for excellent overviews). Nowadays argumentation theory is a vibrant subfield of Artificial Intelligence, covering aspects of knowledge representation, multi-agent systems, and also philosophical questions. Among other approaches which have been proposed for capturing representative patterns of inference in argumentation theory[Besnard et al.2014], Dung’s abstract argumentation frameworks (AFs) [Dung1995] play an important role within this research area. At the heart of Dung’s approach lie the so-called argumentation semantics (cf. [Baroni, Caminada, and Giacomin2011] for an excellent overview). Given an AF , which is set-theoretically just a directed graph encoding arguments and attacks between them, a certain argumentation semantics returns acceptable sets of arguments , so-called -extensions. Each of these sets represents a reasonable position w.r.t. and .
Over the last 20 years a series of abstract argumentation semantics were introduced. The motivations of these semantics range from the desired treatment of specific examples to fulfilling a number of abstract principles. The comparison via abstract criteria of the different semantics available is a topic which emerged quite recently in the community ([Baroni and Giacomin2007b] can be seen as the first paper in this line). Our work takes a further step towards a comprehensive understanding of argumentation semantics. In particular, we study the following question: Do we really need the entire AF to compute a certain argumentation semantics ? In other words, is it possible to unambiguously determine acceptable sets w.r.t. , given only partial information of the underlying framework . In order to solve this problem let us start with the following reflections:
As a matter of fact, one basic requirement of almost all existing semantics111See [Jakobovits and Vermeir1999, Arieli2012, Grossi and Modgil2015] for exemptions. is that of conflict-freeness, i.e. arguments within a reasonable position are not allowed to attack each other. Consequently, knowledge about conflict-free sets is an essential part for computing semantics.
The second step is to ask the following: Which information on top on conflict-free sets has to be added? Imagine the set of conflict-free sets given by . Consequently, there has to be at least one attack between and . Unfortunately, this information is not sufficient to compute any standard semantics (except naive extensions, which are defined as -maximal conflict-free sets) since we know nothing precise about the neighborhood of and . The following three AFs possess exactly the mentioned conflict-free sets, but differ with respect to other
The final step is to try to minimize the added information. That is, which kind of knowledge about the neighborhood is somehow dispensable in the light of computation? Clearly, this will depend on the considered semantics. For instance, in case of stage semantics [Verheij1996], which requests conflict-free sets of maximal range, we do not need any information about incoming attacks. This information can not be omitted in case of admissible-based semantics since incoming attacks require counterattacks.
The above considerations motivate the introduction of so-called verification classes specifying a certain amount of information. In a first step, we study the relation of these classes to each other. We therefore introduce the notion of being more informative capturing the intuition that a certain class can reproduce the information of an other. We present a hierarchy w.r.t. this ordering. The hierarchy contains 15 different verification classes only. This is due to the fact that many syntactically different classes collapse to the same amount of information.
We then formally define the essential property of a semantics being verifiable w.r.t. a certain verification class. We present a general theorem stating that any rational semantics is exactly verifiable w.r.t. one of the different verification classes. Roughly speaking, a semantics is rational if attacks inbetween two self-loops can be omitted without affecting the set of extensions. An important aside hereby is that even the most informative class contains indeed less information than the entire framework by itself.
In this paper we consider a representative set of standard semantics. All of them satisfy rationality and thus, are exactly verifiable w.r.t. a certain class. Since the theorem does not provide an answer to which verification class perfectly matches a certain rational semantics we study this problem one by one for any considered semantics. As a result, only
different classes are essential to classify the considered standard semantics.
In the last part of the paper we study an application of the concept of verifiability. More precisely, we address the question of strong equivalence for semantics lying inbetween known semantics, so-called intermediate semantics. Strong equivalence is the natural counterpart to ordinary equivalence in monotonic theories (see [Oikarinen and Woltran2011, Baumann2016] for abstract argumentation and [Maher1986, Lifschitz, Pearce, and Valverde2001, Turner2004, Truszczynski2006] for other nonmonotonic theories). We provide characterization theorems relying on the notion of verifiability and thus, contributing to a more abstract understanding of the different features argumentation semantics offer. Besides these main results, we also give new characterizations for strong equivalence with respect to naive extensions and strong admissible sets.
An argumentation framework (AF) is a directed graph whose nodes (with being an infinite set of arguments, so-called universe) are interpreted as arguments and whose edges represent conflicts between them. We assume that all AFs possess finitely222Finiteness of AFs is a common assumption in argumentation papers. A systematic study of the infinite case has begun quite recently (cf. [Baumann and Spanring2015] for an overview). many arguments only and denote the collection of all AFs by . If we say that attacks . Alternatively, we write as well as, for some , or if there is some attacked by or attacking , respectively. An argument is defended by a set if for each with , . We define the range of (in ) as . Similarly, we use to denote the anti-range of (in ) as . Furthermore, we say that a set is conflict-free (in ) if there is no argument s.t. . The set of all conflict-free sets of an AF is denoted by . For an AF we use and to refer to and , respectively. Furthermore, we use for the set of all self-defeating arguments. Finally, we introduce the union of AFs and as .
