The cf2 argumentation semantics revisited

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The cf2 argumentation semantics revisited

SARAH ALICE GAGGL and STEFAN WOLTRAN, Institute of Information Systems, Database and Artificial Intelligence Group, Vienna University of Technology, Vienna A-1040, Austria.

E-mail: gaggl@dbai.tuwien.ac.at, woltran@dbai.tuwien.ac.at

Abstract

Abstract argumentation frameworks nowadays provide the most popular formalization of argumentation on a conceptual level.

Numerous semantics for this paradigm have been proposed, whereby thecf2semantics has shown to solve particular problems concerned with odd-length cycles in such frameworks. Due to the complicated definition of this semantics it has somehow been neglected in the literature. In this article, we introduce an alternative characterization of thecf2semantics which, roughly speaking, avoids the recursive computation of subframeworks. This facilitates further investigation steps, like a complete complexity analysis. Furthermore, we show how the notion of strong equivalence can be characterized in terms of thecf2 semantics. In contrast to other semantics, it turns out that for thecf2semantics strong equivalence coincides with syntactical equivalence. We make this particular behaviour more explicit by defining a new property for argumentation semantics, called the succinctness property. If a semanticsσ satisfies the succinctness property, then for every frameworkF, all its attacks contribute to the evaluation of at least one frameworkF^!containingF. We finally characterize strong equivalence also for the stage and the naive semantics. Together with known results these characterizations imply that none of the prominent semantics for abstract argumentation, except thecf2semantics, satisfies the succinctness property.

Keywords: Abstract argumentation, cf2 semantics, succinctness, complexity, strong equivalence.

1 Introduction

Abstract argumentation frameworks (AFs), introduced by Dung [11], represent the most popular approach for formalizing and reasoning over argumentation problems on a conceptual level. Dung already introduced different extension-based semantics (preferred, complete, stable, grounded) for such frameworks. In addition, recent proposals tried to overcome several shortcomings observed for those original semantics. For instance, the semi-stable semantics [8], and likewise the stage semantics [23], handles the problem of the possible non-existence of stable extensions, while the ideal semantics [12] is proposed as a unique-status approach (each AF possesses exactly one extension) less skeptical than the grounded extension.

Another family of semantics, the so-called SCC-recursive semantics has been introduced in [7].

Hereby, a recursive decomposition of the given AF along strongly connected components (SCCs) is necessary to obtain the extensions. Among them, thecf2semantics, first proposed in [3] and later discussed in [7], has been introduced in order to solve particular problems arising for AFs with odd- length cycles. It fulfils several requirements such as the symmetric treatment of odd- and even-length cycles, and ensures that attacks from self-defeating arguments have no influence on the selection of other arguments to be included in an extension. Furthermore, thecf2semantics satisfies most of the evaluation criteria proposed in [4]. Basically, only the admissibility- and reinstatement criteria are violated. This is due to the fact that thecf2semantics explicitly gives up on these conditions when it comes to evaluate SCCs. At this point, it has to be mentioned, however, that the property of admissibility turns out to be crucial when abstract argumentation is employed in certain forms of instantiation-based argumentation, see e.g. [9].

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Due to the quite complicated definition of its semantics, the cf2 approach has been somehow neglected in the literature. For instance, a complete complexity analysis is still missing, although Nieveset al. [19] observed that the decision problem of verifying whether a set of arguments is acf2 extension is polynomial-time computable. However,cf2semantics attracted specific attention lately, for example in [1] it has been used to handle loops in Talmudic Logic.

In another branch of research, attention was directed to the investigation of redundant patterns in AFs. Oikarinen and Woltran [20] identified kernels that eliminate those redundant attacks of AFs and introduced the concept of strong equivalence: two AFs are strongly equivalent wrt. a semanticsσ (i.e. they provide the sameσ-extensions no matter how the two AFs are simultaneously extended), if theirσ-kernels coincide. In [2], the notion of equivalence wrt. stable semantics has been studied also for logic-based argumentation systems. To the best of our knowledge, redundancies for thecf2 semantics have not been studied yet. As we show in this article, thecf2semantics has the interesting feature that strong equivalence coincides with syntactical equivalence. In other words, for thecf2 semantics there are no redundant attacks.

The main contributions of this article are the following.

• To simplify further investigation, we first give an alternative characterization for the cf2 semantics. The original definition of Baroniet al. [7] involves a recursive computation of different sub-frameworks. Our aim here is to shift the need of recursion from generating sub- frameworks to arguments. We show that the required set of arguments can be captured via a fixed-point operator. This allows to characterizecf2semantics using only linear recursion.

• With the alternative characterization at hand, we formally prove the following complexity results. (i) Verifying if a given set is acf2extension is in P; (ii) deciding if an argument is contained in somecf2extension (credulous acceptance) is NP-complete; (iii) deciding if an argument is contained in allcf2extension (skeptical acceptance) is coNP-complete; and (iv) checking whether there exists a non-emptycf2extension is inP.

• As the third main contribution we define a new property for argumentation semantics called thesuccinctness property. As outlined above, a semantics satisfies the succinctness property or ismaximal succinctiff no redundant attacks for this semantics exist. It turns out that the cf2semantics is the only one that is maximal succinct, whereas for other semantics we can reuse results about strong equivalence [20] for an analysis on their succinctness. Our results thus provide a new classification for argumentation semantics, namely in terms of redundant attacks.

Parts of this article have been published in proceedings of conferences [17, 18]. Completely novel material is provided by the complexity analysis as well as by the investigations on the succinctness property.

