .
In the following sections we will relate these results to systems of structured argumentation from the literature.
3.2 Semantic Relevance
As for semantic relevance we study in this paper a criterion known from nonmonotonic logic, namely Cumulativity.
Given and , let be the transitive closure of . Given a setting and a semantics let be an abbreviation of and for .
On the level of consequence relations Cumulativity is the following property, intuitively expressing that the consequence set is invariant under the addition of derivable formulas to the premises: [Cumulativity] A setting satisfies Cumulativity for , iff, for all such that we have: .
On the level of Dungextensions, Cumulativity is: [Extensional Cumulativity] A setting satisfies Extensional Cumulativity for if and only if for all such that we have that:
We will show, in Section 5.2, that a setting satisfies Cumulativity for grounded semantics if is pointed:
[Pointed Settings] is pointed iff

for all , (in this case we say that is pointed), and

satisfies Cut w.r.t. for any , i.e., for every , if and .
Where is pointed, satisfies Cumulativity and Extensional Cumulativity for grounded semantics.
Any setting is pointed iff satisfies Cut. For instance, each of the consequence relations in Examples 2 and 3.1 satisfies Cut and thus the corresponding settings are pointed and therefore satisfy Cumulativity.
If we restrict to consistent sets on the left side, denoted by (see Def. 3.2 below) and if satisfies Cut and Contraposition (see Def. 3.2 below), then the setting is cumulative. In more detail:
is contrapositable iff for all , if where then for all , for some . By extension we call contrapositable if is contrapositable.
Where , a set is inconsistent iff there is a and a for which where . is consistent iff it is not inconsistent.
Given , let .
Where is contrapositable and satisfies Cut, is cumulative and extensionally cumulative for .
4 Systems of Structured Argumentation
In this section we take a look at several of the structured argumentation frameworks from the literature and show how they can be represented in our setting.
[LogicBased Argumentation] Logicbased argumentation is closest to our setting from Section 2. Systems can be found in, for instance, [2, 5].^{3}^{3}3There are differences between these presentations: while [5, 6] use classical logic as a core logic, [2] allows for any Tarskian logic with an adequate sequent calculus to serve as core logic. [5, 6] require the support sets of arguments to be consistent and minimal while [2] omit this requirement. In what follows we follow the generalized setting of [2]. Consistency and minimality can easily be captured by changing the underlying relation (see e.g., Ex. 3.1). The core logic is a finitary Tarskian logic with an adequate consequence relation . Given a set , the set of arguments defined by consists of all where and just like in Def. 2. Different attack rules have been proposed, such as: attacks iff …

Defeat (Def): for some .

Undercut (Ucut): for some .

Direct Compact Defeat (DiCoDef): for some .

Direct Undercut (DiUcut): there is a s.t. .

Direct Defeat (DiDef): there is a s.t. .
Dung semantics are defined as usual on top of an attack diagram analogous to Definitions 2 and 2. Consequence relations are defined analogous to Definition 2 iff in all extensions there is an argument .
Systems of logicbased argumentation translate rather directly to our setting. We only need to adjust the definitions of and so that we can use our attack definition to simulate the attack definitions above. The following table shows how:
DiCoDef  

Def  
DiDef  
DiUcut  
Ucut 
The easy proof concerning the adequacy of our representations is omitted for reasons of space.
The definitions for direct attack forms (DiDef, DiUcut, DiCoDef) all give rise to a pointed (namely ) in our representation. Thus, combining these attack forms with core logics for which satisfies Cut, we obtain Cumulativity.
Instantiating logicbased argumentation with a core logic that satisfies PreRelevance (such as the ones in Examples 3.1, 3.1, 3.1) we obtain NonInterference.
[AssumptionBased Argumentation (ABA), [7]] Let be a formal language, a contrariness function, a subset of socalled assumptions, and be a set of rules of the form where and .^{4}^{4}4In this paper we restrict ourselves to socalled flat frameworks that satisfy the latter requirement. There is an deduction from some to iff there is a sequence for which , and for each , is either in or there is a rule where . Given two sets of assumptions , attacks iff there is a for which there is an deduction of some from some . Subsets of assumptions in and attacks between them give rise to an attack diagram where nodes are sets of assumptions and arcs are attacks. Dungstyle semantics are applied to these graphs: is conflictfree if it does not attack itself, is admissible if it defends itself, it is complete if it contains all assumptions it defends, it is preferred if it is maximally admissible and stable if it is admissible and attacks every assumption it does not contain. Given a semantics , a consequence relation is given by iff is derivable from all sets of assumptions that satisfy the requirements of .
In most presentations of ABA, the rules are considered domainspecific strict inference rules that are part of a given knowledge base. They may also be obtained from an underlying core logic with consequence relation by setting iff .
We can translate ABA into our setting as follows. Where represents domainspecific rules that are part of the knowledge base, we define for and :

