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Range-based argumentation semantics as 2-valued models

Characterizations of semi-stable and stage extensions in terms of 2-valued logical models are presented. To this end, the so-called GL-supported and GL-stage models are defined. These two classes of logical models are logic programming counterparts of the notion of range which is an established concept in argumentation semantics.

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1 Introduction

Argumentation has been regarded as a non-monotonic reasoning approach since it was suggested as an inference reasoning approach [Prakken and Vreeswijk (2002)]. Dung showed that argumentation inference can be regarded as a logic programming inference with negation as failure [Dung (1995)]. In his seminal paper [Dung (1995)], Dung introduced four argumentation semantics: grounded, stable, preferred and complete semantics. Currently, it is known that these four argumentation semantics introduced by Dung can be regarded as logic programming inferences by using different mappings, from argumentation frameworks (AFs) into logic programs, and different logic programming semantics (see Section 4).

Following Dung’s argumentation style, several new argumentation semantics have been proposed. Among them, ideal, semi-stable, stage and CF2 have been deeply explored [Baroni et al. (2011)]. Semi-stable and stage semantics were introduced from different points of view; however, they have been defined in terms of the so-called ranges of complete extensions and conflict-free sets, respectively. It seems that by using the concept of range, one can define different classes of argumentation semantics as is the case with the semi-stable and stage semantics. Given that the concept of range seems a fundamental component of definitions of argumentation semantics such as semi-stable and stage semantics, the following question arises:

[Q1] How the concept of range can be captured from the logic programming point of view?

This question takes relevance in the understanding of argumentation as logic programming.

In this paper, we argue that for capturing the idea of range from the logic programming point view, logic programming reductions which have been used for defining logic programming semantics such as stable model [Gelfond and Lifschitz (1988)] and p-stable [Osorio et al. (2006)] semantics are important. To show this, we introduce a general schema which takes as input a logic program and a set of atoms , then considering a function which maps into another logic program, returns a subset of atoms of the signature of 111The formal definition of is presented in Section 3.1. In order to infer ranges from the argumentation point of view using , the logic program has to capture an argumentation framework. Let us observe that there are different mappings from AFs into logic programs which have been used for characterizing Dung’s argumentation semantics as logic programming inferences [Carballido et al. (2009), Dung (1995), Caminada et al. (2013), Strass (2013)]. In this sense, the following question arises:

[Q2] Can the mappings used for characterizing Dung’s argumentation semantics characterize range-based argumentation semantics using ?

In order to give an answer to Q2, we consider the mappings and which have been used for characterizing Dung’s argumentation semantics in terms of logic programming semantics. has been shown to be a flexible mapping to characterize the grounded, stable, preferred, complete and ideal semantics by using logic programming semantics such as, the well-founded, stable, p-stable, Clark’s completion and well-founded semantics, respectively [Carballido et al. (2009), Nieves et al. (2008), Nieves and Osorio (2014)]. has been used to characterize the grounded, stable, preferred, complete, semi-stable and CF2 [Dung (1995), Nieves et al. (2011), Strass (2013)].

Considering and for defining two different instantiations of , we will define the so-called GL-supported and GL-stage models. We will show that GL-supported and GL-stage models characterize the semi-stable and stage extensions, respectively. In these instantiations of , we will instantiate the function with the well-known Gelfond-Lifschitz reduction which is the core of the construction of the stable model semantics [Gelfond (2008)]. Moreover, we will point out that can be instantiated with the reduction, which is the core of the p-stable semantics [Osorio et al. (2006)], getting the same effect in the constructions of the GL-supported and GL-stage models.

To the best of our knowledge, is the first schema designed to capture the range concept from a logic programming point of view. It is worth mentioning that a range-based semantics as semi-stable semantics has been already characterized as logic programming inference [Caminada et al. (2013), Strass (2013)]; however, these characterizations do not offer a schema for capturing the concept of range from a logic programming point of view in order to characterize (or construct) other range-based argumentation semantics such as stage semantics.

The rest of the paper is structured as follows: In Section 2, a basic background about logic programming and argumentation is introduced. In Section 3, by considering a couple of instantiations of , we introduce the so-called GL-supported and GL-stage models; moreover, we show how these models characterize both semi-stable and stage extensions. In Section 4, a discussion of related work is presented. In the last section, our conclusions are presented.

2 Background

In this section, we introduce the syntax of normal logic programs and the p-stable and stable model semantics. After this, some basic concepts of argumentation theory are presented. At the end of the section, the mappings and are introduced.

