A Comparative Study of Some Central Notions of ASPIC+ and DeLP

This paper formally compares some central notions from two well-known formalisms for rule-based argumentation, DeLP and ASPIC+. The comparisons especially focus on intuitive adequacy and inter-translatability, consistency, and closure properties. As for differences in the definitions of arguments and attack, it turns out that DeLP's definitions are intuitively appealing but that they may not fully comply with Caminada and Amgoud's rationality postulates of strict closure and indirect consistency. For some special cases, the DeLP definitions are shown to fare better than ASPIC+. Next, it is argued that there are reasons to consider a variant of DeLP with grounded semantics, since in some examples its current notion of warrant arguably has counterintuitive consequences and may lead to sets of warranted arguments that are not admissible. Finally, under some minimality and consistency assumptions on ASPIC+ arguments, a one-to-many correspondence between ASPIC+ arguments and DeLP arguments is identified in such a way that if the DeLP warranting procedure is changed to grounded semantics, then DeLP notion of warrant and ASPIC+'s notion of justification are equivalent. This result is proven for three alternative definitions of attack. This paper is under consideration in Theory and Practice of Logic Programming (TPLP).

and Defeasible Logic Programming, or DeLP for short (Garcia and Simari 2004), are two well-known rule-based formalisms for argumentation. 'Rule-based' is not about expressiveness but about how arguments are constructed. In a rule-based approach 1 , arguments are formed by chaining applications of inference rules into inference trees or graphs. This approach can be contrasted with approaches defined in terms of logical consequence notions, in which arguments are premises-conclusion pairs where the premises are consistent and imply the conclusion according to the consequence notion of some adopted 'base logic'. Examples of this approach are classical-logic argumentation (Cayrol 1995;Besnard and Hunter 2001;Besnard and Hunter 2008;Gorogiannis and Hunter 2011) and its generalisation into abstract Tarskian-logic argumentation (Amgoud and Besnard 2013). It is important to note that, unlike these logic-based approaches, rule-based approaches in general do not adopt a single base logic but two base logics, one for the strict and one for the defeasible rules. ASPIC + and DeLP are similar in various respects: both have a distinction between strict and defeasible inference rules and both use preferences to resolve attacks into defeats. The shared rule-based approach and these further similarities warrant a detailed comparison between the two frameworks. Such a comparison is the topic of this paper. It will turn out that there are also differences, the main one being that while ASPIC + evaluates arguments with the by now standard Dungean semantics of abstract argumentation frameworks (Dung 1995), DeLP has a special-purpose definition of argument evaluation. Both of ASPIC + and of DeLP various versions exist. As for DeLP, we will discuss the version introduced by Garcia and Simari (2004), which arguably is the standard version. As for ASPIC + we will unless indicated otherwise assume the version of Modgil and Prakken (2013) with defeat-conflict-freeness, no consistency constraints on premise sets and the contrariness relation corresponding to 'strong' or 'symmetric' negation.
We will compare DeLP 2004 with ASPIC + 2013, and we will also study modifications of both systems with ideas from the other systems. In particular, we will consider a version of DeLP with grounded semantics and a version of ASPIC + with DeLP's notion of rebutting attack; either with or without DeLP's consistency constraints on arguments. Just before this paper was finished, we learned that Parsons and Cohen (2018) had also carried out a comparison between AS-PIC + and DeLP. Nevertheless, our investigation can be regarded as complementary to theirs. In their work, they seek to revisit aspects that differentiate DeLP from ASPIC + , analyze the common ground between the two approaches, and study the possibility of establishing conditions that would help bridge the gap between them. The discussion mainly centers on the similarities and differences between ASPIC + and DeLP regarding knowledge representation capabilities, the mechanism adopted for argument construction, and the different types of attack and defeat they consider. Their focus is not on formally proving properties of or relations between the two formalisms.
To summarise our main findings, as for differences in the definitions of arguments and attack, it will turn out that DeLP's definitions are intuitively appealing and in some special cases the DeLP definitions will be shown to fare better than ASPIC + . On the other hand, the DeLP definitions may not fully comply with the rationality postulates of strict closure and indirect consistency introduced by Caminada and Amgoud (2007). In Subsection 4.1, we will include a thorough discussion about these issues. As we will discuss in Section 6, while the DeLP definition of warrant is similar to grounded semantics, there are also differences, caused by the fact that the constraints on the argument evaluation do not coincide with the constraints on games in the game-theoretic proof theory for grounded semantics. For these reasons, we will introduce a special version of DeLP under grounded semantics, since in some examples its current notion of warrant arguably has counterintuitive consequences and may lead to sets of warranted arguments that are not admissible under Dung's definition. Finally, under some minimality and consistency assumptions on ASPIC + arguments, a one-to-many correspondence between ASPIC + arguments and DeLP arguments will be identified in such a way that if the DeLP warranting procedure is changed to grounded semantics, then DeLP's notion of warrant and ASPIC + 's notion of justification are equivalent. This result will be proven for three alternative definitions of attack. This paper is organised as follows. We start with a brief sketch of the history of both frameworks in Section 2 and a summary of the formalisms in Section 3. We then compare the argument and attack definitions of the two formalisms in Sections 4 and 5. We will argue that DeLP's definitions are interesting alternatives to ASPIC + definitions which in some special cases represent possible improvements. Then in Section 6 we compare the different ways in which ASPIC + and DeLP evaluate arguments. We will argue that some differences reveal possible drawbacks of the DeLP semantics. After observing that the motivation behind DeLP's semantics is similar to the intuitions underlying Dung's (1995) grounded semantics, we propose a version of DeLP with grounded semantics, arguing that all the examples given by Garcia and Simari (2004) as reasons for their special semantics are treated as they want by grounded semantics. Finally, in Section 7 we prove correspondence results with respect to arguments, attacks, defeats, and extensions between ASPIC + and DeLP.

History
ASPIC + originated from the European ASPIC project as an attempt to integrate and consolidate the then state-of-the art in formal argumentation (see Amgoud et al. (2006)). It was in particular inspired by the research of Pollock (1987), Pollock (1995) and Vreeswijk (1997). A basic version without preferences or premise attack was used by Caminada and Amgoud (2007) as a vehicle for introducing and studying various so-called rationality postulates for argumentation. Prakken (2010) introduced the first full version of ASPIC + , introducing premise attack and preferences. Since then the framework has been further developed and studied in several publications. For a detailed overview see Section 5 of Modgil and Prakken (2018). In consequence, ASPIC + as it has been developed over the years is not a single framework but a family of frameworks varying on several elements.
DeLP was developed on the basis of Simari and Loui (1992), who presented a rule-based argumentation system with both strict and defeasible inference rules, with specificity as a means to resolve attacks and with an argument evaluation definition taken from Pollock (1987), which was later by Dung (1995) shown to be equivalent to his grounded semantics. Inspired by this work, DeLP was developed in a series of papers, culminating in Garcia and Simari (2004), which is now regarded as the standard paper on DeLP. The idea of argument evaluation in terms of a dialectical tree, now typical for DeLP, was introduced by Garcia et al. (1993) and Simari et al. (1994). The first paper establishing conditions on the construction of the branches of a dialectical tree (called an argumentation line) was Garcia et al. (1998); thus, this was the paper that gave up grounded semantics for DeLP.

Formal Preliminaries
In this section, we summarise the formal systems used throughout the paper. More details can be found in the papers already mentioned above and in Baroni et al. (2011) and Baroni et al. (2018) for abstract argumentation frameworks, Modgil and Prakken (2014) and Modgil and Prakken (2018) for ASPIC + , and Garcia and Simari (2014) and Garcia and Simari (2018) for DeLP. It is relevant at this point to remark that the presentations of ASPIC + and DeLP contained in this paper heavily rely on earlier presentations of these systems, such as the ones cited.

Abstract Argumentation Frameworks
An abstract argumentation framework (AF) is a pair A, attack , where A is a set of arguments and attack ⊆ A × A. The theory of AFs (Dung 1995) identifies sets of arguments (called extensions) which are internally coherent and defend themselves against attack. An argument A ∈ A is defended by a set by S ⊆ A if for all B ∈ A: if B attacks A, then some C ∈ S attacks B. A set S of arguments is conflict-free if no argument in S attacks an argument in S. Then, relative to a given AF, • E is admissible if E is conflict-free and defends all its members; • E is a stable extension if E is admissible and attacks all arguments outside it; • E ⊆ A is the grounded extension if E is the least fixpoint of operator F, where F(S) returns all arguments defended by S.
Finally, for T ∈ {complete, preferred, grounded, stable}, X is sceptically or credulously justified under the T semantics if X belongs to all, respectively at least one, T extension.
In ASPIC + the attack relation is renamed to defeat to distinguish it from a more basic notion of conflict between arguments, which in ASPIC + is called attack. Moreover, the following terminology is used. Argument A strictly defeats argument B if A defeats B and B does not defeat A. Argument A weakly defeats argument B if A defeats B and B defeats A.
In the comparisons with DeLP we will use grounded semantics. In particular, we will use the following game-theoretic proof theory, which is sound and complete with respect to grounded semantics (Prakken 1999;Modgil and Caminada 2009).