2 Preliminaries

In this section we introduce the basics of abstract argumentation, the semantics we need for further investigations and some properties of the semantics we are mainly interested in this work, thecf2 semantics.

2.1 Abstract argumentation

The definition of abstract argumentation frameworks and the semantics are based on [11, 23].

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Definition 2.1

An argumentation framework (AF)is a pairF=(A,R), where Ais a finite set of arguments and R⊆A×A. The pair (a,b)∈Rmeans thataattacksb. A setS⊆Aof argumentsdefeats b(inF), if there is ana∈S, such that (a,b)∈R. An argumenta∈AisdefendedbyS⊆A(inF) iff, for eachb∈A, it holds that, if (b,a)∈R, thenSdefeatsb(inF).

The inherent conflicts between the arguments are solved by selecting subsets of arguments, where a semanticsσ assigns a collection of sets of arguments to an AFF. The basic requirement for all semantics is that none of the selected arguments attack each other.

Definition 2.2

LetF=(A,R) be an AF. A setS⊆Ais said to beconflict-free(inF), if there are noa,b∈S, such that (a,b)∈R. We denote the collection of sets that are conflict-free (inF) bycf(F). A setS⊆Ais maximal conflict-free ornaive, ifS∈cf(F) and for eachT∈cf(F),S%⊂T. We denote the collection of all naive sets ofFbynaive(F). For the empty AFF₀=(∅,∅), we setnaive(F0)={∅}.

Beside the naive semantics we will consider the following semantics in this work.

Definition 2.3

LetF=(A,R) be an AF. A setS⊆Ais said to be

• a stable extension (of F), i.e. S∈stable(F), if S∈cf(F) and each a∈A\S is defeated bySinF.

• astageextension (ofF), i.e.S∈stage(F), ifS∈cf(F) and there is noT∈cf(F) withT_R⁺⊃S_R⁺, whereS_R⁺=S∪{b|∃a∈S, s. t. (a,b)∈R}.

• anadmissibleextension, i.e.S∈adm(F) ifS∈cf(F) and eacha∈Sis defended byS.

• apreferredextension, i.e.S∈pref(F) ifS∈adm(F) and for eachT∈adm(F),S%⊂T. We illustrate the different behaviour of the introduced semantics in the following example.

Example 2.4

Consider the AFF=(A,R) withA={a,b,c,d,e,f,g}andR={(a,b), (c,b), (c,d), (d,c), (d,e), (e,f), (f,f), (f,g), (g,e)}as in Figure 1. Then, the above defined semantics yield the following extensions.

• stable(F)=∅;

• naive(F)=stage(F)={{a,d,g},{a,c,e},{a,c,g}};

• adm(F)={{},{a},{c},{d},{a,c},{a,d}};

• pref(F)={{a,c},{a,d}}.

!

2.2 The cf2 semantics

The semantics we are mainly interested in this work is based on a decomposition along the SCCs of an AF. Hence, we require some further formal machinery. A directed graph is calledstrongly connected if there is a path from each vertex in the graph to every other vertex of the graph. BySCCs(F), we denote the set ofstrongly connected componentsof an AFF=(A,R), i.e. sets of vertices of the maximal strongly connected sub-graphs ofF;SCCs(F) is thus a partition ofA. Moreover, for an argumenta∈A, we denote byC_F(a) the component of F wherea occurs in, i.e. the (unique) set

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Figure 1. The argumentation frameworkFfrom Example 2.4.

C∈SCCs(F), such thata∈C. AFsF₁=(A1,R₁) andF₂=(A2,R₂) are calleddisjointifA₁∩A₂=∅. Moreover, the union between (not necessarily disjoint) AFs is defined asF₁∪F₂=(A₁∪A₂,R₁∪R₂).

Example 2.5

We consider the frameworkF=(A,R) withA={a,b,c,d,e,f,g,h,i}and R={(a,b), (b,c), (c,a), (b,d), (b,e), (d,f), (e,f), (f,e), (f,g), (g,h), (h,i), (i,f)}as illustrated in Figure 2.Fhas three SCCs, namelyC₁={a,b,c},C₂={d}andC₃={e,f,g,h,i}. For example, the argumentg belongs toC₃,

thusC_F(g)=C₃. !

It turns out to be convenient to use two different concepts to obtain sub-frameworks of AFs. Let F=(A,R) be an AF andSa set of arguments. Then,F|S=((A∩S),R∩(S×S)) is thesub-framework ofFwrt.Sand we also useF−S=F|A\S. We note the following relation (that we use implicitly later on), for an AFFand setsS,S^!:F|S\S^!=F|S−S^!=(F−S^!)|S. In particular, for an AFF, a component C∈SCCs(F) and a setSwe thus haveF|C\S=F|C−S.

For the frameworkFof Example 2.5 and the setS={f}. Then,F|C₃−S=({e,g,h,i},{(g,h),(h,i)}).

We now give the definition of thecf2semantics that slightly differs from (but is equivalent to) the original definition in [3, 7]. (i) We use some of the notation established above, like the concept of sub-frameworks and the corresponding relations; (ii) D_F(S), as introduced next, replaces the set ‘DF(S,E)’ andF|C−D_F(S) replaces ‘F_↓_UP_F(S,E)’; moreover, the set of undefeated arguments

‘UF(S,E)’ as used in the general schema from [7], is not required here, because the base function for thecf2semantics does not make use of this set. Next, we define the set ofcomponent-defeated argumentsD_F(S), which identifies all arguments that are attacked from a given setSfrom outside their SCC.