2.1 Logic Programs: Syntax

A signature is a finite set of elements that we call atoms. A literal is an atom (called a positive literal), or the negation of an atom (called a negative literal). Given a set of atoms , we write to denote the set of literals . A normal clause is written as:

where is an atom, . When the normal clause is called a fact and is an abbreviation of , where is the ever true atom. A normal logic program is a finite set of normal clauses. Sometimes, we denote a clause C by , where contains all the positive body literals and contains all the negative body literals. When , the clause C is called a definite clause. A definite program is a finite set of definite clauses. denotes the set of atoms that occurs in P. Given a signature , denotes the set of all the programs defined over . Given a normal logic program , .

In some cases we treat a logic program as a logical theory. In these cases, each negative literal is replaced by where is regarded as the classical negation in classical logic. Logical consequence in classical logic is denoted by . Given a set of proposition symbols and a logical theory (a set of well-formed formulae) , if .

Given a normal logic program P, a set of atoms is a classical model of if the induced interpretation evaluates to true. If , we write when: and is a classical 2-valued model of the logical theory obtained from (i.e. atoms in are set to true, and atoms not in to false). We say that a model of a program is minimal if a model of different from such that does not exist.

2.2 Stable model and p-stable semantics

Stable model semantics is one of the most influential logic programming semantics in the non-monotonic reasoning community [Baral (2003)] and is defined as follows:

Definition 1

[Gelfond and Lifschitz (1988)] Let P be a normal logic program. For any set , let be the definite logic program obtained from P by deleting

(i)

each clause that has a formula in its body with , and then

(ii)

all formulæ of the form in the bodies of the remaining rules.

Then, S is a stable model of P if S is a minimal model of . denotes the set of stable models of

From hereon, whenever we say Gelfond-Lifschitz (GL) reduction, we mean the reduction . As we can observe GL reduction is the core of the stable model semantics.

There is an extension of the stable model semantics which is called p-stable semantics [Osorio et al. (2006)]. P-stable semantics was formulated in terms of Paraconsistent logics. Like stable model semantics, p-stable semantics is defined in terms of a single reduction, RED, which is defined as follows:

Definition 2

[Osorio et al. (2006)] Let be a normal program and M be a set of atoms. We define .

As we can see, GL reduction and reduction have different behaviors. On the one hand, the output of GL reduction always is a definite program; on the other hand, the output of RED reduction can contain normal clauses.

By considering reduction, the p-stable semantics for normal logic programs is defined as follows:

Definition 3

[Osorio et al. (2006)] Let be a normal program and be a set of atoms. We say that is a p-stable model of if . denotes the set of p-stable models of .

The stable model and p-stable semantics are two particular 2-valued semantics for normal program. In general terms, a logic programming semantics is a function from the class of all programs into the powerset of the set of (2-valued) models.

Before moving on, let us introduce the following notation. Let be a logic program, denotes the 2-valued models of . Given two logic programming semantics and , is stronger than if for every logic program , . Let us observe that the relation stronger than between logic programming semantics is basically defining an order between logic programming semantics.

2.3 Argumentation theory

In this section, we introduce the definition of some argumentation semantics. To this end, we start by defining the basic structure of an argumentation framework (AF).

Definition 4

[Dung (1995)] An argumentation framework is a pair , where AR is a finite set of arguments, and attacks is a binary relation on AR, i.e. attacks .

We say that a attacks b (or is attacked by ) if holds. Similarly, we say that a set of arguments attacks (or is attacked by ) if is attacked by an argument in . We say that defends if and belongs to .

Let us observe that an AF is a simple structure which captures the conflicts of a given set of arguments. In order to select coherent points of view from a set of conflicts of arguments, Dung introduced the so-called argumentation semantics. These argumentation semantics are based on the concept of an admissible set.

Definition 5

[Dung (1995)]

  • A set S of arguments is said to be conflict-free if there are no arguments a, b in S such that a attacks b.

  • An argument is acceptable with respect to a set of arguments if for each argument : If attacks then is attacked by .

  • A conflict-free set of arguments is admissible if each argument in is acceptable w.r.t. .

Let us introduce some notation. Let be an AF and . .

Definition 6

[Caminada (2006), Dung (1995)] Let be an argumentation framework. An admissible set of arguments is:

  • stable if attacks each argument which does not belong to .

  • preferred if is a maximal (w.r.t. set inclusion) admissible set of .