Definition 1
An argument game for grounded semantics is a finite nonempty sequence of moves move i = (Player i , Arg i ) (i > 0), such that 1. Player i = P iff i is odd; and Player i = O iff i is even; 2. If Player i = Player j = P and i = j, then Arg i = Arg j ; 3. If Player i = P, then Arg i strictly defeats Arg i−1 ; 4. If Player i = O, then Arg i defeats Arg i−1 .
The first condition says that the proponent begins and then the players take turns, while the second condition prevents the proponent from repeating its attacks. The last two conditions form the heart of the definition: they state the burdens of proof for P and O. The non-repetition rule and the condition that P moves strict defeaters (as opposed to O being allowed to move any defeater) are not needed for soundness and completeness but they make many otherwise infinite games finite.

Definition 2
A player wins an argument game iff the other player cannot move. An argument A is provably justified iff the proponent has a winning strategy in a game beginning with A.
As is well-known, a strategy for the proponent can be displayed as a tree of games which only branches after the proponent's moves and which then contains all defeaters of this move. A strategy for the proponent is then winning if all games in the tree end with a move by the proponent. We will use these observations below in comparing the grounded argument game with DeLP's dialectical trees.

ASPIC +
We next specify the present paper's instance of the ASPIC + framework. It defines abstract argumentation systems as structures consisting of a logical language L with symmetric negation and two sets R s and R d of strict and defeasible inference rules defined over L. In the present paper L is a language of propositional or predicate-logic literals, since this is also the language assumed by DeLP. Arguments are constructed from a knowledge base (a subset of L) by combining inferences over L. Formally: Definition 3 [Argumentation System] An argumentation system (AS) is a pair AS = (L, R) where: • L is a logical language consisting of propositional or ground predicate-logic literals • R = R s ∪ R d is a set of strict (R s ) and defeasible (R d ) inference rules of the form {ϕ 1 , . . . , ϕ n } → ϕ and {ϕ 1 , . . . , ϕ n } ⇒ ϕ respectively (where ϕ i , ϕ are meta-variables ranging over wff in L), such that R s ∩ R d = / 0. Here ϕ 1 , . . . , ϕ n are called the antecedents and ϕ the consequent of the rule. 2 We write ψ = −ϕ just in case ψ = ¬ϕ or ϕ = ¬ψ. Note that − is not part of the logical language L but a metalinguistic function symbol to obtain more concise definitions. Also, for any rule r the antecedents and consequent are denoted, respectively, with ant(r) and cons(r).
The set R s is said to be closed under transposition if whenever S → ϕ ∈ R s , then S \ {s i } ∪ −ϕ → −s i ∈ R s for any s i ∈ S. This notion is important since many consistency and closure results in the literature depend on the condition that R s is closed under transposition.

Definition 4
[Consistency] For any S ⊆ L, let the closure of S under strict rules, denoted Cl R s (S), be the smallest set containing S and the consequent of any strict rule in R s whose antecedents are in Cl R s (S). Then a set S ⊆ L is directly consistent iff ψ, ϕ ∈ S such that ψ = −ϕ, and indirectly consistent iff Cl R s (S) is directly consistent.
Note that the notion of indirect consistency is relative to a given set of strict rules.

Definition 5 [Knowledge Bases]
A knowledge base over an AS = (L, R) is a set K ⊆ L.
In this paper K corresponds to the 'necessary premises' in other ASPIC + publications, which are intuitively certain and therefore not attackable. Since the 'facts' in DeLP are also not attackable, we assume in this paper that the set of attackable or 'ordinary' premises from other ASPIC + publications is empty. We will, as is also usually done in DeLP, represent what intuitively are uncertain premises ϕ as defeasible rules ⇒ ϕ. In what follows, for a given argument, the function Prem returns all the formulas of K (called premises) used to build the argument, Conc returns its conclusion, Sub returns all its sub-arguments, LDR returns the last defeasible rules used in the argument, Rules and DefRules return, respectively, all rules and all defeasible rules of the argument and TopRule returns the last rule used in the argument. An argument is now formally defined as follows.

Definition 6
[Arguments] A argument A on the basis of a knowledge base K in an argumentation system AS is a structure obtainable by applying one or more of the following steps finitely many times: . . , Conc(A n ) ⇒ ψ and the other notions defined as in (2).
When Conc(A) = ϕ we sometimes say that A is an argument for ϕ. Each of the functions Func in this definition is also defined on sets of arguments S = {A 1 , . . . , A n } as follows: Note that we overload the → and ⇒ symbols to denote an argument while they also denote strict, respectively, defeasible inference rules. This is common practice in argumentation and originates from Vreeswijk (1997).

Example 1
Consider a knowledge base in an argumentation system with L consisting of p, q, r, s,t, u, v, x and their negations, with R s = {s 1 , s 2 } and R d = {d 1 , d 2 , d 3 }, where: x}. An argument A 3 for r (i.e., with conclusion r) with subarguments A 1 for p and A 2 for q is displayed in Figure 1, with the premises at the bottom and the conclusion at the top of the tree. In this and the next figure, strict inferences are indicated with solid lines while defeasible inferences and rebuttable conclusions are displayed with dotted lines. The figure also displays the formal structure of the argument. Note that the argument can also be written as [p, [p ⇒ q] → r].
In general, ASPIC + has three ways of attack: on an argument's uncertain premises (undermining attack), on the conclusion of a defeasible rule (rebutting attack) and on a defeasible rule itself (undercutting attack). However, in this paper we only consider rebutting attack. Example 2 In our running example argument A 3 is rebutted on A 2 by an argument B 2 for ¬q: Note that A 2 does not in turn rebut B 2 on B 2 , since B 2 has a strict top rule while the argument on which an argument is (directly) rebutted has to have a defeasible top rule. For the same reason B 2 can potentially only be rebutted on B 1 . Our argumentation theory allows for such a rebuttal: Note that B 1 in turn rebuts C 4 , since C 4 has a defeasible top rule. All arguments and (direct) attacks in the example are displayed in Figure 2.

Example 3
In our running example this yields one additional rebutting relation, since A 2 u-rebuts B 2 . Furthermore, argument A 3 can be potentially u-rebutted on its final conclusion r. However, C 2 cannot be u-rebutted, since it is not defeasible but strict.
Below we will assume Definition 7 of attack unless specified otherwise.
The ASPIC + counterpart of an abstract argumentation framework is a structured argumentation framework.

Definition 9
[Structured Argumentation Frameworks] Let AT be an argumentation theory (AS, K). A structured argumentation framework (SAF) defined by AT is a triple A, C, where A is the set of all arguments on the basis of AS, is an ordering on A, and (X,Y ) ∈ C iff X attacks Y .

Example 4
In our running example A = {A 1 , A 2 , A 3 , B 1 , B 2 ,C 1 ,C 2 ,C 3 ,C 4 }, while C is such that B 2 attacks both A 2 and A 3 , argument C 4 attacks both B 1 and B 2 and B 1 attacks C 4 .
The attack relation tells us which arguments are in conflict with each other. If an argument A successfully attacks, i.e., defeats, B, then A can be used as a counter-argument to B. Whether a rebutting attack succeeds as a defeat, depends on the argument ordering . In the following definition A ≺ B is defined as usual as A B and B A.

Definition 10
[Defeat]. Argument A defeats argument B if A rebuts B on B and A ≺ B .

Example 5
In our running example, the attack of B 2 on A 2 (and thereby on A 3 ) succeeds if B 1 ≺ A 2 . In that case B 2 strictly defeats both A 2 and A 3 . If B 1 and C 4 are incomparable or of equal priority, then these two arguments defeat each other, while C 4 strictly defeats B 2 . If C 4 ≺ B 1 then B 1 strictly defeats C 4 while if B 1 ≺ C 4 then C 4 strictly defeats both B 1 and B 2 .
AFs are then generated from SAFs by letting the attacks from an AF be the defeats from a SAF.

Definition 11
[AFs corresponding to SAFs] An abstract argumentation framework (AF) corresponding to a SAF = A, C, (where C is ASPIC + 's attack relation) is a pair (A, attack) such that attack is the defeat relation on A determined by SAF.
A nonmonotonic consequence notion can then be defined as follows. Let T ∈ {complete, preferred, grounded, stable} and let L be from the AT defining SAF. A wff ϕ ∈ L is sceptically T -justified in SAF if ϕ is the conclusion of a sceptically T -justified argument, and credulously T -justified in SAF if ϕ is not sceptically T -justified and is the conclusion of a credulously Tjustified argument.

Example 6
In our running example, if B 2 does not defeat A 2 , then all extensions in any semantics contain A 3 so r is sceptically justified. Let us next assume that B 2 defeats A 2 . Then if C 4 strictly defeats B 1 , we have a unique extension in all semantics, namely, {A 1 , A 2 , A 3 ,C 1 ,C 2 ,C 3 ,C 4 }. In both cases this yields that wff r is sceptically justified. Alternatively, if B 1 strictly defeats C 4 , then there again is a unique extension in all semantics, which now is {A 1 , B 1 , B 2 ,C 1 ,C 2 ,C 3 }. Then r is neither sceptically nor credulously justified. Finally, if B 1 and C 4 defeat each other, then the grounded extension is E = {A 1 ,C 1 ,C 2 ,C 3 ,C 4 } while there are two preferred extensions E 1 = {A 1 , A 2 , A 3 ,C 1 ,C 2 ,C 3 ,C 4 } and E 2 = {A 1 , B 2 ,C 1 ,C 2 ,C 3 }. So then r is credulously but not sceptically justified in preferred semantics but is neither sceptically nor credulously justified in grounded semantics.

