On the correspondence between abstract dialectical frameworks and nonmonotonic conditional logics

The exact relationship between formal argumentation and nonmonotonic logics is a research topic that keeps on eluding researchers despite recent intensified efforts. We contribute to a deeper understanding of this relation by investigating characterizations of abstract dialectical frameworks in conditional logics for nonmonotonic reasoning. We first show that in general, there is a gap between argumentation and conditional semantics when applying several intuitive translations, but then prove that this gap can be closed when focusing on specific classes of translations.


Introduction
It is well-known that argumentation and nonmonotonic resp. default logics are closely connected: In [10] it is shown that Reiter's default logic can be implemented by abstract argumentation frameworks, a most basic form of computational model of argumentation to which many existing approaches to formal argumentation refer. On the other hand, it is clear that argumentation allows for nonmonotonic, defeasible reasoning, and in [30] computational models of argumentation are assessed by formal properties that have been adapted from nonmonotonic logics. Nevertheless, argumentation and nonmonotonic reasoning are perceived as two different fields which do not subsume each other, and indeed, often attempts to transform reasoning systems from one side into systems of the other side have been revealing gaps that could not be closed (cf., e.g., [15,17,21,34]). While one might argue that this is due to the seemingly richer, dialectical structure of argumentation, in the end the evaluation of arguments often boils down to comparing arguments with their attackers, and comparing degrees of belief is a basic operation in qualitative nonmonotonic reasoning. Therefore, in spite of the abundance of existing work studying connections between the two fields, the true nature of the relationship between argumentation and nonmonotonic reasoning has not been fully understood.
We aim at deepening the understanding of the relationships between argumentation and nonmonotonic logics and establishing a theoretical basis for integrative approaches by focusing on most fundamental approaches on either side: Abstract Dialectical Frameworks (ADFs) [7] for argumentation, and Conditional Logics 1 (CL) [28,31] for nonmonotonic logics. ADFs are an approach to formal argumentation, which subsumes many other argumentative formalisms in a generic, logic-based way. On the side of nonmonotonic logics, conditionals have been shown (and often used) to implement nonmonotonic inferences and provide expressive formalisms to represent knowledge bases; some of the most popular nonmonotonic inference systems (e. g., system Z [13]) make use of conditionals. Both ADFs and CL can be considered as high-level formalisms implementing properly the basic nature of the respective field without being restricted too much by subtleties of specific approaches, and both are based on 3-valued logics.
In this paper we investigate the correspondence between abstract dialectical frameworks and conditional logic. Syntactically, both frameworks focus on pairs of objects such as (φ, ψ). In conditional logic, these pairs are interpreted as conditionals with the informal meaning "if φ is true then, usually, ψ is true as well" and written as (ψ|φ). In abstract dialectical frameworks, these pairs are interpreted as acceptance conditions, and interpreted as "if φ is accepted then ψ is accepted as well". The resemblance of these informal interpretations is striking, but both approaches use fundamentally different semantics to formalise these interpretations. In this paper, we use this syntactical similarities as the basis of a comparison between abstract dialectical frameworks and conditional logics. In more detail, here we ask the question of whether, and how we can interpret abstract dialectical frameworks in terms of conditional logic so that acceptance in the argumentative system is defined by a nonmonotonic inference relation based on conditionals. We continue work from [22] by considering several translations of ADFs into conditional knowledge bases and applying conditional inference relations based on ordinal conditional frameworks [31], including the Z-inference relation [13], to these knowledge bases. We first show that there is a gap between argumentation and conditional semantics when applying several intuitive translations, but then define a class of translations that are OCF-adequate or Z-adequate-i. e. they preserve the semantics, see Section 3 for the formal definition-for the 2-valued model semantics of ADFs, and for other semantics under certain conditions on the ADFs. Furthermore, we show that none of the translations studied in this paper are Z-adequate for the grounded semantics and for the preferred and stable semantics in general. We furthermore show that our translations satisfy several desirable properties for translations between formalisms for knowledge representation.
The results in this paper are a substantially extended and revised version of the paper [15]. The main contributions beyond [15] are: (1) a new standard of evaluation, dubbed OCF-adequacy (Definition 6); (2) two additional translations 6 and 7 (cf. Section 3); (3) an in-depth study of the consistency of the translations (Section 4.2); (4) additional results on the Zand OCF-adequacy of our translations w.r.t. all of the well-known ADF-semantics (Section 5 and 6).