  • complete if each argument, which is acceptable with respect to , belongs to .

  • semi-stable if is a complete extension such that is maximal w.r.t. set inclusion.

In addition to argumentation semantics based on admissible sets, there are other approaches for defining argumentation semantics [Baroni et al. (2011)]. One of these approaches is the approach based on conflict-free sets [Verheij (1996)]. Considering conflict-free sets, Verheij introduced the so-called stage semantics:

Definition 7

Let be an argumentation framework. is a stage extension if is a conflict-free set and is maximal w.r.t. set inclusion.

Let us observe that both semi-stable and stage semantics are based on the so-called range which is defined as follows: If is a set of arguments, then is called its range. According to the literature, the notion of range was first introduced by Verheij [Verheij (1996)].

2.4 Mappings from argumentation frameworks to normal programs

In this section, two mappings from an AF into a logic program will be presented. These mappings are based on the ideas of conflictfreeness and reinstatement which are the basic concepts behind the definitions of conflict-free sets and admissible sets. In these mappings, the predicate is used, with the intended meaning of being “ is a defeated argument”.

A pair of mapping functions w.r.t. an argument is defined as follows.

Definition 8

Let be an argumentation framework and . We define a pair of mappings functions:

Let us observe that suggests that an argument is defeated when anyone of the arguments which attack is not defeated. In other words, an argument that has an attacker that is not defeated has to be defeated; hence, stands for conflictfreeness. suggests that an argument is defeated when all the arguments that defends are defeated. In other word, any argument that is not defeated has to be defended; therefore stands for admissibility.

One can see that if a given argument has no attacks, then and . This situation happens because an argument that has no attacks is an acceptable argument which means that it belongs to all extensions sets of an .

By considering and , two mappings from an AF into a logic program are introduced.

Definition 9

Let be an argumentation framework. We define their associated normal programs as follows:

Observing Definition 9, it is obvious that is a subset of . However, each mapping is capturing different concepts: is a declarative specification of the idea of conflictfreeness and is a declarative specification of both ideas: conflictfreeness and reinstatement. Indeed, one can see that the 2-valued logical models of characterize the conflict-free sets of an and the 2-valued logical models of characterize the admissible sets of an .

3 Semi-stable and Stage extensions as 2-valued models

This section introduces the main results of this paper. In particular, we will show that the following schema suggests an interpretation of range from the logic programming point of view:

in which is a logic program, is a function which maps a logic program into another logic program considering a set of atoms .

In order to show our results, we will introduce two instantiations of the schema . These instantiations will lead to the so-called GL-supported models and GL-stage models. We will show that the GL-supported models of characterize the semi-stable extensions of a given AF (Theorem 1); moreover, the GL-stage models of characterize the stage extensions of a given AF (Theorem 2).

3.1 Semi-Stable Semantics

We start presenting our results w.r.t. semi-stable semantics. To this end, let us start defining the concept of a supported model.

Definition 10 (Supported model)

Let be a logic program and be a 2-valued model of . is a supported model of if for each , there is such that , and .

As we saw in Definition 6, semi-stable extensions are defined in terms of complete extensions. It has been shown that the supported models of characterize the complete extensions of a given AF [Osorio et al. (2013)]. By having in mind this result, we introduce an instantiation of the schema in order to define the concept of GL-supported-model.

Definition 11 (GL-supported-model)

Let be an argumentation framework and be a supported model of . is a GL-supported-model of if is maximal w.r.t. set inclusion. denotes the GL-supported models of .

In other words, a supported model of is a GL-supported-model if for every supported model of such that is different of , where .

Let us observe that the function of the schema was replaced by the GL-reduction in the construction of a GL-supported model. One of the main constructions of the definition of a GL-supported model is . This part of the construction of a GL-supported model is basically characterizing the set where is a complete extension. We can see that the GL reduction is quite important for this construction. As we saw in Definition 1, GL reduction is the core of the definition of stable models.

We want to point out that the definition of GL-supported models can also be based on the RED reduction which is the reduction used for defining p-stable models (see Definition 3). This similarity between RED and GL reductions argues that both RED and GL reductions can play an important role for capturing the idea of range of an argumentation framework from a logic programming point of view. As we will see in the following theorem, GL-supported models characterize semi-stable extensions; hence, both RED and GL reductions play an important role for capturing semi-stable extension as 2-valued logical models.

In order to simplify the presentation of some results, let us introduce the following notation. Let and where . As we can see, and are basically sets of arguments which are induced by a set of atoms .