Definition 2.6

LetF=(A,R) be an AF andS⊆A. An argumentb∈Aiscomponent-defeated byS (inF), if there exists ana∈S, such that (a,b)∈Randa∈/C_F(b). The set of arguments component-defeated bySin Fis denoted byD_F(S).

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Definition 2.7

LetF=(A,R) be an argumentation framework andSa set of arguments. Then,Sis acf2extension ofF, i.e.S∈cf2(F), iff

• in case|SCCs(F)|=1, thenS∈naive(F),

• otherwise,∀C∈SCCs(F), (S∩C)∈cf2(F|C−D_F(S)).

In words, the recursive definitioncf2(F) is based on a decomposition of the AFF into itsSCCs depending on a given setSof arguments. We illustrate the behaviour of this procedure in the following example.

Example 2.8

Consider the frameworkF from Example 2.5. We want to check whetherS={a,d,e,g,i}is acf2 extension ofF (the arguments of the setS are highlighted in Figure 3). Following Definition 2.7, we first identify the SCCs ofF, henceSCCs(F)={C₁,C₂,C₃}as in Example 2.5. Due to the attack (d,f) andd∈Swe obtainf as the only component-defeated argument, thusD_F(S)={f}. This leads us to the following checks (see also Figure 4 that shows the involved sub-frameworks). Note here that in caseF|Ci−D_F(S)=F|Ciwe only write (S∩C_i)∈cf2(F|Ci).

(1) (S∩C₁)∈cf2(F|C₁):F|C₁consists of a single SCC; hence, we have to check whether (S∩C₁)= {a}∈naive(F|C₁), which indeed holds.

(2) (S∩C₂)∈cf2(F|C2):F|C2consists of a single argumentd(and thus of a single SCC); (S∩C₂)= {d}∈naive(F|C2) thus holds.

(3) (S∩C₃)∈cf2(F|C3−{f}):F|C3−{f}=F|_{e,g,h,i}consists of four SCCs, namelyC₄={e},C₅= {g},C₆={h}andC₇={i}. Hence, we need a second level of recursion forF^!=F|_{e,g,h,i}and S^!=S∩C₃. Note that we haveD_F!(S^!)={h}. The single-argument AFsF^!|C₄=F|_{e},F^!|C₅= F|_{g},F^!|C7=F|_{i} all satisfy (S^!∩C_i)∈naive(F^!|Ci); whileF^!|C6−{h}yields the empty AF.

Therefore, (S^!∩C₆)=∅∈cf2(F|C6−{h}) holds as well.

We thus conclude thatSis acf2extension ofF. Furthercf2extensions ofFare{b,f,h},{b,g,i}and {c,d,e,g,i}. The extensions of the other semantics for this example are as follows:

• stable(F)=∅;

• adm(F)={{},{g,i}};

• pref(F)={{g,i}}.

For the stage semantics we obtain the same result as for thecf2semantics, but this is not the case in

general, as we are going to discuss in the next subsection. !

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Figure 4. Tree of recursive calls for computingcf2(F). from Example 2.5.

Figure 5. The modified AFF^!.

2.3 Properties of the cf2 semantics

Thecf2semantics has some special properties that clearly differ from the admissible based semantics.

Especially the treatment of odd- and even-length cycles is more uniform in the case ofcf2semantics.

For our framework of Example 2.5 we obtain{g,i}as the only preferred extension. This comes due to the fact that in an odd-length cycle, as is the case in this example for the argumentsa,band c, none of these arguments can be defended. Lets modify the framework in the sense that we include a new argumentxwhich makes the cycle even, as illustrated in Figure 5. Then, we obtain totally different preferred extensions, namely {b,x,g,i},{b,x,f,h}and{a,c,d,e,g,i}, which are conform with thecf2extensions of the modified AFF^!. One possible application for thecf2semantics, which makes use of that special behaviour, would be for example that we have three agents, let us call them A,BandC, where agentAdisagrees with agentB,Bdisagrees withCand agentCdisagrees with agentA. Additionally, we have further arguments and attacks as in Figure 2 that are independent from the disagreement of the agents. We would now want to have at least one of the agents to be chosen, which is not possible with the admissible based semantics like preferred. This is exactly what thecf2 semantics does by selecting the maximal conflict-free sets of the SCC{a,b,c}. If now there comes a fourth agent into play, let us call himXlike in Figure 5, the situation of the whole framework does not change that drastically, we just have four in turn of three agents. But now, we obtain for both semantics, the preferred and thecf2semantics, the same results.

One special case of an odd-length cycle is a self-attacking argument.

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Example 2.9

Consider the following AFF:

Then, the empty set is the only preferred extension, whereas{a}is acf2extension. The motivation behind selecting{a}as a reasonable extension is that it is not necessary to defendaagainst the attack

fromb, asbis a self-attacking argument. !

Till now, we only mentioned positive properties of thecf2semantics compared to the admissible based semantics. The next example will show a more questionable behaviour.

Example 2.10 Consider the AFF:

We obtain stage(F)=pref(F)=stable(F)={{a,c,e},{b,d,f}}, but cf2(F)=naive(F)={{a,d}, {b,e},{c,f},{a,c,e},{b,d,f}}. In this example, we have an even-length cycle and thecf2semantics produce three more extensions. This does not really coincide with the motivation for a symmetric treatment of odd- and even-length cycles, as now the results differ significantly for an even-length

cycle. !

One suggestion to repair the undesired behaviour from Example 2.10, could be to check in Definition 2.7 for the case|SCCs(F)|=1 whether S∈stage(F) instead ofS∈naive(F). We leave a formalization of this modification for future work.