Outline of this paper
After stating all the necessary preliminaries in Section 2 on propositional logic (Section 2.1), nonmonotonic conditionals (Section 2.2) and abstract dialectical argumentation (Section 2.3). Then we investigate a family of translations from ADFs into conditional knowledge bases by first introducing the translations and defining OCF-and Z-adequacy for such translations (Section 3), after which we investigate the translations indepth. In more detail, in Section 4 we study the OCF-and Z-adequacy of the translations under the two-valued model semantics by investigating Z-adequacy (Section 4.1), consistency of the translations (Section 4.2) and making some remarks on OCF-adequacy w.r.t. two-valued semantics (Section 4.3). Thereafter, we investigate the Z-and OCF-adequacy w.r.t. the stable and preferred semantics (Section 5) and w.r.t. the grounded semantics (Section 6). Several properties of the translations presented in this paper are discussed in Section 7. We finally discuss related work in Section 8 before concluding (Section 9).

Preliminaries
In the following, we we briefly recall some general preliminaries on propositional logic, as well as technical details on conditional logic and ADFs [7].

Propositional logic
For a set At of atoms let L(At) be the corresponding propositional language constructed using the usual connectives ∧ (and), ∨ (or), ¬ (negation) and → (material implication). A (classical) interpretation (also called possible world) ω for a propositional language L(At) is a function ω : At → {T, F}. Let (At) denote the set of all interpretations for At. We simply write if the set of atoms is implicitly given. An interpretation ω satisfies (or is a model of) an atom a ∈ At, denoted by ω |= a, if and only if ω(a) = T. The satisfaction relation |= is extended to formulas as usual. As an abbreviation we sometimes identify an interpretation ω with its complete conjunction, i. e., if a 1 , . . . , a n ∈ At are those atoms that are assigned T by ω and a n+1 , . . . , a m ∈ At are those propositions that are assigned F by ω we identify ω by a 1 . . . a n a n+1 . . . a m (or any permutation of this). For example, the interpretation ω 1 on {a, b, c} with ω 1 (a) = ω 1 (c) = T and ω 1 (b) = F is abbreviated by abc. For ⊆ L(At) we also define ω |= if and only if ω |= φ for every φ ∈ . Define the set of models Mod(X) = {ω ∈ (At) | ω |= X} for every formula or set of formulas X. A formula or set of formulas X 1 entails another formula or set of formulas X 2 , denoted by X 1 X 2 , if Mod(X 1 ) ⊆ Mod(X 2 ).

Theorem 1 ([13], Theorem 4)
is consistent iff in every non-empty subset ⊆ there exists a rule tolerated by .