Defeasible Logic Programming: DeLP
Defeasible Logic Programming is a formalization of defeasible reasoning in which results of Logic Programming and Argumentation are combined. DeLP has the declarative capability of representing knowledge in a language that extends the language of logic programming with the possibility of representing weak information in the form of defeasible rules, and an argumentationbased inference mechanism for warranting conclusions.
While ASPIC + in general abstracts from the logical language, DeLP chooses a logic-programming language with "strong" negation to represent knowledge in which the antecedents and consequent of a rule (strong or weak) are ground literals. It is possible to employ in DeLP default negation, which is also known as negation as failure, but since this does not crucially change the analysis below, we will for simplicity ignore this extension here. As usual, rules written with free variables are schemes for all their ground instances. Although DeLP and the instance of ASPIC + presented above are similar, they are not fully equivalent. Elements in which DeLP and ASPIC + coincide are the predicate-logic literal language with strong negation, a set of indisputable facts, two sets of strict and defeasible rules, and a binary argument preference relation. However, DeLP's definitions of argument, attack, and defeat are not equivalent to those of AS-PIC + . Moreover, a significant difference with ASPIC + is that DeLP, as defined by Garcia and Simari (2004), does not evaluate arguments by generating abstract argumentation frameworks. Instead, DeLP's notion of warrant is defined in terms of dialectical trees in a way that is similar to the argument game of grounded semantics but with some significant differences, as we will see below.
This section will introduce a description of DeLP's features for knowledge representation (mainly taken from Garcia and Simari (2014)); then, the details concerning its inference mechanism will be explained. Although the work leading to the formalization of DeLP began in the early 90's (Simari et al. 1994a;Simari et al. 1994b) as an evolution of the work of Simari and Loui (1992), its formalization was completed by García (2000) and finally published in Garcia and Simari (2004;Garcia and Simari (2014). Further developments can be found in the following related material: (García et al. 2007;Tucat et al. 2009;Martínez et al. 2012;García et al. 2013;Cohen et al. 2016;Garcia and Simari 2018).
The knowledge representation language of DeLP is determined by a set of atoms. Atoms can be preceded by the strong negation symbol "∼". Atoms that are not preceded by strong negation will also be called positive literals and atoms preceded by strong negation will be called negative literals, the term literal, or sometimes objective literal, will refer to either one. A pair of literals involving a positive and a negative literal over the same atom are called complementary or contradictory. For instance, "∼guilty" and "guilty" are two complementary literals. A defeasible logic program, abbreviated dlp, is a set of facts, strict rules, and defeasible rules defined as follows: -Facts are ground (objective) literals, e.g., guilty, price(100), ∼close. In DeLP facts are used for representing information that is considered to hold in the application domain. Hence, as it will be explained below, a dlp cannot contain two complementary facts. -Strict Rules represent a relation between a ground literal L 0 , or head of the rule, and a set of ground literals {L i } i>0 , or body of the rule, and are denoted L 0 ← L 1 , . . . , L n ; strict rules correspond syntactically to basic rules in Logic Programming (Lifschitz 1996). The use of the adjective 'strict' emphasizes that the relation between the head and the body of the rule is such that if the body is accepted then the head must also be accepted. The examples of strict rules shown below can be understood as expressing that: someone who is guilty cannot be innocent, cats are mammals, and if there are not many surfers then there are few surfers: ∼innocent ← guilty mammal ← cat few surfers ← ∼many surfers -Defeasible Rules are used to represent a weaker connection between pieces of information, they are denoted L 0 − −≺ L 1 , . . . , L n , and like strict rules, the head of the rule L 0 is a ground literal and its body {L i } i≥0 is a set of ground literals. A defeasible rule with empty body is called a presumption; sometimes we will also call the head of such a rule a presumption. Note that initially Garcia and Simari (2004) required defeasible rules to have non-empty bodies. Here and below we follow their extension in Section 6.2 of DeLP with presumptions. Unlike strict rules, acceptance of the body of a defeasible rule does not always lead to the acceptance of the head. Examples of defeasible rules follow. The first one represents that usually, mosquitoes are not dangerous, and the second says that reasons to believe mosquitoes are carrying dengue, justify the belief they are dangerous: Note that, from a syntactic point of view, strict and defeasible rules differ only in the symbol between the head and the body of the rule. It is interesting to remark here that the representational choice between these two forms of relating the head and the body of a rule is ultimately a matter of context, sometimes a rule could change accordingly to the environment in which it is used; for instance, a rule that locally can be considered strict could become defeasible in a larger environment. Defeasible rules allow to represent a weak connection between the body (antecedent) and the head (consequence) of the rule. A defeasible rule H − −≺ B expresses that reasons to believe in B provide a (defeasible) reason to believe in H. As an example, consider an scenario where an agent has to decide how to spend the day. Then, the defeasible rule "nice − −≺ waves" can represent that "reasons to believe that there are big waves at the beach, is a reason to believe that it should be a nice day for surfing". The connection between "waves" and "nice" is weak in the sense that there might be other reasons such as "normally, if it is raining it is not nice for surfing", represented as "∼nice − −≺ rain", that will lead to the contrary conclusion. Suppose that today there are big waves and it is raining, then the acceptance of the body of the rule "nice − −≺ waves" does not lead directly to the acceptance of the head.
Nevertheless, strict rules establish a strict connection between body and head; thus, the rule "∼working ← vacation" represents the fact that in vacation an agent it is not working. Then, as we will show below, due to this strict connection in DeLP if "vacation" is accepted then "∼working" is also accepted.
Note that the symbols " − −≺ " and " ← " denote meta-relations between a literal and a set of literals, and have no interaction with language symbols. As in Logic Programming, strict and defeasible rules are not conditionals nor implications, they are inference rules. Consequently, strict rules do not automatically contrapose or (in Caminada and Amgoud's (2007) terms) 'transpose'. In DeLP a knowledge engineer has to separately determine for each strict rule whether adding its transposition is appropriate for that rule. This is relevant since many positive results in the literature on satisfaction of Caminada and Amgoud's rationality postulates depend on the assumption that the set of strict rules is closed under transposition.

Definition 12
[Defeasible Logic Program] A defeasible logic program (dlp) is set of facts, rules and presumptions. However, when required, a dlp is denoted (Π, ∆), to distinguish the subset Π of facts and strict rules and the subset ∆ of defeasible rules and presumptions. Moreover when we want to refer to just the facts in Π we write Π f and for the strict rules we write Π s . Naturally, Π f ∪ Π s = Π.
Example 7 or there exists a rule R i in (Π, ∆) (strict or defeasible) with head L i and body B 1 , B 2 , . . . , B m such that every literal B j , 1 ≤ j ≤ m, of the body is an element L k already appearing in the sequence preceding L i (k < i). If L has a derivation that only uses facts and strict rules from Π and no defeasible rules, in this case we say that L has a strict derivation.
In the program (Π 7 , ∆ 7 ) shown in Example 7, the literal surf has a defeasible derivation: vacation, ∼working, ∼busy, spare time, waves, nice, surf , which contains two facts (vacation and waves), and the use of a strict rule (∼working ← vacation) and four defeasible rules. Note that every fact of a DeLP has a defeasible derivation; however, not every head of a rule has a derivation, for instance, neither cold nor ∼surf have a defeasible derivation. Note that, ∼working has a strict derivation from Π 7 . Note that literals that have a strict derivation must be facts or the head of a strict rules; however, a literal can be the head of a strict rule and might have a defeasible derivation, but not a strict derivation. For instance, few surfers has no strict derivation from Π 7 , although it has a defeasible derivation from (Π 7 , ∆ 7 ) that uses a defeasible rule for the derivation of ∼many surfers.
It is important to note that in DeLP the set Π is used to represent non-defeasible information, consequently it is required that the set be representationally coherent. Therefore, for any program (Π, ∆) we assume that Π is non-contradictory: no pair of contradictory literals can be derived from Π, i.e., no strict derivation for complementary literals can be obtained from a DeLP-program. Saying that Π is non-contradictory is equivalent to saying in ASPIC + that K is indirectly consistent relative to R s .

Definition 14
[Argument] Let (Π, ∆) be a dlp and L a ground literal. We say that A is an argument for the conclusion L from (Π, ∆), denoted A, L , if A is a set of defeasible rules (A ⊆∆), such that: 1. there exists a defeasible derivation for L from Π ∪ A, and 2. no pair of contradictory literals can be defeasibly derived from Π ∪ A. 3. A is minimal in that there is no proper subset of A satisfying conditions (1) and (2). Observe that although facts and strict rules are used in the defeasible derivation, the argument structure only mentions the defeasible rules, i.e., facts and strict rules are not part of an argument. Note also that unlike in Definition 6 of ASPIC + -arguments, the set of defeasible rules of a DeLP argument has to be minimal and its set of 'conclusions' has to be indirectly consistent. (Strictly speaking, the set of conclusions of a DeLP argument is not formally defined, but the set of all literals in the defeasible derivation corresponding to an argument can be seen as such.) Note that it could happen that a literal L has a defeasible derivation from a dlp but there is no argument for L from that dlp. For instance, consider the dlp (Π 7 , ∆ 7 ) of Example 7, from the fact monday and the defeasible rule working − −≺ monday, there is a defeasible derivation for the literal working. However, note that there is a strict derivation for ∼working, and hence from the set Π 7 ∪ {working − −≺ monday} both literals working and ∼working can be defeasibly derived, for that reason there is no argument for the literal working from (Π 7 , ∆ 7 ). Consider Observe that S ∪ Π 7 is non contradictory and allows for the defeasible derivation of nice; however, S is not an argument for nice because it is not minimal. Observe that A 2 ⊂ S is an argument for nice: A 2 = {nice − −≺ waves}. Attack on arguments is in DeLP defined in terms of disagreement between literals. Two literals L and Q are said to disagree in the context of the program (Π, ∆) if the set Π ∪ {L, Q} is contradictory, i.e., from Π ∪ {L, Q} is possible to strictly derive a literal and its complementary. For example, given Π = {(h ← a), (∼h ← b)}, the literals a and b disagree. This notion of disagreement allows us to find direct and indirect conflicts between arguments. This is equivalent to saying in ASPIC + that K ∪ {L, Q} is indirectly inconsistent relative to R s . Note that two complementary literals always disagree (e.g., nice and ∼nice). Since for any program (Π, ∆) it is required that Π be non-contradictory, the disagreement cannot come from Π.
Given a dlp (Π, ∆) and two arguments A, L and B, Q obtained from it, if B ⊆ A then we say that B, Q is a subargument of A, L and that A, L is the superargument of B, Q (note that trivially every argument is a subargument/superargument of itself).

