RP-DeLP: a weighted defeasible argumentation framework based on a recursive semantics

Inthisarticle,weproposearecursivesemanticsforwarrantedformulasinageneraldefeasiblelogicargumentationframeworkbyformalizinganotionofcollective(non-binary)conﬂictamongarguments.Therecursivesemanticsforwarrantedformulas isbasedontheintuitivegroundsthatifanargumentisrejected,thenfurtherargumentsbuiltontopofitshouldalsoberejected.Themaincharacteristicofourrecursivesemanticsisthatanoutput(orextension)ofaknowledgebaseisapairconsistingofa setofwarrantedandasetofblockedformulas.Argumentsforbothwarrantedandblockedformulasarerecursivelybasedonwarrantedformulasbut,whilewarrantedformulasdonotgenerateanycollectiveconﬂict,blockedconclusionsdo.Formulas thatareneitherwarrantednorblockedcorrespondtorejectedformulas.Thenweextendtheframeworkbyattachinglevelsofpreferencetodefeasibleknowledgeitemsandbyprovidingalevel-wisedeﬁnitionofwarrantedandblockedformulas.After weconsiderthewarrantrecursivesemanticsfortheparticularframeworkofPossibilisticDefeasibleLogicProgramming(RP-DeLPforshort).SinceRP-DeLPprogrammesmayhavemultipleoutputs,wedeﬁnethe maximal ideal output of an RP-DeLP programme as the set of conclusions which are ultimately warranted, and we present an algorithm for computing it in polynomial space and with an upper bound on complexity equal to P NP . Finally, we propose an efﬁcient and scalable implementation of this algorithm using SAT encodings, and we provide an experimental evaluation when solving test sets of instances with single and multiple preference levels for defeasible knowledge.


Introduction and motivation
Defeasible argumentation is a natural way of identifying relevant assumptions and conclusions for a given problem which often involves identifying conflicting information, resulting in the need to look for pros and cons for a particular conclusion [56].This process may involve chains of reasoning, where conclusions are used in the assumptions for deriving further conclusions and the task of finding pros and cons may be decomposed recursively.Logic-based formalizations of argumentation that take pros and cons for some conclusion into account assume a set of formulas and then lay out arguments and counterarguments that can be obtained from these assumed formulas [26].
Defeasible Logic Programming (DeLP) [42] is a formalism that combines techniques of both logic programming and defeasible argumentation.As in logic programming, knowledge is represented in DeLP using facts and rules; however, DeLP also provides the possibility of representing defeasible knowledge under the form of weak (defeasible) rules, expressing reasons to believe in a given conclusion.In DeLP, a conclusion succeeds if it is warranted, i.e., if there exists an argument (a consistent sets of defeasible rules) that, together with the non-defeasible rules and facts, entails the conclusion, and moreover, this argument is found to be undefeated by a warrant procedure which builds a dialectical tree containing all arguments that challenge this argument, and all counterarguments that challenge those arguments, and so on, recursively.Actually, dialectical trees systematically explore the universe of arguments in order to present an exhaustive synthesis of the relevant chains of pros and cons for a given conclusion.In fact, the interpreter for DeLP [41] (http://lidia.cs.uns.edu.ar/DeLP)takes a knowledge base (programme) P and a conclusion (query) Q as input, and it then returns one of the following four possible answers: YES, if Q is warranted from P; NO, if the complement of Q is warranted from P; UNDECIDED, if neither Q nor its complement are warranted from P; or UNKNOWN, if Q is not in the language of the programme P.
Possibilistic Defeasible Logic Programming (P-DeLP) [5] is an extension of DeLP in which defeasible rules are attached with weights (belonging to the real unit interval [0,1]) expressing their relative belief or preference strength.As many other argumentation frameworks [32,56], P-DeLP can be used as a vehicle for facilitating rationally justifiable decision making when handling incomplete and potentially inconsistent information.Actually, given a P-DeLP programme, justifiable decisions correspond to warranted conclusions (to some necessity degree), i.e., those which remain undefeated after an exhaustive dialectical analysis of all possible arguments for and against.
In [30] Caminada and Amgoud proposed three rationality postulates which every rule-based argumentation system should satisfy.One of such postulates (called Indirect Consistency) requires that the set of warranted conclusions must be consistent (w.r.t. the underlying logic) with the set of strict facts and rules.In [30] a number of rule-based argumentation systems were identified in which such postulate does not hold (including DeLP [42] and Prakken and Sartor's [55], among others).As a way to solve this problem, the use of transposed rules is proposed in [30] to extend the representation of strict rules.Recently, in [6] Amgoud proposes a new rationality postulate (called Closure under Subarguments) which rule-based argumentation systems should satisfy.This postulate claims that the acceptance of an argument should imply also the acceptance of all its subarguments which reflect the different premises on which the argument is based.
Since the dialectical analysis-based semantics of (P-)DeLP for warranted conclusions does not satisfy the Indirect Consistency postulate, our aim in this article is to provide (P-)DeLP with a new semantics satisfying the above mentioned postulates.To this end, we consider recursive semantics for defeasible argumentation as defined by Pollock in [53], where recursive definitions of conflict between arguments were characterized by means of inference-graphs, representing (binary) support and attack (pros and cons) relations among the conclusions of arguments.Recursive semantics are based on the fact that if an argument is rejected, then all arguments built on top of it should also be rejected.On the other hand, as stated in [53], recursive definitions of conflict among arguments can lead to different outputs (extensions) for warranted conclusions.
The first contribution of this article is to define a recursive semantics for warranted conclusions in a quite general framework (without levels of strength) by formalizing a new collective (nonbinary) notion of conflict between arguments.The main characteristic of our recursive semantics is that an output (extension) of a knowledge base is now a pair of sets, a set of warranted and a set of blocked formulas.Arguments for both warranted and blocked formulas are recursively based on warranted formulas but, while warranted formulas do not generate any conflict with the set of already warranted formulas and the strict part of the knowledge base (information we take for granted they hold true), blocked formulas do.Formulas that are neither warranted nor blocked correspond to rejected formulas.The key feature that our warrant recursive semantics addresses corresponds with the closure under subarguments postulate recently proposed by Amgoud [6], claiming that if an argument is excluded from an output, then all the arguments built on top of it should also be excluded from that output.
The second contribution of this article is to extend the recursive semantics to a general argumentation framework with defeasibility (preference) levels, by providing a level-wise definition of warranted and blocked conclusions.We characterize the properties of outputs in terms of some propagation criteria between defeasibility levels of warranted and blocked conclusions.
The third contribution of this article is to specialize the warrant recursive semantics with defeasibility levels to the particular framework of P-DeLP, we refer to this formalism as Recursive P-DeLP (RP-DeLP for short).Following the approach of Pollock [53], in RP-DeLP inferences from some propositions to others and definitions of conflict are characterized by means of what we call Warrant Dependency Graphs, representing support and (collective) conflict relations between argument conclusions.For example, let Jones and Smith be two experts on economy discussing about whether in our country the economy is improving or not.Jones' opinion is that 'the economy improves (E) since the export surplus is increasing (ES)', while Smith believes 'the economy is not improving (∼E) since the income taxes have decreased (T )'.Assuming Smith and Jones are equally reliable, what should one believe?It seems clear that one should accept neither E nor ∼E.This situation is represented in the warrant dependency graph of Figure 1 where dashed arrows represent inferences and continuos arrows represent conflicts.In RP-DeLP, the direct binary attack between E and ∼E expresses that E blocks ∼E, and vice versa.
The situation may turn more complex when conflict loops among arguments appear in the warrant dependency graph.In such a case, the recursive semantics for warranted conclusions may result in multiple outputs for RP-DeLP programmes.For example, assume Jones' opinion is that 'if the economy improves (E) then the salaries will increase (S) and, as a consequence, the income taxes will increase (T ) as well'.On the other hand, Smith's point of view is that 'if the income taxes decrease (∼T ) then the public investment will decrease (I), and thus the economy will eventually shrink (∼E)'.Assume further that some indices point out that economy is improving (E) and that the government wants to decrease taxes (∼T ), and assume again that Jones and Smith are equally reliable: the question is what one should expect to happen in the near future?This situation is represented in the warrant dependency graph of Figure 2, where the basis of Smith's argument is attacked by the conclusion of Jones' one, and vice versa.This cycle expresses that we have in fact two incompatible assessments, each one leading to a different output or extension: either we should accept E and block both T and ∼T , or accept ∼T and block both E and ∼E.
For RP-DeLP programmes with multiple outputs, we consider the problem of deciding the set of conclusions that can be ultimately warranted.The usual sceptical approach consists of adopting the intersection of all possible outputs.However, in addition to the computational limitation, as stated in [53], adopting the intersection of all outputs may lead to an inconsistent output.Intuitively, for a conclusion, to be in the intersection does not guarantee the existence of an argument that is recursively based on ultimately warranted conclusions.To this end, based on an alternative sceptical semantics for defining collections of justified arguments in abstract argumentation frameworks proposed by Dung, Mancarella and Toni [35,36], we introduce the notion of maximal ideal output for an RP-DeLP programme.This is based on a recursive, level-wise definition, considering at each level the maximum set of conclusions based on warranted information and not involved in either a conflict or in a circular definition of conflict.The maximal ideal output for the previous example should accept neither E nor ∼T , and block both E and ∼T .
The fourth contribution of the article is the development and experimental validation of a procedure to compute the maximal ideal output for an RP-DeLP programme.To this end, first we define an algorithm that computes the maximal ideal output in polynomial space and with an upper bound on complexity equal to P NP .Second, we present SAT encodings for the two main combinatorial subproblems that arise when computing warranted and blocked conclusions of the maximal ideal output for an RP-DeLP programme, so that we can take profit of existing state-of-the-art SAT solvers for solving instances of big size.It is worth pointing out that the way of computing warranted and blocked conclusions with our SAT encodings allows us not only to eventually compute the maximal ideal output, but also to produce an argument for each warranted conclusion and a blocking set of conclusions (with their arguments) for each blocked literal.Thus, our system not only computes answers using SAT encodings but also explains them, in accordance with one of the desirable features of argumentation systems based on Answer Set Programming (ASP) put forward in Toni and Sergot's survey [58] for non-abstract argumentation frameworks.
Finally, we present empirical results obtained with an implementation of our algorithm that uses these SAT encodings.The results show that, at least on randomly generated instances, the practical complexity is strongly dependent on the size of the strict part of the programme.Indeed, for a same number of variables, RP-DeLP programmes with different size for their strict part can range from trivially solvable to exceptionally hard.Moreover, the experimental results also show that the fraction of defeasible knowledge considered at each defeasible level is also relevant in assessing the tractability and scalability of RP-DeLP programmes.In a recent work [3], we have developed an alternative implementation of our algorithm but based on ASP encodings, but that version is preliminary and works only with one defeasible level, so the evaluation of an ASP based version that works with multiple levels is left as future work.
This article extends our previous work in [1,2] by providing the characterization of the properties of the framework, the algorithm for the computation of the maximal ideal output for an RP-DeLP programme based on SAT encodings, experimental results and proofs for all outcomes.We also provide new running examples that may help the reader to understand the different notions discussed in the article.The rest of the article is organized as follows.In Section 2, we define a general defeasible argumentation framework with recursive semantics.In Section 3, we introduce several levels of defeasibility or preference among different pieces of defeasible knowledge.In Section 4, we particularize the recursive warrant semantics to the case of the P-DeLP programmes and we provide some examples in the context of political debates.In Section 5, we define the maximal ideal output for RP-DeLP programmes and in Section 6, we present an algorithm for its computation.In Section 7, we present SAT encodings for the two main queries performed in the algorithm and in Section 8, we study the scaling behaviour of the (average) computational cost of our implementation.Section 9 discusses related work, mainly in the areas of preference-based argumentation and SAT-based encodings for argumentation frameworks.Finally, in Section 10, we present some concluding remarks.