Abstract dialectical frameworks
We now recall some technical details on ADFs following loosely the notation from [7]. We can depict an ADF D as a directed graph whose nodes represent statements or arguments which can be accepted or not. With links we represent dependencies between nodes. A node s is dependant on the status of the nodes with a direct link to s, denoted parent nodes par D (s). With an acceptance function C s we define the cases when the statement s can be accepted (truth value ), depending on the acceptance status of its parents in D.
An ADF D is a tuple D = (S, L, C) where S is a set of statements, L ⊆ S × S is a set of links, and C = {C s } s∈S is a set of total functions C s : 2 par D (s) → { , ⊥} for each s ∈ S with par D (s) = {s ∈ S | (s , s) ∈ L}. By abuse of notation, we will often identify an acceptance function C s by its equivalent acceptance condition which models the acceptable cases as a propositional formula.  Fig. 1.
Informally, the acceptance conditions can be read as "a is accepted if b is not accepted", "b is accepted if a is not accepted" and "c is accepted if a is not accepted or b is not accepted". (false, rejected), or u (unknown). A 3-valued interpretation v can be extended to arbitrary propositional formulas over S via strong Kleene semantics: V consists of all three-valued interpretations whereas V 2 consists of all the two-valued interpretations (i.e. interpretations such that for every We define the information order ≤ i over { , ⊥, u} by making u the minimal element: u < i and u < i ⊥ and this order is lifted pointwise as follows (given two valuations v, w over S) The truth ordering ≤ t over { , ⊥, u} is defined as ⊥ ≤ t u ≤ t and is lifted to interpretations similarly. The set of two-valued interpretations extending a valuation v is defined as We denote by 2mod(D), complete(D), preferred(D), grounded(D) respectively stable(D) the sets of 2-valued models and complete, preferred, grounded respectively stable interpretations of D. We will sometimes denote the grounded interpretation by v G .
We finally define consequence relations for ADFs: Definition 2 Given sem ∈ {2mod, preferred, grounded, stable}, an ADF D = (S, L, C) and s ∈ S we define: 3 is the grounded interpretation whereas v 1 and v 2 are both preferred, stable and 2valued.
We recall the following relationships between the semantics defined above: 2 We notice that [7] showed the grounded extension to be unique for any ADF. 3 Since [7] showed the grounded extension to be unique for any ADF, we will omit ∩ from | ∼ grounded .

Theorem 2 ([7])
Given any ADF D, the following relationships hold: Below we will make use of the following semantic ADF-subclasses: Notice that any coherent ADF is also semi-coherent in view of Theorem 2 and transitivity of ⊆.
We furthermore recall some syntactic ADF-subclasses. We first have to distinguish between different kinds of links: if it is both attacking and supporting. 4. dependent (in D) it if is neither attacking nor supporting.
The set of supporting, respectively attacking links of an ADF D = (S, L, C) will be denoted by L + , respectively L − .
The corresponding graph can be found in Fig. 2.
(a, b) is a supporting link, (a, c) is an attacking link, (a, a) is a redundant and (a, d) is dependent.
We can now define the following syntactic subclasses of ADFs:  is acyclic.
We will furthermore denote the class of SFADFs that do not contain any odd-length cycles as ASSADF OLC s.
The following results on syntactic subclasses of ADFs will prove useful below:

Translations from ADFs to conditional logics
The general aim of this paper is to study translations of ADFs in CL. In more detail, where S is a set of atoms and D S is the set of all ADFs defined on the basis of S (i.e. all ADFs D = (S, L, C)), and (L(S)|L(S)) is the set of all condtionals over the propositional language generated by S, we investigate mappings T : D S → ℘ ((L(S)|L(S))) (for arbitrary S).