Definition 15
[Counterargument / Attack] In DeLP, an argument B, Q is a counterargument for A, L at literal P, if there exists a subargument C, P of A, L such that P and Q disagree. The literal P is referred to as the counterargument point and C, P as the disagreement subargument. If B, Q is a counterargument for A, L , then we also say that B, Q attacks A, L , and that B, Q and A, L are in conflict.
Except for the disagreement check instead of a simple syntactic check for complementariness, this definition is similar to Definition 7 of rebutting attack in ASPIC + in that an argument can attack a subargument of its target and does so on a specific point. On the other hand, unlike in ASPIC + , in DeLP the attacking point can be the consequent of a strict rule. For instance, we include below Example 4 "Married John" from Caminada and Amgoud (2007) in terms of DeLP syntax.

Example 8
John wears something that looks like a wedding ring", go = "John often goes out until late with his friends", hw = "John has a wife", b = "John is a bachelor", m = "John is married". The following arguments can be constructed: In DeLP arguments A 1 and A 2 have no defeaters; argument A 5 defeats A 6 and vice versa; and argument A 3 defeats A 4 and vice versa. Note also that argument A 3 defeats A 6 and argument A 4 defeats A 5 . Consequentely, b = "John is a bachelor" and m = "John is married" are not warranted (justified).
As a further example, consider the dlp (Π 7 , ∆ 7 ) of Example 7 and the sets Then, A 1 , ∼nice is a counterargument for A 2 , nice and vice versa because in this particular case the conclusion of both arguments disagree. As another example, A 4 , ∼rain is a counterargument for A 1 , ∼nice at the counterargument point rain and A 3 , rain is the disagreement subargument. Note that in DeLP a counter-argument for an argument A is also a counter-argument for any super-argument of A. Also note that in DeLP there is no possible counterargument for a claim having a strict derivation, see Garcia and Simari (2004) for the proof. Observe that from (Π 7 , ∆ 7 ) there is a strict derivation for ∼working, however, although there is a derivation for working, no argument for working can exists, and hence, no counter-argument for ∼working.
The argument comparison criterion is modular in DeLP; hence, it is possible to use any preference criterion established over the set of arguments (see Garcia and Simari (2014) for details and Teze et al. (2015) for an application). This allows the user to select the most appropriate criterion for the application domain that is being represented. For the rest of the presentation we will assume an abstract preference criterion ≺ of strict comparison on the set of arguments, where A ≺ B means that argument B is strictly better than argument A.

Definition 16
[Defeaters] Argument A 1 , L 1 is a proper defeater of argument A 2 , L 2 iff there exists a subargument A, L of A 2 , L 2 such that A 1 , L 1 attacks A 2 , L 2 at literal L and A, L ≺ A 1 , L 1 . Argument A 1 , L 1 is a blocking defeater of argument A 2 , L 2 iff there exists a subargument A, L of A 2 , L 2 such that A 1 , L 1 attacks A 2 , L 2 at literal L and A, L ≺ A 1 , L 1 and A 1 , L 1 ≺ A, L . Argument A is a defeater of argument B iff A is a proper or a blocking defeater of B.
In the context of program (Π 7 , ∆ 7 ), A 4 , ∼rain is a counterargument for A 1 , ∼nice at the counterargument point rain and A 3 , rain is the disagreement subargument; therefore, A 4 is compared with A 3 to determine if it is a defeater.
To facilitate comparison of both approaches, in the rest of the paper a DeLP strict rule a ← b can also be denoted using ASPIC + notation as b → a, and defeasible rule a − −≺ b as b ⇒ a. Also, a dlp (Π, ∆) can be denoted (K, R s , R d ) assuming Π= K ∪ R s and ∆ = R d .
The notions of proper and blocking defeater are not equivalent to the ASPIC + notions of strict and weak defeater (see Section 3.1). Consider the following example: Then we have the following DeLP arguments: And let A 1 ≺ B 2 and B 1 ≺ A 2 (a preference based on strict specificity). Note that A 1 is a subargument of A 2 and B 1 is a subargument of B 2 . Then, A 2 and B 2 are proper defeaters of each other, while the ASPIC + relation of strict defeat is asymmetric. Note also that A 2 and B 2 weakly defeat each other while they are not blocking defeaters of each other.
An argumentation line for an argument A 1 , L 1 is a sequence of arguments from a dlp, denoted Λ = [ A 1 , L 1 , A 2 , L 2 , A 3 , L 3 , . . .], where each element of the sequence A i , L i , i > 1, is a defeater of its predecessor A i−1 , L i−1 . The first element, A 1 , L 1 , becomes a supporting argument for the conclusion L 1 , A 2 , L 2 an interfering argument, A 3 , L 3 a supporting argument, A 4 , L 4 an interfering one, continuing in that manner. Thus, an argumentation line can be split into two disjoint sets: Λ S = { A 1 , L 1 , A 3 , L 3 , A 5 , L 5 , . . .} of supporting arguments for the conclusion L 1 , and Λ I = { A 2 , L 2 , A 4 , L 4 , . . .} of interfering arguments for L 1 .

Definition 17
Given a program (Π, ∆), a set of arguments 2. The set Λ S of supporting arguments (resp. Λ I ) is concordant.
Given a program, there can be more than one argumentation line starting with the same argument A, L . Therefore, analyzing a single acceptable argumentation line for A, L will not be enough to determine whether A, L is an undefeated argument. In the general situation, there might be several defeaters B 1 , Q 1 , B 2 , Q 2 , . . ., B k , Q k for A 1 , L 1 , and for each defeater B i , Q i there could be in turn several defeaters; thus, a tree structure is defined which is called a dialectical tree. In this tree, the root is labeled with A, L and every node (except the root) represents a defeater (proper or blocking) of its parent. Each branch in the tree, i.e., each path from a leaf to the root, corresponds to a different acceptable argumentation line.

Definition 19
[Dialectical Trees] Let A 1 , L 1 be an argument obtained from a DeLP-program P, a dialectical tree for A 1 , L 1 from P is denoted T A 1 , L 1 and is constructed as follows: 1. The root of the tree is labeled with A 1 , L 1 . 2. Let N be a node labeled A n , L n , and [ A 1 , L 1 , . . . , A n , L n ] be the sequence of labels of the path from the root to N. Let { B 1 , Q 1 , B 2 , Q 2 , . . . , B k , Q k } be the set of all the defeaters for A n , L n from P. For each defeater B i , Q i (1 ≤ i ≤ k), such that the argumentation line Λ = [ A 1 , L 1 , . . . , A n , L n , B i , Q i ] is acceptable, the node N has a child N i labeled B i , Q i . If there is no defeater for A n , L n or there is no B i , Q i such that Λ is acceptable, then N is a leaf.
A dialectical tree provides a useful structure for considering all possible acceptable argumentation lines that can be generated for deciding whether the starting argument is defeated. Given a literal L and an argument A, L , to decide whether the literal L is warranted, every node in the dialectical tree T A,L is recursively marked as "D" (defeated) or "U" (undefeated), obtaining a marked dialectical tree T * A,L . Nodes are marked by a bottom-up procedure that starts marking all leaves in T * A,L as "U"s. Then, for each inner node B, Q of T * A,L , either: (a) B, Q will be marked as "U" iff every child of B, Q is marked as "D", or (b) B, Q will be marked as "D" iff it has at least a child marked as "U".
This marking procedure provides an effective way of determining if a DeLP-query L is warranted. It is important to note that given a DeLP-query L, there can be several arguments that support L; therefore, L will be warranted if there exists at least one argument A for L such that the root of a dialectical tree for A, L is marked as "U". Given an argument A, L obtained from a program P, we will write Mark(T * A,L ) = U to denote that the root of T * A,L is marked as "U"; otherwise we will write Mark(T * A,L ) = D (if the root of T * A,L is marked as "D"). Thus, we can define warrant in terms of the marking procedure Mark: Definition 20 [Warrant] Let (Π, ∆) be a dlp and L a ground literal. We say that L is warranted from (Π, ∆) if there exist at least one argument A, L from (Π, ∆), such that Mark(T * A,L ) = U, we also say that T * A,L warrants L and that A is a warrant for L. When no such argument exists the literal L is said to be unwarranted.
Each acceptable argumentation line can be seen as a two-player argument game like the grounded game except that the rules of the game are given by the conditions of Definition 18 on acceptable argument lines. Thus there is an equivalence between the above definition of warrant and the notion of a winning strategy for the proponent in the corresponding argument game. Given a dialectical tree in which the root is labelled U, the proponent in the game has a winning strategy by for each defeater moved by the opponent picking a reply from the tree that is labelled U. Conversely, if the proponent has a winning strategy in a game for argument A, then A will clearly have to be labelled U in its dialectical tree, since this tree contains the winning strategy as a subtree that contains all legal defeaters of any supporting argument in the tree. This equivalence will be exploited below in Section 6.4 in the proposal to consider a version of DeLP where the conclusions will be obtained through grounded semantics, we distinguish this particular system by denoting it as DeLP (GR) .