A general defeasible argumentation framework with recursive semantics
We will start by considering a rather general framework for defeasible argumentation based on a propositional logic (L, ) with a special symbol ⊥ for contradiction 1 .For any set of formulas A, if A ⊥ we will say that A is contradictory, while if A ⊥ we will say that A is consistent.A knowledge base (KB) is a triplet P = ( , , ), where , , ⊆ L, and ⊥. is a finite set of formulas representing strict knowledge (formulas we take for granted they hold to be true), is another finite set of formulas representing the defeasible knowledge (formulas for which we have reasons to believe they are true) and denotes the set of formulas (conclusions) over which arguments can be built.In many argumentation systems, e.g. in rule-based argumentation systems, is taken to be a set of literals.
The notion of argument is the usual one.Given a KB P, an argument for a formula ϕ ∈ is a pair A = A,ϕ , with A ⊆ such that: (1) ∪A ⊥, and (2) A is minimal (with respect to set inclusion) such that ∪A ϕ.
If A =∅, then we will call A an s-argument (s for strict), otherwise it will be a d-argument (d for defeasible).The notion of subargument is referred to d-arguments and expresses an incremental proof relationship between arguments which is formalized as follows.
More generally, we say that B,ψ is a subargument of a set of arguments G, written B,ψ < G, if B,ψ < A,ϕ for some A,ϕ ∈G.
Notice that if ( , , ) = ({r},{r → p∧q},{p,q,p∧q}) and A ={r → p∧q} then A 1 = A,p , A 2 = A,q and A 3 = A,p∧q are arguments for different formulas with a same support and thus, in our framework, A 3 < A 1 and A 3 < A 2 are the subargument relations between arguments A 1 , A 2 and A 3 since p∧q p, p∧q q, p p∧q and q p∧q.
A formula ϕ ∈ will be called justifiable conclusion with respect to P if there exists an argument for ϕ, i.e. there exists A ⊆ such that A,ϕ is an argument.
The usual notion of attack or defeat relation in an argumentation system is binary.However in certain situations, the conflict relation among arguments is hardly representable as a binary relation, mainly (but not only) when =∅.For instance, consider the following KB P 1 = ( , , ) with ={¬a∨¬b∨¬p}, ={a,b,p} and ={a,b,p}.
Clearly, A 1 = {a},a , A 2 = {b},b and A 3 = {p},p are arguments that justify a, b and p respectively, and which do not pair-wisely generate a conflict.Indeed, ∪{a,b} ⊥, ∪{a,p} ⊥ and ∪ {b,p} ⊥.However, the three arguments are collectively conflicting since ∪{a,b,p} ⊥, hence in P 1 there is a non-binary conflict relation among several arguments.Notice that a collective conflict can also happen when the strict part of a knowledge base is empty.For instance, consider now that ¬a∨¬b∨¬p is also a defeasible formula and not of a strict formula, and that ¬a∨¬b∨¬p is a justifiable conclusion, i.e. consider the following modified KB: =∅, ={a,b,p,¬a∨¬b∨¬p} and ={a,b,p,¬a∨¬b∨¬p}.
Then, A 1 ,A 2 ,A 3 and A 4 = {¬a∨¬b∨¬p},¬a∨¬b∨¬p are arguments that justify a, b, p and ¬a∨¬b∨¬p respectively, and which do not generate a conflict neither pair-wisely nor three to three.However, the four arguments together are collectively conflicting since {a,b,p,¬a∨¬b∨¬p} ⊥.
In the following, we formalize this notion of collective conflict in a set of arguments which captures the idea of an inconsistency arising from a consistent set of justifiable conclusions W together with the strict part of a knowledge base and the set of conclusions of those arguments.
Notice that if a set of arguments G ={ A 1 ,ϕ 1 ,..., A k ,ϕ k } minimally conflicts with respect to a set of conclusions W , then the arguments A i ,ϕ i cannot be s-arguments, i.e. for each i, A i =∅.Indeed, if A i =∅ for some i, then ϕ i , and hence G would not satisfy the minimality Condition (M).Consider the previous KB P 1 , and the set of arguments {A 1 ,A 2 ,A 3 } for a, b and p, respectively, and let W =∪ i=1,...,3 {ψ | B,ψ < A i }=∅.According to the previous definition, it is clear that the set of arguments {A 1 ,A 2 ,A 3 } minimally conflicts with respect to ={¬a∨¬b∨¬p}.The intuition is that this collective conflict should block the conclusions a,b and p to be warranted.Now, this general notion of conflict is used to define a recursive semantics for warranted conclusions of a knowledge base.Actually we define an output of a KB P = ( , , ) as a pair (Warr,Block) of subsets of of warranted and blocked conclusions respectively all of them based on warranted information but while warranted conclusions do not generate any conflict, blocked conclusions do.Definition 2.3 (Output for a KB) An output for a KB P = ( , , ) is any pair (Warr,Block), where Warr∩Block =∅, Warr∪Block ⊆ and {ϕ ∈ | ϕ}⊆Warr, satisfying the following recursive constraints: (1) ϕ ∈Warr∪Block iff there exists an argument A,ϕ such that for every B,ψ < A,ϕ , ψ ∈Warr.
In this case we say that the argument A,ϕ is valid with respect to Warr.(2) For each valid argument A,ϕ : • ϕ ∈ Block whenever there exists a set of valid arguments G such that (i) A,ϕ < G, and (ii) { A,ϕ }∪G minimally conflicts with respect to the set W ={ψ | B,ψ < G∪{ A,ϕ }}.
The intuition underlying this definition is as follows: an argument A,ϕ is either warranted or blocked whenever for each subargument B,ψ of A,ϕ , ψ is warranted; then, it is eventually blocked if ϕ is involved in some conflict, otherwise it is warranted.
Notice that if an argument A,ϕ is warranted, and A,ψ is another argument, then A,ψ is warranted as well.
According to Definition 2.3 Warr =∅ and the arguments {a},a , {b},b , {c},c and {¬c},¬c are valid.Now, for every such valid argument there exists a set of valid arguments which minimally conflicts: indeed both sets of valid arguments { {a},a , {b},b , {c},c } and { {c},c , {¬c},¬c } minimally conflict (since ∪{a,b,c} ⊥ and ∪{c,¬c} ⊥).Therefore a, b, c and ¬c are blocked conclusions.On the other hand, the arguments {a,b},¬c , {a},y and {b,c},¬y are not valid since they are based on conclusions which are not warranted.Hence y and ¬y are considered as rejected conclusions.Thus, the (unique) output for P 2 is the pair (Warr,Block) = (∅, ).Intuitively, this output for P 2 expresses that all conclusions in are (individually) valid; however, all together are contradictory with respect to .
It can be proven that if (Warr,Block) is an output for a KB ( , , ), the set Warr of warranted conclusions is indeed non-contradictory and satisfies indirect consistency with respect to the strict knowledge.
Proposition 2.5 (Indirect consistency) Let P = ( , , ) be a KB and let the pair (Warr,Block) be an output for P.Then, ∪Warr ⊥.
Proof.By Definition 2.3, for every ϕ ∈Warr there does not exist a set W ⊆Warr such that ∪W ∪ {ϕ} ⊥, and therefore, ∪W ⊥ for all W ⊆Warr.
In the following, we will see that the satisfaction of the closure postulate (in the sense of Caminada and Amgoud [30]) with respect to the strict knowledge actually depends on how the set of formulas (over which arguments can be built up) is defined.For instance, consider the KB P 4 = ( , , ), with ={a∧b → y}, ={a,b} and ={a,b}.
A different case occurs when ∪Warr ϕ with ϕ ∈ and there does not exist a consistent proof for ϕ with respect to the set of warranted conclusions.For instance, consider now the KB P 5 = ( , , ) with ={a∧b → y}, ={s → a,¬s → b,s,¬s} and ={a,b,y}.
Again, (Warr,Block) = ({a,b},∅) is the only output for P 5 and thus, although ∪Warr y and y ∈ , y ∈Warr.The problem here is that there does not exist an argument for y with respect to P 5 since {s,¬s} ⊥ and the proof of y should be based on a and b which are respectively based on s and ¬s.Moreover, note that since neither s nor ¬s are in , they cannot be used to attack the arguments for b and a respectively, and hence a and b are (surprisingly) warranted.
Finally, it can be the case that there exists an argument for a conclusion ϕ ∈ but, the argument is not valid with respect to Warr.For instance, consider now the KB P 6 = ( , , ) with ={a∧b → y}, ={s,¬s,s → a,¬s → b,p,¬p,p → y} and ={a,b,p,¬p,y}.
Then, (Warr,Block) = ({a,b},{p,¬p}) is the only output for P 6 and, again, we can see that ∪Warr y and y ∈ but, y ∈Warr.The problem here is that although there exists an argument for y based on p, {p,p → y},y , this argument is not valid with respect to Warr since p is a blocked conclusion and, as it occurs with programme P 5 , the proof of y based on a and b is not consistent.Proposition 2.6 (Closure) Let P = ( , , ) be a KB and let the pair (Warr,Block) be an output for P. If ∪Warr ϕ then ϕ ∈Warr whenever there exits a valid argument for ϕ with respect to Warr.
Proof.Assume ϕ ∈ and ∪Warr ϕ, but ϕ, otherwise it is clear that ϕ ∈Warr.Further, suppose there exists a valid argument A,ϕ with respect to Warr.By way of contradiction, let us suppose ϕ ∈Warr.Then, there would exist a set of valid arguments G such that A,ϕ is not a subargument of any argument in G and that G∪{ A,ϕ } minimally conflicts with respect to the set W ⊆Warr of conclusions of all subarguments of arguments in G∪{ A,ϕ }.If G∪{ A,ϕ } minimally conflicts with respect to the set W , ∪W ∪{ϕ}∪{ψ | B,ψ ∈G} ⊥ (Condition (C)), and ∪W ∪ S ⊥, for all set S ⊂{ϕ}∪{ψ | B,ψ ∈G} (Condition (M)).Then, ∪W ϕ, and thus, there would exist a set W ⊆Warr such that W ∩W =∅ and ∪W ∪W ϕ.Now, since the conclusions of all subarguments of G are in W and all conclusions in Warr have valid arguments, there would exist a conclusion φ ∈ W such that its valid argument C,φ and G∪{ D,χ |χ ∈ W ∪(W \{φ})} minimally conflict, and thus, φ ∈Warr.
Remark that the particular behaviour of above KBs P 4 , P 5 and P 6 can be avoided with a suitable definition of the set of justifiable conclusions .For instance, if we extend the set of justifiable conclusions of P 4 with {y} and of P 5 and P 6 with {s,∼s}, we get that the pair (Warr,Block) = ({a,b,y},∅) is the only output for the new definition of P 4 , the pair (Warr,Block) = (∅,{s,∼s}) is the only output for the new definition of P 5 and the pair (Warr,Block) = (∅,{s,¬s,p,¬p}) is the only output for the new definition of P 6 .
Given a set of strict and defeasible formulas and respectively, we define its set of justifiable conclusions as Conc( , ) ={ϕ | ∪A ϕ for some set A ⊆ such that ∪A ⊥}.Then KBs of the form ( , , ) where the set of formulas over which arguments can be built includes Conc( , ) enjoy the following proper Closure property.
Corollary 2.7 (Closure) Let P = ( , , ) be a KB such that Conc( , ) ⊆ .For any output (Warr,Block) of P, if ∪ Warr ϕ, then ϕ ∈Warr.Proof.For every ψ i ∈Warr, there exists a valid argument C i ,ψ i .Then, for every B ⊆ C i such that ∪B φ and ψ i φ, we have that B,φ < C i ,ψ i , φ ∈ and φ ∈Warr.It is clear that ∪(∪ i C i ) ϕ, and let A be a minimal subset of ∪ i C i such that ∪A ϕ.Then, it easily follows that A,ϕ is a valid argument with respect to Warr.