The relation between the introduced semantics is illustrated in Figure 6, an arrow from semantics σ to semanticsτ encodes that eachσ-extension is also aτ-extension. The relations between thecf2 semantics and the stable, resp. the naive semantics, are due to [6].

As pointed out in Example 2.8, there is no particular relation between thecf2and the preferred semantics, but the stage and thecf2semantics coincide for this framework. The following examples will show that there is no particular relation between stage andcf2extensions as well.

Example 2.11

Consider the following AFF:

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Figure 6. Relations between Semantics.

Here{a,c}is the only stage extension ofF(it is also stable). Concerning thecf2semantics, note that Fis built from a singleSCC. Thus, thecf2extensions are given by the maximal conflict-free sets of F, which are{a,c}and{a,d}. Thus, we havestage(F)⊂cf2(F).

As an example for a frameworkGsuch thatcf2(G)⊂stage(G), consider the following AF:

ThenGconsists of two SCCs namelyC₁={a}andC₂={b,c}. The conflict-free sets ofGareE₁={a} andE₂={b}. Now it remains to check ifE₁andE₂are alsocf2extensions ofG. First we make the check forE₁. Due to the attack (a,b) anda∈E₁we obtainD_G(E₁)={b}. We have the following two cases:

• (E1∩C₁)∈cf2(G|C1), which holds, since{a}∈naive(G|C1).

• (E1∩C₂)∈cf2(G|C₂−D_G(E1)), which holds, since∅∈naive(G|_{c}).

ForE₂we obtainD_G(E2)=∅. The check (E2∩C₁)∈cf2(G|C1) does not hold, sincenaive(G|C1)={a}.

Hence,E₂is not acf2extension ofG. Thus,cf2(G)={E₁}butstage(G)={E₁,E₂}. !

3 An alternative characterization for the cf2 semantics

In the original definition of the SCC-recursive semantics in [7], the computation is based on checking recursively whether a set of arguments fulfils a base function (depending on the semantics) in a single SCC. Thus, the computation is based on a decomposition of the framework along its SCCs. Our alternative characterization is based on the idea to decompose the framework as well, but differently to the original approach the decomposition is only recursive in terms of a certain set of arguments, for which we provide a fixed-point operator. This modification allows us to avoid the recursive computation of several sub-frameworks. Instead we only compute one, possibly not connected,

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Figure 7. Separation of the AFFfrom Example 2.5.

framework where we eliminate the arguments and corresponding attacks which are, what we call,

‘recursively component defeated’. We start with the following concept.

Definition 3.1

An AF! F=(A,R) is called separated if for each (a,b)∈R, C_F(a)=C_F(b). We define [[F]]=

C∈SCCs(F)F|Cand call[[F]]theseparationofF.

In words, an AF is separated if there are no attacks between different SCCs. Thus, the separation of an AF always yields a separated AF.

The separation of the frameworkFof Example 2.5 is depicted in Figure 7. The following technical lemma will be useful later.

Lemma 3.2

For any AFFand setSof arguments,!

C∈SCCs(F)[[F|C−S]]=[[F−S]].

Proof. We first note that for disjoint AFsFandG,[[F]]∪[[G]]=[[F∪G]]holds. Moreover, for a setSof arguments and arbitrary frameworksFandG, (F−S)∪(G−S)=(F∪G)−Sis clear. Using these observations, we obtain

"

C∈SCCs(F)

[[F|C−S]]=[[ "

C∈SCCs(F)

(F|C−S)]]=[[( "

C∈SCCs(F)

F|C)−S]]=[[[[F]]−S]].

It remains to show that[[[[F]]−S]]=[[F−S]]. Obviously, both AFs possess the same arguments A. Thus, letRbe the attacks of[[[[F]]−S]]andR^!the attacks of[[F−S]].R⊆R^!holds by the fact that each attack in[[F]]is also contained inF. To showR^!⊆R, let (a,b)∈R^!. Thena,b∈/S, and C_F₋_S(a)=C_F₋_S(b). From the latter,C_F(a)=C_F(b) and thus (a,b) is an attack in[[F]]and also in [[F]]−S. Again usingC_F₋_S(a)=C_F₋_S(b), shows (a,b)∈R. ! Next, we define the level of recursiveness a framework shows with respect to a setSof arguments and then the aforementioned set of recursively component defeated arguments (byS) in an AF.

Definition 3.3

For an AFF=(A,R) and a setS of arguments, we recursively define thelevel#_F(S) ofF wrtSas follows:

• if|SCCs(F)|=1 then#_F(S)=1;

• otherwise,#_F(S)=1+max({#_F_|_C₋_D_F_(S)(S∩C)|C∈SCCs(F)}).

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For our running example we obtain the level#_F(S) wrt the setS={a,d,e,g,i}as follows.#_F(S)= 1+max({#_F_|_C₋_D_F_(S)(S∩C)|C∈SCCs(F)}), where D_F(S)={f} and SCCs(F)={C₁,C₂,C₃} with C₁={a,b,c},C₂={d}andC₃={e,f,g,h,i}. This leads to the following recursive calls:

• #_F_|_C

1(S∩C₁)=1,

• #_F_|_C

2(S∩C₂)=1,

• #_F!(S^!)=1+max({#_F!|C!−D_F!(S!)(S^!∩C^!)|C^!∈SCCs(F^!)}). WhereF^!=F|C3−D_F(S),S^!=S∩ C₃={e,g,i}andD_F!(S^!)={h}, furthermoreSCCs(F^!)={C₄,C₅,C₆,C₇}withC₄={e},C₅= {g},C₆={h}andC₇={i}. As all those SCCs ofF^!are single SCCs, we obtain in each recursive call level 1.