A semantics assigns to each a set where the elements are called -extensions. Numerous semantics are available. Each of them captures different intuitions about how to reason about conflicting knowledge. We consider for admissible, naive, stable, preferred, complete, grounded, semi-stable, stage, ideal, and eager semantics [Dung1995, Caminada, Carnielli, and Dunne2012, Verheij1996, Dung, Mancarella, and Toni2007, Caminada2007].
Given an AF and let .
iff and each is defended by ,
iff and there is no s.t. ,
iff and ,
iff and there is no s.t. ,
iff and for any defended by , ,
iff and there is no s.t. ,
iff and there is no s.t. ,
iff and there is no s.t. ,
iff , and there is no satisfying s.t. ,
iff , and there is no satisfying s.t. .
For two semantics , we use to indicate that for each AF . If we have and for semantics , we say that is --intermediate. Well-known relations between semantics are , meaning, for instance, that ss is stb--intermediate.
We call a semantics rational if self-loop-chains are irrelevant. That is, for every AF it holds that , where .
Indeed, all semantics introduced in Definition 1 are rational. A prominent semantics that is based on conflict-free sets, but is not rational is the cf2-semantics [Baroni, Giacomin, and Guida2005], since here chains of self-loops can have an influence on the SCCs of an AF (see also [Gaggl and Woltran2013]).
2.2 Equivalence and Kernels
The following definition captures the two main notions of equivalence available for non-monotonic formalisms, namely ordinary (or standard) equivalence and strong (or expansion) equivalence. A detailed overview of equivalence notion including their relations to each other can be found in [Baumann and Brewka2013, Baumann and Brewka2015].
Given a semantics . Two AFs and are
standard equivalent w.r.t. () iff ,
expansion equivalent w.r.t. () iff for all AFs :
Expansion equivalence can be decided syntactically via so-called kernels [Oikarinen and Woltran2011]. A kernel is a function mapping each AF to another AF (which we may also denote as ). Consider the following definitions.
Given an AF and a semantics . We define -kernels whereby
We say that a relation is characterizable through kernels if there is a kernel , s.t. iff . Moreover, we say that a semantics is compatible with a kernel if iff . All semantics (except naive semantics) considered in this paper are compatible with one of the four kernels introduced above. In the next section, we will complete these results taking naive semantics and strong admissible sets into account.
3 Complementing Previous Results
In order to provide an exhaustive analysis of intermediate semantics (confer penultimate section) we provide missing kernels for naive semantics as well as strongly admissible sets. We start with the so-called naive kernel characterizing expansion equivalence w.r.t. naive semantics. As an aside, the following kernel is the first one which adds attacks to the former attack relation.
Given an AF . We define the naive kernel whereby
The following example illustrates the definition above.
Consider the AFs and . Note that . Consequently, .
In accordance with Definition 5 we observe that both AFs possess the same naive kernel .
The following theorem proves that possessing the same kernels is necessary as well as sufficient for being strongly equivalent, i.e. .
For all AFs ,,
In [Baumann and Woltran2014] it was already shown that iff jointly and . Consequently, it suffices to prove that implies as well as and vice versa.
() Given . By Definition 5 we immediately have . Assume now that and without loss of generality let . Obviously, for any AF , . Hence, there is an , s.t. . Thus, there are , s.t. . Furthermore, and since for any AF , we obtain . Consequently, we have to consider . Since , we obtain . Since is assumed we derive . By Definition 5 we must have contradicting the conflict-freeness of in .
() We show the contrapositive, i.e. implies or . Observe that for any AF , . Consequently, if , then . Assume now . Without loss of generality let . Since for any AF , we obtain . Furthermore, implies and consequently, for any , . Since we deduce . Hence, and thus, there exists a set , s.t. (compare [Baumann and Spanring2015, Lemma 3]) witnessing . ∎
We turn now to strongly admissible sets (for short, sad) [Baroni and Giacomin2007b]. We will show that, beside grounded [Oikarinen and Woltran2011] and resolution based grounded semantics [Baroni, Dunne, and Giacomin2011, Dvořák et al.2014], strongly admissible sets are characterizable through the grounded kernel. Consider the following self-referential definition taken from [Caminada2014].
Given an AF . A set is strongly admissible, i.e. iff any is defended by a strongly admissible set .
The following properties are needed to prove the characterization theorem. The first two of them are already shown in [Baroni and Giacomin2007a]. The third statement is an immediate consequence of the former.
Given two AFs and , then
if we have: for all , and
The following definition provides us with an alternative criterion for being a strong admissible set. In contrast to the former it allows one to construct strong admissible sets step by step. Thus, a construction method is given.
For the proof we use as a shorthand for in the sense of Definition .
Given . Hence, there is a finite partition, s.t. , and . Observe that for any . Let . Consequently, there is an index , s.t. . Furthermore, since by definition, we deduce that defends . We have to show now that (the smaller set w.r.t. ) . Note that . Since we are dealing with finite AFs we may iterate our construction. Hence, no matter which elements are chosen we end up with a -chain, s.t. and defends for some index , set and element . This means, the question whether can be decided positively by proving . Since the empty set does not contain any elements we find concluding .