Extending the framework with a preference ordering on arguments
In the previous section, we have considered knowledge bases containing formulas describing knowledge at two epistemic levels, strict and defeasible.A natural extension is to introduce several levels of defeasibility or preference among different pieces of defeasible knowledge, as it has first proposed in default reasoning by Brewka [29] and then extensively used in different approaches reasoning under inconsistency, e.g.[20][21][22][23][24] (see a discussion in this issue in Section 9).
A stratified knowledge base (sKB) is a tuple P = ( , , , ), such that ( , , ) is a KB (in the sense of the previous section) and is a total pre-order on ∪ representing levels of defeasibility: ϕ ≺ ψ means that ϕ is more defeasible than ψ.Actually, since formulas in are not defeasible, is such that all formulas in are at the top class of the ordering.For the sake of a simpler notation, we will often refer in the paper to numerical levels for defeasible formulas and arguments rather than to the pre-ordering , so we will assume a mapping N : ∪ →[0,1] such that N(ϕ) = 1 for all ϕ ∈ and N(ϕ) < N(ψ) iff ϕ ≺ ψ. 2 Then we define the strength of an argument A,ϕ , written s( A,ϕ ), as follows: Since we are considering several levels of strength among arguments, the intended construction of the sets of conclusions Warr and Block is done level-wise, starting from the highest level and iteratively going down from one level to next level below.If 1 >α 1 >...>α p ≥ 0 are the strengths of d-arguments that can be built within a sKB P = ( , , , ), we define: Warr =Warr(1)∪{∪ i=1,p Warr(α i )} and Block =∪ i=1,p Block(α i ), where Warr(1) ={ϕ | ϕ}∩ , and Warr(α i ) and Block(α i ) are respectively the sets of the warranted and blocked justifiable conclusions of strength α i .Then, we will also write Warr(≥ α i ) and Warr(>α i ) to denote ∪ β≥α i Warr(β) and ∪ β>α i Warr(β), respectively, and analogously for Block(>α i ), assuming Block(>α 1 ) =∅. Definition 3.1 (Output for a sKB) An output for a sKB P = ( , , , ), with 1 >α 1 >...>α p ≥ 0 as set of strengths of d-arguments, is any pair (Warr,Block), where the sets Warr(α i )'s and Block(α i )'s are required to satisfy the following recursive constraints: 3(1) ϕ ∈Warr(α i )∪Block(α i ) iff there exists an argument A,ϕ of strength α i satisfying the following three conditions: In this case we say that A,ϕ is valid with respect to the sets Warr(≥ α i ) and Block(>α i ).
(2) For every valid argument A,ϕ of strength α i we have that ϕ ∈ Block(α i ) whenever there exists a set G of valid arguments of strength α i such that (i) A,ϕ < G, and (ii) G∪{ A,ϕ } minimally conflicts with respect to the set There are two main remarks when considering several levels of strength among arguments.On the one hand, a d-argument A,ϕ of strength α i is valid whenever (V1) it is based on warranted conclusions; (V2) there does not exist a valid argument for ϕ with strength greater than α i ; and (V3) ϕ is consistent with both each blocked argument with strength greater than α i and the set of already warranted conclusions Warr(>α i )∪{ψ | B,ψ < A,ϕ }.On the other hand, a valid argument A,ϕ of strength α i becomes blocked as soon as it leads to some conflict among arguments with strength α i with respect to the set of warranted conclusions with higher strengths.
Notice that Conditions (V2) and (V3) define how warranted and blocked conclusions at higher levels are taken into account in lower levels.In particular, blocked conclusions play a key role in the propagation mechanism between defeasibility levels.In our approach, if a conclusion ϕ is blocked at level α, then for any lower level than α, not only the conclusion ϕ is rejected but also every conclusion ψ such that {ϕ,ψ} ⊥.
The following examples show how warranted and blocked conclusions at higher levels are taken into account in lower levels.

Example 3.4
Consider now that the KB P 2 is extended with two defeasible formulas p and ¬p∨a as follows: ={¬a∨y,¬b∨¬c∨¬y}, ={a,b,c,¬c,p,¬p∨a}, and ={a,b,c,¬c,y,¬y,p}, and stratified into two levels of defeasibility as follows: {p,¬p∨a,¬c}≺{a,b,c}.Assume α 1 is the highest level and α 2 is the lowest level.Notice that there exist two different arguments for conclusions ¬c and a: {a,b},¬c and a,a of strength α 1 , and {¬c},¬c {p,¬p∨a},a of strength α 2 .Again, Warr(1) =∅ but now, at level α 1 we have that the conclusions a, b and c have valid arguments all involved in conflicts, and thus, {a,b,c}⊆Block(α 1 ) and arguments {a},y and {a,b},¬c of strength α 1 are not valid because the conclusions a and b are not warranted at level α 1 (Condition (V1)).
Hence, Warr(α 1 ) =∅ and Block(α 1 ) ={a,b,c}.At level α 2 , we have arguments for p, a, ¬c and y: {p},p , {p,¬p∨a},a , {¬c},¬c and {p,¬p∨a},y .The argument for p is valid and does not conflict with any set of arguments, and thus, p ∈Warr(α 2 ).However, the argument for a, which is based on {p},p , is not valid because the conclusion a has been blocked at level α 1 (Condition (V2)), and the argument for ¬c is not valid because the conclusion c has been blocked at level α 1 (Condition (V3)).Finally, the argument for y, which is based on {p,¬p∨a},a , is not valid because the conclusion a is not warranted at level α 2 (Condition (V1).Therefore, Warr(α 2 ) ={p} and Block(α 2 ) =∅ and hence, the output for the sKB is (Warr,Block) = ({p},{a,b,c}).
The above examples show that blocked conclusions play a key role in the propagation mechanism between defeasibility levels.In our approach, if a conclusion ϕ is blocked at level α, then for any lower level than α, not only the conclusion ϕ is rejected but also every conclusion ψ such that {ϕ,ψ} ⊥, as has happened in Example 3.4 with conclusions a and ¬c at level α 2 .Intuitively, our mechanism is based on the idea that, if a conclusion is warranted at level α, so it could also be at any higher level.A different approach could have been to consider that blocked conclusions at one level are not propagated to lower levels.In such a case, it could happen to have a conclusion ϕ blocked at a given level and to have ϕ or ψ, with {ϕ,ψ} ⊥, warranted at a lower level.For instance, in Example 3.4, if Conditions (V2) and (V3) are not checked, conclusions a and ¬c would be warranted at level α 2 although arguments for conclusions a and c are valid at level α 1 but involved in a conflict.
The following results provide an interesting characterization of the relationship between warranted and blocked conclusions in stratified knowledge bases.Proposition 3.5 Let P = ( , , , ) be a sKB and let (Warr,Block) be an output for P. Then: (1) If ϕ ∈Warr(α)∪Block(α), then there exists an argument A,ϕ of strength α such that for all subargument B,ψ < A,ϕ of strength β, ψ ∈Warr(β).