To sum up the level ofFwrtSis#_F(S)=3. One can compare the tree of recursive calls in Figure 3 with the computation of#_F(S). When theheight hof a tree is the length of the path from the root to the deepest node in the tree, we denote the height of the computation tree for thecf2semantics for an AFFwrtSash_F(S), then#_F(S)=h_F(S)+1.

Definition 3.4

LetF=(A,R) be anAF andSa set of arguments. We define the set of argumentsrecursively component defeatedbyS(inF) as follows:

• if|SCCs(F)|=1 thenRD^F(S)=∅;

• otherwise,RD^F(S)=D_F(S)∪!

C∈SCCs(F)RDF|C−DF(S)(S∩C).

We are now prepared to give our first alternative characterization, which establishes acf2extension Sof a given AFFby checking whetherSis maximal conflict-free in a certain separated framework constructed fromFusingS.

Lemma 3.5

LetF=(A,R) be an AF andSbe a set of arguments. Then,

S∈cf2(F) iffS∈naive([[F−RDF(S)]]).

Proof. We show the claim by induction over#_F(S).

Induction base. For #_F(S)=1, we have|SCCs(F)|=1. By definitionRD^F(S)=∅and we have [[F−RDF(S)]]=[[F]]=F. Thus, the assertion states thatS∈cf2(F) iffS∈naive(F), which matches the original definition for the cf2 semantics in case the AF has a single strongly connected component.

Induction step. Let #_F(S)=n and assume the assertion holds for all AFs F^! and setsS^! with

#_F!(S^!)<n. In particular, we have by definition that, for eachC∈SCCs(F),#_F_|_C₋_D_F_(S)(S∩C)<n.

By the induction hypothesis, we thus obtain that, for eachC∈SCCs(F), the following holds:

(S∩C)∈cf2(F|C−D_F(S)) iff (S∩C)∈naive#

[[(F|C−D_F(S))−R^!F,C,S]]$

(3.1)

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whereR^!F,C,S=RDF|C−D_F(S)(S∩C). Let us fix now aC∈SCCs(F). Since for each further C^!∈ SCCs(F) (i.e.C%=C^!), no argument fromRD^F|C!−D_F(S)(S∩C^!) occurs inF|C, we have

(F|C−D_F(S))−R^!F,C,S=

#(F|C−D_F(S))−R^!F,C,S

$− "

C^!∈SCCs(F);C%=C^!

RDF|C!−DF(S)(S∩C^!)=

#F|C−D_F(S)$

− "

C∈SCCs(F)

RD^F|C−D_F(S)(S∩C)=

F|C−#

DF(S)∪ "

C∈SCCs(F)

RDF|^C−DF(S)(S∩C)$

=F|C−RDF(S).

Thus, for anyC∈SCCs(F), relation (3.1) amounts to

(S∩C)∈cf2(F|C−DF(S)) iff (S∩C)∈naive%

[[F|C−RDF(S)]]&

. (3.2)

We now prove the assertion. Let S∈cf2(F). By definition, for each C∈SCCs(F), (S∩C)∈ cf2(F|C−D_F(S)). Using (3.2), we get that for eachC∈SCCs(F), (S∩C)∈naive([[F|C−RDF(S)]]).

By the definition of components and the semantics of being maximal conflict-free, the following relation thus follows:

"

C∈SCCs(F)

(S∩C)∈naive# "

C∈SCCs(F)

[[F|C−RDF(S)]]$ .

SinceS=!

C∈SCCs(F)(S∩C) and, by Lemma 3.2,!

C∈SCCs(F)[[F|C−RDF(S)]]=[[F−RDF(S)]], we arrive atS∈naive([[F−RDF(S)]]) as desired. The other direction is by essentially the same

arguments. !

Next, we provide an alternative characterization forRDF(S) via a fixed-point operator. In other words, this yields a linearization in the recursive computation of this set. To this end, we require a parametrized notion of reachability.

Definition 3.6

LetF=(A,R) be an AF, argumentsa,b∈AandB⊆A. We say thatbisreachableinFfroma modulo B, in symbolsa⇒^B_Fb, if there exists a path fromatobinF|B, i.e. there exists a sequencec₁,...,c_n (n>1) of arguments such thatc₁=a,c_n=b, and (ci,c_i₊₁)∈R∩(B×B), for alliwith 1≤i<n.

Definition 3.7

For an AFF=(A,R),D⊆A, and a setSof arguments,

$_F,S(D)={a∈A|∃b∈S:b%=a,(b,a)∈R,a%⇒^A_F^\^Db}.

The operator is clearly monotonic, i.e.$F,S(D)⊆$F,S(D^!) holds forD⊆D^!. As usual, we let$⁰_F,S=

$_F,S(∅) and, fori>0,$ⁱ_F,S=$($ⁱ_F,S⁻¹). Due to monotonicity the least fixed-point (lfp) of the operator exists and, with slight abuse of the notation, will be denoted as$F,S. The$F,Soperator applied to the empty-set computes recursively the arguments that are defeated from outside their component.

Hence, it also takes into account that the SCCs of the framework may change during the computation.

We need two more lemmata before showing that$_F,ScapturesRD^F(S).

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Lemma 3.8

For any AFF=(A,R) and any setS⊆A,$⁰_F,S=D_F(S).

