Formalization and Logical Properties of the Maximal Ideal Recursive Semantics for Weighted Defeasible Logic Programming

Possibilistic Defeasible Logic Programming (P-DeLP) is a logic programming framework which combines features from argumentation theory and logic programming, in which defeasible rules are attached with weights expressing their relative belief or preference strength. In P-DeLP a conclusion succeeds if there exists an argument that entails the conclusion and this argument is found to be undefeated by a warrant procedure that systematically explores the universe of arguments in order to present an exhaustive synthesis of the relevant chains of pros and cons for the given conclusion. Recently, we have proposed a new warrant recursive semantics for P-DeLP, called Recursive P-DeLP (RP-DeLP for short), based on the claim 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. This paper explores the relationship between the exhaustive dialectical analysis based semantics of P-DeLP and the recursive based semantics of RP-DeLP, and analyzes a non-monotonic inference operator for RP-DeLP which models the expansion of a given program by adding new weighted facts associated with warranted conclusions. Given the recursive based semantics of RP-DeLP, we have also implemented an argumentation framework for RP-DeLP that is able to compute not only the output of warranted and blocked conclusions, but also explain the reasons behind the status of each conclusion. We have developed this framework as a stand-alone application with a simple text-based input/output interface to be able to use it as part of other AI systems.


Introduction and motivations
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 (Prakken & Vreeswijk, 2002). 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 formalisations 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 (Besnard & Hunter, 2008).
proposal by governments), and then finally allow policy-makers to understand the relevant opinions on the ongoing debates. A relevant example of this last kind of tools is Parmenides (Atkinson, Bench-Capon, & McBurney, 2007), which follows a structured approach to propose arguments in a debate and uses valued abstract argumentation frameworks (Bench-Capon, 2003) for analysing the results of the debate. However, as a first step, it is important to understand the characteristics of our argumentation framework when compared with other existing ones.
In this paper, we explore the relationship between the exhaustive dialectical analysis-based semantics of P-DeLP and the maximal ideal output of RP-DeLP, and we analyse a nonmonotonic inference operator for RP-DeLP which models the expansion of a given program by adding new weighed facts associated with warranted conclusions. Finally, considering our final aim of using RP-DeLP in argumentation applications such as those related to the social web, we briefly explain the characteristics of an implementation of an argumentation framework for RP-DeLP we have recently developed as a command-line application that can be easily used by third-party final applications or services.

The language of P-DeLP and RP-DeLP
In order to make this paper self-contained, we present next the main definitions that characterise P-DeLP and RP-DeLP frameworks. For details the reader is referred to  and Alsinet et al. (2014).
The language of P-DeLP and RP-DeLP, denoted L, 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 consist of rules of the form QˆP 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. P-DeLP and RP-DeLP frameworks are based on the propositional logic ðL; rÞ where the inference operator r is defined by instances of the modus ponens rule of the form: fQˆP 1^. . .^P k ; P 1 ; . . . ; P k } r Q. A set of clauses G will be deemed as contradictory, denoted G r', if, for some atom q, G r q and G r,q.
In both frameworks a program P is a tuple P ¼ ðP; D; WÞ over the logic ðL; rÞ, where P; D # L, and P Z'. P is a finite set of clauses representing strict knowledge (information we take for granted they hold true), D is another finite set of clauses representing the defeasible knowledge (formulas for which we have reasons to believe they are true). Finally, W is a total preorder on P < D representing levels of defeasibility: w a c means that w is more defeasible than c. Actually, since formulas in P are not defeasible, W is such that all formulas in P are at the top of the ordering. For the sake of a simpler notation we will often refer in the paper to numerical levels for defeasible clauses and arguments rather than to the pre-ordering W, so we will assume a mapping N : P < D ! ½0; 1 such that NðwÞ ¼ 1 for all w [ P and NðwÞ , NðcÞ iff w a c. 1 The notion of argument is the usual one inherited from similar definitions in the argumentation literature (Chesñevar et al., 2000;Prakken & Vreeswijk, 2002;Simari & Loui, 1992). Given a program P, an argument for a literal (conclusion) Q of L is a pair A ¼ kA; Ql, with A # D such that P < A Z', and A is minimal (w.r.t. set inclusion) such that P < A r Q. If A ¼ Y, then we will call A a s-argument (s for strict), otherwise it will be a d-argument (d for defeasible). We define the strength of an argument kA; Ql, written sðkA; QlÞ, as follows: The notion of sub-argument is referred to d-arguments and expresses an incremental proof relationship between arguments which is defined as follows. Let kB; Ql and kA; Pl be two d-arguments such that the minimal sets (w.r.t. set inclusion) P Q # P and P P # P such that P Q < B r Q and P P < A r P verify that P Q # P P . Then, kB; Ql is a sub-argument of kA; Pl, written kB; Ql f kA; Pl, when either B , A (strict inclusion for defeasible knowledge), or B ¼ A and P Q , P A (strict inclusion for strict knowledge). A literal Q of L is called justifiable conclusion w.r.t. P if there exists an argument for Q, i.e. there exists A # D such that kA; Ql is an argument.
As in most argumentation formalisms (see, e.g. Chesñevar et al., 2000;Prakken & Vreeswijk, 2002), in P-DeLP and RP-DeLP frameworks it can be the case that there exist arguments supporting contradictory literals, and thus, there exist sets of conflicting arguments. Since arguments can rely on defeasible information, conflicts among arguments may be resolved in both frameworks by comparing their strength. In this sense, the aim of both frameworks is to provide a useful warrant procedure in order to determine which conclusions are ultimately accepted (or warranted) on the basis of a given program. The difference between the two frameworks lies in the way in which this procedure is defined and the type of conflicts are handled.
In P-DeLP, warranted conclusions are justifiable conclusions which remain undefeated after an exhaustive dialectical analysis of all possible arguments for an against and only binary attacks or defeat relations are considered. In RP-DeLP semantics for warranted conclusions is based on a collective (non-binary) notion of conflict between arguments and if an argument is excluded from an output, then all the arguments built on top of it are excluded from that output. In the following sections we describe both mechanisms.