Comparing the Argument Definitions
In this section we compare the argument definitions of DeLP and ASPIC + . At first sight, the deduction nature of DeLP arguments would seem to allow a straightforward many-to-one mapping onto ASPIC + arguments in that DeLP arguments would capture one possible ordering of the inferences in an ASPIC + arguments. A mapping of this kind was by Prakken (2010) established between the arguments of assumption-based argumentation and ASPIC + arguments. However, two features of DeLP arguments prevent a straightforward mapping onto ASPIC + arguments: the minimality requirement and the consistency requirement. We discuss both requirements in turn.

On Rationality Postulates
In this and the following sections we will report several positive and negative results on satisfaction of the rationality postulates of Caminada and Amgoud (2007). We now make some introductory remarks on these postulates, in order to put the later results into perspective. While it is hard to disagree that the postulates of direct consistency and closure under subarguments should be satisfied, this is different for indirect consistency and closure under strict rules. 5 Various positions can be adopted. One position (which is the one of Caminada & Amgoud) is that strict closure and indirect consistency should always be satisfied, given the intuitive reading of strict rules S strictly implies p as "If S then always, or without exception, p".
Another position (which is the one of the first and third author of this paper), is that these properties are only desirable for sets of arguments that are not attackable (for instance, in DeLP or ASPIC + sets of arguments that only use facts and strict rules). By contrast, if an antecedent of a strict rule is provided by a defeasible rule, then it may be reasonable to not accept the consequent of the strict rule even if all its antecedents are accepted. This is one reason why a knowledge engineer in DeLP has to determine separately for each strict rule whether it transposes (cf. Section 3.3 above).
A third position (adopted by the second author of this paper in (Prakken 2016)) is that what is decisive is the properties of the argument ordering. Prakken (2010) defined when an argument is "reasonable" (in a technical sense) and showed that (together with indirect consistency of the necessary part of the knowledge base and closure of the strict rules under contraposition or transposition) both strict closure and indirect consistency are satisfied for ASPIC + if the argument ordering is 'reasonable'. Modgil and Prakken (2013) showed the same for a weaker definition of reasonable argument orderings. Prakken (2016) agrees with the second position that strict closure should only hold in general for sets of arguments that are not attackable. He then argues that whether strict closure and indirect consistency should hold for other cases depends on whether it makes sense to require that the argument ordering is 'reasonable', and, so he argues, this depends on the nature of the knowledge and inference rules.
With this in mind, in the remainder of this paper several results on (non-)satisfaction of strict closure and indirect consistency will be reported in a neutral way, without taking a stance on whether these results are good or bad for the investigated system(s). This posture is to leave the door open for more research on this crucial topic.

Minimality
DeLP requires arguments to be subset-minimal in their sets of defeasible rules. The general ASPIC + framework imposes no explicit minimality conditions on arguments, although the definition of an ASPIC + argument is such that it cannot contain 'unused' premises or rules: if an argument A contains inferences, then all premises in Prem(A) and all rules in Rules(A) are used in at least one inference. In various publications the addition of minimality conditions on arguments has been studied. Modgil and Prakken (2013) study a minimality requirement on the set of premises in order to establish relations with classical argumentation as studied by Gorogiannis and Hunter (2011). DeLP does not impose minimality of premise sets (which in DeLP are the facts used in an argument). It can even be the case that a DeLP argument which is minimal in its set of defeasible rules is non-minimal in its sets of premises or strict rules. Consider: A: { f 1 ⇒ p; p, f 2 → q}, q B: { f 1 ⇒ p; f 1 ⇒ r; p, r ⇒ q}, q where K = { f 1 , f 2 }. Argument A is minimal in its set of defeasible rules but B is minimal in its sets of premises and strict rules. The ASPIC + definition of an argument allows arguments that are non-minimal in their set of defeasible rules. The just-given example illustrates this, since ASPIC + counterparts of both A and B can be constructed. Requiring arguments to be minimal in their set of defeasible rules makes sense on the assumption that arguments with a non-minimal set of defeasible rules can never be stronger than a minimal version with the same conclusion (this assumption does not hold in general for ASPIC + ). For DeLP this assumption is reasonable, since defeasible rules are the only fallible elements in a DeLP argument. Given this assumption, the above example shows that if arguments are required to be minimal in their set of defeasible rules, they cannot be required to be also minimal in their sets of premises and/or strict rules.

Consistency
DeLP arguments have to be consistent in that no pair of complementary literals should be derivable from the set of all facts and strict rules of the program plus the defeasible rules used in the argument. In ASPIC + this would amount to saying that the set Conc(Sub(A)) ∪ K is indirectly consistent (since as noted above, all rules of an ASPIC + argument are used to derive conclusions). Only one publication on ASPIC + has studied the same constraint, namely, Prakken (2016). However, in that paper the constraint was combined with a different notion of rebutting attack, in order to capture forms of probabilistic reasoning. Wu and Podlaszewski (2015) study a slightly weaker constraint, namely, that Sub(A) is indirectly consistent.
The DeLP consistency constraint is intuitively appealling. However, Wu and Podlaszewski (2015) remark that their slightly weaker constraint for ASPIC + arguments induces counterexamples to indirect consistency. A very similar example can be constructed in DeLP.

Example 10
This enables the following DeLP arguments: In DeLP both A 2 and A 5 and A 3 and A 4 attack each other. If these conflicts are resolved with a last-link ordering on arguments as defined by Modgil and Prakken (2013), then the following sets of rules have to be compared: If the rules are in increasing order of priority ordered as f 1 ⇒ p < f 2 ⇒ ¬r < p ⇒ q then with the last-link ordering, by comparing sets on their minimal elements, we obtain that A 3 ≺ A 4 and A 5 ≺ A 2 . Then, in DeLP, the attacks of A 2 on A 5 and A 4 on A 3 succeed as proper defeats, so A 1 , A 2 and A 4 are warranted while A 3 and A 5 are not warranted. Thus, the set of warranted arguments is not strictly closed and not indirectly consistent.
We can conclude from this example that DeLP's strong consistency requirement on arguments does not in general suffice for satisfying strict closure and indirect consistency.

Comparing the attack relations
In order to compare the attack relations of DeLP and ASPIC + , we first define an ASPIC + counterpart of DeLP rebuttal.

Definition 21
[dlp-rebutting attack] A dlp-rebuts B iff for some B ∈ Sub(B) it holds that Conc(A)∪Conc(B)∪ K is indirectly inconsistent.
It is easy to verify for ASPIC + that rebut implies unrestricted rebut and unrestricted rebut implies dlp-rebut. Counterexamples to the converse implications can easily be constructed.
Next we address the question whether adopting dlp-rebut (but not DeLP's strong consistency condition) could improve ASPIC + . So in the remainder of this section we assume all definitions of ASPIC + except that rebutting attack is replaced with dlp-rebutting attack. In particular, we do for now not require that arguments are non-contradictory in the sense of Definition 14(2).
For complete, preferred and stable semantics the answer to our question arguably is negative, as can be shown with the following example, inspired by an example of Caminada and Wu (2011), who read it as 'any two of three persons can ride on a tandem together, but they cannot ride the tanden together all three of them'.

Example 11
Consider an ASPIC + AT with K = { f 1 , f 2 , f 3 } and R s = {p, q → ¬r; p, r → ¬q; q, r → ¬p} and R d consisting of the defeasible rules in the following arguments: With DeLP rebut, A + B and C rebut each other, A + C and B rebut each other and B + C and A rebut each other. If all arguments are incomparable in the argument ordering, then all these attacks succeed as defeats, so there exists an admissible set containing all of A, B and C, which violates strict closure and indirect consistency. Note that R s is closed under transposition. On the other hand, the grounded extension is { f 1 , f 2 , f 3 }, which is strictly closed and indirectly consistent.
This example shows that if satisfying strict closure and indirect consistency is regarded as desirable, then adopting dlp-rebut in ASPIC + is not in general an improvement but may be an improvement in special cases. The example also shows that adopting dlp-rebut affects the set of extensions in at least complete, preferred and stable semantics even if the strict rules are closed under transposition. Caminada et al. (2014) prove for unrestricted rebut (see Definition 8 above) that for a limited case with a total preference ordering on the set of defeasible rules and a weakest-or last-link argument ordering, the grounded extension satisfies both strict closure and indirect consistency (under the assumption that the set of strict rules is closed under transposition). Since unrestricted rebut and dlp-rebut are similar, it is interesting to see if similar results can be obtained for dlprebut. We first investigate this for the so-called simple argument ordering, which is such that A B iff A is defeasible and B is strict. Direct consistency can be easily shown on the assumption that K is indirectly consistent.

Proposition 12
Suppose attack in ASPIC + is dlp-rebut, K is indirectly consistent and the argument ordering is simple. Then for any AF corresponding to a SAF with grounded extension E, it holds that there is no ϕ such that both ϕ and ¬ϕ are in Concs(E).