Definition 6 Let S be a set of atoms and T : D S → ℘ ((L(S)|L(S))) be a translation from
ADFs to conditional knowledge bases. T is: -Z-adequate with respect to semantics sem if: for every D = (S, L, C) and every s ∈ S it holds that: We notice that, given some semantics sem, any translation T that is Z-adequate (w.r.t. sem) is also OCF-adequate (w.r.t. sem). The other direction, however, does not necessarily hold.
There is a whole family of translations from ADFs to conditional logics which are prima facie apt to express the links between nodes s and their acceptance conditions C s : Notice that all of these translations are based on the idea that there is a strong connection between the acceptance of an acceptance condition C s and the acceptance of the corresponding node s. Indeed, as [7] puts it: "each node s has an associated acceptance condition C s specifying the exact conditions under which s is accepted". However, in this formulation, it is not specified (1) when a formula is true according to a three-valued interpretation (i.e. is a ∨ ¬a true according to an interpretation v with v(a) = u? Different three-valued logics give different answers to this question), (2) what to accept when there are conflicts between different acceptance conditions (e.g. if C a = ¬b and C b = ¬a) and (3) under which conditions we are justified in rejecting a node. Therefore, we systematically investigate different forms of conditionals based on the common idea that "the influence a node may have on another node is entirely specified through the acceptance condition" [7].
We now explain in more detail every translation. 1 formalizes the intuition that whenever the condition of a node s is believed, normally, s should be believed as well. Likewise, Proof Suppose is a set of conditionals. In what follows, we will denote κ Z ∪{(ψ|φ)} by κ and κ Z ∪{(φ→ψ| )} by κ . We show that κ −1 (0) = κ −1 (0), which implies the proposition. For this, suppose towards a contradiction that ω ∈ κ −1 (0) yet ω ∈ κ −1 (0). By Lemma 1 this means that there is some (λ|δ) ∈ ∪ {(ψ|φ)} s.t. (λ|δ)(ω) = 0.Since κ accepts , (λ|δ) = Table 1 Schematic summary of the results on Z-adequacy and OCF-adequacy of the translations in this paper means that the selected form of adequacy w.r.t. the semantics of the respective column is satisfied for the translation in the respective row. × means that the translation in the respective row is Zresp. OCFinadequate w.r.t. the semantics in the respective column. ×( [xADF]), finally, means that the translation in the respective row is Zresp. OCF-inadequate w.r.t. the semantics in the respective column in general, but is Zresp. OCF-adequate w.r.t. the semantics for the class of xADFs in square brackets This contradicts κ (ω) = 0 and the assumption that κ accepts ∪ {(φ → ψ| )} and thus we have shown that The above proposition thus establishes that within our perspective, it does not matter if we consider the conditional "ψ is plausible if φ is the case" or the conditional "φ → ψ is plausible". Notice that this does not imply that we can equivalently consider φ → ψ to be true. However, the above proposition does not generalize for arbitrary κ, i.e. there might be an OCF κ that accept (ψ → φ| ) but not (φ|ψ): Example 5 Consider an OCF over the signature {p, q} with: ω pq pq qp p q κ 1 whereas κ(q ∧ p) = 1 = κ(q ∧ ¬p). Therefore, κ accepts (q → p| ) but not (p|q).

Remark 1
We have implemented a reasoner in java by use of the TweetyProject 4 library which calculates the translations 1 , . . . , 7 and compares | ∼ Z -inference of these translations with inferences of the translated ADF under the grounded, preferred, stable and two-valued model-semantics.

Two-valued semantics
In this section we discuss the adequacy of our translations w.r.t. the 2mod-semantics. We will show the Z-inadequacy of the translations 1 and 2 w.r.t. the 2mod-semantics and the Z-adequacy of 3 ,. . . , 7 w.r.t. the 2mod-semantics in Section 4.1. The results in Section 4.2 establish conditions for the consistency of these translations. In Section 4.3, we finally make some observations on the OCF-adequacy of the translations 3 ,. . . , 7 .

Z-adequacy w.r.t. two-valued semantics
In this section we study Z-adequacy with respect to the 2mod-semantics for the translations suggested in the previous section. In particular, we will show that 1 and 2 are not Zadequate whereas 3 , 4 , 5 , 6 and 7 are in fact Z-adequate for the 2mod-semantics.
We first observe that 1 and 2 are not Z-adequate w.r.t. two-valued semantics.
Example 6 (Z-Inadequacy of 1 w.r.t. 2mod) We consider the following ADF D 1 from Example 2. Notice that 1 (D 1 ) = {(b|¬a), (a|¬b), (c|¬a ∨ ¬b)}, which is the conditional knowledge base considered in Example 1. We therefore see that 1 (D 1 ) | ∼ Z c even though D | ∼ ∩ 2mod c and thus 1 is not Z-adequate with respect to the 2mod-semantics.
We will now show that the translations 3 , 4 , 5 , 6 and 7 are Z-adequate for 2valued models. For these results, the following conditions on translations will prove useful: Proof We show the claim for i = 3 and C1, the proofs for i ∈ {4, 5, 6, 7} and C2 are analogous. Suppose towards a contradiction that there is some ADF D = (S, L, D) and some s ∈ S s. t. κ Z 3 (D) (C s ∧ ¬s) = 0 or κ Z 3 (D) (¬C s ∧ s) = 0. Suppose the former.