Proof. We have$⁰_F,S=$_F,S(∅)={a∈A|∃b∈S:b%=a,(b,a)∈R,a%⇒^A_Fb}. Hence,a∈$⁰_F,S, if there exists ab∈S, such that (b,a)∈Randadoes not reachbinF, i.e.b%∈C_F(a). This meets exactly the

definition ofD_F(S). !

Lemma 3.9

For any AFF=(A,R) and any setS∈cf(F),

$_F,S=D_F(S)∪ "

C∈SCCs(F)

$_F_|_C₋_D_F_(S),(S_∩_C).

Proof. Let F=(A,R). For the ⊆-direction, we show by induction over i≥0 that $ⁱ_F,S⊆ D_F(S)∪!

C∈SCCs(F)$_F_|_C₋_D_F_(S),(S_∩_C). To ease notation, we write $¯_F,S,C as a shorthand for

$_F_|_C₋_D_F_(S),(S_∩_C), whereC∈SCCs(F).

Induction base. Fori=0,$⁰_F,S⊆D_F(S)∪!

C∈SCCs(F)$¯_F,S,C follows from Lemma 3.8.

Induction step. Leti>0 and assume$^j_F,S⊆D_F(S)∪!

C∈SCCs(F)$¯_F,S,C holds for allj<i. Let a∈$ⁱ_F,S. Then, there exists ab∈S, such that (b,a)∈Randa%⇒^D_Fb, whereD=A\$ⁱ_F,S⁻¹. Ifb∈/C_F(a), we have also a%⇒^A_Fb and thus a∈DF(S). Hence, suppose b∈CF(a). Then, a∈/DF(S) and, since S∈cf(F) andb∈S, alsob∈/D_F(S). Thus, bothaandbare contained in the frameworkF|C−D_F(S) (and so is the attack (b,a)) forC=C_F(a). Moreover,b∈(S∩C). Towards a contradiction, assume now a∈/$¯_F,S,C. This yields thata⇒^D_F_|^!_C₋_D_F_(S)bforD^!=A\ ¯$_F,S,C, i.e. there exist argumentsc₁,...,c_n (n>1) inF|C−D_F(S) but not contained in$¯_F,S,C, such thatc₁=a,c_n=b, and (ci,c_i₊₁)∈R, for alliwith 1≤i<n. Obviously all thec_i’s are contained inF as well, but sincea%⇒^D_Fb (recall that D=A\$ⁱ_F,S⁻¹), it must hold that at least one of thec_i’s, sayc, has to be contained in$ⁱ_F,S⁻¹. By the induction hypothesis, we getc∈$¯_F,S,C, a contradiction.

For the⊇-direction of the claim we proceed as follows. By Lemma 3.8, we know thatD_F(S)=$⁰_F,S and thus D_F(S)⊆$_F,S. It remains to show that !

C∈SCCs(F)$_F_|_C₋_D_F_(S),(S_∩_C)⊆$_F,S. We show by induction overi that $ⁱ_F_|

C−DF(S),(S∩C)⊆$_F,S holds for eachC∈SCCs(F). Thus, let us fix a C∈SCCs(F) and use$¯ⁱ_F,S,C as a shorthand for$ⁱ_F_|

C−DF(S),(S∩C).

Induction base. Leta∈$¯⁰_F,S,C. Then, there is ab∈(S∩C), such thatbattacksainF^!=F|C−D_F(S) anda%⇒^A_F^!!b, whereA^! denotes the arguments ofF^!, i.e.A^!=C\D_F(S). SinceF|C is built from a SCCCofF, it follows thata%⇒^A_F^\^D^F^(S)b. Sinceb∈S, (b,a)∈R, andD_F(S)=$⁰_F,S(Lemma 3.8), we geta∈$¹_F,S⊆$_F,S.

Induction step. Leti>0 and assume$¯^j_F,S,C⊆$_F,Sfor allj<i. Leta∈$¯ⁱ_F,S,C. Then, there is a b∈(S∩C), such thatbattacksainF^!anda%⇒^D_F^!!b, whereD^!=A^!\ ¯$ⁱ_F,S,C⁻¹ . Towards a contradiction, supposea∈/$_F,S. Sinceb∈S and (b,a)∈R, it follows that there exist argumentsc₁,...,c_n (n>1) inF\$_F,S, such thatc₁=a,c_n=b, and (ci,c_i₊₁)∈R, for alliwith 1≤i<n. All thesec_i’s are thus contained in the same component asa, and moreover thesec_i’s cannot be contained inD_F(S), since D_F(S)⊆$_F,S. Thus, they are contained inF|C−D_F(S), but sincea%⇒^D_F^!!b, there is at least one such c_i, sayc, contained in$¯ⁱ_F,S,C⁻¹ . By the induction hypothesis,c∈$_F,S, a contradiction. ! We now are able to obtain the desired relation.

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Lemma 3.10

For any AFF=(A,R) and any setS∈cf(F),$_F,S=RDF(S).

Proof. The proof is by induction over#_F(S).

Induction base. For#_F(S)=1,|SCCs(F)|=1 by Definition 3.3. From this and Definition 3.4, we obtain RD^F(S)=DF(S)=∅. By Lemma 3.8,$⁰_F,S=DF(S)=∅. By definition, $F,S=∅ follows from$⁰_F,S=∅.

Induction step. Let #_F(S)=n and assume the claim holds for all pairs F^!, S^!∈cf(F^!), such that #_F!(S^!)<n. In particular, this holds for F^!=F|C−D_F(S) and S^!=(S∩C), with C∈ SCCs(F). Note that (S∩C) is indeed conflict-free in F|C−D_F(S). By definition we have, RD^F(S)=D_F(S)∪!