Proof
Suppose for contradiction ϕ, ¬ϕ ∈ Conc(E). Then there exist two arguments A and B in E such that Conc(A) = ϕ and Conc(B) = ¬ϕ. Since K is assumed to be indirectly consistent, at least one of A and B is defeasible. Assume without loss of generality that B is defeasible. Then A ≺ B so A defeats B. But then E is not conflict free.
However, unlike in the case studied by Caminada et al. (2014) there are counterexamples to strict closure and indirect consistency for ASPIC + with dlp-rebut even if K is indirectly consistent and R s is closed under transposition.

Example 13
Let K = {t} and R d = {⇒ a 1 , ⇒ a 2 , ⇒ q} while R s consists of the following rules: a 1 , a 2 → p p, q → r p, q → ¬r t, r → s t, r → ¬s a 1 , ¬p → ¬a 2 p, ¬r → ¬q p, r → ¬q t, ¬s → ¬r t, s → ¬r a 2 , ¬p → ¬a 1 q, ¬r → ¬p q, r → ¬p r, ¬s → ¬t r, s → ¬t t, ¬r → u t, ¬r → ¬u t, ¬u → r t, u → r ¬r, ¬u → ¬t ¬r, u → ¬t We first show that arguments ⇒ a 1 and ⇒ a 2 are in the grounded extension but ⇒ a 1 , ⇒ a 2 → p is not. Consider first argument ⇒ a 1 . This argument has several defeaters. All of them combine the arguments for p and q to conclude either r or ¬r and then combine the resulting argument for either r or ¬r with the argument for q into an argument for ¬p. This argument then is with ⇒ a 2 combined into an argument for ¬a 1 . These complex arguments all have a strict defeater, namely, t, since t together with r implies s and ¬s while t together with ¬r implies u and ¬u. Note that t is strictly preferred over all arguments it dlp-rebuts, sine t is strict while all arguments it dlp-rebuts are defeasible. Since t, being strict, has no defeaters by consistency of K, there is a winning strategy in the grounded game for ⇒ a 1 , so ⇒ a 1 is in the grounded extension.
The proof that ⇒ a 2 is in the grounded extension is entirely similar. We next show that ⇒ a 1 , ⇒ a 2 → p is not in the grounded extension. Call this argument A. It is dlp-rebutted by argument C of the form ⇒ q. Since both A and C are defeasible, they defeat each other. Next, observe that there is no strict defeater of C, since all dlp-rebuttals of C need the argument for p as a subargument, which is defeasible. So there is no winning strategy for A in the grounded game, so A is not in the grounded extension.
Note that this also yields a counterexample if the set of defeasible rules is totally ordered and arguments are compared with the weakest-or last-link argument ordering as in Caminada et al. (2014), since we can then give all four defeasible rules equal priority.
Note that various arguments in the example are contradictory in the sense of Definition 14(2), so imposing DeLP's consistency condition on arguments (by requiring that the set of all conclusions of all their subarguments is indirectly consistent), excludes this counterexample for DeLP. However, other counterexamples for DeLP exist, for instance, Example 10 from Section 4.3. If strict closure and indirect consistency are regarded as desirable, then this is worrying not just for DeLP but also for ASPIC + since with the original ASPIC + definition of rebuttal strict closure and indirect consistency can, as noted above, for the full case with preferences not be shown without allowing inconsistent arguments. We have now seen that replacing rebut with dlp-rebut does not change this, so as regards strict closure and indirect consistency the current state-of-the art for ASPIC + is still suboptimal.

Differences in argument evaluation
While the DeLP definition of warrant is similar to grounded semantics, there are also differences, caused by the fact that the constraints on argument lines (Definition 18) do not coincide with the constraints on games in the game-theoretic proof theory for grounded semantics (Definition 1). The choices made on the definition of argumentation lines by Garcia and Simari (2004) establish constraints based on particular intuitions that were shown in examples. However, we believe this definition of argumentation lines has some arguably counterintuitive consequences over the set of warranted literals. We will discuss them below and then show that adopting grounded semantics avoids these counterintuitive consequences.

Having to move a proper defeater after a blocking defeater
Condition (4) of the definition of an acceptable argument line requires the move of a proper defeater if the previous argument was a blocking defeater, regardless whether the previous argument was a supporting or an interfering argument. This differs from the grounded game, in which the proponent must move strict defeaters while the opponent can move weak defeaters.

Example 14
Consider a dlp with Π f = {p, s, u}, Π s = / 0 and ∆ consisting of the rules of the following arguments: A 2 and B 2 attack each other on ¬r and r, while C attacks B 1 and thus also B 2 on t. Assume an argument ordering that makes A 2 and B 2 as well as C and B 1 blocking defeaters of each other (for example, by not assigning any priority to the rules). Then A 2 is not warranted. Its dialectical tree consists of just one argument line, namely, A 2 , B 2 and A 2 is marked D in this tree. Note that the line cannot be extended with C, since B 2 is a blocking defeater of A 2 while C is not a proper defeater of B 2 . Assume now an argument ordering in which C and B 1 are still blocking defeaters of each other but in which B 2 is a proper defeater of A 2 (for instance, by giving all rules in B 2 priority over all rules in A 2 ). Then A 2 is warranted, since its dialectical tree again consists of just one argument line but now it is A 2 , B 2 ,C and A 2 is marked U in this tree. Note that the line cannot be extended with B 1 since C is a blocking defeater of B 2 while B 1 is not a proper defeater of C.
So by strengthening B's attack on A, A turns from not warranted into warranted, which seems counterintuitive. This example could be analyzed from the reverse perspective of A being warranted and debilitating B to a blocking defeater will lead to A not being warranted. This clash of intuitions becomes an interesting issue to study, and a possible avenue for doing so is to adopt the grounded game because supporting arguments have to be strict defeaters while interfering arguments can be weak defeaters. According to the grounded game, A 2 is not justified with the second argument ordering, since the line A 2 , B 2 ,C can be extended with B 1 after which C is not allowed since it is not a strict defeater of B 1 .

The non-repetition rule
Suppose that in line with our analysis of Example 14 condition (3) of acceptable argument lines is changed to the effect that supporting arguments must be strict defeaters while interfering arguments can be weak defeaters. Then in Example 14 argument A 2 is still warranted with the second argument ordering, since the line A 2 , B 2 ,C cannot be extended with B 1 for another reason: B 1 is a subargument of an argument already moved in the line, namely, B 2 , so condition (2) of acceptable argument lines prevents extending the line with B 1 . So this condition could also be changed by adopting the rule of the grounded game that only the proponent cannot repeat its arguments. Moreover, this non-repetition rule might not be extended to proper subarguments of an already-moved argument, as shown by the following example.

Example 15
Consider a dlp with Π f = { f 1 , f 2 , f 3 , f 4 }, Π s = / 0 and R d consists of the rules of the following arguments. Assume also that arguments are ordered according to strict specificity relations between the conflicting rules.

A:
{p Note that A and B weakly defeat each other, C strictly defeats B on its subargument for q, and D strictly defeats C by weakly defeating its subargument for s. Now there is a strict defeater of D, namely, However, E is a subargument of C, so if constraint (3) on argument lines is adopted in the grounded game, then the game loses completeness, since A and C are in the grounded extension. Note that the non-repetition rule of the grounded game does not prevent the moving of E, since E is not identical to C.
We next show that DeLP's non-repetition rule can in combination with the other DeLP constraints on argument lines make that the set of warranted arguments is not admissible in the sense of Dung (1995).

Example 16
Consider a dlp with Π f = { f 1 , f 2 , f 3 , f 4 , f 5 }, Π s = / 0 and R d consists of the rules of the following arguments. Assume also that arguments are ordered according to strict specificity relations between the conflicting rules.

A:
{p According to specificity, B is a proper defeater of A, C is a blocking defeater of B, D is a proper defeater of C and E is a proper defeater of D. So A, B,C, D, E is an acceptable argument line.
Here the argument line terminates, while there is a proper defeater of E, namely F: But F cannot be appended to the argument line since it is a subargument of B. So A is warranted. Moreover, it is easy to see that C, which is a supporting argument for A, is not warranted, because of the argument line Note that here F can be appended to C, D, E, since A is not in this line.
In sum, we have that A is warranted even though there is no warranted supporting argument that defends A against its defeater B. So we end up with a set of warranted arguments that is not admissible in the sense of Dung (1995). If admissibility is accepted as a minimum rationality constraint on argument evaluation, then this is another reason to adopt grounded semantics for DeLP.
García and Simari motivate their non-repetition rule with an example that has essentially the same structure as Example 9. In this example they want to prevent infinite argumentation lines. The grounded game indeed prevents this: the only possible game for A 2 is P 1 : A 2 , O 1 : B 2 and the game terminates with a win by O since P cannot repeat its argument A 2 .
García and Simari motivate the subargument part of their non-repetition rule with a schematic example of the following form (leaving the rules implicit):

Example 17
Consider the following arguments, where each argument X 1 is a subargument of X 2 : Assuming that the other constraints on argumentation lines are satisfied, they want to prevent the infinite line A 2 , B 2 ,C 2 , D 2 , A 1 , B 2 , . . .. In the grounded game this is indeed achieved, since the proponent is not allowed to repeat C 2 in attack on B 2 .
We conclude that the non-repetition rule of the grounded game treats all of García and Simari's examples in the way they want and avoids the arguably counterintuitive outcomes of DeLP in other examples.