A particular case: recursive P-DeLP
In this section, we particularize the recursive warrant semantics we have presented above for stratified knowledge bases to the case of the P-DeLP programmes.As mentioned in Section 1, P-DeLP is a rule-based argumentation system extending the well-known DeLP system [42] in which weights are attached to defeasible rules expressing their belief or preference strength.For a detailed description of the P-DeLP argumentation system based on dialectical trees the reader is referred to [5].
Although the original syntax and inference of P-DeLP are a bit different (e.g. the weights are explicit in the formulas and arguments), here we will present them in a way so to adapt them to the framework introduced in the previous sections.We will refer to this particular framework as RP-DeLP (recursive P-DeLP).Hence we define the logic (L R , R ) underlying RP-DeLP as follows.
The language of RP-DeLP is inherited from the language of logic programming, including the notions of atom, literal, rule and fact.Formulas are built over a finite set of propositional variables p,q,... which is extended with a new (negated) atom '∼p' for each original atom p. Atoms of the form p or ∼p will be referred as literals, and if P is a literal, we will use ∼P to denote ∼p if P is an atom p, and will denote p if P is a negated atom ∼p.Formulas of L R consist of rules of the form Q ← P 1 ∧...∧P k , where Q,P 1 ,...,P k are literals.A fact will be a rule with no premises.We will also use the name clause to denote a rule or a fact.The inference operator R is defined by instances of the modus ponens rule of the form: {Q ← P 1 ∧...∧P k ,P 1 ,...,P k } R Q.A set of clauses is contradictory, denoted ⊥, if , for some atom q, R q and R ∼q.An RP-DeLP programme P is just a stratified knowledge base ( , , , ) over the logic (L R , R ), where consists of the set of all literals of L R .As already pointed out, we will assume that is representable by a mapping N : ∪ →[0,1] such that N(ϕ) = 1 for all ϕ ∈ and N(ϕ) < N(ψ) iff ϕ ≺ ψ, so we will often refer to numerical weights for defeasible clauses and arguments rather than to the pre-ordering .Also, for the sake of a simpler notation we will get rid of of a programme specification.
In the rest of this section, we first provide an extensive example of use of RP-DeLP in a scenario of political debates, showing how RP-DeLP can deal with different hypothesis about the scenario.In particular in one of them it is shown a situation where the argumentation system provides several outputs as result.Then in the second subsection, we characterize those RP-DeLP programmes with a single output.

Using RP-DeLP: a practical example
In this section, we explore the application of the RP-DeLP argumentation framework to the extraction of consistent information out of political debates.Suppose we have two opposite parties of the sphere of Spanish politics: a left-wing party (PSOE) and a right-wing party (PP).The idea is to see whether one can find consistent information based on solid arguments both in agreement with the law and their particular beliefs.In the following, we prefix the rules with a label to make easier to mention them in arguments.
First suppose the parties are involved in a debate about possible ways to increase the Gross domestic product (GDP) of Spain (target represented by the literal GDP_UP).To reach this goal, the government may consider to undertake the following policies: G1: increase the education expenditure G2: increase the infrastructures expenditure G3: decrease taxes for private companies Assume further the current Spanish law only allows two of the previous actions to be executed at most, so executing all three actions is forbidden by law.So, at the strict level we have the following hard constraints: Since they are only possible policies, we consider {G1,G2,G3} as defeasible facts.Moreover, the left-wing party believes that executing G1 and G2 will increase the GDP, and that the same result will happen if executing G1 and G3.On the other hand, the right-wing party believes that executing G2 and G3 will increase the GDP.So we have the following defeasible rules: Next we consider different scenarios arising when different defeasibility levels are assigned to the above defeasible facts and rules.

Scenario 1.
A first scenario results from considering that all the above set of defeasible facts and rules are at the same defeasibility level, so we have a programme P 1 = ( , 1 ) with only one defeasibility level, where 1 ={G1,G2,G3,PSOE1,PSOE2,PP1 }.
As a consequence, all the arguments for the literal GDP_UP are rejected and the target GDP_UP cannot be warranted.Scenario 2. Suppose we have a stronger preference for implementing action G1 than actions G2 and G3.In this case, we can stratify 1 in two defeasibility levels: α 1 and α 2 with 1 >α 1 >α 2 > 05 as follows: Let us denote by 2 this two-level defeasible knowledge base, and we define P 2 = ( , 2 ).So in this case G1 is the only warranted action at level α 1 (i.e.Warr(α 1 ) ={G1}), but the actions G2 and G3 become blocked at level α 2 because ∪Warr(α 1 )∪{G2,G3} ⊥.But, even if now the action G1 is warranted, this is not enough to have a valid argument for GDP_UP with any of the rules (i.e.all the arguments for GDP_UP are based on some blocked argument), and thus, we finally get for P 2 : Warr ={G1} and Block ={G2,G3}.Scenario 3. Suppose now there is a stronger preference for implementing the actions G1 and G2 than action G3.In this case, the defeasible knowledge 3 becomes: It is clear that in the programme P 3 = ( , 3 ), G1 and G2 are warranted actions at level α 1 , and consequently so are the literals GDP_UP and ∼G3 because the arguments {G1,G2,PSOE1},GDP_UP and {G1,G2},∼G3 are valid at level α 1 , and thus, the action G3 becomes invalid (rejected) at level α 2 .Therefore, for P 3 we have the following output: Warr ={G1,G2,GDP_UP,∼G3} and Block =∅.Scenario 4. Finally suppose that the right-wing party introduces a new argument (in the informal sense) into the debate by claiming that 'increasing the education expenditure will cause the GDP to not increase'.This new information is represented by the defeasible rule PP2 :∼GDP_UP ← G1 and is incorporated with the same strength than the previous rules into the debate.So, the new defeasible knowledge 4 becomes: level α 1 : {G1,G2,PSOE1,PSOE2,PP1,PP2} level α 2 : {G3}.

RP-DeLP programmes with single outputs
As we have mentioned in Section 2, in some cases the output (Warr,Block) for a knowledge base in general, and for an RP-DeLP programme in particular, is not unique, due to some recursive definitions of conflict that emerge when considering inference (support) and conflict relations among arguments.The above example in the above Scenario 5 shows such a case.In this section, we first identify what are the recursive definitions of conflict in RP-DeLP that ultimately cause programmes having multiple outputs and then, based on this, we provide necessary and sufficient condition for a programme to have a single output.We call this Unique Output Property.
Actually we characterize recursive definitions of conflict by means of what we call Warrant Dependency Graphs.In [53] similar graph structures, called inference-graphs, were defined to represent inference (support) and defeat relations among arguments allowing to detect recursive defeat relations when considering recursive semantics for defeasible reasoning.The main difference between both approaches is that in our case we handle collective conflicts among arguments in order to preserve indirect consistency and closure among warranted conclusions with respect to the strict knowledge.
Intuitively, the characterization of the unique output property for an RP-DeLP programme P = ( , , ) is done level-wise, starting from the highest level and iteratively going down from one level to the next level below.At each level α, it consists of checking whether the warranty of a literal L recursively depends on itself, based on the topology of a warrant dependency graph built for a given suitable set of valid arguments of strength α and a given suitable set of what we call almost valid arguments of strength α.A valid argument captures the idea of a non-rejected argument (i.e.warranted or blocked, but not rejected) while an almost valid argument captures the idea of an argument whose rejection is conditional to the warranty of some other valid argument.
The following definition makes use of a slight generalization of the notion of valid argument as introduced in (1) of Definition 3.1, 6 where we use this notion relative to an arbitrary subset W ⊆Warr of warranted conclusions and a subset of B ⊆ Block of blocked conclusions.We also use the same abbreviations W (α), W (≥ α) and W (>α) to denote the subsets of warranted conclusions from W with strength α, greater or equal than α and greater than α respectively.Analogous abbreviations are also used for B. argument of L 1 , L 1 ∈{∼R,R 1 ,...,R p } and, either L 2 ∈{∼R,R 1 ,...,R p } or, L 2 is a subargument of the argument of L 3 , for some L 3 ∈{P 1 ,...,P n } such that L 3 ∈{∼R,R 1 ,...,R p }.9 (5) Elements of V and E are only obtained by applying the above construction rules.
Intuitively, the warrant dependency graph for {A 1 ,...,A k } and {F 1 ,...,F n } represents conflict and support relationships among these sets of arguments of strength α with respect to the set W (≥ α) of warranted conclusions of equal or higher strength.
The following example shows a recursive definition of conflict which arises from the strict knowledge.
Obviously, A 1 and A 2 are valid arguments with respect to W (≥ α) and B(>α), and F 1 and F 2 are almost valid arguments with respect to {A 1 ,A 2 }. Figure 4 shows the warrant dependency graph for {A 1 ,A 2 } and {F 1 ,F 2 }.The cycle of the graph expresses that (1) the warranty of p depends on a (possible) conflict with r; (2) the support of r depends on q; (3) the warranty of q depends on a (possible) conflict with s; and (4) the support of s depends on p. Proposition 4.4 (RP-DeLP programme with unique output) Let P = ( , , ) be an RP-DeLP programme and let (Warr,Block) be an output for P. (Warr,Block) is the unique output for P iff, for any defeasibility level α and literal L ∈Warr(α), there is no cycle in the warrant dependency graph for the set of arguments A and the set of arguments F, where -A is the set of all d-arguments of strength α that are valid with respect to Warr(≥ α)\{L} and Block(>α), and -F is the set of all d-arguments of strength α that are almost valid with respect to A.
Proof.Suppose that (Warr,Block) is the unique output for P and there is a cycle in the graph for some literal L ∈Warr(α).On the one hand, if (Warr,Block) is the unique output for P, there does not exist a pair (Warr ,Block ) that satisfies Definition 3.1 and Warr =Warr or Block = Block, and thus, every literal is either warranted, or blocked, or rejected.On the other hand, given L ∈Warr(α), A is the set of arguments of strength α which are valid with respect to Warr(≥ α)\{L} and Block(>α), hence, arguments in A do not depend on L and there is an argument for L in A. Similarly, F is the set of arguments of strength α that are almost valid with respect to A, hence, the support of arguments in F depends on L or some argument in A. Now, according to Definition 4.2, if there is a cycle in the warrant dependency graph, it must be that the warranty of the argument for L depends on the validity of at least an argument F,P ∈F, which depends on the warranty of some argument A,L ∈A with L = L , which depends on the validity of at least an argument F ,P ∈F with P = P, which in turn depends on the warranty of L.Then, according to Definition 3.1, either L is warranted and L is blocked, or L is warranted and L is blocked, and therefore, there exists at least two different outputs for P. Finally, if for all defeasibility level α and literal L ∈Warr(α), there is no cycle in the the warrant dependency graph with respect to Warr(≥ α)\{L} and Block(>α), there exists a unique warranty evaluation order between arguments, and thus, there exists a unique output for P.  1)∪Warr(α), Warr(1) ={y}, Warr(α) ={p} and Block = Block(α) = {q,∼s}, is an output for P R1 .Then, as p ∈Warr(α), defining W =Warr(≥ α)\{p}=Warr (1) ={y} and B = Block(>α) =∅, we get that A ={A 1 ,A 2 } is the set of all valid arguments with respect to W and B, and F ={F 1 ,F 2 } is the set of all almost valid arguments with respect to A. Moreover, the warrant dependency graph for A and F contains a cycle (see Figure 4), proving that the output for P R1 is not unique.Indeed, notice that Output = (Warr ,Block ) with Warr =Warr(1)∪Warr (α), Warr (α) ={q} and Block = Block (α) ={p,r}, is also an output for P R1 .Moreover, as Warr (≥ α)\{q}=Warr(≥ α)\{p}=Warr( 1) ={y} and Block (>α) = Block(>α) =∅, the warrant dependency graph for q ∈Warr (α) also corresponds to the graph in Figure 4.
In the rest of the article, we tackle the problem of which output one should consider for an RP-DeLP programme with multiple outputs.To this end, in Section 5, we define the maximal ideal output of an RP-DeLP programme as the set of conclusions which are ultimately warranted, and in Section 6 we design an algorithm for computing them in polynomial space and with an upper bound on complexity equal to P NP .