Proposition 3 For any that satisfies
Proof Suppose that (D) satisfies C1 for the ADF D = (S, L, C) and that ω ∈ (κ Z (D) ) −1 (0). We show that ω is a two-valued model of D. Indeed suppose towards a contradiction that ω(s) = ω(C s ) for some s ∈ S. This means that ω |= s ∧¬C s or ω |= ¬s ∧C s .
Since ω ∈ (κ Z (D) ) −1 (0), this contradicts (D) satisfying C1 for D. Thus, it has to be the case that ω is a model of D. That ω ∈ V 2 is clear from the fact that ω |= s ∨ ¬s for every s ∈ S.

Consistency of translations
We now discuss the requirement in Theorem 4 of i (D) being consistent. 5  For the first two conditionals, there is no κ that accepts these conditionals, since this would mean that κ(a ∧ ¬a ∧ ¬b) < κ(¬a ∧ ¬b) respectively κ(a ∧ ¬a ∧ ¬b) < κ(a ∧ (a ∨ b)). It can easily be seen that also for i ∈ {4, 6}, there is no κ that accepts i (D).
We now show that also for 7 , there might be ADFs D for which the translation is inconsistent: There is no κ that accepts (⊥|¬a) since this would mean that κ(⊥ ∧ ¬a) < κ( ∧ ¬a).
Observe that 5 (D) is consistent for D as in Example 9. In fact, we can show the following proposition, which not only establishes consistency of 5 (D) whenever 2mod(D) = ∅, but also ascertains that consistency of 5 (D) guarantees 2mod(D) = ∅:
We can now show the Z-adequacy of 5 w.r.t. two-valued model semantics (without having to assume the consistency of 5 ):
One could ask now, for the translations 3 , 4 , 6 and 7 , whether there are conditions under which they are consistent. One conjecture could be that ADFs without self-attacking nodes make 3 consistent.
Definition 7 An ADF D = (S, L, C) contains no self-attacking nodes iff for no s ∈ S, C s ¬s.
We now give an example of an ADF without self-attacking nodes for which 3 is inconsistent. For this it is convenient to define an exclusive disjunction φ∨ψ := (φ ∨ ψ) ∧ ¬(ψ ∧ φ). The node c in the above example is meant to take away the presumption that the inconsistency of 3 (D) is caused by there being no two-valued model for which some node is validated (i.e. there existing no ω ∈ 2mod(D) s.t. for some s ∈ S, ω(s) = ). Observe that for D = ({a, b}, L, C) with C a = C b = a∨b, this would be the case.
Unfortunately, non-refutingness of D is not a necessary condition for the consistency of 3
The following example shows that non-refutingness of D is not a sufficient condition for consistency of 4 (D), 6

Theorem 8 Given an ADF D, 6 (D) is consistent if D is non-validating and non-refuting.
Proof The proof is similar to that of Theorem 7.

Theorem 9 Given an ADF D, 7 (D) is consistent if D is non-validating.
Proof The proof is similar to that of Theorem 7.
These theorems allow us to make the following statements about the Z-adequacy of the translations 3 , 4 , 6 and 7 : Proof Follows from Theorems 6, 7, 8 and 9. We close this sub-section by making some obeservations on both the consequence relations resulting from non-refuting and non-validating ADFs and their corresponding translations. We first observe that from a conditional logic perspective, non-refuting and non-validating ADFs are rather simple

Proposition 5 For any ADF D:
Proof Notice that in the proof of Theorem 7, we have actually established that for every s ∈ S, (s|C s ) and (C s |s) are tolerated by 3 (D). Similarly for 4 , 5 and 7 , using Theorems 7, 8 respectively 9 instead of Theorem 7.
We furthermore observe that ADFs that are both non-validating and non-refuting are inconclusive, in the sense that they do not allow to infer a conclusive judgement about any node:

Proposition 6 If D is non-refuting and non-validating, D |∼ 2modṡ for any s ∈ S.
Proof Suppose D is non-refuting and non-validating. Then i 2mod(s) = u for any s ∈ S and thus D |∼ ∩ 2modṡ for any s ∈ S.