Theorem 1

Let be an argumentation framework and . is a GL-supported model of iff is a semi-stable extension of .

Let us start introducing the following result from [Osorio et al. (2013)]:

:

Let be an argumentation framework. is a supported model of iff is a complete extension of .

The proof goes as follows:

=>

Let be a GL-supported model of and . Then by definition of a GL-supported model, is maximal w.r.t. set inclusion. Moreover, is a supported model. Therefore, by , is a complete extension. Hence, it is not hard to see that is a range with respect to the complete extension . Since is maximal w.r.t. set inclusion, is also maximal w.r.t. set inclusion. Hence, is a semi-stable extension.

<=

Let us suppose that is a semi-stable extension of . By definition is maximal w.r.t. set inclusion and is a complete extension. By , there exists a supported model of such that ; moreover, any supported model of has the property that is maximal w.r.t. set inclusion. Then is maximal w.r.t. set inclusion. Then is a GL-supported model of .

An interesting property of GL-supported models is that they can be characterized by both the set of p-stable models of and the set of 2-valued models of .

Proposition 1

Let be an argumentation framework.

  1. is a GL-supported model of iff is maximal w.r.t. set inclusion where is a 2-valued model of .

  2. is a GL-supported model of iff is maximal w.r.t. set inclusion where is a p-stable model .

We start introducing the following observations from the state of the art:

  1. Let . is a 2-valued model of iff is an admissible extension of [Nieves and Osorio (2014)].

  2. According to Proposition 4 by [Caminada et al. (2012)] the following statements are equivalent:

    1. is a complete extension such that is maximal (w.r.t. set inclusion).

    2. is an admissible set such that is maximal (w.r.t. set inclusion).

  3. Let . is a p-stable model of iff is a preferred extension of [Carballido et al. (2009)].

Now let us prove each of the points of the proposition:

  1. is a GL-supported model of iff is maximal w.r.t. set inclusion and is a supported model. By Theorem 1, is maximal w.r.t. set inclusion and is a supported model iff is maximal and is a complete extension of . By Observation 2, is maximal and is a complete extension of iff is maximal and is an admissible extension of . Hence, the result follows by Observation 1 which argues that any 2-valued model of characterizes an admissible set of .

  2. Let us start by observing that semi-stable extensions can be characterized by preferred extensions with maximal range which means: is a semi-stable extension iff is maximal (w.r.t. set inclusion) and is a preferred extension (see Proposition 13 from [Baroni et al. (2011)]). Hence, the result follows by Observation 3 and Theorem 1.

A direct consequence of Proposition 1 and Theorem 1 is the following corollary which introduces a pair of characterizations of semi-stable extensions as 2-valued models and p-stable models of .

Corollary 1

Let be an argumentation framework.

  1. Let be a p-stable model of . is a semi-stable extension of iff is maximal w.r.t. set inclusion.

  2. Let be a 2-valued model of . is a semi-stable extension of iff is maximal w.r.t. set inclusion.

Observing Corollary 1, we can see that there is an interval of logic programming semantics which characterizes semi-stable extensions. This interval of logic programming semantics is defined by the order-relation between logic programming semantics: stronger than. This result is formalized by the following corollary.

Corollary 2

Let be an argumentation framework and be a logic programming semantics such that is stronger than and is stronger than . If , then is a semi-stable extension of iff is maximal w.r.t. set inclusion.

Given the relation of semi-stable extensions with the stable and preferred extensions, we can observe some relations between GL-supported models w.r.t. the stable model semantics [Gelfond and Lifschitz (1988)] and p-stable semantics.

Proposition 2

Let be an argumentation framework.

  1. If is a stable model of then is a GL-supported model of .

  2. If is a GL-supported model of then is a p-stable model of .

  1. It follows from Theorem 1 and Theorem 2 by [Caminada et al. (2012)].

  2. It follows from Theorem 1 and Theorem 3 by [Caminada et al. (2012)].

Proposition 3

Let be an argumentation framework such that . Then, .

We know that is a stable extension of iff where is a stable model of (Theorem 5 by [Carballido et al. (2009)]). Hence, the result follows from Theorem 1 and Theorem 5 by [Caminada et al. (2012)].

3.2 Stage Semantics

We have seen that the idea of range w.r.t. complete extensions can be captured by instantiating the schema considering supported-models, the GL-reduction and .

In Section 2.4, the mappings and were introduced. We have observed that is basically a declarative specification of conflict-free sets. Given that stage semantics is based on conflict-free sets, we will consider for instantiating and defining the so-called GL-stage models:

Definition 12

Let be an argumentation framework and be a 2-valued model of . is a GL-stage model of if is maximal w.r.t. set inclusion.

In other words, a 2-valued model of is a GL-stage-model if for every 2-valued model of such that is different of , where .

In this characterization of , once again we are replacing the function of by the GL-reduction; however, one can use RED reduction for defining GL-stage models.

One can observe that GL-stage models characterize stage extensions. In order to formalize this result, the following notation is introduced: Let and where . Like and , and return sets of arguments given a set of atoms from .

Theorem 2

Let be an argumentation framework. is a GL-stage model of iff is a stage extension of .

Let us start with one observation:

O1:

Let be an argumentation framework. is a conflict-free set of iff is a 2-valued model of .

=>

Let be a GL-stage model of and . Then by definition of a GL-stage model, is maximal w.r.t. set inclusion and is a 2-valued model. Hence, by , is a conflict-free set. One can see that is a range with respect to the conflict-free set . Since is maximal w.r.t. set inclusion, is also maximal w.r.t. set inclusion. Hence, is a stage extension.

<=

Let us suppose that is a stage extension of . By definition is maximal w.r.t. set inclusion and is a conflict-free set. By , there exists a 2-valued model of such that ; moreover, any 2-valued model of has the property that is maximal w.r.t. set inclusion. Then is maximal w.r.t. set inclusion. Then is a GL-stage model of .

Let us observe that , which is the key construction of GL-stable models, is basically characterizing ranges w.r.t. conflict-free sets.

Dvorák and Woltran have shown that the decision problems of the credulous and sceptical inferences are of complexity -hard and -hard, respectively, for both semi-stable and stage semantics [Dvorák and Woltran (2010)]. Hence it is straightforward to observe that the decision problems of the credulous and sceptical inferences are of complexity -hard and -hard, respectively, for both GL-supported models and GL-stage models. Let us remember that GL-supported models and GL-stage models are defined under the resulting class of programs of the mappings and , respectively.

4 Related work

Dung showed that argumentation can be viewed as logic programming with negation as failure and vice versa. This strong relationship between argumentation and logic programming has given way to intensive research in order to explore the relationship between argumentation and logic programming [Caminada et al. (2013), Carballido et al. (2009), Dung (1995), Nieves et al. (2005), Nieves et al. (2008), Osorio et al. (2013), Nieves et al. (2011), Strass (2013), Wu et al. (2009)]. A basic requirement for exploring the relationship between argumentation and logic programming is to identify proper mappings which allow us to transform an argumentation framework into a logic program and vice versa. The flexibility of these mappings will frame the understanding of argumentation as logic programming (and vice versa). Therefore, defining simple and flexible mappings which regard argumentation as logic programming (and vice versa) will impact the use of logic programming in argumentation (and vice versa). Currently, we can find different mappings for regarding argumentation as logic programming (and vice versa) [Caminada et al. (2013), Carballido et al. (2009), Dung (1995), Gabbay and d’Avila Garcez (2009)]. All of them offer different interpretations of argumentation as logic programming (and vice versa). Depending on these interpretations, one can identify direct relationships between argumentation inferences and logic programming inferences.

In this paper, we have limited our attention to the interpretation of argumentation as logic programming. In this sense, there are some characterizations of semi-stable inference as logic programming inference [Caminada et al. (2013), Strass (2013)]. Caminada et al., [Caminada et al. (2013)], showed that the semi-stable semantics can be characterized by the L-stable semantics and the mapping which is defined as follows: Given an argumentation framework :

Unlike GL-supported models which are 2-valued models, the models of the L-stable semantics are 3-valued. Moreover, unlike which is a declarative specification of admissible sets, is a declarative specification of conflict-free sets.

Strass [Strass (2013)] has also showed that the semi-stable semantics can by characterized by both the so-called L-supported models and L-Stable models. Unlike Caminada’s characterization and our characterizations, Strass considered the mapping . As we have observed in Section 2.4, the clauses of are a subset of which is the mapping that we considered in both Theorem 1 and Corollary 2. It is worth mentioning that the mapping introduced by Dung [Dung (1995)] can be transformed into .

We cannot argue that one characterization is better than the other; however, we can observe that all these characterizations, including the ones introduced in this paper, offer different interpretations of semi-stable inference. Moreover, given that semi-stable inference has been characterized in terms of both L-stable semantics and L-supported modes, it seems that these logic programming semantics are related to GL-supported semantics.

In the literature, there are different characterizations of argumentation semantics in terms of logic programming semantics. A summary of these characterization is presented in Table 1.

Argumentation semantics Logic programming semantics using Logic programming semantics using Logic programming semantics using
Grounded Semantics Well-founded semantics [Caminada et al. (2013)], the Kripke-Kleene model [Strass (2013)] Well-founded semantics [Carballido et al. (2009)] Well-founded semantics [Dung (1995)], the Kripke-Kleene model [Strass (2013)]
Stable Semantics Stable model semantics [Caminada et al. (2013), Nieves et al. (2005)], Supported models [Strass (2013)] Stable model semantics [Carballido et al. (2009)] Stable models semantics [Dung (1995)], Supported models [Strass (2013)]
Preferred Semantics Regular semantics [Caminada et al. (2013)], M-supported models, M-stable models [Strass (2013)] P-stable Semantics [Carballido et al. (2009)] M-supported models, M-stable models [Strass (2013)]
Complete Semantics 3-valued stable semantics [Wu et al. (2009), Strass (2013)], 3-valued supported models [Strass (2013)] Supported Models [Osorio et al. (2013)] 3-valued stable semantics, 3-valued supported models [Strass (2013)]
Semi-stable Semantics L-Stable [Caminada et al. (2013), Strass (2013)], L-Supported models [Strass (2013)] GL-supported models (Theorem 1) L-supported models, L-stable models [Strass (2013)]
Ideal Semantics [Nieves and Osorio (2014)]
CF2 Semantics [Nieves et al. (2011)]
Stage Semantics GL-stage models (Theorem 2)
Table 1: Characterization of argumentation semantics as logic programming inferences.

Table 1 argues for a strong relationship between argumentation inference and logic programming inference. Moreover, we can observe that the argumentation semantics which have been characterized by logic programming semantics have been studied from different points of view, e.g., Labellings [Baroni et al. (2011)]. This evidence argues that any well-defined argumentation semantics must be characterized by a logic programming semantics. However, further research is required in order to identify the necessary conditions which could support a basic definition of a Well-defined Non-monotonic Inference of any argumentation semantics. These conditions can be identified in terms of non-monotonic reasoning properties which have been explored in both fields argumentation and logic programming, e.g., the property of relevance [Caminada (2006), Nieves et al. (2011)].

The exploration of argumentation as logic programming inference is not limited to the characterization of argumentation semantics in terms logic programming semantics. Since Dung’s presented his seminal paper [Dung (1995)], he showed that logic programming can support the construction of argumentation-based systems. Currently there are quite different logic-based argumentation engines which support the inference of argumentation semantics [Charwat et al. (2015), Egly et al. (2010), Toni and Sergot (2011)]. It is well-known that the computational complexity of the decision problems of argumentation semantics ranges from NP-complete to -complete. In this setting, Answer Set Programming has consolidated as a strong approach for building argumentation-based systems [Charwat et al. (2015), Egly et al. (2010), Toni and Sergot (2011), Nieves et al. (2005)].

5 Conclusions

Currently, most of the well accepted argumentation semantics have been characterized as logic programming inference (Table 1). This evidence argues that whenever a new semantics appears, it is totally reasonable to search for a characterization of it as a logic programming inference.

According to Theorem 1, semi-stable semantics can share the same mapping (i.e. ) with grounded, stable, preferred, complete and ideal semantics for being characterized as logic programming inference. This result argues that all these argumentation semantics can share the same interpretation of an argumentation framework as a logic program. Certainly, the logic programming semantics which are considered for characterizing these argumentation semantics share also a common interpretation of the argumentation inference which is restricted to the class of programs defined by . We have also showed that stage semantics can be also characterized by a logic programming semantics (Theorem 2). This result argues that stage semantics has also logic programming foundations. Considering Theorem 1 and Theorem 2, we can give a positive answer to Q2.

An interesting observation, from the results of this paper, is that the concept of range which is fundamental for defining semi-stable and stage semantics can be captured from the logic programming point of view by considering which can be based on well-acceptable reductions from logic programming. It is worth mentioning that reductions as GL and RED suggest some general rules for managing negation as failure. This evidence suggests that defines an approach for answering .

We argue that suggests a generic approach for exploring the concept of range in two directions: to explore ranges as logic programming models and to explore new argumentation semantics based on both logic programming models and ranges.

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