, iff, there is an deduction of from making use of the rules in (and only of these).^{5}^{5}5For this the language underlying the original ABA framework is enriched by so that . This is important to track syntactic relevance.
Where is generated from a given core logic , we define for :

, iff, .
In both cases, we use the definition of from ABA, let . Clearly, in our setting attacks iff for some . For reasons of space we omit the proof that the setting [resp. ] adequately represents the ABA framework based on and for in () [resp. ()] so that iff [resp. ].
It is easy to see that for representation () the underlying consequence relation satisfies PreRelevance and if () for all , we obtain NonInterference. For the representation it depends on the logic . In case satisfies PreRelevance and if () we obtain NonInterference.
Our representation of ABA makes use of the pointed (namely ) and derivability satisfies Cut. Note that [resp. ] adequately represents the ABA framework based on for in () [resp. for in ()]. Thus we obtain Cumulativity. [ASPIC, [19, 20]] In ASPIC we work with a formal language , a contrariness function , a set of defeasible rules and a set of strict rules of the form resp. . Similarly as was the case for ABA, the strict rules may reflect domainspecific knowledge or be generated in view of an underlying core logic . We assume that contains for each defeasible rule a logical atom that serves as name of . An deduction of from is given by a tree

whose leaves are labeled by elements in (so that each occurs as label of a leaf),

for every nonroot node labeled by there is a rule or and its childnodes are labeled by (if has an empty body, the single childnode is unlabeled). The edges connecting the childnodes with the parent are labeled .^{6}^{6}6Usually edges are not labeled with rules in ASPIC (and so in cases of rules with empty bodies, there are usually no childnodes either). We introduce these labels since they enable us to define our representation in a simpler way. We also simplify the presentation in that we do not assume there to be defeasible premises.

the root of the tree is labeled by .
Given a derivation , [] is the set of all node labels to which an edge labeled with a defeasible [strict] rule leads and [] is the set of all edge labels that are defeasible [strict] rules.
An argumentation theory is a triple where is a set of premises, is a set of strict rules and is a set of defeasible rules. The set is the set of all derivations of some from some finite . Given two arguments , rebuts iff there is a such that ; undercuts iff for some . Attack diagrams, underlying Dungsemantics and consequence relations are then defined in the usual way.
To represent ASPIC in our setting we first need to define our derivability relation and then translate the ASPIC attacks. In case the set of strict rules presents domainspecific knowledge we define:

iff there is a derivation of from where .^{7}^{7}7Similar as in the case of ABA we enrich the language for to track syntactic relevance. See Fn. 5.
If is generated via an underlying core logic we define:

iff there is a derivation of from where .
For reasons of space we omit the proof that, where and ,^{8}^{8}8For the variants ASPIC [10] and ASPIC [15] where rebut is unrestricted we need to add to in () and (). For generalized rebut in ASPIC we can proceed analogous to Ex. 2. the setting [resp. ] represents the ASPIC theory for in () [resp. in ()] so that iff [resp. ].
Analogous to Remark 4, if () holds, we obtain NonInterference for the presentation and for if additionally the underlying logic satisfies PreRelevance.
Our representation of ASPIC makes use of the pointed (namely ) and derivability satisfies Cut. Note that [resp. ] adequately represents the ASPIC argumentation theory for in () [resp. for in ()] and as specified in Example 4. Thus we obtain Cumulativity for grounded semantics.
5 MetaTheory
5.1 Syntactic Relevance
In this section we prove Theorem 3.1. In the following we suppose that is a setting that satisfies PreRelevance (see Def. 3.1). We start with some notations:
Where and , we write iff .
Where and , let be the set of all arguments that are defended by arguments in .
In view of the monotonicity of we have: Where , if attacks then attacks .
Complete extensions are closed under : Where , , , and , then if .
Where , if attacks , there is an that attacks .
Proof.
Suppose attacks . Then, for some . Where , , , , and , with Def. 3.1, where , and . Thus, attacks . ∎
Where , , , .
Proof.
Suppose , and . We now show that is admissible.
Conflictfree: Assume for a contradiction that there are such that attacks . By the conflictfreeness of and it is not the case that or . Wlog. suppose and . By Lemma 5.1, there is a that attacks . Thus, is trivially defended by and by the completeness of , . This is a contradiction to the conflictfreeness of .
Where , , and attacks ,

some attacks ;

if , some attacks .
Proof.
Let . Suppose attacks . Thus, there is a s.t. . By Def. 3.1 (ii), there are , , and s.t. . By Def. 3.1 (iii), and hence attacks .
For Item 2 note that when setting and in Def. 3.1. ∎
Where , , and ,

;

.
Proof.
Where , , , ,

and .

.
Proof.
Ad 1. Suppose . Thus, it is defended by
Comments
There are no comments yet.