C∈SCCs(F)RDF|C−DF(S)(S∩C) and by Lemma 3.9 we know that $_F,S= D_F(S)∪!

C∈SCCs(F)$_F_|_C₋_D_F_(S),S_∩_C. Using the induction hypothesis, i.e. $_F_|_C₋_D_F_(S),S_∩_C=

RDF|^C−DF(S)(S∩C), the assertion follows. !

We finally reached our main result in this section, i.e. an alternative characterization forcf2 semantics, where the need for recursion is delegated to a fixed-point operator.

Theorem 3.11

For any AFF,cf2(F)={S|S∈cf(F)∩naive([[F−$_F,S]])}.

Proof. The result holds by the following observations. By Lemma 3.5,S∈cf2(F) iffS∈naive([[F− RD^F(S)]]). Moreover, from Lemma 3.10, for any S∈cf(F),$F,S=RDF(S). Finally,S∈cf2(F)

impliesS∈cf(F) (see [7], Proposition 47). !

Example 3.12

To exemplify the behaviour of$_F_,S and[[F−$_F,S]], we consider the AFF andS={a,d,e,g,i} from Example 2.8. In the first iteration of computing the lfp of$_F_,S, we have$_F,S(∅)={f}because the argumentf is the only one that is attacked byS but its attackerd is not reachable byf inF.

In the second iteration, we obtain$_F,S({f})={f,h}, and in the third iteration we reach the lfp with

$_F,S({f,h})={f,h}. Hence,[[F−$_F,S]]of the AFFwrtSis given by [[F−$_F,S]]=%

{a,b,c,d,e,g,i},{(a,b),(b,c),(c,a)}&

.

Figure 8 shows the graph of[[F−$_F_,S]]. It is easy to see thatS∈naive([[F−$_F,S]]) as expected, sinceS∈cf2(F). For comparison, Figure 9 shows the graph of[[F−$_F,S!]]wrt thecf2extension

S^!={b,f,h}consisting of two SCCs. !

Figure 8. Graph of instance[[F−$_F_,S]]of Example 3.12.

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Figure 9. Graph of instance[[F−$_F,S!]]of Example 3.12.

4 Complexity analysis

In this section, we investigate the computational complexity of thecf2semantics. We consider the following decision problems for givenF=(A,R),a∈AandS⊆A:

• Vercf2: isS∈cf2(F)?

• Credcf2: isacontained in at least onecf2extension ofF?

• Skept_cf2: isacontained in everycf2extension ofF?

• NE_cf2: is there anyS∈cf2(F) for whichS%=∅?

So far, the only mentionable reference in this context is the article of Nieveset al. [19], where the authors state that the decision problem Vercf2is in P. In the following, we prove this statement with the help of our alternative characterization.

Theorem 4.1 Ver_cf2is in P.

Proof. For any AFF=(A,R) and a setS⊆A, to check ifS∈cf2(F) can be computed in polynomial time. We show that all steps in Definition 3.11 are in P. Verifying ifS∈cf(F) andS∈naive(F) can be done in polynomial time. Given$F,S, computing the instance[[F−$F,S]]can be done efficiently;

this follows from known results about graph reachability and efficient algorithms for computing SCCs [22]. It remains to show that the operator$_F,S(D) reaches its fixed-point after a polynomial number of iterations. The operator is clearly monotonic, and it is easy to see that in every iteration less or equal connections between the arguments do exist. Hence, the computation terminates when

no argumentais attacked from anyb∈S, anda%⇒^A_F^\^Db. !

For the hardness proofs of Credcf2and Skept_cf2we use the standard reduction from propositional formulas in conjunctive normal form (CNF) to AFs as in [10, 13].

Definition 4.2

Given a 3-CNF formulaϕ='_m

j=1C_jover atomsZwithC_j=l_j1∨l_j2∨l_j3(1≤j≤m) the corresponding AFF_ϕ=(Aϕ,R_ϕ) is built as follows:

A_ϕ=Z∪Z¯∪{C₁,...,C_m}∪{ϕ}∪{¬ϕ}

R_ϕ= {(z,¯z),(¯z,z)|z∈Z}∪{(Cj,ϕ)|j∈{1,...,m}}∪{(ϕ,¬ϕ)}∪

{(z,C_j)|j∈{1,...,m},z∈{l_j1,l_j2,l_j3}}∪ {(¯z,C_j)|j∈{1,...,m},¬z∈{l_j1,l_j2,l_j3}}

Figure 10 illustrates the AFF_ϕfor the formulaϕ=(z₁∨z₂∨z₃)∧(¬z₂∨¬z₃∨¬z₄)∧(¬z₁∨z₂∨z₄).

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Figure 10. AFF_ϕfor the example 3-CNFϕ.

Lemma 4.3

For anycf2extensionEof the AFF_ϕ=(Aϕ,R_ϕ) andz_i∈Zfori∈{1,...,n}, eitherz_i∈Eorz¯_i∈E.

Proof. The AFF_ϕhas the following singleton SCCs{ϕ},{¬ϕ}, andCi(1≤i≤m). The remaining SCCs areC_l_i∈{C_l₁,...,C_l_n}, withC_l_i={z_i,z¯_i}.As allC_l_iare not attacked from outside their component they remain unchanged in[[F_ϕ−$_F_ϕ_,E]]andnaive(Fϕ|C_li)={{z_i},{ ¯z_i}}. Hence, eitherz_i∈Eorz¯_i∈E

(but never both). !

Theorem 4.4

Cred_cf2is NP-complete.