Maximal ideal output
In the previous section, we have characterized the unique output property for the particular framework of RP-DeLP programmes.Now in this section we are interested in the problem of deciding the set of conclusions that can be ultimately warranted in RP-DeLP programmes with multiple outputs.The usual sceptical approach corresponds to adopt the intersection of all possible outputs.However, in addition to the computational limitation, as stated in [53], adopting the intersection of all outputs may lead to an inconsistent output (in the sense of violating the base of the underlying recursive warrant semantics) in case some particular recursive situation among literals of a programme occurs.Intuitively, for a conclusion, to be in the intersection does not guarantee the existence of an argument for it that is recursively based on ultimately warranted conclusions.
For instance, consider the following situation involving three conclusions P, Q and T , where P can be warranted whenever Q is blocked, and vice versa.Moreover, suppose that T can be warranted when either P or Q are warranted.Then, according to the warrant recursive semantics, we would get two different outputs: one where P and T are warranted and Q is blocked, and the other one where Q and T are warranted and P is blocked.Then, adopting the intersection of both outputs we would get that T would be ultimately warranted, however T should be in fact rejected since neither P nor Q are ultimately warranted conclusions.
According to this example, one could take then as the set of ultimately warranted conclusions of RP-DeLP programmes those conclusions in the intersection of all outputs which are recursively based on ultimately warranted conclusions.However, as in RP-DeLP there might be different levels of defeasibility, this approach could lead to an incomplete solution, in the sense of not being the biggest set of ultimately warranted conclusions with maximum strength.
For instance, consider the above example extended with two defeasibility levels as follows.Suppose that P can be warranted with strength α whenever Q is blocked, and vice versa.Moreover, suppose that T can be warranted with strength α whenever P is warranted at least with strength α, and that T can be warranted with strength β, with β <α, independently of the status of conclusions P and Q.Then, again we get two different outputs: one output warrants conclusions P and T with strength α and blocks conclusion Q, and the other one warrants conclusions Q and T with strengths α and β, respectively, and blocks P. Now, by adopting conclusions of the intersection which are recursively based on ultimately warranted conclusions, we get that conclusion T is finally rejected, since T is warranted with a different argument and strength in each output.However, as we are interested in determining the biggest set of warranted conclusions with maximum strength, it seems quite reasonable to reject T at level α but to warrant it at level β.
Therefore, we are led to define the maximal ideal output for an RP-DeLP programme P = ( , , ) as a pair (Warr,Block) of respectively warranted and blocked conclusions, with a maximum strength, such that: (i) the arguments of all conclusions in Warr∪Block are recursively based on warranted conclusions; (ii) a conclusion is warranted (at level α) if it does not generate any conflict with the set of already warranted conclusions (at a level β >α) and it is not involved in any cycle of a warrant dependency graph; otherwise, it is blocked; and (iii) a conclusion is rejected if it can be neither warranted nor blocked to any level.
In fact, in a different context, this idea corresponds to the maximal ideal extension defined by Dung, Mancarella and Toni [35,36] as an alternative sceptical basis for defining collections of justified arguments in the abstract argumentation frameworks promoted by Dung [34] and Bondarenko et al. [28].

Definition 5.1 (Maximal ideal output)
The maximal ideal output for an RP-DeLP programme P = ( , , ), with defeasibility levels 1 > α 1 >...>α p , is a pair (Warr,Block) such that, for every valid argument A,Q of strength α i with respect to Warr(≥ α i ) and Block(>α i ), the following recursive constraint is satisfied: (1) Q ∈ Block(α i ) whenever one of the two following cases holds: Case 1 There exists a set G of valid arguments of strength α i such that the two following conditions hold: (i) A,Q < G, and (ii) G∪{ A,Q } minimally conflicts with respect to the set W =Warr(>α i )∪{P | B,P < G∪{ A,Q }}.
Case 2 There exists a set H of valid arguments of strength α i such that the three following conditions hold: (i) A,Q < H.
(ii) There exists a set of arguments F of strength α i that are almost valid with respect to H∪ A,Q and such that there is a cycle in the warrant dependency graph (V ,E) for H∪ A,Q and F, and any argument C,R ∈H is such that R is either a vertex of the cycle or C,R does not satisfy Case 1. (iii) For some vertex v ∈ V of the cycle, either v is the vertex of conclusion Q or v is the vertex of some other conclusion in H, and there exists a path from v to the the vertex of conclusion Q.
The intuition underlying the maximal ideal output definition is as follows.The conclusion of every valid (not rejected) argument A,Q of strength α i is either warranted or blocked.Then, it is eventually blocked if either (Case 1) it is involved in some conflict with respect to Warr(>α i ) and a set G of valid arguments whose supports do not depend on A,Q , or (Case 2) the warranty of A,Q depends on some circular definition of conflict between a set of valid arguments H whose supports do not depend on A,Q and a set of almost valid arguments F whose supports depend on some argument in H∪ A,Q .In fact, the idea here is that if the warranty of A,Q depends on some circular definition of conflict between the arguments of H and F, one could consider two different outputs (status) for conclusion Q: one with Q warranted and another one with Q blocked.Therefore, conclusion Q is blocked for the maximal ideal output.In general, the arguments of H and F involved in a cycle are respectively warranted and rejected for the maximal ideal output.For instance, consider again the recursion case in the example of Scenario 5 from Section 4.1.Figure 3 showed the warrant dependency graph for the set of valid arguments and the set of almost valid arguments Then, since for every valid argument in H there is a a cycle, the maximal ideal output for the RP-DeLP programme is Warr =∅ and Block ={G1,G2,G3} and the goal GDP_UP is rejected.
As a matter of another example, consider the RP-DeLP programme P R1 from Example 4.3.Figure 4 showed the warrant dependency graph for the set of valid arguments H = { {p},p , {q},q } and the set of almost valid arguments F = { {q,r ← q},r , {p,s ← p},s }.
Again, since for every valid argument there is a cycle, the maximal ideal output for P R1 , is Warr ={y} and Block ={p,q}.
Next proposition shows that the maximal ideal output for an RP-DeLP programme is unique.Moreover, as Q ∈Warr (α), A,Q is not involved in a conflict nor in a cycle with respect to Warr (>α).As all sets G and H of valid arguments of strength α whose supports do not depend on A,Q are also valid with respect to (Warr ,Block ), and all sets G and H of valid arguments of strength α which supports do not depend on A,Q are also valid with respect to (Warr,Block), there should exist at least an argument B,P such that (i) it is almost valid with respect to a set H of valid arguments that satisfy Condition (b) for argument A,Q and output (Warr,Block), and (ii) it is not almost valid with respect to H.
Therefore, B,P should violate Condition (AV5) with respect to H and Warr and Block , and thus, for some subargument C,R < B,P of strength α it must hold that R ∈Warr (α) and C,R ∈ H and R or ∼R ∈ Block (≥ α).Now, as C,R ∈ H and B,P is almost valid with respect to H, either R ∈Warr(α), or R,∼R ∈ Block(≥ α).If R ∈Warr(α), because of the recursive warrant semantics, A,Q < C,R , and thus, R ∈Warr (α).If R ∈Warr(α), we have R,∼R ∈ Block(≥ α) and R or ∼R ∈ Block (≥ α).As Block(β) = Block (β) for all β >α, R,∼R ∈ Block(α) and R or ∼R ∈ Block (α).Then either R ∈Warr(α) or C,R is not valid with respect to (Warr,Block), and thus, A,Q < C,R .Now, as the warranty of A,Q depends on a circular definition of conflict between the set H and a set F of almost valid arguments which supports depend on some argument in H∪ A,Q with B,P ∈F, there is a cycle in the warrant dependency graph (V ,E) for H and F and any argument C ∈ H is such that the conclusion of C is either a vertex of the cycle or C does not satisfy Condition (a).Then, if R or ∼R ∈ Block (α) and C,R ∈ H, R or ∼R ∈ Block(α).Hence, Warr(α) =Warr (α) and Block(α) = Block (α) for all defeasibility level α.
Next we show that for the case of RP-DeLP programmes with unique output, the maximal ideal output corresponds with the (unique) output.