In P-DeLP warranted conclusions are formalised in terms of an exhaustive dialectical analysis of all possible argumentation lines rooted in a given argument. An argumentation line starting in an argument kA 0 ; Q 0 l is a sequence of arguments l ¼ ½kA 0 ; Q 0 l; kA 1 ; Q 1 l; . . . ; kA n ; Q n l; . . . such that each kA i ; Q i l defeats the previous argument kA i21 ; Q i21 l in the sequence, i . 0. In order to avoid fallacious reasoning, additional constraints are imposed, namely: (1) Non-contradiction: given an argumentation line l, the set of arguments of the proponent (respectively, opponent) should be non-contradictory w.r.t. P. 3 (2) Progressive argumentation: (i) every blocking defeater kA i ; Q i l in l with i . 0 is defeated by a proper defeater 4 kA iþ1 ; Q iþ1 l in l; and (ii) each argument kA i ; The non-contradiction condition disallows the use of contradictory information on either side (proponent or opponent). The first condition of progressive argumentation enforces the use of a proper defeater to defeat an argument which acts as a blocking defeater, while the second condition avoids non-optimal arguments in the presence of a conflict. An argumentation line satisfying the above restrictions is called acceptable, and can be proven to be finite. The set of all possible acceptable argumentation lines results in a structure called dialectical tree. Given a program P and a goal Q, Q is warranted w.r.t. P with maximum strength a iff there exists an argument kA; Ql with sðkA; QlÞ ¼ a such that (i) every acceptable argumentation line starting with kA; Ql has an odd number of arguments and (ii) there is no other argument of the form kB; Ql, with sðkB; QlÞ . a, satisfying the above condition. In the rest of the paper we will write P j , w kA; Q; al to denote this fact and we will write C w DT ðPÞ to denote the set of warranted conclusions of P based on dialectical trees, i.e. C w DT ðPÞ ¼ fQ j P j, w kA; Q; al}. Caminada and Amgoud (2007) 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 clauses. This means that the warrant semantics of P-DeLP satisfies the indirect consistency postulate iff given a program P ¼ ðP; D; WÞ its set of warranted conclusions C w DT ðPÞ is such that P < C w DT ðPÞ Z'. The defeat relation in P-DeLP, as occurs in most rule-based argumentation systems, is binary and, in some cases, the conflict relation among arguments is hardly representable as a binary relation when we compare them with the strict part of a program. For instance, consider the following program P ¼ ðP; D; aÞ with P ¼ fp; ,pˆa^b^c}, D ¼ fa; b; c} and a single defeasibility level a for D. Clearly, A 1 ¼ kfa}; al, A 2 ¼ kfb}; bl and A 3 ¼ kfc}; cl are arguments that justify conclusions a, b and c; respectively, and A 1 , A 2 and A 3 have no defeaters, and thus, fa; b; c} are warranted w.r.t. the P-DeLP program P. Indeed, conclusions a, b and c do not pair wisely generate a conflict since P < fa; b} Z', P < fa; c} Z' and P < fb; c} Z'. However, these conclusions are collectively conflicting w.r.t. the strict part of program P since P < fa; b; c} r', and thus, the warrant semantics of P-DeLP does not satisfy the indirect consistency postulate.
In order to characterise such situations we proposed in Alsinet et al. (2014) the RP-DeLP framework, a new warrant semantics for P-DeLP based on a general notion of collective (nonbinary) conflict among arguments ensuring the three rationality postulates defined by Caminada and Amgoud.

Recursive warrant semantics of RP-DeLP
The warrant recursive semantics of RP-DeLP is based on the following general 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 program and the set of conclusions of those arguments. Let P ¼ ðP; D; aÞ be a program and let W # L be a set of conclusions. We say that a set of arguments fkA 1 ; Q 1 l; . . . ; kA k ; Q k l} minimally conflicts with respect to W iff the two following conditions hold: (i) the set of argument conclusions fQ 1 ; . . . ; Q k } is contradictory with respect to W, i.e. it holds that P < W < fQ 1 ; . . . ; Q k } r' and (ii) the set fkA 1 ; Q 1 l; . . . ; kA k ; Q k l} is minimal with respect to set inclusion satisfying (i), i.e. if S C fQ 1 ; . . . ; Q k }, then P < W < S Z'.
This general notion of conflict is used to define an output of an RP-DeLP program P ¼ ðP; D; aÞ as a pair (Warr, Block) of subsets of L of warranted and blocked conclusions, respectively. 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 the next level below. If 1 . a 1 . . . . . a p $ 0 are the strengths of d-arguments that can be built within P, we define Warr ¼ Warrð1Þ < f< i¼1;p Warrða i Þ} and Block ¼ < i¼1;p Blockða i Þ, where Warrð1Þ ¼ fQ j P r Q}, and Warrða i Þ and Blockða i Þ are, respectively, the sets of the warranted and blocked justifiable conclusions of strength a i . Intuitively, an argument kA; Ql of strength a i is valid whenever (i) it is based on warranted conclusions, (ii) there does not exist a valid argument for Q with strength . a i and (iii) Q is consistent with both blocked conclusions of strength . a i and the strict knowledge extended with warranted conclusions of strength . a i . Then, a valid argument kA; Ql becomes blocked as soon as it leads to some conflict among valid arguments of the same strength and the set of already warranted conclusions, otherwise, it is warranted.