Proof. For hardness, we show that any 3-CNF formulaϕis satisfied iff the corresponding AFF_ϕ as in Definition 4.2 has acf2extension containingϕ.

For the if direction, letϕbe a 3-CNF formula overZ andM⊆Z a model ofϕ. We show thatE= {{zi|zi∈M}∪{ ¯zi|zi∈Z\M}∪{ϕ}}is acf2extension ofF_ϕ. We need to show that (i)Eis conflict-free inF_ϕand (ii)E∈naive([[F_ϕ−$_F_ϕ_,E]]). As to (i), from Lemma 4.3 we know that for alli∈{1,...,n} eitherz_i orz¯_i is inE, so there are no conflicts between the arguments inZ andZ¯. The argument ϕ is not attacked by anyz_i at all. Hence,E∈cf(Fϕ). As to (ii), let us first compute$_F_ϕ_,E, where

$_F_ϕ_,E(∅)={x∈A_ϕ|∃l∈E:l%=x,(l,x)∈R_ϕ,x%⇒l}. AsMis a model ofϕ, all clauses inϕare satisfied, hence,∀C_j∃l_isuch that (li,C_j)∈R_ϕ, wherel_i∈{z_i,z¯_i}forj={1,...,m}andi={1,...,n}. Furthermore, ϕ∈E, (ϕ,¬ϕ)∈R_ϕand¬ϕ%⇒ϕ. Therefore, we obtain$_F_ϕ_,E(∅)={C₁,...,C_m,¬ϕ}, which is also the lfp$_F_ϕ_,E. Finally, we compute the instance[[F_ϕ−$_F_ϕ_,E]]=(Aϕ\{C₁,...,C_m,¬ϕ},{(z,¯z),(¯z,z)|z∈ Z}). It is easy to see thatE∈naive([[F_ϕ−$_F_ϕ_,E]]) holds.

Only if: LetE∈cf2(F_ϕ) such thatϕ∈E. We show thatM={z_i|z_i∈E}∪{¬z_i| ¯z_i∈E}is a model of ϕ. Asϕ∈Ewe know that it is not attacked by anyd∈$_F_ϕ_,E. Assume there exists aC_j%∈$_F_ϕ_,Ewith (C_j,ϕ)∈R_ϕ. We know C_j%∈E because E∈cf(F_ϕ), hence from Definition 3.7 we conclude there is no x∈E such that (x,Cj)∈R_ϕ. In this case, the argument C_j is contained in [[F_ϕ−$_F_ϕ_,E]], but this is a contradiction to E∈naive([[F_ϕ−$_F_ϕ_,E]]), because the set E^!=E∪{C_j}is conflict- free in[[F_ϕ−$_F_ϕ_,E]]. It follows that for eachC_j there exists a l_i∈{z_i,z¯_i}such that (li,C_j)∈R_ϕ, for j={1,...,m}. This means that for every clause C_j there exists a literall_i∈M. Hence, M is a model ofϕ.

For membership one can construct an algorithm as follows. For any AFF=(A,R) anda∈A, guess S⊆Awitha∈Sand checkS∈cf2(F). As Vercf2∈P, this yields an NP algorithm. !

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Table 1. Complexity of decision problems (C-c denotes completeness for classC)

cf2 stable stage adm pref

Ver_σ in P in P coNP-c in P coNP-c

Cred_σ NP-c NP-c &₂^P-c NP-c NP-c

Skept_σ coNP-c coNP-c '^P₂-c Trivial '^P₂-c

NEσ in P NP-c in L NP-c NP-c

Theorem 4.5

Skept_cf2is coNP-complete.

Proof. For hardness, we show that a given 3-CNF formulaϕ is unsatisfiable iff¬ϕ is contained in every cf2 extension of F_ϕ, whereF_ϕ is constructed following Definition 4.2. From the proof of Theorem 4.4, we already know that ϕ is contained in acf2 extension iff ϕ is satisfiable. By definition of thecf2semantics, it is easily seen that eachcf2extension ofF_ϕwhich does not contain argumentϕ, has to contain¬ϕ. Thus, in caseϕis unsatisfiable, argument¬ϕis indeed skeptically accepted.

Membership can be shown as follows via the complementary problem. Thus, for given AFF= (A,R) anda∈Awe guess a setSwitha∈/S and checkS∈cf2(F). As Ver_cf2∈P, this yields an NP algorithm for the complementary problem of Skept_cf2. Hence, Skept_cf2is in coNP. ! Theorem 4.6

NEcf2∈P.

Proof. Recall, that for every AFF it holds that eachcf2extension ofFis also a naive extension ofF. Thus, in case we have thatFpossesses only the empty set as itscf2extension, we know that the empty set is also the only naive extension ofF. However, this is only the case if all arguments of F are self-attacking. Thus to decide whether there exists a non-emptycf2extension, of an AF F=(A,R), it is sufficient to check if there exists any argumenta∈Asuch that (a,a)%∈R. This can be

done in polynomial time. !

Our results are summarized in the first column of Table 1 together with results of the other semantics used in this context ([10, 13–15]). We observe that the complexity of thecf2semantics behaves slightly different to these semantics.

5 Strong equivalence of argumentation semantics

So far, we have focused exclusively on thecf2semantics. In this section, we will show a distinguished feature of the cf2 semantics, which separates it from all other important semantics proposed for abstract argumentation. In a nutshell, this particular property states that each attack in an AF has a potential ‘meaning’ under thecf2semantics, while this is not the case for other semantics where attacks may be redundant as the following example illustrates.