Proposition 5.3 (Maximal ideal and unique outputs)
If an RP-DeLP programme has a single output then it coincides with the maximal ideal output.
Proof.The proof is straightforward from Proposition 4.4.If a RP-DeLP programme has a unique output, for any defeasibility level there does not exist a warrant dependency graph with a cycle, and thus, Definition 5.1 and Definition 3.1 are equivalent.
When we restrict ourselves to the case of RP-DeLP programmes with a single defeasibility level, we get the following property of the maximal ideal output.
Proposition 5.4 (Programmes with a single defeasibility level) Let P be an RP-DeLP programme with a single defeasibility level, and let (Warr,Block) be the maximal ideal output for P.Then, for each output (Warr ,Block ) for P, we have Warr ⊆Warr and Block ⊆Warr ∪Block .
Proof.Obviously, Warr( 1) =Warr (1), for each output (Warr ,Block ) for P. Since we are considering a single defeasibility level, A,Q is a valid argument with respect to Warr iff P ∈Warr for all B,P < A,Q .Suppose that A,ϕ is valid with respect to Warr and not valid with respect to Warr .Then, there should exist an argument B,P such that B,P < A,Q and B,P ∈Warr but, B,P ∈Warr and B,P is valid with respect to Warr .Hence, there should exist a set of arguments G valid with respect to Warr such that B,P < G and { B,P }∪G minimally conflicts with respect to the set W ={R | C,R < G∪{ B,P }}.If each argument in G was valid with respect to Warr, then { B,P } ∈Warr.Then, there should exist an argument C,R ∈G such that C,R is valid with respect to Warr and not valid with respect to Warr, and thus, there should exist an argument D,T such that D,T < C,R and D,T ∈Warr but, D,T ∈Warr and B,P is valid with respect to Warr.Hence, there should exist a cycle in a warrant dependency graph and vertices for B,P and C,R should be vertices of the cycle and there should exist a path from some vertex of the cycle to the vertex of B,P , and thus, B,P ∈Warr.Hence, Warr ⊆Warr .Finally, as each argument A,Q valid with respect to Warr is also valid with respect to Warr and each valid argument is either warranted or blocked, Block ⊆Warr ∪Block .
The following example shows that in case we consider multiple defeasibility levels for , a conclusion can be warranted for the maximal ideal output at some level α and, due to the set of warranted conclusions at higher levels, rejected for each output (extension).
Let us consider now the maximal ideal output for P R3 , in which valid arguments involved in cycles are blocked and almost valid arguments involved in cycles are rejected.Obviously Warr maximal (1) ={y}, and at level α 1 the maximal ideal output for P R3 is: Warr maximal (α 1 ) =∅, Block maximal (α 1 ) ={p,q}.Now, at level α 2 we have that the argument {s},s is valid and it is not involved in a cycle nor in a conflict, and thus, s is warranted at level α 2 for the maximal ideal output.Hence, the maximal ideal output for P R3 is: Warr maximal ={y,s}, Block maximal ={p,q}.
Therefore, in the programme P R3 , s is not warranted in any of their two outputs Warr 1 and Warr 2 , but still s is warranted in the maximal ideal output.
In Section 3, we have seen that in case we consider multiple defeasibility levels, the set of conclusions that are warranted and blocked at each level is decisive for determining which arguments are valid at lower levels.Then, since the maximal ideal output for an RP-DeLP programme corresponds to a sceptical criterion regarding warranted conclusions, it is very interesting to analyse the status of the Closure Postulate for the maximal ideal output for RP-DeLP programmes with multiple defeasibility levels.
then ∪W ∪{P | B,P ∈G} ⊥, and thus, either Q is warranted at level α i or Q is rejected at level α i because Q or ∼Q are blocked at a level β with β >α i .In other words, either Q ∈Warr(α i ) , or Case 2 There is a set of valid arguments H of strength α i such that (i) there is a set of arguments F of strength α i that are almost valid with respect to H∪{ A,Q }, (ii) there is a cycle in the warrant dependency graph (V ,E) for H∪{ A,Q } and F, and any argument C,R ∈H is either a vertex of the cycle or C,R does not generate any conflict, and (iii) the vertex v Q for Q is a vertex of the cycle or there is a path from a vertex for some conclusion in H to v Q .Then, according to Definition 4.2, there is an almost valid argument for conclusion ∼Q in F or an strict rule ∼Q ← L 1 ∧...∧L m ∈ such that {L 1 ,...,L m }⊆Warr(≥ α i )∪{H | E,H ∈H}∪{F | J,F ∈F}, and thus, there is an almost valid argument D,∼Q for conclusion ∼Q in F, and an edge from the vertex v ∼Q to the vertex v Q .Now, since ∪Warr(≥ α i ) R Q and Q ∈Warr(≥ α i ), there exists a strict rule Q ← L 1 ∧...∧L p ∈ with all the L j 's in Warr(≥ α i ).Moreover, as ∪Warr(>α i ) R Q, there is at least one literal L ∈{L 1 ,...,L p } such that L ∈Warr(α i ), and thus, there is a valid argument J,L for L of strength α i and A,Q < J,L .Then, there is a cycle in the warrant dependency graph (V ,E ) for H∪{ A,Q }∪{ J,L } and F and an edge from the vertex v ∼Q to the vertex v L , and thus, L ∈Warr(α i ).Hence, either Q ∈Warr(α i ) , or Q ∈ Block(>α i ), or ∼Q ∈ Block(>α i ).
As a direct consequence, we have the following simpler form of the Closure Postulate for the particular case of programmes with a single defeasibility level.
Corollary 5.7 (Closure for RP-DeLP programmes with a single defeasibility level) Let P be an RP-DeLP programme with a single defeasibility level and let (Warr,Block) be the maximal ideal output for P.Under this hypothesis, if ∪Warr R Q, then Q ∈Warr.
The following example shows the closure result for the maximal ideal output.
Obviously, Warr(1) =∅.Then, at level α 1 , we have two valid arguments: and four almost valid arguments with respect to {H 1 ,H 2 }: Figure 7 shows the warrant dependency graph for {H 1 ,H 2 } and {F 1 ,F 2 ,F 3 ,F 4 }.The cycles express that either r or s can be warranted, but not both.Hence, at level α 1 , we have two possible outputs for P R4 : Then at level α 2 , all arguments are rejected in both outputs, and thus, Warr 1 (α 2 ) =Warr 2 (α 2 ) =∅ and Block 1 (α 2 ) = Block 2 (α 2 ) =∅.Therefore, the two possible outputs for P R3 are: Consider now the maximal ideal output for P R4 in which valid arguments involved in cycles are blocked and almost valid arguments involved in cycles are rejected.Obviously, Warr(1) maximal =∅ and, at level α 1 , the maximal ideal output for P R4 is: Warr maximal (α 1 ) =∅, Block maximal (α 1 ) ={r,s}.Now, at level α 2 we have that arguments {q},q and {h},h are valid and none of them is involved in a cycle neither in a conflict, and thus, q and h are warranted conclusions at level α 2 (i.e.{q,h}⊆Warr maximal (α 2 )).Finally, although arguments {q,∼s ← q},∼s and {h,∼r ← h},∼r are recursively based on warranted conclusions, both violate Condition (V3) (i.e.s,r ∈ Block maximal (≥ α 1 )), and thus, both arguments are rejected since they are not valid.Hence, at level α 2 , s and r are rejected for the maximal ideal output: Hence, the maximal ideal output for P R4 is: Warr maximal ={q,h}, Block maximal ={r,s}.
For any level α, the procedure level_computing first computes the set VA of valid arguments with respect to W (>α) and B(>α).Then, this set of valid arguments is dynamically updated depending on new warranted and blocked conclusions with strength α.The procedure level_computing finishes when the status for every valid argument is computed.The status of a valid argument is computed by means of the four following auxiliary functions.The empty set value is used to detect conflicts between the argument A,Q and the arguments in VA, and thus, every valid argument involved in a conflict is blocked.On the other hand, the value set of almost valid arguments which do not depend on argument A,Q is used to detect possible conflicts between the argument A,Q and the arguments in VA∪ND, and thus, every valid argument involved in a possible conflict remains as valid.In fact, the function almost_valid computes the set of almost valid arguments that satisfies Conditions (AV1)-(AV6) with respect to the current set of valid arguments.The function not_dependent considers almost valid arguments with respect to the current set of valid arguments which do not depend on A,Q .Finally, the function cycle checks the existence of a cycle in the warrant dependency graph for the current set of valid arguments and its set of almost valid arguments, and verifies whether the vertex of argument A,Q is in the cycle or there exists a path from a vertex of the cycle to it.
One of the main advantages of the maximal ideal warrant recursive semantics for RP-DeLP is from the implementation point of view.Warrant semantics based on dialectical trees, like DeLP [31,33], might consider an exponential number of arguments with respect to the number of rules of a given programme.The previous algorithm can be implemented to work in polynomial space, with a complexity upper bound equal to P NP .This can be achieved because it is not actually necessary to find all the valid arguments for a given literal Q, but only one witnessing a valid argument for Q is enough.Analogously, function not_dependent can be implemented to generate at most one almost valid argument, not dependent on A,Q , for a given literal.The only function that in the worst case can need an exponential number of arguments is cycle, but next we show that whenever cycle returns true for A,Q , then a conflict will be detected with the almost valid arguments not dependent on A,Q , so warranted literals can be detected without function cycle.Also, blocked literals detected by function cycle can also be detected by checking the stability of the set of valid arguments after two consecutive iterations, so it is not necessary to explicitly compute warrant dependency graphs.Proposition 6.1 (Optimization) Let P = ( , , ) be an RP-DeLP programme with defeasibility levels 1 >α 1 >...>α p > 0 for , and let W and B be two sets of warranted and blocked conclusions with strength ≥ α i , respectively.If VA is the set of all d-arguments of strength α i that are valid with respect to (W ,B) and AV is the set of all d-arguments of strength α i that are almost valid with respect to VA, we get the following results: (i) If there is a cycle in the warrant dependency graph for VA and AV, and A,Q ∈VA is such that the vertex of conclusion Q is a vertex of the cycle or there exists a path from a vertex of the cycle to the the vertex of conclusion Q, then there exists a set ND ⊆AV such that A,Q < R,P for all R,P ∈ND, and there exists a set S ⊆VA\{ A,Q } such that ∪W ∪{P | B,P ∈S}∪ND ⊥ and ∪W ∪{P | B,P ∈S}∪ND∪{Q} ⊥. (ii) If for all A,Q ∈VA there exists a set ND ⊆AV such that A,Q < R,P , for all R,P ∈ND, and there exists a set S ⊆VA\{ A,Q } such that ∪W ∪{P | B,P ∈S}∪ND ⊥ and ∪W ∪{P | B,P ∈S}∪ND∪{Q} ⊥, then there is at least a cycle in the warrant dependency graph for VA and AV, and every A,Q ∈VA is such that the vertex of conclusion Q is a vertex of a cycle or there exists a path from a vertex of a cycle to the the vertex of conclusion Q.Finally, observe that the following queries can be implemented with NP algorithms: (1) Whether a literal P is a conclusion of some argument returned by not_dependent(α, A,Q , VA, W , B).To check the existence of an almost valid argument C,P not dependent on A,Q , we can nondeterministically guess a subset of literals with valid arguments and a subset of rules appearing in an almost valid argument of strength α and needed to generate literals not yet in W and not yet valid (we call the latter α− rules 10 ), and check in polynomial time whether, together with set of all the warranted literals W , they actually generate the desired argument for P, as all the conditions for an almost valid argument can be checked in polynomial time, as well as the condition of not being dependent on the literal Q.This is because: (i) all the conditions of an α−rule can be checked in linear time; and (ii) checking that the guessed subset of α−rules and subset of valid literals, together with the current set of warranted literals W , generates the desired conclusion P and no contradiction can be checked by the repeated application of the modus ponens inference rule up to saturation.
(2) Whether the function conflict(in α, A,Q , VA, W , ND) returns true.To check the existence of a conflict, we can non-deterministically guess a subset of literals S from {P | C,P ∈VA\{ A,Q }∪ND} and check in polynomial time whether (i) ∪W (≥ α)∪S ⊥ and (ii) ∪W (≥ α)∪S ∪{Q} ⊥.This is again because both conditions can be checked by the repeated application of the modus ponens rule up to saturation in polynomial time.
Then, as the maximum number of times these queries need to be executed before the set of conclusions associated with VA becomes stable is polynomial in the size of the input programme, the P NP upper bound follows.