In Alsinet et al. (2014) we show that, in case of some circular dependences among arguments, the output of an RP-DeLP program may be not unique, that is, there may exist several pairs (Warr, Block) satisfying the above conditions for a given RP-DeLP program. The following example shows a circular relation among arguments involving strict knowledge. Consider the RP-DeLP program P ¼ ðP; D; aÞ with P ¼ fy; ,yˆp^r; ,yˆq^s}, D ¼ fp; q; rˆq; sˆp} and a single defeasibility level a for D. Then, Warrð1Þ ¼ fy} and A 1 ¼ kfp}; Pl and A 2 ¼ kfq}; ql are valid arguments for conclusions p and q, respectively, and thus, conclusions p and q may be warranted or blocked but not rejected. Moreover, since arguments B 1 ¼ kfq; rˆq}; rl and B 2 ¼ kfp; sˆp}; sl are valid whenever q and p are warranted, respectively, and P < fp; r} r' and P < fq; s} r', we get that p can be warranted iff q is blocked and that q can be warranted iff p is blocked. Hence, in that case we have two possible outputs: (Warr 1 , Block 1 ) and (Warr 2 , Block 2 ), where Warr 1 ¼ fy; p}; Block 1 ¼ fq; s} and Warr 2 ¼ fy; q}; Block 2 ¼ fp; r}. Figure 1 shows the circular dependences among fA 1 ; A 2 } and fB 1 ; B 2 }. Conflict and support dependencies among arguments are represented as dashed and solid arrows, respectively. 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 (i.e. r is valid whenever q is warranted), (3) the warranty of q depends on a (possible) conflict with s and (4) the support of s depends on p (i.e. s is valid whenever p is warranted).
In Alsinet et al. (2014) we analyse the problem of deciding the set of conclusions that can be ultimately warranted in RP-DeLP programs with multiple outputs. The usual sceptical approach would be to adopt the intersection of all possible outputs. However, in addition to the computational limitation, as stated in Pollock (2009), 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, 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. 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. Instead, the set of ultimately warranted conclusions we are interested in for RP-DeLP programs is characterised by means of a recursive level-wise definition considering at each level the maximum set of conclusions based on warranted information and not involved in neither a conflict nor a circular definition of warranty. We refer to this output as maximal ideal output of an RP-DeLP program.
Intuitively, a valid argument kA; Ql becomes blocked in the maximal ideal output, as soon as (i) it leads to some conflict among valid arguments of same strength and the set of already warranted conclusions or (ii) the warranty of kA; Ql depends on some circular definition of conflict between arguments of the same strength; otherwise, it is warranted. Consider again the previous program P. According to Figure 1, valid arguments for conclusions p and q are involved in a circular definition of conflict, and thus, conclusions p and q must be blocked in the maximal ideal output of P and arguments for conclusions r and s are rejected. Hence, in that case, we have the following maximal ideal output of P: ðWarr max ; Block max Þ, where Warr max ¼ fy} and Block max ¼ fp; q}.

Dialectical analysis and recursive warrant semantics
In Alsinet et al. (2014) we prove that the maximal ideal output of an RP-DeLP program is unique and satisfies the indirect consistency property defined by Caminada and Amgoud with respect to the strict knowledge. Next we show that given a program P and its set C w DT ðPÞ of warranted conclusions based on dialectical trees, C w DT ðPÞ contains each warranted conclusion in the maximal ideal output of P.
Proposition 5.1. Let P ¼ ðP; D; aÞ be a program with levels of defeasibility 1 . a 1 . . . . . a p $ 0. If ðWarr; BlockÞ is the maximal ideal output of P, for each level a i it holds that Warrða i Þ # fQ j P j, w kA; Q; a i l}. Obviously, Warrð1Þ ¼ fQ j P j, w kA; Q; 1l}.
Proof. If Q [ Warrða i Þ, there exits an argument kA; Ql of strength a i such that (i) it is based on warranted conclusions, (ii) there does not exist an argument for Q of strength . a i based on warranted conclusions, (iii) Q is consistent with both blocked conclusions of strength . a i and the strict knowledge extended with warranted conclusions of strength . a i , (iv) it does not lead to any conflict among arguments of strength a i and (v) it is not involved in any circular definition of conflict between arguments of strength a i . Hence, every acceptable argumentation line w.r.t. P starting in kA; Ql has an odd number of arguments and there is no other argument of the form kB; Ql w.r.t. P, with sðkB; QlÞ . a i , satisfying the above condition, and thus, P j, w kA; Q; a i l. A Notice that the inverse of Proposition 5.1 does not hold since the dialectical analysis-based semantics of P-DeLP does not satisfy the indirect consistency property defined by Caminada and Amgoud with respect to the strict knowledge. Because we are interested in exploring the relationship between the dialectical analysis-based semantics of P-DeLP and the maximal ideal recursive semantics of RP-DeLP, we have to extend the P-DeLP framework with some mechanism ensuring this property. Caminada and Amgoud (2007) propose as a solution the definition of a special transposition operator Cl tp for computing the closure of strict rules. This accounts for taking every strict rule . . . ; f n ! : f i can be obtained (transpositions of r). If S is a set of strict rules, Cl tp is the minimal set such that (i) S # Cl tp ðSÞ and (ii) If s [ Cl tp ðSÞ and t is a transposition of s, then t [ Cl tp ðSÞ.
Computing the closure under transposition of strict rules allows the indirect consistency property to be satisfied in the case of rule-based argumentation systems such as DeLP or P-DeLP as it was proved in Caminada and Amgoud (2007). In fact, in some sense, it allows to perform forward reasoning from warranted conclusions, and thus, to evaluate collective conflicts among arguments. However, P-DeLP is a Horn-based system, so that strict rules should be read as inference rules rather than as material implications. In this respect, the use of transposed rules might lead to unintuitive situations in a logic programming context. Consider, e.g. the program P ¼ ðP; D; WÞ with P ¼ fqˆp^r; sˆ,r; p; ,q; ,s}, D ¼ Y. In P-DeLP, p, ,q and ,s would be warranted conclusions, i.e. C w DT ðPÞ ¼ fp; ,q; ,s}. However, the closure under transposition Cl tp ðPÞ would include the rule ,rˆp^,q, resulting in inconsistency since both s and ,s can be derived, so that the whole program would be deemed as invalid.