Concordance
Constraint (2) on argument lines requires both the set of all supporting and the set of all interfering arguments in the line to be concordant, which means that the set of all rules of all these arguments must be consistent with the facts. If the dlp satisfies Caminada and Amgoud's (2007) rationality postulate of strict closure (saying that the set of all conclusions of all arguments in an extension must be indirectly consistent), then for the set of supporting arguments the requirement of concordance is a semantically redundant but computationally desirable addition to the grounded game: redundant since if an argument is warranted/justified, then all of proponent's arguments in a winning strategy for the argument will be in the grounded extension and will thus satisfy concordance; and desirable since it may prune the search space. However, otherwise concordance for supporting arguments can make a difference. In such cases there seems to be no clear reason why either adopting or not adopting concordance is better. However, for the set of interfering arguments requiring concordance is arguably undesirable, as the following example shows.

Example 18
Consider a dlp with Π f = { f 1 , f 2 , f 3 , f 4 }, Π s = / 0 and R d consists of the rules of the following arguments. Assume also that arguments are ordered according to strict specificity relations between the conflicting rules.

A:
{p The dialectical tree for A has just one line, viz. A, B,C. Note that D cannot be appended to the line since B and D support contradictory (sub)conclusions q and ¬q. So A is warranted. Yet C, which supports A, is not warranted, since the line C can be extended with D. Note also that {A} is not an admissible set since it does not defend A against B while {A,C} is not an admissible set, since it does not defend C against D.
If admissibility is adopted as a minimum constraint on sets of warranted arguments, then this example shows that concordance for the set of interfering arguments is undesirable. In other words, arguing in favour of concordance for the set of interfering arguments requires arguing against admissibility as a minimum requirement on sets of warranted arguments. A less controversial solution is to adopt the game for grounded semantics, which, as shown above, is arguably also a good idea for other reasons.

Example 19
Consider the following arguments, built from the rules in the previous example, where each argument X 1 is a subargument of X 2 : Assuming that the other constraints on argumentation lines are satisfied, they want to prevent the line A 2 , B 2 ,C 2 , A 2 . In the grounded game this not prevented but this does not make A 2 warranted or the line infinite. The line can (as an argument game) only be continued with P 3 : B 2 , O 3 : C 2 after which the game terminates with a loss by the proponent since he cannot repeat A 2 . So the grounded game satisfies García and Simari's intuitions about this example.

Reformulating DeLP with grounded semantics
We have seen that DeLP's definition of warrant has some arguably counterintuitive outcomes, due to the particular constraints on acceptable argumentation lines. We have also seen that adopting grounded semantics both avoids these outcomes and treats García and Simari's motivating examples in the way they want. Therefore and because of the similarities between DeLP's notion of warrant and the grounded argument game, it seems a good idea to develop a version of DeLP with its current account of warrant replaced with grounded semantics. This can be done by replacing the current constraints on acceptable argument lines with the rules of the grounded game (where proponent and opponent arguments are, respectively, equated with supporting and interfering arguments), and by replacing proper and blocking defeat with strict defeat and defeat. Then given the equivalence noted at the end of Section 3.3, either the grounded game or an accordingly modified definition of dialectical trees can be used to redefine warrant. Thus adopting grounded semantics for DeLP would also establish clear links between DeLP and a large body of other research on formal argumentation. Among other things, it would facilitate a study of the satisfaction of rationality postulates in DeLP. Finally, a version of DeLP with grounded semantics would in fact adopt the semantics of Simari and Loui (1992), which paper was the original source of inspiration for the development of DeLP and which, as noted above, proposes a notion of warrant that was by Dung (1995) shown to be equivalent to grounded semantics. The question arises whether the move to grounded semantics and leaving all other definitions as they are (let us call the resulting system DeLP (GR) ) changes anything as regards strict closure and indirect consistency. As it turns out, the answer is no, since Example 10 from Section 4.3 is also for DeLP (GR) a counterexample to satisfaction of strict closure and indirect consistency. The point is that neither A 2 nor A 5 has a defeater, so the grounded game for these arguments ends after the first move. So while the move to DeLP (GR) ensures admissibility of the set of warranted arguments, it does not guarantee strict closure and indirect consistency of the set of warranted conclusions. On the other hand, Proposition 12 applies to DeLP (GR) , so the set of warranted conclusions is, as in original DeLP, guaranteed to be directly consistent.

Correspondence results
We next study correspondence results between aspects of DeLP, DeLP (GR) and ASPIC + . Of all these results, only Proposition 26 will depend on DeLP (GR) ; all other results hold for both DeLP and DeLP (GR) . Throughout this section we will implicitly assume corresponding defeasible logic programs and argumentation theories with the same language and the same sets of rules and facts. That is, we assume that f ∈ K just in case f ∈ Π f , that S → ϕ ∈ R s just in case ϕ ← S ∈ Π s and that S ⇒ ϕ ∈ R s just in case ϕ − −≺ S ∈ ∆. Below we will leave the translation between the ASPIC + and DeLP notations implicit.
We first address the problem of finding a correspondence between DeLP arguments and AS-PIC + arguments. To find such a correspondence, several assumptions on ASPIC + argumentation theories are needed, since DeLP has unlike ASPIC + minimality and consistency conditions on arguments. Accordingly, we define simplified ASPIC + argumentation theories as those AT in which all arguments A are minimal in that if Conc(A) = p then there exists no argument A such that Conc(A ) = p and DefRules(A ) ⊂ DefRules(A); and in which for all arguments A the set Conc(Sub(A)) ∪ R s ∪ K is indirectly consistent.

Lemma 20
Let AT = (AS, K) be any ASPIC + argumentation theory. For any argument A based on AT it holds that Conc(Sub(A)) equals the set of all antecedents and consequents of any rule in Rules(A).

Proof
Immediate from the definition of an ASPIC + argument.

Proposition 21
Let (Π, ∆) be any defeasible logic program with a corresponding ASPIC + argumentation theory (AS, K). For any DeLP argument D = R, p given (Π, ∆) there exists an ASPIC + argument A for p on the basis of (AS, K) with DefRules(A) = R.

Proof
Let D = R, p be any DeLP argument given (Π, ∆). Then there exists a defeasible derivation Dd = L 1 , . . . , L n = p of p given (Π, R). Assume without loss of generality that Dd is minimal. We prove by induction on the definition of defeasible derivations that for any element of Dd there exists an ASPIC + argument A on the basis of (AS, K) with DefRules(A) ⊆ R.
There are two base cases. If L i is a fact, then L i ∈ K so L i is an ASPIC + argument with The induction hypothesis is that for all elements L i of Dd such that there exists a rule r in Π ∪ R with body B 1 , . . . , B m and head L i and such that all of B 1 , . . . , B m precede L i in Dd there exists an ASPIC + argument C j for any B j (1 ≤ j ≤ m) with DefRules(C j ) ⊆ R. For the induction step consider any such a rule r and let the ASPIC + arguments for B 1 , . . . , B m be C 1 , . . . ,C m . Then if r ∈ Π then r ∈ R s , so C = C 1 , . . . ,C m → L i is an ASPIC + argument with DefRules(C) = DefRules(C 1 ) ∪ . . . DefRules(C m ) ⊆ R. Otherwise, r ∈ R so r ∈ R d , so C 1 , . . . ,C m ⇒ L i is an ASPIC + argument with DefRules(C) = DefRules(C 1 ) ∪ . . . DefRules(C m ) ∪ {r} ⊆ R.
Finally, to prove that DefRules(A) = R, assume for contradiction that there exists a rule r ∈ R such that r ∈ DefRules(A). Then by Lemma 20 it holds that r's head is not in Conc(Sub(A)). But consider then the sequence Dd obtained by listing all elements of Conc(Sub(A)) in any order such that the bodies of any rule precede the rule's head. It is easy to verify that Dd is a defeasible derivation of p given (Π, ∆). But Dd ⊂ Dd, so Dd is not minimal: contradiction.

Proposition 22
Let (Π, ∆) be any defeasible logic program with a corresponding ASPIC + argumentation theory (AS, K) that is simplified. For any ASPIC + argument A for p on the basis of (AS, K) there exists a DeLP argument D = DefRules(A), p given (Π, ∆).

Proof
Suppose A is an ASPIC + argument for p. There are two base cases. Assume first A = p. Then p ∈ Π f and p is a defeasible derivation since p is a fact. Moreover, since the ASPIC + AT is simplified, {p} ∪ R s ∪ K is indirectly consistent, so (since Π = R s ∪ K), no pair of contradictory literals can be derived from Π ∪ / 0. Finally, / 0 is obviously a minimal subset of ∆ satisfying all this. So D = / 0, p is a DeLP argument for p given (Π, ∆). Assume next A is of the form ⇒ p. Then p is a defeasible derivation since ⇒ p ∈ ∆ so p is a presumption. Moreover, since AT is simplified, there exists no strict ASPIC + argument for p so A is minimal in its set of defeasible rules. Then the proof that D = {⇒ p}, p is a DeLP argument for p given (Π, ∆) is similar as for facts.
The induction hypothesis is that for any ASPIC + argument {A 1 , . . . , A m } →/⇒ p there exist DeLP arguments for Conc(A 1 ), . . . , Conc(A m ) given (Π, ∆). For the induction step, consider any such ASPIC + argument. Then there exist defeasible derivations Dd j for all these conclusions Conc(A j ) (1 ≤ j ≤ m). Then clearly Dd 1 , . . . , Dd m , p is a defeasible derivation for p. Moreover, since the ASPIC + AT is assumed to be simplified, Conc(Sub(A)) ∪ K is indirectly consistent and since the heads of all rules in DefRules(A) are in Conc(Sub(A)), no pair of contradictory literals can be derived from Π ∪ DefRules(A) (recall that Π = R s ∪ K). If there existed a DeLP argument D = R, p for p such that R ⊂ DefRules(A), then by Proposition 21 there would exist an ASPIC + argument A for p with DefRules(A ) = R. But then A would not be minimal in its set of defeasible rules (note that Conc(Sub(A )) ∪ K is indirectly consistent since D satisfies the consistency constraint on DeLP arguments). So then AT would not be simplified, which contradicts our assumption that it is simplified. Thus D = DefRules(A), p is a DeLP argument for p given (Π, ∆).
There are counterexamples to Proposition 22 for non-simplified ASPIC + argumentation theories. Some counterexamples are due to DeLP's consistency constraint on arguments. Consider an ASPIC + AT with K = {p} and R d = {⇒ ¬p}. Then there are ASPIC + arguments p for p and ⇒ ¬p for ¬p but the latter has no corresponding DeLP argument, since both p and ¬p can be defeasibly derived from Π ∪ {⇒ ¬p} (recall that K ∈ Π). Other counterexamples are due to DeLP's minimality constraint on arguments. Consider an ASPIC + AT with K = {p}, R s = {p → q} and R d = {⇒ q}. Then the ASPIC + argument ⇒ p for p has no corresponding DeLP argument, since / 0, q is a DeLP argument for q, so {⇒ q}, q is not minimal. It can be shown that the correspondence between ASPIC + and DeLP arguments is many-toone.