SAT encodings for finding warranted literals
From a computational point of view, the maximal ideal output for an RP-DeLP programme can be computed by means of a level-wise procedure, starting from the highest level and iteratively going down from one level to next level below.At every level, it is necessary to determine the status (warranted or blocked) of each valid argument by checking the existence of both conflicts between arguments, and cycles at the warrant dependence graphs.In the previous section, we showed that this level-wise procedure can be implemented to work in polynomial space.On the one hand, this can be achieved because it is not actually necessary to find all the valid arguments for a given literal, it is enough to find only one.Actually, in our implementation to explain the existence of a valid argument for a literal Q we simply record the last rule of the argument, i.e. a rule with Q as conclusion, and with all the literals of its body as warrants.To give a full explanation for a valid argument, we recursively give explanations for all the warrants of the body of the rule.Something similar applies to the computation of at most one almost valid argument for a given literal.This will be done with the first of the two SAT encodings we present next, and it allows also to explicitly give an almost valid argument for a literal, not only to check the existence.On the other hand, the existence of cycles in the warrant dependency graph among valid and almost valid arguments can be validated by checking the stability of conflicts between valid and almost valid arguments, so it is not necessary to explicitly compute the warrant dependency graphs.Hence, the procedure to find warranted literals needs to compute two main queries during its execution: (i) whether an argument is almost valid, and (ii) whether there is a conflict among valid and almost valid arguments.
In this section, we present SAT encodings for these two main combinatorial queries.The input and output specification of each query is as follows:

Looking for almost valid arguments
The idea for encoding the problem of searching almost valid arguments is based on the same behind successful SAT encodings for solving STRIPS planning problems [48].In a STRIPS planning problem, given an initial state, described with a set of predicates, the goal is to decide whether a desired goal state can be achieved by means of the application of a suitable sequence of actions.Each action has a set preconditions, when they hold true the action can be executed and as a result certain facts become true and some others become false (its effects).Hence executing an action changes the current state, and the application of a sequence of actions creates a sequence of states.
The planning problem is to find a sequence of actions such that, when executed, the obtained final state satisfies the goal state.
In our case, the search for an almost valid argument C,P can be seen as the search for a plan for producing P, taking as the initial set of facts some subset of a set of literals in which we already trust.We call such initial set the base set of literals, 11 and we say that they are true at the first step of the argument.For looking for an almost valid argument C,P , we will consider what rules should be executed, such that starting from the initial set will finally obtain the desired goal P. We say that a rule R can be executed starting from a set of literals S, when Body(R) ⊆ S, and that when it is executed we obtain a new set S ∪{Head(R)}.We have to consider only some rules for looking for almost valid arguments of strength α for literals not yet warranted, as we have explained in the previous section, that is, the α−rules we have defined before.
We use the following sets of literals and rules to define our SAT encoding.Consider first the initial set S 0 : which is the base set of warranted and valid literals.If we execute all the α−rules that can be executed from S 0 , that is: we obtain a new state S 1 that contains S 0 plus the heads of all the executed rules.This process can be repeated iteratively, obtaining a sequence of sets of literals S ={S 0 ,S 1 ,...,S t } and a sequence of sets finding two arguments starting from a subset of W ∪G and forcing the inclusion of {Q}.That is, the SAT clauses of this first part are as follows: (1) A clause that states that the literal Q must be true at the first step.
(2) A clause that states that at least one conflict variable c L must be true.
(3) For every conflict variable c L , a clause that states that if c L is true then literals L and ∼L must be true at the final step of the argument.(4) The rest of clauses are the same ones described in the first part of the previous encoding, except the clauses of the item 5 that are not included, but now considering as possible literals and rules at every step the ones computed from the base set W ∪G∪{Q} and using only strict rules.
The process for computing the possible literals and rules that can be potentially applied in every step of the argument is the same forward reasoning process presented for the previous encoding.This same process is used for discovering the set of conflict variables c L that need to be considered, because we can potentially force the conflict c L if at the end of this process both L and ∼L appear as reachable literals.
A second part of the SAT formula is devoted to checking that the selected set of variables and clauses S at the first step, without using Q, does not cause any conflict with the strict rules.So this second part of the formula contains a variable for any literal that appears in the logical closure of G∪W with respect to the strict rules.Actually, this second part of the formula is analogous to the second part of the formula for the previous encoding.
Observe that this encoding for searching conflicts for Q not only allows to check the existence of conflicts, but it also gives an explicit conflict set: the variables set to true that represent the chosen set S, together with almost valid arguments for those literals in S that have arguments in ND.So, we can explain the reasons for each conflict detected.

Average computational cost and easy/hard problem instances
To study the scaling behaviour of the (average) computational cost of our P NP algorithm as the size increases, as well as how different characteristics of the problem instances affect its computational cost, we have implemented our algorithm and conducted a series of experiments.
The main algorithm has been implemented with python, but for solving the SAT formulas presented in the previous section, the algorithm uses a SAT solver, that can be either MiniSAT [39] or Glucose [18].However, our architecture easily allows to use any other SAT solver that appears in the future.Minisat is one of the publicly available SAT solvers which implements most of the current state-of-the-art solving techniques such as conflict-clause recording and conflict-driven backjumping, among others.Glucose, that has some common parts with MiniSAT, implements some new learning mechanisms which made it award winning on SAT 2011 competition.As we have mentioned in the Introduction, we have a preliminary version of the algorithm that works with ASP encodings [3] instead of with SAT encodings, although the currentASP based version only works with one defeasible level.In the near future, we plan to improve the ASP based version to be able to work with multiple levels.
In the experiments, the algorithm solves different test-sets of problem instances obtained with a random generation algorithm.To study and analyse how our RP-DeLP algorithm behaves as different characteristics of the problem change, we generated our instances using one and two levels of defeasibility and changing the other parameters of the problem instances.

Random generation of RP-DeLP problem instances
We used different parameters to control the generation of random RP-DeLP problem instances with different sizes, defeasibility levels and other characteristics.We focused or experimented first on one set of problems with only one defeasible level and then on another set with two defeasible levels.In both cases, we were interested in how the resolution time differs when the ratio of clauses to the number of variables increases.Then, in the first case with only one defeasible level, we were also interested in the results when the fraction of clauses of the programme at the strict knowledge level is modified, ranging from no strict knowledge at all to all clauses at the strict knowledge level.For the case of two defeasible levels, we have investigated the effect of modifying the fraction of clauses between the two defeasible levels.We next explain the generation of our problem instances.
Generation of instances with one defeasible level: Given a number of variables (V ), a maximum clause length (ML), a ratio of clauses to variables (C/V ), and a fraction (f ), between 0.0 an 1.0, of strict knowledge, the algorithm generates an RP-DeLP problem instance by generating C clauses, such that the length of every clause is selected uniformly at random from [1,ML] (clauses with length 1 are facts).The variables of the literals of a clause are selected uniformly at random without repetition, and are negated with probability 0.5.From the C clauses, f •C clauses are in the strict knowledge and the rest in the defeasible set.
Two defeasible levels instance generation: Similar to the previous instance generator with a number of variables (V ), a maximum clause length (ML), a ratio of clauses to variables (C/V ), now we fix the fraction of strict knowledge (f ) to 0.1.Then two defeasible levels are built assigning a fraction l between 0.0 and 1.0 of the total number of defeasible clauses to the first defeasible level and 1-l to the second defeasible level.

Test instances considered
We generated two different groups of test sets: test sets with one defeasible level and test sets with two defeasible levels.In both groups, test instances were created with a number of variables (V ) selected from {20,30}, 12 and with maximum clause length (ML) selected from {2,4}.
In the case of one defeasible level, for each combination (V ,ML), different test sets of instances were created by selecting a number of total clauses, such that the ratio C/V ranged from 1 to 12 in steps of 1, and the fraction of clauses in the strict knowledge ranged from 0 to 0.9 in steps of 0.1.So, the total number of test sets for each combination (V ,ML) was 90.The number of instances generated in each test set was 50.
In the case of two defeasible levels, for each combination (V ,ML) and an strict knowledge fraction set to 0.1, different test sets of instances were created by selecting a number of total clauses, such that the ratio C/V ranged from 1 to 12 in steps of 1, and the fraction of clauses in the first level l ranged from 0.1 to 0.9 in steps of 0.1.

Empirical results
For one defeasible level, we analyse the results for instances with 30 variables and maximum clause length 2. The left plot of Figure 8 shows the median time to solve the instances with our algorithm argumentation systems this would still lead to inconsistency problems, and they show that in order to satisfy the consistency postulates an attack relation should be valid, in the sense that when two arguments have jointly inconsistent premises, they should attack each other.
Regarding the kind of logical language used, following the terminology used in [30], RP-DeLP can be seen as a member of the family of rule-based argumentation systems, as it is based on a language defined over a set of literals and of strict and (weighed) defeasible rules.In this sense, RP-DeLP is very close in spirit to the well-known ASPIC argumentative framework [9,30], that was developed in response to the fact that the abstract nature of Dung's theory gives no guidance as to what kinds of instantiations satisfy intuitively rational properties.ASPIC adopts an intermediate level of abstraction between Dung's fully abstract level and concrete instantiating logics, by making some minimal assumptions on the nature of the logical language and the inference rules, and then providing abstract accounts of the structure of arguments, the nature of attack, the use of preferences and rationality postulates a well-behaved system should satisfy [30].Prakken [54] further develops the ASPIC framework (ASPIC + ) as a general abstract model of argumentation with, among other features, structured arguments and preferences on , showing that under some assumptions, rationality postulates were satisfied.when applying preferences to resolve attacks.ASPIC + has been further generalized by Modgil and Prakken [50,51] to accommodate classical logic instantiations extended with preferences.
The output for an RP-DeLP programme is a rank-ordered set of warranted and blocked conclusions which satisfy the consistency postulates [30].In contrast to DeLP and other argument-based approaches, the RP-DeLP semantics is based on a (not necessarily binary) general notion of collective conflict among arguments and on the fact that if an argument is warranted it must be that all its subarguments also are warranted.
Collective conflicts has also been considered in several papers, e.g. in [52], while in [7,8] discuss to some extent the problems binary attacks can cause.On the other hand, the idea of defining a warrant semantics on the basis of conflicting sets of arguments was proposed in [59] and [52].The difference between these approaches and our notion of collective conflict is that in [59] the notion of conflict is not relative to a set of already warranted conclusions and [52] defines a generalization of Dung's abstract framework with sets of attacking arguments not relative to the strict part of the knowledge base.Although the RP-DeLP semantics for warranted conclusions is sceptical, circular definitions of conflict between sets of arguments can lead to situations in which multiple evaluation orders exist, giving rise to different outputs of warranted and blocked conclusions.Following Pollock's recursive semantics for defeasible argumentation [53], circular definitions of conflict between sets of arguments have been characterized by means of dependency graphs representing support and collective conflict relations between the conclusions of arguments and the strict part of the knowledge base.
RP-DeLP recursive semantics draws from the so-called 'ideal semantics' promoted by Dung, Mancarella and Toni [35,36] as an alternative basis for sceptical reasoning within abstract argumentation settings.Informally, ideal acceptance not only requires an argument to be sceptically accepted in the traditional sense but further insists that the argument is in an admissible set all of whose arguments are also sceptically accepted.While the original proposal was couched in terms of the socalled preferred semantics for abstract argumentation, in [38] the notion of 'ideal acceptability' has been extended to arbitrary semantics, showing that standard properties of classical ideal semantics, e.g.unique status, continue to hold in some extension-based semantics (see also [37] for an analysis of the computational complexity of the ideal semantics within abstract argumentation frameworks and assumption-based argumentation frameworks).In RP-DeLP, the maximal ideal output for an RP-DeLP programme is defined in terms of the maximum rank-ordered set of warranted and blocked conclusions recursively based on warranted information and not involved in neither a conflict nor a circular definition of conflict.The idea is that if a conclusion is warranted at a given level β, so it could also be at any higher level.A different approach could have been to consider that blocked conclusions at one level are not propagated to lower levels.In such a case, an alternative semantics for our system could therefore be defined following a similar line to the one in [44].
Finally, the use of SAT/ASP technology for solving reasoning problems in argumentation frameworks was first advocated in [25], where the authors present different ways to solve the acceptability problem of extensions in abstract argumentation frameworks using propositional logic encodings.A related approach can be found in [40] where, in the context of logic-based argumentation frameworks (built on top of a generic monotonic deductive system), the authors propose to implement them based on the satisfiability problem of quantified Boolean formulas (QBFs).For the specific case of abstract argumentation frameworks, a QBF approach has also been recently proposed in [17].Finally, in [27], in the context of logic-based argumentation, the authors use quantified Boolean formulae (QBFs) to characterize various problems (arguments, undercut, argument trees) in argumentation based on classical logic, and use them to obtain new computational complexity results.