Apart from the above limitation, when extending a P-DeLP with all possible transpositions of every strict rule, the system can possibly establish as warranted goals conclusions which are not explicitly expressed in the original program. Consider, e.g. the program P ¼ ðP; D; WÞ with P ¼ f,yˆa^b; y}, D ¼ fa; b} and two levels of defeasibility for D as follows: fb} a fa}. Assume a 1 is the level of fa} and a 2 is the level of fb}, with 1 . a 1 . a 2 . 0. Transpositions of the strict rule ,yˆa^b are ,aˆy^b and ,bˆy^a. Then, the argument A ¼, f,bˆa^y; a}; ,b . with strength a 1 justifies conclusion ,b. Moreover, as there is neither a proper nor a blocking defeater of A, we conclude that ,bis warranted w.r.t. P w ¼ ðP < Cl tp ðPÞ; D; WÞ, although no explicit information is given for literal ,b in P. Moreover, notice that C w DT ðPÞ ¼ fy; a; b} and C w DT ðP w Þ ¼ fy; a; ,b}. Next we show that if ðWarr; BlockÞ is the maximal ideal output of a program P ¼ ðP; D; WÞ such that P < Cl tp ðPÞ Z', the set Warr of warranted conclusions contains indeed each literal Q satisfying that P w j , w kA; Q; al whenever P < A r Q, with P w ¼ ðP < Cl tp ðPÞ; D; WÞ.
Proposition 5.2. Let P ¼ ðP; D; aÞ be a program with levels of defeasibility 1 . a 1 . . . . . a p $ 0 and such that P < Cl tp ðPÞ Z'. If ðWarr; BlockÞ is the maximal ideal output of P and P w ¼ ðP < Cl tp ðPÞ; D; WÞ, for each level a i it holds that fQ j P w j , w kA; Q; a i l and P < A r Q} # Warrða i Þ. Obviously, fQ j P w j , w kA; Q; 1l and P < A r Q} ¼ fQ j P r Q} ¼ Warrð1Þ.
Proof. We distinguish between two cases: (a) P w j , w kA; Q; a i l and there is no defeater for kA; Ql w.r.t. P w , i.e. ½kA; Ql is the only acceptable argumentation line starting in kA; Ql, and thus, there does not exist any argument kD; ,Ql w.r.t. P w such that sðkD; ,QlÞ $ a i . Moreover, if kB; Pl f kA; Ql, there is no defeater of kB; Pl w.r.t. P w , and thus, there does not exist any argument kC; ,Pl w.r.t. P w such that sðkC; ,PlÞ $ sðkB; PlÞ. Since P w contains the closure of strict rules Cl tp ðPÞ, if P < A r Q we have that kA; Ql of strength a i is valid w.r.t. the recursive warrant semantics for P. Moreover, kA; Ql is not involved in a conflict nor in a cycle among valid arguments of strength a i and the set of warranted conclusions of strength greater than a i , and thus, argument kA; Ql of strength a i is warranted in the maximal ideal output of P. Hence, if P < A r Q, Q [ Warrða i Þ. (b) P w j , w kA; Q; a i l and kA; Ql has at least one defeater w.r.t. P w , i.e. every acceptable argumentation line starting in an argument kA; Ql is of the form l ¼ ½kA; Ql; . . . ; A 2n21 ; A 2n with n $ 1. Again we distinguish between two cases: (a) Argument A 2n is a blocking defeater for argument A 2n21 . In this case, conclusions of arguments A 2n and A 2n21 are not warranted w.r.t. P w and the argumentation line ½kA; Ql; . . . ; A 2n21 ; A 2n can be omitted in the sense that it is subsumed by the rest of acceptable argumentation lines starting in argument kA; Ql and containing argument A 2n21 . (b) Argument A 2n is a proper defeater for argument A 2n21 . In this case, A 2n has no defeaters w.r.t. P w . Then, if A 2n ¼ kB; Pl and P < B r P, P [ WarrðsðkB; PlÞÞ. Moreover, if A 2n21 ¼ kC; Rl, R Ó WarrðbÞ for all b $ sðkC; RlÞ. Since the above reasoning can be applied recursively for every argument A 2ðn2iÞ in l with i ¼ 0; . . . ; n, if P < A r Q, Q [ Warrða i Þ.

Logical properties of the maximal ideal recursive semantics of RP-DeLP
Following the approach we made in  for dialectical semantics, next we study the behaviour of the maximal ideal output of an RP-DeLP in the context of non-monotonic inference relationships. In order to do this, we define an inference operator 1 w RS that computes the expansion of a program including all new facts which correspond to warranted conclusions in the maximal ideal output.
Notice that by definition, operator 1 w RS is well defined (i.e. given an RP-DeLP program as input, the associated output is also an RP-DeLP program). Moreover, 1 w RS satisfies inclusion: given an RP-DeLP program P ¼ ðP; D; WÞ with levels of defeasibility 1 . a 1 . . . . . a p $ 0 and maximal ideal output ðWarr; BlockÞ, P # P < Warrð1Þ, D # D < ð< i¼1; ...; p Warrða i ÞÞ and W 0 preserves the total pre-order W on P < D.
In what follows, given an RP-DeLP program P ¼ ðP; D; WÞ, a clause w and a set of clauses G, we will write w [ P and G # P to denote that w [ P < D and G # P < D, respectively. Moreover, given a different RP-DeLP program P 0 , if w [ P in defeasibility level a, we will write P 0 < fw} to denote that program P 0 is extended with the clause w in defeasibility level a.