OCF-adequacy w.r.t. two-valued semantics
In this section, we generalize some of the results from the previous section in order to show OCF-adequacy of translations based on 3 ,. . . , 7 w.r.t. the two-valued model semantics.
We first show the OCF-adequacy of 1 and 2 : Proposition 7 1 (respectively 2 ) is OCF-adequate w.r.t. 2mod for the class of ADFs for which 1 (D) (respectively 2 (D)) is consistent.
We now show the following result on the relationship between OCFs induced by a Zpartitioning and other OCFs.
For any OCF κ that accepts 3 , 4 , 5 or 6 , the most plausible worlds according to κ will be a subset of the two-valued models of the translated ADF: Notice that this result is, in a sense, stronger than just establishing OCF-adequacy. In fact, OCF-adequacy of 3 , . . . , 7 follows from the Z-adequacy of these translations. What Proposition 9 establishes is that any κ that accepts i (for 3 ≤ i ≤ 7) will give rise to a set of beliefs that is determined by 2mod(D), in the sense that the beliefs also follow from a subset of the two-valued models. Furthermore, | ∼ ∩ 2mod forms a lower bound on Bel (κ) in the sense that everything that is derivable using | ∼ ∩ 2mod from D will be in Bel (κ). These two insights are shown in the following proposition: Proof Both statements follow immediately from Proposition 9.

Z-adequacy and OCF-adequacy w.r.t. stable and preferred semantics
In this section, we study the Z-and OCF-adequacy of the translations 3 ,. . . , 7 w.r.t. the stable and preferred semantics.
We can strengthen Theorem 4 to obtain Z-adequacy with respect to the stable and preferred semantics for specific subclasses of ADFs: Theorem 10 For any i ∈ {3, 4, 5, 6, 7} the following results hold:

1.
i is Z-adequate w.r.t. the stable semantics for the class of weakly coherent ADFs for which i (D) is consistent.
These results can now be rephrased for syntactic subclasses of ADFs as follows: Corollary 3 For any i ∈ {3, 4, 5, 7} the following results hold:

1.
i is Z-adequate w.r.t. the stable semantics for the class of ASSADFs and the class of SFSADFs, whenever i (D) is consistent.

2.
i is Z-adequate w.r.t. the preferred semantics for the class of SFADFs that do not contain any odd-length cycles, whenever i (D) is consistent.
Proof This follows from Theorem 3 and Theorem 10.
In general, however, any translation based on 3 , . . . , 7 will be OCF-inadequate w.r.t. preferred semantics as well: A critical reader might remark that the above example is pathological since it depends on the node c being "self-attacking" in the sense that C c ¬c. The following alternative yet more involving example shows that a similar behaviour can be created with an odd cycle (notice that an example without an odd-length cycle cannot be found, since any SFADF without odd-length cycle is coherent and therefore any preferred interpretation will also be a two-valued interpretation): Example 13 We consider the following ADF D = ({a, b, c, d, e}, L, C) where: C a = ¬b; C b = ¬a; C c = ¬b ∧ ¬e; C d = ¬c; C e = ¬d; Thus, by Proposition 9, this implies that if κ |= i (D) for some 3 ≤ i ≤ 7 then κ |= d.
Remark 3 Notice that these propositions also imply the Z-inadequacy of 3 , 4 and 5 w.r.t. preferred semantics.
One proposal to avoid the impossibility result of Proposition 11 above would be to add some conditionals to i (for 3 ≤ i ≤ 7). Unfortunately, such an escape route does not hold much promise: Proof To show this proposition we first show the following Lemma:
We can now reproduce the inadequacy results we had before (Proposition 11) for any ⊇ i (for 3 ≤ i ≤ 7): Proof The proof of claim 1 respectively claim 2 is identical to the proofs of Proposition 15 respectively 11 except that instead of Proposition 9 we use Proposition 12.
Remark 4 Again these propositions imply the Z-inadequacy of any ⊇ i (for 3 ≤ i ≤ 7).