Conclusions and future work
In this article, we have introduced a new recursive semantics for determining the warranty status of arguments in defeasible argumentation.The distinctive features of this semantics, e.g. with respect to Pollock's critical link semantics, are: (i) it is based on a non-binary notion of conflict in order to preserve consistency with the strict knowledge and (ii) besides the set of warranted and rejected conclusions, we introduce the set of blocked conclusions, which are those conclusions which are based on warranted information but they generate a conflict with other already warranted arguments of the same strength.
We have also contributed an efficient implementation of the algorithm, which computes the maximal ideal output for an RP-DeLP programme and is based on SAT encodings for the two NP queries that need to be resolved during the computation of the output: looking for almost valid arguments and looking for collective conflicts.So far, with this implementation we have studied the behaviour of RP-DeLP programmes on randomly generated instances, where parameters such as number of clauses, number of variables and size of levels (strict and defeasible), have been changed on different test sets to try to understand how these parameters affect the problem complexity.For instances with only one defeasible level, we have seen that as the fraction of clauses on the strict part increases, more instances become inconsistent for the same total number of clauses.When we look only at the consistent instances, the computation time increases when more conflicts arise (the number of blocked literals at the output increases).However, only when the fraction of clauses on the strict part is very small we observe really computationally challenging instances with many blocked literals.For instances with two defeasible levels, we have a similar situation.As more blocked literals appear at the output, the computation time increases.But we have also seen that the balance on the number of clauses between defeasible levels affects the output and its computation time.That is, the more balanced the defeasible levels are, the more warranted literals, the less blocked literals and the less computation time we have.
As future work, we plan to improve the efficiency of the algorithm we have already designed and implemented by minimising the effective number of NP queries that have to be made during its execution.Also, with the aim of obtaining an algorithm able to scale up with problem size, we will improve a preliminary implementation based on ASP encodings [3], to be able to solve problems with multiple defeasible levels, as we have done with the SAT-based version we have presented here.We believe that using ASP encodings in the most general case of multiple defeasible levels will be helpful to improve the performance of the system, given that the preliminary results in the above mentioned work indicate that ASP encodings can be competitive with SAT encodings for our problem, at least for the case of a single defeasible level.Regarding the efficiency of our implemented algorithm, it is worth to mention that the scaling behaviour we have observed in our experiments could be very specific to the random generator of instances we have used.Results found in the AI literature about the relation of problem instances structure with algorithm efficiency for NP-hard problems, see e.g.[43,45,47], suggest that in our problem we could observe significant differences in solving performance between random instances and instances with some particular structure.That is, instances generated from some particular application domain.So, we plan to study the generation of instances obtained from some particular domains, in order to check whether instances with particular structure could be solved more efficiently with our algorithm or with specialized new versions of our algorithm.

Figure 3 .
Figure 3. Recursion case from the example in Scenario 5.

Function
almost_valid(in α, VA, W , B) return AV: set of arguments AV := { C,P with strength α | C,P satisfies Conditions (AV1)-(AV6) wrt VA} end function Function not_dependent(in α, A,Q , VA, W , B) return ND: set of almost valid arguments which do not depend on Q AV := almost_valid(α, VA, W , B) ND := { C,P ∈AV | A,Q < C,P } end function Function conflict(in α, A,Q , VA, W , ND) return con: Boolean con := ∃ S ⊆VA\{ A,Q }∪ND such that ∪W (≥ α)∪{P | C,P ∈S} ⊥ and ∪W (≥ α)∪{P | C,P ∈S}∪{Q} ⊥ end function Function cycle(in α, A,Q , VA, W , AV ) return cy: Boolean cy := there is a cycle in the warrant dependency graph for VA and AV and the vertex for A,Q is a vertex of the cycle or there exists a path from a vertex in VA of the cycle to the vertex for A,Q end function The function conflict checks (possible) conflicts among the argument A,Q and the set VA of valid arguments extended with the set ND of arguments.The set ND of arguments takes two different values: the empty set and the set of almost valid arguments whose supports depend on some argument in VA\{ A,Q }.
If the vertex of conclusion Q is a vertex of the cycle, because of the warranty dependency graph definition, we can consider the set ND ⊆AV such that the vertex of each conclusion in ND is a vertex of the cycle and A,Q < R,P for all R,P ∈ND, and then, there should exist a set S ⊆VA\{ A,Q } such that ∪W ∪{P | B,P ∈S}∪ND ⊥ and ∪W ∪{P | B,P ∈ S}∪ND∪{Q} ⊥.If the vertex of conclusion Q is not a vertex of the cycle and there exists a path from a vertex of the cycle to the vertex of conclusion Q, we can consider the set ND ⊆AV such that the vertex of each conclusion in ND is a vertex of the cycle.Now, because of the warranty dependency graph definition, A,Q < R,P for all R,P ∈ND and there should exist a set S ⊆VA\{ A,Q } such that ∪W ∪{P | B,P ∈S}∪ND ⊥ and ∪W ∪{P | B,P ∈ S}∪ND∪{Q} ⊥.(ii)We have that for all S ⊆VA, ∪W ∪{P | R,P ∈S} ⊥ and that for all A,Q ∈VA there exists a set ND ⊆AV such that A,Q < R,P for all R,P ∈ND, and there exists a set S ⊆VA\{ A,Q } such that ∪W ∪{P | B,P ∈S}∪ND ⊥ and ∪W ∪{P | B,P ∈S}∪ND∪{Q} ⊥.Then, for all A,Q ∈VA, we have that the warranty of Q depends on a possible conflict with a set S ⊆VA\{ A,Q } and a set ND ⊆AV such that A,Q < R,P for all R,P ∈ND.Therefore, because of the warranty dependency graph definition, there should exists a cycle in the warrant dependency graph (V ,E) for VA and AV such that the vertexes of conclusions of ND are vertexes of the cycle and the vertexes of conclusions of S and { A,Q } are vertexes of the cycle or there exists a path from a vertex of the cycle to the the vertex of these conclusions.

( i )
Almost valid argument: It takes as input a defeasibility degree α, a literal P, sets W and B of warranted and blocked literals of strength ≥ α, respectively, a set VA of valid arguments of strength α, and an argument A,Q ∈VA.It computes an almost valid argument C,P of strength α that does not depend on A,Q .(ii) Conflict: It takes as input a defeasibility degree α, a set W of warranted literals of strength ≥ α, a set VA of valid arguments of strength α, a valid argument A,Q of strength α, and a set ND of almost valid arguments of strength α that do not depend on A,Q .It checks (possible) conflicts among the argument A,Q and the set VA of valid arguments extended with the set ND of almost valid arguments.
If ϕ ∈Warr(α)∪Block(α), by Condition (V2), ϕ ∈Warr(β)∪Block(β), for all β >α.Suppose that there exists an argument A,ϕ of strength β, with β >α, verifying Condition (V1).and, if β <α, ϕ ∈ Block(β).Then, it must be that ϕ ∈Warr(α) and ϕ ∈ Block(α), and thus, there exists two valid arguments of strength α such that one is involved in a conflict and the other is not.Suppose that A,ϕ is a valid argument involved in a conflict.Then, there should exist a set G of valid arguments of strength α such that A,ϕ is not a subargument of arguments in G and G∪{ A,ϕ } minimally conflicts.Hence, every valid argument B,ϕ is not a subargument of arguments in G, and thus, B,ϕ is involved in a conflict.Proof that ψ ∈ Figure 5. Warrant dependency graph for P R2 at level α 1 .Proposition 5.2 (Unicity of the maximal ideal output) Let P = ( , , ) be an RP-DeLP programme.The pair (Warr,Block) of warranted and blocked conclusions that satisfies the maximal ideal output characterization for P of Definition 5.1 is unique.Proof.Suppose that (Warr,Block) and (Warr ,Block ) are pairs of warranted and blocked conclusions that satisfy the maximal ideal output characterization for P stated in Def.5.1.Obviously, Warr(1) = (1)r(1).Suppose that for some α, Warr(α) =Warr (α) and Warr(β) =Warr (β), for all β >α.As Warr(α) =Warr (α), suppose that A,Q of strength α is valid with respect to (Warr,Block) and (Warr ,Block ) but Q ∈Warr(α) and Q ∈Warr (α).Then, Q ∈ Block(α) and A,Q is either Case 1: involved in a conflict with respect to Warr(>α) and a set G of valid arguments of strength α which supports do not depend on A,Q , or Case 2: the warranty of A,Q depends on a circular definition of conflict between a set H of valid arguments which supports do not depend on A,Q and a set F of almost valid arguments which supports depend on some argument in H∪ A,Q .