Besides, monotonicity does not hold for 1 w RS , as expected. It is satisfied if all warranted conclusions from a given program are preserved when the program is augmented with new clauses. As a counterexample consider the program P ¼ ðP; D; WÞ with P ¼ fq}, D ¼ fpˆq} and a single level of defeasibility a for D. Then, Warrð1Þ ¼ fq} and WarrðaÞ ¼ fp}, and thus, fq; p} # 1 w RS ðPÞ. However, if we extend program P with the strict fact ,p, we get the following program P 0 ¼ ðP 0 ; D; W 0 Þ with P 0 ¼ fq; ,p} and N 0 ð,pÞ ¼ 1. Then, Warrð1Þ ¼ fq; ,p} and WarrðaÞ ¼ Y in the maximal ideal output of P 0 . Hence, p Ó 1 w RS ðP 0 Þ but p [ 1 w RS ðPÞ. Semi-monotonicity is an interesting property for analysing non-monotonic consequence relationships. It is satisfied if all defeasible warranted conclusions are preserved when the program is augmented with new defeasible clauses. Semi-monotonicity does not hold for 1 w RS , as adding new defeasible clauses cannot invalidate already valid arguments, but it can enable new ones that were not present before, thus introducing new conflicts or new circular dependences among arguments. Arguments that were warranted may therefore no longer keep that status. Consider a variant of the previous counterexample: we consider the fact ,p as defeasible information, i.e. we define the following program P 0 ¼ ðP; D 0 ; W 0 Þ with D 0 ¼ fpˆq; ,p} N 0 ð,pÞ ¼ N 0 ðpˆqÞ. Now, Warrð1Þ ¼ fq}, WarrðaÞ ¼ Y and BlockðaÞ ¼ fp; ,p} for the maximal ideal output of P 0 . Hence, p Ó 1 w RS ðP 0 Þ but p [ 1 w RS ðPÞ. Next we define some relevant logical properties that operator 1 w RS satisfies.
. Operator 1 w RS satisfies idempotence: Then, if ðWarr 0 ; Block 0 Þ is the maximal ideal output of P 0 ¼ 1 w RS ðPÞ, we have to show that Warr 0 # Warr. Suppose that Warr 0 Ü Warr, then there should exist an argument involved in some conflict or some cycle w.r.t. P and not involved in any conflict and any cycle w.r.t. P 0 , and thus, there should exist at least an argument A valid w.r.t. P but not valid w.r.t. P 0 and such that all sub-arguments of A are warranted w.r.t. both P and P 0 . Therefore, it must be the case that A is inconsistent with either blocked conclusions of strength greater than A or the strict knowledge extended with warranted conclusions of strength greater than A. However, as the sets of blocked and warranted conclusions of strength greater than A are the same in both cases, we have that every valid argument w.r.t. P is also valid w.r.t. P 0 , and thus, Warr 0 # Warr.
Cummulativity property follows directly from idempotence. Finally, as fQ j P r Q} ¼ Warrð1Þ # 1 w RS ðPÞ, (Horn) supraclassicality holds for RP-DeLP programs. A Finally, the operator 1 w rs satisfies (somewhat softened) right weakening with respect to the set of strict rules. Indeed, it is satisfied in the full sense for RP-DeLP programs with a single defeasibility level: let P ¼ ðP; D; WÞ be an RP-DeLP program with a single defeasibility level for D, if QˆP 1^. . .^P k [ P and fP 1 ; . . . ; P k } # 1 w RS ðPÞ, then Q [ 1 w RS ðPÞ. The key point here is how warranted and blocked conclusions at higher levels of the maximal ideal output are taken into account in lower levels. In particular blocked conclusions play a key role in the propagation mechanism between defeasibility levels. In the RP-DeLP approach if a conclusion w is blocked at level a, then for any lower level than a, not only the conclusion w is rejected but also every conclusion c such that fw; c} r'.
The following example shows the propagation mechanism between defeasibility levels for the maximal ideal recursive semantics. Consider the RP-DeLP program P ¼ ðP; D; WÞ with P ¼ f,sˆq; ,rˆh} and D ¼ fqˆr; hˆs; r; s; a; b; qˆa; hˆb}; and two defeasibility levels for D: a 1 and a 2 with 1 . a 1 . a 2 . 0. Consider that D is stratified as follows: level a 1 : fqˆr; hˆs; r; s; a; b} level preposition and a 2 : fqˆa; hb }: Obviously, Warrð1Þ ¼ Y. Then, at level a 1 , we have four valid arguments: and four almost valid arguments 5 : F 1 ¼ kfr; qˆr}; ql; F 2 ¼ kfs; hˆs}; hl F 3 ¼ kfr; qˆr}; ,sl F 4 ¼ kfs; hˆs}; ,rl: Figure 2 shows a graphical representation of support and conflict dependences between fH 1 ; H 2 ; H 3 ; H 4 } and fF 1 ; F 2 ; F 3 ; F 4 }. Obviously, a and b can be warranted since they are not involved in any conflict nor in any cycle. The cycles express that either r or s can be warranted, but not both. Since valid arguments and almost valid arguments involved in cycles must be blocked and rejected, respectively, the maximal ideal output at level a 1 is Warrða 1 Þ ¼ fa; b}; Blockða 1 Þ ¼ fr; s}: Then, at level a 2 we have that arguments kfqˆa; a}; ql and kfhˆb; b}; hl are valid and they are not involved in any cycle nor in any conflict, and thus, q and h are warranted conclusions at level a 2 (i.e. fq; h} # Warrða 2 ÞÞ. Finally, although arguments kfqˆa; a}; ,sl and kfhˆb; b}; ,rl are recursively based on warranted conclusions, both are inconsistent with blocked conclusions at level a 1 , and thus, s and r are rejected for the maximal ideal output: Warrða 2 Þ ¼ fq; h}; Blockða 2 Þ ¼ Y: Hence, the maximal ideal output for P isWarr ¼ fa; b; q; h}; Block ¼ fr; s}: Therefore, due to the fact that the set of conclusions that are warranted and blocked at each level determines which arguments are valid at lower levels, we get that P < Warr r,s and P < Warr r,r, but ,s; ,r Ó Warr since s; r [ Blockða 1 Þ. Then, for the general case we have the following right weakening logical property for operator 1 w RS . Proposition 6.2. (Right weakening) Let P ¼ ðP; D; WÞ be an RP-DeLP program with defeasibility levels 1 . a 1 . . . . . a p . 0, and let ðWarr; BlockÞ be the maximal ideal output of P. If QˆP 1^. . .^P k [ P and fP 1 ; . . . ; P k } # 1 w RS ðPÞ, then either Q [ 1 w RS ðPÞ and N 0 ðQÞ $ minfN 0 ðP i Þ j P i [ fP 1 ; . . . ; P k }}, or Q, or ,Q [ BlockðbÞ for some b . min fN 0 ðP i Þ j P i [ fP 1 ; . . . ; P k }}.
Proof. We prove that if P < Warrð$ a i Þ r Q and P < Warrð. a i Þ Z Q, then either Q [ Warrða i Þ, or Q [ Blockð. a i Þ, or ,Q [ Blockð. a i Þ. 6 Suppose that for some a i , P < Warrð$ a i Þ r Q, P < Warrð. a i Þ Z Q, Q Ó Warrða i Þ and Q; ,Q Ó Blockð. a i Þ. Then, since P < Warr Z', P < Warrð$ a i Þ < fQ} Z', there exists a valid argument kA; Ql for Q of strength a i . Now, since Q Ó Warrða i Þ we get two possible cases: Case 1: There exists a conflict between kA; Ql and a set of valid arguments G, with kA; Ql y G, w.r.t. W ¼ Warrð. a i Þ < fP j kB; Pl f G < fkA; Ql}}. Thus P < W < fQ} < fP j kB; Pl [ G} r' and P < W < S Z', for all S , fQ} < fP j kB; Pl [ G}. Consider now W 0 ¼ fR j, B; R .f kA; Ql}. Then, as W 0 # W and P < W 0 r Q, if P < W < fQ}< fP j kB; Pl [ G} r', then P < W < fP j kB; Pl [ G} r', and thus, either Q is warranted at level a i or Q is rejected at level a i because Q or ,Q is blocked at a level b with b . a i . In other words, either Case 2: There is a cycle in a warrant dependency graph at level a i and either kA; Ql is a vertex of the cycle or there exists a path from some vertex of the cycle to kA; Ql. If H is the set of valid arguments of the cycle and F is the set of almost valid arguments w.r.t. H < fkA; Ql}, there is an almost valid argument for conclusion ,Q in F or an strict rule ,QˆL 1^. . .^L m [ P such that fL 1 ; . . . ; L m } # Warrð$ a i Þ < fH j kE; Hl [ H} < fF j kJ; Fl [ F}. Hence, there is an almost valid argument kD; ,Ql for conclusion ,Q in F, and an edge from the vertex of ,Q to the vertex of Q. Now, since P < Warrð$ a i Þ r Q and Q Ó Warrð$ a i Þ, there exists a strict rule QˆL 0 1^. . .^L 0 p [ P with all the L 0 j 's in Warrð$ a i Þ. Moreover, as P < Warrð. a i Þ Z Q, there is at least one literal L 0 [ fL 0 1 ; . . . ; L 0 p } such that L 0 [ Warrða i Þ, and thus, there is a valid argument kJ; L 0 l for L 0 of strength a i and kA; Ql y kJ; L 0 l. Then, there is a cycle in the warrant dependency graph for H < f, A; Q . } < fkJ; L 0 l} and F and an edge from the vertex of ,Q to the vertex of L 0 , and thus, A Let us briefly discuss the most relevant results provided by the properties presented before. When analysing warranted conclusions under the operator 1 w RS , idempotence shows us that adding warranted conclusions as new facts to a given program does not add any new warrant capabilities. Cummulativity shows us that any warranted conclusion obtained from a program P can be considered as a strict warranted conclusion since in some sense it is an intermediate proof (lemma) to be used in building more complex warranties. (Horn) Supraclassicality indicates that every conclusion that follows via traditional Selective Linear Definite clause inference (involving only certain clauses) can be considered as a special form of argument (namely, an empty argument), whereas right weakening tells us that RP-DeLP preserves the usual semantics for Horn rules: the existence of a strict rule XˆY causes that the warranty of Y is also a warranty for X.

Implementation of the RP-DeLP framework
As we have said in the introduction, our final aim is to investigate the use of an argumentation framework based on RP-DeLP in application domains such as debates in social networks. As a first step, we have implemented the algorithm for computation of the maximal ideal output (Alsinet et al., 2014) and also the algorithm for computation of the multiple outputs semantics (Alsinet, Béjar, Godo, & Guitart, 2013). Both algorithms are based on solving a sequence of queries, with complexity within Non-deterministic Polynomial-time (NP), related to the discovery of valid arguments and conflicts between sets of arguments. We have made available both algorithms in a single system, and the implementation of the main code of both algorithms has been done with the programming language python. However, in order to have a good scaling behaviour when solving RP-DeLP programs of big size, the NP queries are solved trough reductions to propositional satisfiability problem (SAT) encodings or ASP encodings. Our system is freely available as a command-line application at the GitHub repository 7 and can be installed on Linux and Mac OSX systems, but a web-based version in our web server http://arinf. udl.cat/rp-delp is also available. In order to install and use the command-line version in your system, apart of having python 2.7 installed, one needs to have also installed the SAT solver minisat (Eén & Sörensson, 2003) 8 and the answer set programming (ASP) solver clingo (Gebser et al., 2011). 9 The web version can be used through the html interface provided, but final applications can also use it through the http POST method.
One important feature of our system is that due to the recursive nature of our semantics, the computation of the maximal ideal output allows not only to provide warranted and blocked conclusions, but also easily explain the reasons for a warranted or blocked conclusion. Remember that for the recursive semantics from Section 4, the main notion for characterising warranted and blocked conclusions is the one of valid argument. The recursive definition for valid argument indicates that an argument is valid if it is based on warranted sub-arguments and it does not have conflicts with warranted conclusions and blocked conclusions of greater strength.
In practical terms, a valid argument for a conclusion Q is simply explained in our system by a rule QˆW 1^. . .^W n , where each W i is a warranted conclusion, and the strength a with which Q has been found valid. To fully understand the argument for Q, the user has simply to recursively follow the arguments provided for each W i . Then, if our system finds a conflict between Q and other valid conclusions of the same strength, it blocks Q and gives that set of conflicting valid conclusions as the explanation for blocking Q. More complex is the situation of blocked conclusion due to a circular definition of conflict between valid arguments. In Alsinet et al. (2014), this situation was characterised through the definition of dependency graphs, like those shown in Figures 1 and 2. However, for obtaining a more efficient implementation, we have followed an approach that avoids the explicit computation of dependency graphs. Instead, we follow an alternative characterisation of circular conflicts based on the detection of sets of valid arguments that after two iterations of the main loop of our algorithm cannot be decided to be either warranted or blocked because of dependencies that cannot be resolved. In that case, our system identifies such valid conclusions also as blocked, but due to a circular conflict, and informs of such set of valid conclusions as being part of a circular conflict. The problem of choosing this, more indirect, way of detecting circular conflicts is that the information provided to the user is not as clear as in the case of the information for valid and warranted conclusions. But we believe that it is a good trade-off between the computation cost and the quality of the information provided to the user to understand the output of the argumentation system.
Consider the following example of application of RP-DeLP to obtain the arguments that are consistently supported by all the participants in a political debate. The debate is about possible policies to increase the gross domestic product (GDP; represented by literal G). There are different policies being discussed: increase the public infrastructures expenditure (A 1 ), reduce the public debt (A 2 ), increase the taxes on the richer (A 3 ) and increase the education expenditure (A 4 ). All the participants agree that, no matter what policies are chosen, the public debt should be in any case decreased and that if we have increase in infrastructures and education expenditure, then the public debt will not decrease. We represent these two arguments as the strict knowledge, because all the participants agree that this should hold in any possible scenario: For the rest of the arguments, not all participants agree with the same strength, so we turn to have two defeasible levels. At the stronger one (more participants support this opinion), we have the belief that increase in the education expenditure should be performed, so we have: D 1 ¼ fA 4 }: At the next defeasible level, we have other opinions that have less support from the participants. Participants believe that increase in the public infrastructures expenditure and increase in the taxes on the richer should be performed. Also, they believe that the GDP would be increased if there is reduction in public debt, and taxes on the richer are not increased, but another equally strong opinion is that the GDP would be increased if there is increase of taxes on the richer and increase in the public infrastructures expenditure. Finally, another opinion is that if the reduction of the public debt is to happen, then necessarily this cannot be done without not increasing the taxes on the richer. So, at the second defeasible level we have this knowledge: D 2 ¼ fA 1 ; A 3 ; GˆA 2^, A 3 ; GˆA 3^A1 ; ,A 3ˆA2 }: Our argumentation system will give the following output for this program: . Meaning that the valid argument for A 3 given by the fact A 3 , has a conflict with a valid argument for ,A 3 and that the valid argument for ,A 3 given by the rule ,A 3ˆA2 and the warranted fact A 2 , has a conflict with a valid argument for A 3 , at the second defeasible level.
This information is discovered in an iterative process, starting from simpler arguments and moving towards more complex arguments. First, A 2 and A 4 are found to be warranted (no conflicts with P up to level D 1 . Then, at level D 2 A 1 is found to be not valid, as it produces a conflict with P and the current set of warrants. Finally, valid arguments for A 3 and ,A 3 are found, but with the same strength, so they are blocked because they generate a conflict.
The command-line version of our argumentation system gives this output formatted in a JSON object, 10 so that web-based and similar applications can easily process the output. Regarding the input format for programs, it is a simple ASCII file format that follows a syntax similar to the one used by standard Prolog interpreters, but with labels that distinguish the strict and the different defeasible parts of the program.

Related work
RP-DeLP, as well as its predecessor P-DeLP (Alsinet, Chesñevar, Godo, Sandri, & Simari, 2008;, builds on top of DeLP argumentative system (García & Simari, 2004), and extends it by introducing different levels of preference or priority at the object language level by means of weights. In this approach, arguments are sets of weighted formulas that support a goal, and weights are used to compute the strength of an argument and then to resolve conflicts among contradictory conclusions.
Actually, introducing preferences in argumentation frameworks goes back to Simari and Loui (1992). In that work the authors defined an argumentation framework in which arguments are built from a propositional knowledge base. The arguments grounded on specific information are considered stronger than those built from more general information. This preference relation is used then to solve conflicts between a pair of conflicting arguments. Nonetheless, the way RP-DeLP (and P-DeLP) makes use of preferences, to define the strength of arguments by stratifying the formulas in a program (or knowledge base), directly stems from Brewka's preferred subtheories-based approach (Brewka, 1989) to non-monotonic reasoning, where the different levels in which default theories are stratified represent different degrees of reliability. This idea has been used in many other approaches to reasoning with inconsistent information, mainly in those related to possibilistic logic (Benferhat, Cayrol, Dubois, Lang, & Prade, 1993;, 1995, 1996. In particular, the notion of argumentative inference introduced in  is based on a measure of strength of arguments that is in fact the one used in RP-DeLP. On the other hand, Prakken and Sartor (1997) formalised the role of preferences in the underlying logical formalisms that instantiate Dung's seminal theory of argumentation (Dung, 1995).
Other approaches have formalised the role of preferences at an abstract level. In Amgoud and Cayrol's preference-based argumentation frameworks (PAFs) (Amgoud & Cayrol, 2002a, 2002b), Dung's framework is augmented with a preference ordering on the set of arguments, so that an attack by an argument X on an argument Y is successful only if Y is not preferred to X.
In Bench-Capon's (2003) value-based argumentation frameworks, Dung's framework is augmented with values and value orderings, so that an attack by X on Y is successful only if the value promoted by Y is not ranked higher than the value promoted by X according to a given ordering on values. Recently, Kaci (2010) and Amgoud and Vesic (2009, 2010a, 2010b have addressed the issue of how consistency postulates (Caminada & Amgoud, 2007) can be ensured for instantiations of PAFs. They all argue that instantiations of standard PAFs have problems with unsuccessful asymmetric binary attacks. Kaci (2010) argues that all attacks should therefore be symmetric. However, Amgoud andBesnard (2009, 2010) show that for logic-based 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.
The output for an RP-DeLP program is a rank-ordered set of warranted and blocked conclusions which satisfy the consistency postulates (Caminada & Amgoud, 2007). 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 sub-arguments also are warranted.
Collective conflicts have also been considered in several papers, e.g. in Nielsen and Parsons (2006), while Amgoud andBesnard (2009, 2010) 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 Vreeswijk (1997) and Nielsen and Parsons (2006). The difference between these approaches and our notion of collective conflict is that in Vreeswijk (1997), the notion of conflict is not relative to a set of already warranted conclusions, and Nielsen and Parsons (2006) define a generalisation 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 (2009) recursive semantics for defeasible argumentation, circular definitions of conflict between sets of arguments have been characterised by means of dependency graphs representing support and collective conflict relationships 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 Toni (2006, 2007) 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 also 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 so-called preferred semantics for abstract argumentation, in Dvorák, Dunne, and Woltran (2011), 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 Dunne (2009) 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 program 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 b, 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 line similar to that in Governatori, Maher, Antoniou, and Billington (2004).
Research in logical properties for defeasible argumentation can be traced back to Benferhat et al. ( , 1998 and Vreeswijk (1997Vreeswijk ( , 1993. In the context of his abstract argumentation systems, Vreeswijk showed that many logical properties for non-monotonic inference relationships turned out to be counter-intuitive for argument-based systems.  were the first who studied argumentative inference in uncertain and inconsistent knowledge bases. They defined an argumentative consequence relationship taking into account the existence of arguments favouring a given conclusion against the absence of arguments in favour of its contrary. In  we defined a nonmonotonic expansion operator for P-DeLP which modelled the expansion of a program by adding new weighed facts associated with warranted literals according to the dialectic analysisbased semantics of P-DeLP, and we showed that right-weakening does not hold for P-DeLP. Challenges in the field of argumentation have been recently addressed towards the implementation of argumentation frameworks. The system ASPARTIX (Egly, Gaggl, & Woltran, 2008, 2010) is a tool for computing acceptable extensions for some formalisations of Dung's abstract argumentation framework and relies on a fixed disjunctive datalog program which takes an instance of an argumentation framework as input, and uses an ASP solver for computing the extension specified by the user. The work of Nieves, Cortés, and Osorio (2008) also suggests to use ASP for computing extensions of argumentation frameworks. Dungine (South, Vreeswijk, & Fox, 2008) is a Java reasoner capable of reasoning with grounded and preferred extensions. CASAPI (Gaertner & Toni, 2008) is a Prolog implementation that combines abstract and assumption-based argumentation. ArguLab (Podlaszewski, Caminada, & Pigozzi, 2011) is a library of components that provide basic functionalities for agent-related argumentation tasks and DIAMOND (Ellmauthaler & Strass, 2014) uses ASP encodings to compute interpretations of Brewka and Woltran's abstract dialectical frameworks (Brewka & Woltran, 2010).
In our case, we have implemented the algorithm for computation of the maximal ideal output (Alsinet et al., 2014) and also the algorithm for computation of the multiple outputs semantics (Alsinet et al., 2013). We have made available both algorithms in a single system and the implementation is based on reductions to SAT encodings and ASP encodings.

Conclusions and future work
In this paper we have analysed the relationship between the exhaustive dialectical analysisbased semantics of P-DeLP and the recursive-based semantics of RP-DeLP, and we have shown that the maximal ideal semantics of RP-DeLP provides a useful framework for making a formal analysis of logical properties of warrant in defeasible argumentation. In this sense, we have defined a non-monotonic expansion operator for RP-DeLP which modelled the expansion of a program by adding new weighed facts associated with warranted literals, and we showed that idempotence, cummulativity and right-weakening properties hold for RP-DeLP. We have also developed an argumentation framework for RP-DeLP that is able to compute not only the output of warranted and blocked conclusions, but also explain the reasons behind the status of each conclusion. Our system is freely available as a command-line application and as a web-based service in order to use it as part of other AI systems. Argumentation web services can also be used as part of bigger systems, like in the BDI system described in Schlesinger, Errecalde, and Aguirre (2010), where they use an available argumentation web system (García, Rotstein, Tucat, & Simari, 2007) for the DeLP argumentation framework (García & Simari, 2004).
Our current research work in RP-DeLP will follow two main directions: on the one hand, we are concerned with characterising a lower bound on complexity for computing the warranty status of arguments according to the maximal ideal recursive semantics. For this goal, we plan to study reductions from ideal semantics for abstract argumentation frameworks, given that its complexity is well understood (Dunne, 2009), or also reductions from the more recent framework of probabilistic abstract frameworks (Fazzinga, Flesca, & Parisi, 2013). Studying reductions from these systems to ours is not only useful for obtaining lower bounds on complexity, but also to understand the relationship between the semantics of these argumentation systems with the maximal ideal semantics of RP-DeLP. On the other hand, we are concerned with developing a graphical representation framework of the maximal ideal recursive semantics. This representation could be used as a mechanism for refinement of strict and defeasible information. As more concrete application domains, we have already started to consider the use of our system to encourage users to domains such as debates in social networks (Heras et al., 2013) and discuss political actions (Wardeh, Wyner, Atkinson, & Bench-Capon, 2013), through the use of argumentation structures, following the line of an existing tool for that purpose that is based also on argumentation structures: the Parmenides system for deliberative democracy (Atkinson et al., 2007;Cartwright & Atkinson, 2009). For using our system in other application domains such as in legal reasoning, another interesting line of work would be the study of relationships between RP-DeLP and deontic logics, considering recent approaches to characterise deontic logics beyond classic deontic logic, like, for example the alternative semantics (Hansson, 2013) or the input/output logic (Parent & van der Torre, 2013).