OCF-and Z-inadequacy w.r.t. the grounded semantics
In this section we show the OCF-and Z-inadequacy of all translations 1 , . . . , 7 w.r.t. the grounded semantics.
We start by showing the OCF-and Z-inadequacy of 1 w.r.t. the grounded semantics: We can again reproduce the inadequacy results we had before (Proposition 15) for any ⊇ i (3 ≤ i ≤ 7):

Proposition 16
There is an ADF D s.t. for any 3 ≤ i ≤ 7, for no ⊇ i (D) is OCF-adequate w.r.t. grounded semantics.
Proof The proof of claim 1 respectively claim 2 is identical to the proofs of Proposition 15 respectively 11 except that instead of Proposition 9 we use Proposition 12.
Nevertheless, we can report on some classes of ADFs for which 3 ,. . . , 7 are Zadequate. In particular, for acyclic ADFs, Z-adequacy w.r.t. the grounded semantics is guaranteed:

Properties of the translations
In this section, we study several general properties of our translations. In Section 7.1, we study several desirable properties for translations between non-monotonic formalisms, originally proposed by [14]. In Section 7.2, we make some remarks on the properties of the translations from a conditional perspective.

Properties for translations between non-monotonic formalisms
In this section we want to look at several desirable properties, proposed by [14] for translations between non-monotonic formalisms like adequacy, polynomiality and modularity. In Section 3 we already discussed adequacy in-depth and we have shown, that translations 1 and 2 are never OCF-or Z-adequate where as 3 , 4 , 5 , 6 and 7 are OCF-and Zadequate for 2mod-semantics (Section 4.1). However these translations are not inadequate for preferred and grounded semantics as shown in Proposition 11 and Proposition 15.
A translation satisfies polynomiality if the translation is computable with reasonable bounds, i.e. within time bounded by a polynomial of the input. It is easy to see, that our translations are polynomial in the number of statements.
For modularity we follow the formulation of [32] for a translation from ADFs to a target formalism, even though modularity was originally defined for translations between circumscription and default logic [19]. In more detail, a translation being modular means that "local" changes in the translated ADF results in "local" changes in the translation. A minimal notion of modularity of a translation would be that given two syntactically disjoint ADFs D 1 and D 2 , i.e. two ADFs D 1 = (S 1 , L 1 , C 1 ) and . Clearly the translations presented in this paper are modular in this sense.
None of our translations needs a language extensions therefore they are languagepreserving.
Finally, it is clear, that the translations are syntax-based, in the sense that the translations i (D) (for any 1 ≤ i ≤ 7) can be derived purely on the basis of the syntactic form of the ADF D.

Properties from a conditional perspective
In this section we make some observations on the conditional structure of the translations suggested in this paper. In particular, we observe that for any consistent translation, the conditional structure is flat in the sense that all conditionals get assigned the same Z-rank 0. We first show this for 5 : Proposition 17 5 (D) = ( 5 (D)) 0 for any ADF D for which 2mod(D) = ∅.
A similar result has already been shown in Theorem 5 for i for any 3 ≤ i ≤ 7 for which i (D) is consistent. We summarize these results in the following corollary:

Related works
Our aim in this paper is to lay foundations of integrative techniques for argumentative and conditional reasoning. There are previous works, which have similar aims or are otherwise related to this endeavour. We will discuss those in the following.
First, there is huge body of work on structured argumentation (see e. g. [3]). In these approaches, arguments are constructed on the basis of a knowledge base possibly consisting of conditionals. An attack relation between these arguments is constructed based on some syntactic criteria. Acceptable arguments are then identified by applying argumentation semantics to the resulting argumentation frameworks. Thus, even though structured argumentation syntactically uses conditional knowledge bases, it relies semantically on formal argumentation.
There have been some attempts to bridge the gap between specific structured argumentation formalisms and conditional reasoning. For example, in [21] conditional reasoning based on system Z [13] and DeLP [12] are combined in a novel way. Roughly, the paper provides a novel semantics for DeLP by borrowing concepts from system Z that allows using plausibility as a criterion for comparing the strength of arguments and counterarguments. Our approach differs both in goal (we investigate the correspondence between argumentation and conditional logics instead of integrating insights from the latter into the former) and generality (DeLP is specific and arguably rather peculiar argumentation formalism whereas ADFs are the most general formalism around).
Several works investigate postulates for nonmonotonic reasoning known from conditional logics [23] for specific structured argumentation formalisms, such as assumptionbased argumentation [1,8,16,18] and ASPIC + [25]. These works revealed gaps between nonmonotonic reasoning and argumentation which we try to bridge in this paper.
Besnard et al. [4] develop a structured argumentation approach where general conditional logic is used as the base knowledge representation formalism. Their framework is constructed in a similar fashion as the deductive argumentation approach [5] but they also provide with conditional contrariety a new conflict relation for arguments, based on conditional logical terms. Even though insights from conditional logics are used in that paper, this approach stays well within the paradigm of structured argumentation. In [35] a new semantics for abstract argumentation is presented, which is also rooted in conditional logical terms. In more detail, a ranking interpretation is provided for extensions of arguments instantiated by strict and defeasible rules by using conditional ranking semantics. Thus, Weydert presupposes a conditional knowledge base that is used to contruct an argumentation framework whereas we investigate what are sensible translations of ADFs into conditional knowledge bases. In [33] Strass presents a translation from an ASPIC-style defeasible logic theory to ADFs. While actually Strass embeds one argumentative formalism (the ASPIC-style theory) into another argumentative formalism (ADFs) and shows how the latter can simulate the former, the process of embedding is similar to our approach.

Conclusion
In this paper we investigated the correspondence between abstract dialectical frameworks and conditional logics based on the syntactic similarities between the two frameworks. We have investigated seven different translations from ADFs into conditional logics and were able to show the OCF-and Z-adequacy of five of these translations under the two-valued semantics. Furthermore, we have shown that for certain classes of ADFs, these results carry on to the preferred and stable semantics, whereas for the grounded semantics there is a signifcant difference between the semantics of ADFs and conditional logics. Furthermore we have shown several desirable properties of these translations.
Since this paper investigates connections between two highl-level formalisms implementing the basic nature of two fields which have co-existed peacefully but largely independent of each-other for at least 25 years, it sheds important light on the exact relationship between these two fields by showing precisely where these two formalisms behave similarly and where these approaches are actually different. As such, this paper provides a foundation for cross-fertilization between the two fields as well as a justification for the adaption of ideas from conditional reasoning into ADFs. For example, in view of the OCF-and Z-adequacy of five of the presented translations with respect to the two-valued semantics, we can look at other inference relations for CL (e.g. c-representations [20], lexicographic closure [24] or disjunctive rational entailment [6]) and compare these with | ∼ ∩ 2mod . On the other hand, our results showed that for preferred, stable and grounded semantics in general, the translations are neither OCF-nor Z-adequate. This can be used as a justification for incorporating ideas from conditional reasoning into argumentative formalisms. For example, one might extend ADFs to allow for conditional acceptance conditions (e.g. " if φ then normally s is accepted if and only if C s is accepted"). Furthermore, in view of the findings in Section 7.2, we plan to take the investigation of the correspondences between ADFs and conditional logics beyond the level of beliefs (i.e. beyond κ(0)). We plan to do this by defining conditional derivations in ADFs based on the Ramsey-test [29], which says that a conditional (φ|ψ) is valid in a context if φ is believed after revision of the knowledge context by ψ. To model such conditionals, we will make use of work on revision of ADFs [26].