Proposition 23
Let (Π, ∆) be any defeasible logic program with a corresponding ASPIC + argumentation theory (AS, K) that is simplified.
1. For some DeLP arguments given (Π, ∆) there exist more than one corresponding ASPIC + argument on the basis of (AS, K). 2. For all ASPIC + arguments (AS, K) there exist a unique corresponding DeLP argument given (Π, ∆).

Proof
For (1)  (2) follows since each ASPIC + argument has a unique set of defeasible rules and uses all these rules, while DeLP arguments are defined by minimal sets of defeasible rules. Then the construction in the proof of Proposition 22 clearly induces a unique DeLP argument.
We next prove correspondences with respect to the attack relations. To this end, we first define DeLP-counterparts of ASPIC + 's rebutting and unrestricted rebutting attacks.

Definition 22
[a-rebutting and ua-rebutting attack] A DeLP argument A, p a-rebuts a DeLP argument B, q on B , q if B , q is a subargument of B, q and p = −q and q was derived in B , q with a defeasible rule.
A DeLP argument A, p ua-rebuts a DeLP argument B, q on B , q if B , q is a subargument of B, q and p = −q and B = / 0.
In the proofs of the following propositions we overload the symbol ⊆ by writing for two DeLP arguments D 1 = S 1 , p and D 2 = S 2 , q that D 1 ⊆ D 2 to mean that S 1 ⊆ S 2 . Likewise for other set-theoretic notations.

Proposition 24
For all ASPIC + arguments A and A on the basis of a simplified argumentation theory it holds that if A is (rebutted, u-rebutted, dlp-rebutted) by A then the DeLP argument D corresponding to A is (a-rebutted, ua-rebutted, rebutted) by the DeLP argument D corresponding to A .

Proof
Suppose ASPIC + argument A for p rebuts, u-rebuts or dlp-rebuts ASPIC + argument A on A for p. Consider the corresponding DeLP arguments D for p and D for p, which exist and are unique by For dlp-rebut, K ∪ {p, p } is indirectly inconsistent. Then in DeLP the set Π ∪ {p, p } is contradictory. Then D rebuts D . But then D rebuts D since D ⊆ D.

Proposition 25
For all DeLP arguments D and D based on any defeasible logic program it holds that if D is (rebutted, a-rebutted, ua-rebutted) by D then all ASPIC + arguments A corresponding to D are (dlp-rebutted, rebutted, u-rebutted) by any ASPIC + argument A corresponding to D .

Proof
Suppose D for p rebuts, a-rebuts or ua-rebuts D on D for p. Consider any ASPIC + argument A corresponding to D and A corresponding to D , which exist by Proposition 22. By the same proposition, DefRules(A) = D and DefRules(A ) = D .
For rebut, Π ∪ {p, p } is contradictory, i.e., there exists a defeasible derivation for two literals h and −h from this set. Note that such a defeasible derivation is strict since the set contains no defeasible rules. Then in ASPIC + the set K ∪ {p, p } is indirectly inconsistent. By Lemma 20 and the fact that DefRules(A) = D, we have that p ∈ Conc(Sub(A)) and p ∈ Conc(Sub(A )). Then A dlp-rebuts A.
For a-rebut, p and p are complementary literals while p was derived in D with a defeasible rule. Then p = −p . By Lemma 20 and the fact that DefRules(A) = D, we have that p ∈ Conc(Sub(A)). Moreover, by construction of A, p is derived in A with a defeasible rule. Then A a-rebuts some subargument of A so A a-rebuts A.
For ua-rebut, p and p are complementary literals while D = / 0. Then p = −p . By Lemma 20 and the fact that DefRules(A) = D, we have that p ∈ Conc(Sub(A)). Then A a-rebuts some subargument of A so A a-rebuts A.
Propositions 21, 22, 24 and 25 together imply the following proposition (recall that DeLP (GR) is the variant of DeLP modified with grounded semantics).

Proposition 26
Let (Π, ∆) be any defeasible logic program with a corresponding ASPIC + argumentation theory (AS, K) that is simplified. Let the DeLP (GR) and ASPIC + orderings coincide in that for all DeLP (GR) arguments D 1 and D 2 it holds that D 1 ≺ D 2 iff for all corresponding ASPIC + arguments A 1 and A 2 it holds that A 1 ≺ A 2 and for all ASPIC + arguments A 1 and A 2 it holds that A 1 ≺ A 2 iff for the corresponding DeLP arguments D 1 and D 2 it holds that D 1 ≺ D 2 . Let also DeLP (GR) attack be rebut (a-rebut, ua-rebut) iff ASPIC + attack is dlp-rebut (rebut, u-rebut). Then any DeLP (GR) argument D is warranted iff all corresponding ASPIC + arguments A are justified and any ASPIC + argument A is justified iff the corresponding DeLP argument D is warranted.

Proof (Sketch)
From left to right, consider any DeLP argument D that is warranted. Then there exists a dialectical tree T D for D with D labelled U. Consider any corresponding dialectical tree T A of ASPIC + arguments obtained by replacing any supporting argument in T D by some corresponding ASPIC + argument and replacing any interfering argument in T D by all corresponding ASPIC + arguments. By Proposition 25 and our assumptions on the argument orderings, the defeat relations between DeLP arguments in T A are preserved as defeat relations between the corresponding ASPIC + arguments in T D . It is left to prove that T A contains all defeaters of any of its supporting arguments. Assume for contradiction that some defeater B of some supporting argument A in T A is not in T A . By Proposition 22 B has a unique corresponding DeLP argument B d . Then by Proposition 24 and our assumptions on the argument orderings it holds that B d defeats the DeLP argument A d corresponding to A. But then B d is in T D so B is in T A . Contradiction.
From right to left the proof is similar but simpler since any ASPIC + argument has just one corresponding DeLP argument.

Conclusion
In this paper we have made detailed comparisons between DeLP and ASPIC + as formalisms for rule-based argumentation. The comparisons especially focussed on inter-translatability, consistency and closure properties and intuitive adequacy. Computational and implementational issues were not discussed but from other sources, such as Modgil and Prakken (2018) and Garcia and Simari (2018), it is clear that these issues have received substantially more attention for DeLP than for ASPIC + . Our comparisons have hopefully contributed to a better understanding of the two formalisms and their relations, similarities and differences.
To summarise our main findings, we have first seen that DeLP's notion of rebutting attack and its consistency and minimality constraints on arguments are intuitively appealing and in some special cases more so than there ASPIC + counterparts. However, we have also seen that the DeLP definitions may not fully comply with Caminada and Amgouds rationality postulates of strict closure and indirect consistency in cases where ASPIC + satisfies these postulates. In Subsection 4.1 we have included a thorough discussion about these issues.
Furthermore, we have argued that there are reasons to consider a variant of DeLP with grounded semantics, since its current notion of warrant arguably has counterintuitive consequences in some examples and in general leads to sets of warranted arguments that are not admissible. We have seen that both problems can be avoided by adopting the argument game for grounded semantics in DeLP. A version of DeLP with grounded semantics would also be a return to the semantics of Simari and Loui (1992), which paper was the original source of inspiration for the development of DeLP. Arguing in defence of DeLP's current notion of warrant requires arguing, first, that its treatment of Example 14 is not counterintuitive and second, that admissibility is not a minimum requirement for sets of warranted arguments. Alternatively, if one agrees that Example 14 is treated incorrectly by DeLP but not that admissibility must be satisfied by warrant, then less substantial changes in the definition of acceptable argument lines might suffice.
Finally, we have under some minimality and consistency assumptions on ASPIC + arguments identified a one-to-many mapping between DeLP arguments and ASPIC + arguments in such a way that if DeLP is modified with grounded semantics, then the resulting DeLP (GR) 's notion of warrant is equivalent to ASPIC + 's notion of justification. This result was proven for three alternative definitions of attack.
As for future research, since DeLP (GR) generates abstract argumentation frameworks, it can be investigated to which extent properties of DeLP (GR) depend on grounded semantics or are inherited by other semantics for abstract argumentation frameworks. Future research could also investigate whether incorporating DeLP's notion of rebutting attack and/or its consistency and minimality requirements on arguments in ASPIC + can be done in a way that fully preserves the current results on how ASPIC + respects the various rationality postulates. Given the results in this paper, this would require further changes in the ASPIC + framework.