Given , consider the following sets : , , , …, . Since we are dealing with finite AFs there has to be a natural , s.t. . Consider now the union of these sets, i.e. . We show now that and . By construction we have . Moreover, . This can be seen as follows. By definition . This means, . Since contains all elements defended by we obtain . Obviously, . In order to derive a contradiction we suppose . This means there is a nonempty set , s.t. . Let . Observe that no element is defended by (*). Since we obtain a set , s.t. and defends . We now iterate this procedure ending up with a set , s.t. and defends contradicting (*) and concluding the proof. ∎
The following example shows how to use the new construction method.
Consider the following AF .
We have . Hence, for all , . Furthermore, , and . This means, additionally . Finally, justifying the last missing set .
The following corollary is an immediate consequence of Definition 7. It is essential to prove the characterization theorem for strongly admissible sets.
Given an AF and two sets . If defends , then is strong admissible if is.
The following lemma shows that the grounded kernel is insensitive w.r.t. strong admissible sets.
For any AF , .
The grounded kernel is node- and loop-preserving, i.e. and . Furthermore, and as shown in [Oikarinen and Woltran2011, Lemma 6].
() Given . The proof is by induction on indicating the number of sets forming a suitable (according to Definition 7) partition of . Let . In consideration of the grounded kernel we observe , i.e. the set of unattacked arguments does not change. Since is assumed we are done. Assume now that the assertion is proven for any -partition. Let be a -partition, i.e. . According to induction hypothesis as well as Corollary 1 it suffices to prove defends in . Assume not, i.e. there are arguments , s.t. and for all , (*). Since defends in we deduce the existence of an argument s.t. . Thus, is redundant w.r.t. the grounded kernel. According to Definition 4 and due to the conflict-freeness of we have and . Consequently, . Since is a strong admissible -partition in we obtain by induction hypothesis that is strong admissible in and therefore, admissible in (Proposition 1). Hence there has to be an argument , s.t. , contradicting (*).
() Assume . We show by induction on indicating that is a -partition in . Due to the base case is immediately clear. For the induction step let be a -partition, i.e. . By induction hypothesis we may assume that is strongly admissible in . Using Corollary 1 it suffices to prove defends in . Assume not, i.e. there are arguments , s.t. and for all , . We even have since . Consequently, has to be deleted in . Definition 4 requires contradicting the conflict-freeness of in . ∎
For any two AFs and we have,
() Given . Since expansion equivalence is a congruence w.r.t. we obtain for any AF . Consequently, . Due to Lemma 1 we deduce , concluding the proof. ∎
In this section we study the question whether we really need the entire AF to compute the extensions of a given semantics. Let us consider naive semantics. Obviously, in order to determine naive extensions it suffices to know all conflict-free sets. Conversely, knowing only does not allow to reconstruct unambiguously. This means, knowledge about is indeed less information than the entire AF by itself. In fact, most of the existing semantics do not need information of the entire framework. We will categorize the amount of information by taking the conflict-free sets as a basis and distinguish between different amounts of knowledge about the neighborhood, that is range and anti-range, of these sets.
We call a function () which is expressible via basic set operations only neighborhood function. A neighborhood function induces the verification class mapping each AF to
We coined the term neighborhood function because the induced verification classes apply these functions to the neighborhoods, i.e. range and anti-range of conflict-free sets. The notion of expressible via basic set operations simply means that (in case of ) the expression is in the language generated by the following BNF:
Consequently, in case of , we may distinguish eight set theoretically different neighborhood functions, namely
A verification class encapsulates a certain amount of information about an AF, as the following example illustrates.
Consider the following AF :
Now take, for instance, the verification class induced by , that is , storing information about conflict-free sets together with their associated ranges w.r.t. . It contains the following tuples: , , , and . The verification class induced by contains the same tuples but instead of .
Intuitively, it should be clear that the set suffices to compute stage extensions (i.e., range-maximal conflict-free sets) of . This intuitive understanding of verifiability will be formally specified in Definition 10. Note that a neighborhood function may return -tuples. Consequently, in consideration of the eight listed basic function we obtain (modulo reordering, duplicates, empty set) syntactically different neighborhood functions and therefore the same number of verification classes. As usual, we will denote the -ary combination of basic functions as with .
With the following definition we can put neighborhood functions into relation w.r.t. their information. This will help us to show that actually many of the induced classes collapse to the same amount of information.
Given neighborhood functions and returning -tuples and -tuples, respectively, we say that is more informative than , for short , iff there is a function such that for any two sets of arguments , we have .
We will denote the strict part of by , i.e. iff and . Moreover in case and , we say that represents and vice versa.
We begin by showing that all neighborhood functions are represented in Figure 1. Clearly, each neighborhood function represents itself, i.e. . All neighborhood functions for are are depicted in Figure 1. We turn to . Consider the neighborhood functions , , and , defined as , , and for . Observe that . Hence, we can easily define functions in the spirit of Definition 9 mapping the images of the function to one another: