Block Argumentation

TL;DR

This paper introduces block argumentation, a higher-level bipolar abstract argumentation framework that models complex, non-elementary arguments as coherent blocks, addressing internal consistency and semantics beyond classical approaches.

Contribution

It formulates block argumentation as a Dung-style bipolar framework, explores its semantics under constraints, and introduces semi-grounded semantics to address limitations of classical semantics.

Findings

Classical acceptability semantics do not always hold with constraints.

Unattacked arguments may not always be acceptable.

Semi-grounded semantics ensures minimality within complete semantics.

Abstract

We contemplate a higher-level bipolar abstract argumentation for non-elementary arguments such as: X argues against Ys sincerity with the fact that Y has presented his argument to draw a conclusion C, by omitting other facts which would not have validated C. Argumentation involving such arguments requires us to potentially consider an argument as a coherent block of argumentation, i.e. an argument may itself be an argumentation. In this work, we formulate block argumentation as a specific instance of Dung-style bipolar abstract argumentation with the dual nature of arguments. We consider internal consistency of an argument(ation) under a set of constraints, of graphical (syntactic) and of semantic nature, and formulate acceptability semantics in relation to them. We discover that classical acceptability semantics do not in general hold good with the constraints. In particular, acceptability of unattacked arguments is not always warranted. Further, there may not be a unique minimal member in complete semantics, thus sceptic (grounded) semantics may not be its subset. To retain set-theoretically minimal semantics as a subset of complete semantics, we define semi-grounded semantics. Through comparisons, we show how the concept of block argumentation may further generalise structured argumentation.

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1 Introduction

In higher-level argumentation [12], or temporal/modal argumentation networks [3], an argumentation $F_{1}$ may be substituted into an argument of a given argumentation $F_{2}$. In the case of higher-level argumentation, the key is to find out which arguments in $F_{1}$ may be attacking or attacked by other arguments in $F_{2}$. In meta-level argumentation [15, 22], properties of argumentation at object level

• such as whether an object-level argument attacks another object-level argument, or the trustworthiness of an arguer - may be discussed in an argumentation about the object-level argumentation.

In a realistic argumentation, it can of course happen that some agent argues about some attack of an argument $a_{1}$ on an argument $a_{2}$. However, that may not be at meta-level; the agent’s argument could be in the same argumentation with $a_{1}$ and $a_{2}$, interacting with any arguments in the argumentation. Thus, the clear-cut distinction between object-level and meta-level argumentations does not always apply. Further, it may be that some argumentation $F$ itself as an argument $a_{F}$ attacks an argument, in which case it is not the case that we must seek the origin of the attack in arguments in $F$, which does not conform to the principle of higher-level argumentation.

Seeing the gap, in this work, we formulate block argumentation, where an argumentation may be an argument and vice-versa, for non-elementary arguments such as: X argues against Y’s sincerity with the fact that Y has presented his argument to draw a conclusion C, by omitting other facts which would not have validated C. This argument is in the form: “The argument:[the argument:[other facts attack C] attacks the argument:[some facts support C]] attacks Y’s sincerity.”, indicating an argument may be a block of argumentation.

Having such an argument blurs the boundary between an argumentation and an argument, leaving the differentiation only up to the perspective one employs. While block argumentation as we show is a specific instance of Dung-style bipolar abstract argumentation (see practical motivation for bipolar argumentation in, e.g. [9, 1, 20]) with the dual nature in arguments, it propels us to contemplate internal consistency of an argument. For example, one may (or may not) find “The conclusion by X in the past that Y was a terrorist corroborates X’s sinister personality” factually inconsistent if there was no such conclusion by X.111A possible-world-semantic observation that some arguments may be unusable was made [3]. Ergo, we posit a set of constraints, of graphical (syntactic) and semantic nature, which one may choose to impose on a block argumentation, and we characterise its acceptability semantics in accordance with them. As we are to show, classical Dung acceptability semantics do not in general hold good once these constraints are taken into consideration. In particular, unattacked arguments may not be outright acceptable. Further, there may be more than one minimal member in complete semantics, thus the sceptic (grounded) semantics may not form its subset; such situation has already arisen in the context of weighted argumentation [5, 4]. To retain set-theoretically minimal semantics as a subset of complete semantics, we define semi-grounded semantics.

The paper has the following structure: in Section 2 we motivate our approach with a real legal example from a popular case. Section 3 reports the necessary technical preliminaries. Section 4 presents block (bipolar) argumentation with formal results (e.g. on the existence of semantics). Finally in Section 5, we wrap up the paper with related work, where we discuss ABA-style structured argumentation [11], noting how the concept of block argumentation may further generalise the formalism.

2 Motivation for Block Argumentation

During the still ongoing trial over the death of Kim Jong-Nam222https://en.wikipedia.org/wiki/Assassination_of_Kim_Jong-nam., a certain argumentation was deployed by a suspect’s defence lawyer as he cast a blame on Malaysian authorities for having released only portions of CCTV footage of the fatal attack. Broadly:

Prosecutor:

the CCTV footage released by Malaysian Police shows a suspect walking quickly to an airport restroom to wash hands after attacking the victim with VX, which produces an impression that the suspect, contrary to her own statement that she thought she was acting for a prank video, knew what was on her hands.

Defence Lawyer:

however, the CCTV footage in its entirety shows the suspect adjusting her glasses after the attack, with VX on her hands, which counter-evidences her knowledge of the substance. Since Malaysian authorities know of the omitted footage, they are clearly biased against the suspect, intentionally tampering with evidence.

Assume the following arguments (with attacks and supports):

$a_{1}$:

After the victim was attacked with VX, the suspect walked quickly to a restroom for washing hands.

$a_{2}$:

The suspect knew VX was on her hands.

$a_{3}$:

The suspect was acting for a prank video.

$a_{4}$:

The suspect adjusted her glasses with VX on her hands before walking to restroom.

$a_{5}$:

Malaysian authorities are biased against the suspect, tampering with evidence by intentional omission of relevant CCTV footage.

$a_{6}$:

$a_{1}$ supports $a_{2}$.  $a_{7}$:   $a_{4}$ attacks $a_{2}$.  $a_{8}$:  $a_{7}$ attacks $a_{6}$.

Then we can model the example argumentation as in A. Malaysian Police uses $a_{1}$ for $a_{2}$ ($a_{1}$ supports $a_{2}$) to dismiss $a_{3}$ ($a_{2}$ attacks $a_{3}$). All these three arguments are made available to the audience. The defence lawyer uses $a_{4}$ to counter $a_{2}$. $a_{4}$ is also available to the audience as attacking $a_{2}$. He then uses $a_{7}$, which is itself an argumentation, to attack Malaysian Police’ argumentation $a_{6}$. This is also presented to the audience. Finally, he uses $a_{8}$, an argumentation, for $a_{5}$.

Arguments of the kinds of $a_{6}$, $a_{7}$ and $a_{8}$ are themselves argumentations, so “$a_{7}$ attacks $a_{6}$” could be detailed as in B, and “$a_{8}$ supports $a_{5}$” as in C.

2.0.1 Constraints.

These non-elementary arguments occur in natural language constructs such as “The fact: [something criticises (supports) something] is at odds with (in support of) something”, “The sentiment: [something is being defeated by something] enforces something”, and so on, which thus appear rather commonly in practice. By recognising these arguments, however, we seem to face some challenges to the semantic account given in Dung classic abstract argumentation theory [10]. Assume, for instance, that argumentation as in B: “$a_{7}$ attacks $a_{6}$”, i.e. “The fact: [the suspect adjusted her glasses with VX on her hands before walking to restroom counter-evidences her knowledge of the substance on her hands] attacks the argument: [she knew VX was on her hands because, after attacking the victim with VX, she walked quickly to a restroom for washing hands].”, is given without any prior mention of arguments $a_{1}$, $a_{2}$ and $a_{4}$. Then, we do not even know what relations may hold between them. Regardless, any argument not attacked is always justified in classic theory, according to which $a_{7}$, i.e. that $a_{4}$ attacks $a_{2}$, is an acceptable argument, while in the first place nothing is known about $a_{2}$ and $a_{4}$ that would enable us to see $a_{4}$’s attack on $a_{2}$ [21]. Essentially, the problem boils down to one’s interpretation of acceptance of an argument. When we accept an argument, are we only accepting the fact that it has resisted refutations, or are we also accepting the information in the argument?

On many occasions, acceptance of an argument means the latter, for which the classic prediction could appear hasty and possibly unsafe, more so in block argumentation, for an argument can be explicitly seen appearing multiple times, and one may like to impose certain constraints to enforce argument’s dependency on the same argument that occurs in a given argumentation. For B, the constraint to be imposed could be the presence of the argumentation involving $a_{1},a_{2}$ and $a_{4}$ outside the blocks, as in A. With it, it is immediate whether the argumentation in the argument $a_{6}$ or $a_{7}$ actually refers to a part of an already known argumentation. We envision both graphical (syntactic) and semantic constraints in this paper (more details are provided in Section 4).

3 Technical Preliminaries

Let $\mathcal{A}$ be a class of abstract entities we understand as arguments. We refer to a member of $\mathcal{A}$ by $a$ with or without a subscript and/or a superscript, and a finite subset of $\mathcal{A}$ by $A$ with or without a subscript. A bipolar argumentation framework (e.g. see [9]) is a tuple $(A,R,R^{\mathbf{s}})$ with two binary relations $R$ and $R^{\mathbf{s}}$ over $A$. For any $(A,R,R^{\mathbf{s}})$, $A_{1}\subseteq A$ is said to attack, or support, $A_{2}\subseteq A$ if and only if, or iff, there exist $a_{1}\in A_{1}$ and $a_{2}\in A_{2}$ such that $(a_{1},a_{2})\in R$ (for attack), or $(a_{1},a_{2})\in R^{\mathbf{s}}$ (for support).

An extension-based acceptability semantics of $(A,R,R^{\mathbf{s}})$ is a family of $A$ (i.e. a subset of power set of $A$). When $R^{\mathbf{s}}$ is not taken into account, a semantics of $(A,R,R^{\mathbf{s}})$ is effectively that of $(A,R)$, a Dung abstract argumentation framework [10]. In Dung’s, $A_{1}\subseteq A$ is said to defend $a_{x}\in A$ iff each $a_{y}\in A$ attacking $a_{x}$ is attacked by at least one member of $A_{1}$, and said to be conflict-free iff $A_{1}$ does not attack $A_{1}$. $A_{1}\subseteq A$ is said to be: admissible iff it is conflict-free and defended; complete iff it is admissible and includes all arguments it defends; preferred iff it is a maximal complete set; and grounded iff it is the set intersection of all complete sets. Complete / preferred / grounded semantics is the set of all complete / preferred / grounded sets. The grounded set defined as the set intersection may not be a complete set [5] in non-classic setting; it, however, reflects the attitude of sceptic acceptance, the whole point of the grounded semantics, which would not be necessarily fulfilled if the grounded set were defined the least complete set.

A label-based acceptability semantics [8] for Dung’s $(A,R)$ makes use of the set $\mathcal{L}$ of three elements, say $\{+,-,?\}$, and the class $\Lambda$ of all functions from $\mathcal{A}$ to $\mathcal{L}$. While, normally, it is $\{\textsf{in},\textsf{out},\textsf{undec}\}$, by $\{+,-,?\}$ we avoid direct acceptability readings off them. $\lambda\in\Lambda$ is said to be a complete labelling of $(A,R)$ iff:

• •

$\lambda(a)=+$ iff $\lambda(a_{p})=-$ for every $a_{p}\in A$ that attacks $a$.

• •

$\lambda(a)=-$ iff there is some $a_{p}\in A$ with $\lambda(a_{p})=+$ that attacks $a$.

$A_{1}\subseteq A$ is the set of all arguments that map into $+$ under complete labelling iff $A_{1}$ is a complete set, thus a label-based semantics provides the same information as an extension-based semantics does, and more because it classifies the remaining arguments in $-$ and ?.

In bipolar argumentation where $R^{\mathbf{s}}$ properly matters for an acceptability semantics, the notion of support is given a particular interpretation which influences the semantics.333See [9] for a survey of three popular types: deductive support [6] (for any member of a semantics, if $(a_{1},a_{2})\in R^{\mathbf{s}}$ and if $a_{1}$ is in the member, then $a_{2}$ is also in the member); evidential support [19, 1] (for any member of a semantics, if $a$ is in the member, then it can be traced back through $R^{\mathbf{s}}$ to some indisputable arguments in the member) and necessary support [17, 18] (for any member of a semantics, if $(a_{1},a_{2})\in R^{\mathbf{s}}$ and if $a_{2}$ is in the member, then so must $a_{1}$ also be). $R^{\mathbf{s}}$ can be used also for the purpose of expressing premise-conclusion relation in structured argumentation [16, 11] among arguments, which will be separately discussed in Section 5.

4 Block (Bipolar) Argumentation

Let $\mathbb{N}$ be the class of natural numbers including 0. We refer to its member by $n$ with or without a subscript. Let $\mathcal{X}$ be a class of an uncountable number of abstract entities. We refer to a member of $\mathcal{X}$ by $x$ with or without a subscript. It will be assumed that every member of $\mathcal{X}$ is distinguishable from any other members. Further, every member of $\mathcal{X}$ has no intersection with any others. Lack of these assumptions is not convenient if one wants to know equality of two arguments for graphical and semantic constraints.444Alternatively, we can assume a domain of discourse and interpretation as customary in formal logic; see [14] or any other standard text for the foundation of mathematical logic.

Definition 1 (Arguments and argumentations)

We define a (block) argument $a\in\mathcal{A}$ to be either $(\{x\},\emptyset,\emptyset)$ for some $x\in\mathcal{X}$ or $(\{a_{1},\ldots,a_{n}\},R_{p},$ $R^{\mathbf{s}}_{p})$ for some $a_{1},\ldots,a_{n}\in\mathcal{A}$ and some binary relations $R_{p}$ and $R^{\mathbf{s}}_{p}$ over $\{a_{1},\ldots,a_{n}\}$. We say $a\in\mathcal{A}$ is unitary iff $a$ is some $(\{x\},\emptyset,\emptyset)$.

We define a Block (Bipolar) Argumentation (BBA) to be some argument $(A,R,R^{\mathbf{s}})\linebreak\in\mathcal{A}$. We say that it is finite iff the number of occurrences of symbols is finite in $A$.555 As in set theory (see [13]), this is not equivalent to finiteness in the number of members of $A$. We denote the class of all finite BBAs by $\mathcal{A}^{\small\mbox{{BBA}}}$, a subclass of $\mathcal{A}$, and refer to its member by $a^{\small\mbox{{BBA}}}$ with or without a subscript.

Example 1 (BBA argumentation)

Denote the argumentation A in Section 2 by $a^{\small\mbox{{BBA}}}$.

• $a^{\small\mbox{{BBA}}}=(\{a_{1},\ldots,a_{8}\},\{(a_{4},a_{2}),(a_{2},a_{3}),(a_{7},a_{6})\},\{(a_{1},a_{2}),(a_{8},a_{5})\})$.
• $a_{i}=(\{x_{i}\},\emptyset,\emptyset)$ for $1\leq i\leq 5$.     $a_{6}=(\{a_{1},a_{2}\},\emptyset,\{(a_{1},a_{2})\})$.
• $a_{7}=(\{a_{2},a_{4}\},\{(a_{4},a_{2})\},\emptyset)$.       $a_{8}=(\{a_{6},a_{7}\},\{(a_{7},a_{6})\},\emptyset)$.

4.1 Representation of Argument(ation)s

To refer to arguments in a specific position in $a^{\small\mbox{{BBA}}}$, we make use of:

Definition 2 (Flat representation)

Let $\langle\mathbb{N}\rangle$ denote the class of all sequences of natural numbers (an empty sequence is included), whose member is referred to by $\langle m\rangle$ with or without a subscript, or by a specific sequence of natural numbers. We use ‘.’ for sequence concatenation. Let $\varpi:\mathcal{A}\rightarrow 2^{\mathcal{A}\times\langle\mathbb{N}\rangle}$ be such that $\varpi(a)$ is the least set that satisfies all the following.

$(a,0)\in\varpi(a)$. 2. 2.

For any $(a_{p},\langle m\rangle)\in\varpi(a)$, if $a_{p}$ is not unitary and is some $\{a_{1},\ldots,a_{n_{p}}\}$, then $(a_{i},\langle m\rangle.n_{i})\in\varpi(a)$ for every $1\leq i\leq{n_{p}}$ such that all $n_{1},\ldots,n_{n_{p}}$ are distinct.

For any $a$ and any $n$, we say that $\varpi(a)$ is its flat representation.

Example 2 (Flat representation)

(Continued) For the same example argumentation $a^{\small\mbox{{BBA}}}$ in A, we can define its flat representation $\varpi(a^{\small\mbox{{BBA}}})$ to be the set of all the following. Just for disambiguation, we demarcate the constituents in a sequence of natural numbers with ‘.’.

• $(a^{\small\mbox{{BBA}}},0)$.        $(a_{i},0.{i})$ for $1\leq i\leq 8$.        $(a_{1},0.6.{1})$ and $(a_{2},0.6.{2})$.
• $(a_{2},0.7.{1})$ and $(a_{4},0.7.2)$.                             $(a_{6},0.8.{1})$ and $(a_{7},0.8.{2})$.
• $(a_{1},0.8.1.1)$ and $(a_{2},0.8.1.2)$.                     $(a_{2},0.8.2.1)$ and $(a_{4},0.8.2.2)$.

Definition 3 (Order in flat representation)

Let $\sqsupseteq:({\mathcal{A}}\times\langle\mathbb{N}\rangle)\times({\mathcal{A}}\times\langle\mathbb{N}\rangle)$ be such that $((a_{p},\langle m_{p}\rangle),(a_{q},\langle m_{q}\rangle))\in\ \sqsupseteq$, synonymously written as $(a_{p},\langle m_{p}\rangle)\sqsupseteq(a_{q},\langle m_{q}\rangle)$, iff $\langle m_{p}\rangle.\langle m_{r}\rangle=\langle m_{q}\rangle$ for some $\langle m_{r}\rangle$. We write $(a_{p},\langle m_{p}\rangle)\sqsupset(a_{q},\langle m_{q}\rangle)$ iff $(a_{p},\langle m_{p}\rangle)\sqsupseteq(a_{q},\langle m_{q}\rangle)$ and $(a_{q},\langle m_{q}\rangle)\not\sqsupseteq(a_{p},\langle m_{p}\rangle)$.666“and” instead of “and” is used in this paper when the context in which it appears strongly indicates truth-value comparisons. It follows the semantics of classical logic conjunction.

4.2 Characterisation of Complete Sets with No Constraints

We characterise complete sets with no constraints initially.

Definition 4 (Arguments, attacks and supports in $\langle m\rangle$)

Let $\textsf{Arg}:\mathcal{A}^{\small\mbox{{BBA}}}\times\langle\mathbb{N}\rangle\rightarrow 2^{\mathcal{A}}$ be such that $\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)=A_{p}$ iff $((A_{p},R,R^{\mathbf{s}}),\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ for some $R$ and $R^{\mathbf{s}}$. Let $\textsf{Attck},\textsf{Spprt}:\mathcal{A}^{\small\mbox{{BBA}}}\times\langle\mathbb{N}\rangle\rightarrow 2^{\mathcal{A}\times\mathcal{A}}$ be such that:

• $\textsf{Attck}(a^{\small\mbox{{BBA}}},\langle m\rangle)=\{(a_{1},a_{2})\in R\ |\ \exists((A,R,R^{\mathbf{s}}),\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}}).\ a_{1},a_{2}\in A\}$
• $\textsf{Spprt}(a^{\small\mbox{{BBA}}},\langle m\rangle)=\{(a_{1},a_{2})\in R^{\mathbf{s}}\ |\ \exists((A,R,R^{\mathbf{s}}),\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}}).\ a_{1},a_{2}\in A\}$

For any $a^{\small\mbox{{BBA}}}$ and any $\langle m\rangle$, we say:

• $A_{p}$ is the set of arguments in $\langle m\rangle$ iff $A_{p}=\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$.
• $a_{1}$ attacks $a_{2}$ in $\langle m\rangle$ iff $(a_{1},a_{2})\in\textsf{Attck}(a^{\small\mbox{{BBA}}},\langle m\rangle)$.
• $a_{1}$ supports $a_{2}$ in $\langle m\rangle$ iff $(a_{1},a_{2})\in\textsf{Spprt}(a^{\small\mbox{{BBA}}},\langle m\rangle)$.

While there are three typical interpretations (deductive, necessary, evidential) of support in the literature, they enforce a strong dependency between arguments and the arguments that support them concerning their acceptance. In light of the example in Section 2, our interpretation of support here is weaker, almost supplementary, as in the following definitions. Briefly, it is not necessary that an argument be in a complete set when its supporter/supportee is in the set. A supporter can, however, prevent an argument attacked by an attacker from being strongly rejected (with labels, it concerns the difference of whether the argument gets $-$ (which leads to strong rejection) or ?). We look at extension-based complete set characterisation first.

Definition 5 (Extension-based complete set when no constraints)

For any $a^{\small\mbox{{BBA}}}$ and any $\langle m\rangle$, we say: $A_{1}$ defends $a$ in $\langle m\rangle$ iff $A_{1}\subseteq\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ and $a\in\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ and every $a_{1}$ attacking $a$ in $\langle m\rangle$ is: attacked by at least some $a_{2}\in A_{1}$; and not supported by any $a_{3}\in A_{1}$.

We say that $A_{1}$ is conflict-free in $\langle m\rangle$ iff $A_{1}\subseteq\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ and $(a_{1},a_{2})\not\in\textsf{Attck}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ for any $a_{1},a_{2}\in A_{1}$.

We say that $A_{1}$ is standard complete in $\langle m\rangle$ iff $A_{1}\subseteq\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ and $A_{1}$ is conflict-free and includes all arguments it defends in $\langle m\rangle$.

Example 3 (Complete set with no constraints)

(Continued) For $a^{\small\mbox{{BBA}}}$ in A, we have only one standard complete set in 0 with this characterisation, namely $\{a_{1},a_{4},a_{5},a_{7},a_{8}\}$. To explain the role of a supporter to prevent strong rejection of an argument, notice that $a_{1}$, which is in the standard complete set in 0, supports $a_{2}$, which is attacked by $a_{4}$ in the same standard complete set in 0. If it were not for the supporter, $a_{3}$ would be in the standard complete set in 0.

We can also have a label-based characterisation with $\mathcal{L}(=\{+,-,?\})$.

Definition 6 (Complete labelling when no constraints)

Let $\Lambda$ be the class of all $\lambda:\mathcal{A}\times\langle\mathbb{N}\rangle\rightarrow\mathcal{L}$ such that $\lambda((a,\langle m\rangle))=l$ for some $l$. For any $a^{\small\mbox{{BBA}}}$ and any $\lambda\in\Lambda$, we say that $\lambda$ is a standard complete labelling of $a^{\small\mbox{{BBA}}}$ iff every $(\{a_{1},\ldots,a_{n}\},\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ satisfies all the following.

• •

$\lambda((a_{i},\langle m\rangle.n_{i}))=+$, $1\leq i\leq n$, iff every $a_{j}$, $1\leq j\leq n$, attacking $a_{i}$ in $\langle m\rangle$ satisfies $\lambda((a_{j},\langle m\rangle.n_{j}))=-$.

• •

$\lambda((a_{i},\langle m\rangle.n_{i}))=-$, $1\leq i\leq n$, iff there exists some $1\leq j\leq n$ such that $\lambda((a_{j},\langle m\rangle.n_{j}))\\ =+$ and that $a_{j}$ attacks $a_{i}$ in $\langle m\rangle$ and there is no $1\leq k\leq n$ such that $\lambda((a_{k},\langle m\rangle.n_{k}))=+$ and that $a_{k}$ supports $a_{i}$ in $\langle m\rangle$.

Theorem 4.1 (Correspondence between standard complete sets and standard complete labellings)

For any $a^{\small\mbox{{BBA}}}$ and any $\langle m\rangle$, $A_{1}\subseteq\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ is standard complete in $\langle m\rangle$ only if there is some standard complete labelling $\lambda$ of $a^{\small\mbox{{BBA}}}$ such that $\lambda((a_{p},\langle m\rangle.n))=+$ is equivalent to $a_{p}\in A_{1}$ for any $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$. Conversely, $\lambda$ is a standard complete labelling of $a^{\small\mbox{{BBA}}}$ only if, for every $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$, $\{a_{p}\in\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)\ |\linebreak\exists n\in\mathbb{N}.\lambda((a_{p},\langle m\rangle.n))=+\}$ is a standard complete set in $\langle m\rangle$.

On the basis of this correspondence, we will work mainly with labels, as they simplify referrals of arguments that do not get $+$.

4.3 Graphical (Syntactic) Constraints

We now characterise constraints, which can be graphical (syntactic) or semantic. The former enforces that any argumentation within an argument(ation) has already occurred, while the latter enforces that acceptability statuses of arguments in any argumentation respect in a certain way those of the same arguments that have already occurred. We begin with a graphical one, for which we need to be able to tell equality of two arguments.

Definition 7 (Argument equality)

Let $\textsf{Eq}:\mathcal{A}\times\mathcal{A}$ be such that $\textsf{Eq}(a_{1},a_{2})$ iff one of the following conditions is satisfied.

• •

$a_{1}=a_{2}=(\{x\},\emptyset,\emptyset)$ for some $x\in\mathcal{X}$.

• •

If $a_{1}$ is some $(\{a^{\prime\prime}_{1},\ldots,a^{\prime\prime}_{n}\},R_{1},R^{\mathbf{s}}_{1})$, then $a_{2}$ is some $(\{a^{\prime}_{1},\ldots,a^{\prime}_{n}\},R_{2},R^{\mathbf{s}}_{2})$ such that:

• –

$\textsf{Eq}(a^{\prime\prime}_{i},a^{\prime}_{i})$;

• –

$(a^{\prime\prime}_{i},a^{\prime\prime}_{j})\in R_{1}$ iff $(a^{\prime}_{i},a^{\prime}_{j})\in R_{2}$;

• –

and $(a^{\prime\prime}_{i},a^{\prime\prime}_{j})\in R^{\mathbf{s}}_{1}$ iff $(a^{\prime}_{i},a^{\prime}_{j})\in R^{\mathbf{s}}_{2}$

for $1\leq i,j\leq n$.

Definition 8 (Sub-argumentation)

Let $\searrow:\mathcal{A}\times\mathcal{A}$ be such that $((A_{1},R_{1},R^{\mathbf{s}}_{1}),(A_{2},$ $R_{2},R^{\mathbf{s}}_{2}))\in\searrow$, written synonymously as $(A_{1},R_{1},R^{\mathbf{s}}_{1})\searrow(A_{2},R_{2},R^{\mathbf{s}}_{2})$ iff $A_{2}=\{a^{\prime}_{1},\ldots,$ $a^{\prime}_{n_{2}}\}$ and $A_{1}=\{a_{1},\ldots,a_{n_{1}}\}$ are such that:

• •

$|A_{2}|\leq|A_{1}|$.

• •

For each $a^{\prime}_{i}$, $1\leq i\leq n_{2}$, there exists some $a_{j}$, $1\leq j\leq n_{1}$, such that $\textsf{Eq}(a^{\prime}_{i},a_{j})$.

• •

For each two $a^{\prime}_{i_{1}},a^{\prime}_{i_{2}}$, $1\leq i_{1},i_{2}\leq n_{2}$, if $(a^{\prime}_{i_{1}},a^{\prime}_{i_{2}})\in R_{2}$, then there exist some $a_{j_{1}},a_{j_{2}}$, $1\leq j_{1},j_{2}\leq n_{1}$ such that $(a_{j_{1}},a_{j_{2}})\in R_{1}$, that $\textsf{Eq}(a^{\prime}_{i_{1}},a_{j_{1}})$, and that $\textsf{Eq}(a^{\prime}_{i_{2}},a_{j_{2}})$.

• •

For each two $a^{\prime}_{i_{1}},a^{\prime}_{i_{2}}$, $1\leq i_{1},i_{2}\leq n_{2}$, if $(a^{\prime}_{i_{1}},a^{\prime}_{i_{2}})\in R^{\mathbf{s}}_{2}$, then there exist some $a_{j_{1}},a_{j_{2}}$, $1\leq j_{1},j_{2}\leq n_{1}$ such that $(a_{j_{1}},a_{j_{2}})\in R^{\mathbf{s}}_{1}$, that $\textsf{Eq}(a^{\prime}_{i_{1}},a_{j_{1}})$, and that $\textsf{Eq}(a^{\prime}_{i_{2}},a_{j_{2}})$.

We say that $a_{2}$ is a sub-argumentation of $a_{1}$ iff $a_{1}\searrow a_{2}$.

Definition 9 (Graphical (syntactic) constraints)

For any $a^{\small\mbox{{BBA}}}$, we say that$(a_{2},\langle m_{2}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ satisfies G iff both of the conditions below hold.

If $\langle m_{2}\rangle\not=0$ and if $a_{2}$ is not unitary, then there exists some $(a_{1},\langle m_{1}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ such that $(a_{1},\langle m_{1}\rangle)\sqsupset(a_{2},\langle m_{2}\rangle)$ and that $a_{1}\searrow a_{2}$. 2. 2.

Every $(a_{p},\langle m_{2}\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$ satisfies G.

Example 4 (Graphical (syntactic) constraints)

(Continued) As has been the case so far, let $a^{\small\mbox{{BBA}}}$ be the argumentation in A. and let its flat representation be as given in Example 2. Then $a^{\small\mbox{{BBA}}}$ clearly satisfies G for any $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$, since:

• •

$(a^{\small\mbox{{BBA}}},0)$: there is nothing to show, as the sequence is 0.

• •

$(a_{i},0.i)$, $1\leq i\leq 5$: there is nothing to show, as $a_{1,\ldots,5}$ are unitary.

• •

$(a_{6},0.6)$: $a^{\small\mbox{{BBA}}}\searrow(\{a_{1},a_{2}\},\emptyset,\{(a_{1},a_{2})\})$.

• •

$(a_{i},0.6.i)$, $1\leq i\leq 2$: there is nothing to show, as $a_{1}$ and $a_{2}$ are unitary.

• •

$(a_{7},0.7)$: $a^{\small\mbox{{BBA}}}\searrow(\{a_{2},a_{4}\},\{(a_{4},a_{2})\},\emptyset)$.

• •

$(a_{2},0.7.1),(a_{4},0.7.2)$: there is nothing to show, as $a_{2}$ and $a_{4}$ are unitary.

• •

$(a_{8},0.8)$: $a^{\small\mbox{{BBA}}}\searrow(\{a_{6},a_{7}\},\{(a_{7},a_{6})\},\emptyset)$.

• •

$(a_{6},0.8.1)$: see above for $(a_{6},0.6)$.

• •

$(a_{7},0.8.2)$: see above for $(a_{7},0.7)$.

• •

$(a_{i},0.8.1.i),1\leq i\leq 2$: there is nothing to show.

• •

$(a_{2},0.8.2.1)$ and $(a_{4},0.8.2.2)$: there is nothing to show.

For comparison, however, suppose that $a^{\small\mbox{{BBA}}}$ is the argumentation in B, i.e. $a^{\small\mbox{{BBA}}}=(\{a_{6},a_{7}\},\{a_{7},a_{6}\},\emptyset)$, $a_{6}=(\{a_{1},a_{2}\},\emptyset,\{(a_{1},a_{2})\})$, and $a_{7}=(\{a_{2},a_{4}\},\{(a_{4},a_{2})\},\emptyset)$. Assume $(a_{6},0.1),(a_{7},0.2)\in\varpi(a^{\small\mbox{{BBA}}})$, then neither of them satisfies G, because none of $a_{1},a_{2},a_{4}$ are in $\textsf{Arg}(a^{\small\mbox{{BBA}}},0)$.

Since the graph structure of a given $a^{\small\mbox{{BBA}}}$ never changes, violation of the graphical constraint monotonically propagates up to $\langle m\rangle=0$ (0 excluded) from longer sequences. Thus, it is rather straightforward to handle graphical constraint satisfaction.

4.4 Semantic constraints

By contrast, semantic constraint satisfaction depends on what labels are assigned to arguments, which adds to technical subtlety. We define a partial order on $\mathcal{L}$, and characterise semantic constraints based on them.

Definition 10 (Order in labels)

Let $\succeq:\mathcal{L}\times\mathcal{L}$ be $\{(?,+),(?,-),(+,+),(-,-),(?,?)\}$. We write $(l_{1},l_{2})\in\succeq$ alternatively as $l_{1}\succeq l_{2}$.

Definition 11 (Semantic constraints)

For any $a^{\small\mbox{{BBA}}}$ and for any $\lambda\in\Lambda$, we say that $(a_{2},\langle m_{2}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ and $\lambda$ satisfy:

S

iff for every $(a_{1},\langle m_{1}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$, if $\textsf{Eq}(a_{1},a_{2})$ and $(a_{1},\langle m_{1}\rangle)\sqsupset(a_{2},\langle m_{2}\rangle)$, then $\lambda((a_{1},\langle m_{1}\rangle))\succeq\lambda((a_{2},\langle m_{2}\rangle))$.

$\star$

iff, for any $(a_{1},\langle m_{1}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$, if there exist some $\langle m_{3}\rangle$, $n_{1}$ and $n_{2}$ such that $\langle m_{3}\rangle.n_{1}=\langle m_{1}\rangle$ and that $\langle m_{3}\rangle.n_{2}=\langle m_{2}\rangle$, and if $\textsf{Eq}(a_{1},a_{2})$, then $\lambda((a_{1},\langle m_{1}\rangle))\linebreak=\lambda((a_{2},\langle m_{2}\rangle))$.

To speak of the use of $\succeq$, there should be no oddity from a semantic coherency perspective when $\lambda((a_{1},\langle m_{1}))=\lambda((a_{2},\langle m_{2}\rangle))$ for every $(a_{1},\langle m_{1}\rangle),(a_{2},\langle m_{2}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ such that $\textsf{Eq}(a_{1},a_{2})$. That just means that same arguments in $a^{\small\mbox{{BBA}}}$ are assigned the same label. However, consider our example A. There, $a_{2}$ in $\textsf{Arg}(a^{\small\mbox{{BBA}}},0)$ which is assigned $?$ by a standard complete labelling is given $-$ in $\textsf{Arg}(a_{6},0.6)$ by the Malaysian authorities and $+$ in $\textsf{Arg}(a_{7},0.7)$ by the defence lawyer. $?$ assigned to an argument in $\langle m\rangle$, therefore, can be interpreted flexibly in argumentation in $\langle m\rangle.\langle m_{p}\rangle$, that it can be any of $+$, $-$, $?$ depending on which part of the argumentation in $\langle m\rangle$ is selected to be included in the argumentation in $\langle m\rangle.\langle m_{p}\rangle$ (for some non-empty $\langle m_{p}\rangle$). This is the intuition for $\succeq$ and its use in S.

The $\star$ symbol denotes a semantic constraint among the members of $\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ for some $\langle m\rangle$, to prevent the same arguments from being assigned a different label.

Example 5 (Semantic constraints)

Denote the argumentation in D by $a^{\small\mbox{{BBA}}}$. Assume that $a_{0}=(\{a_{3},a_{4}\},\{(a_{3},a_{4})\},\emptyset)$ as shown in E, and assume that $a_{1},\ldots,a_{4}$ are all unitary such that $\textsf{Eq}(a_{i},a_{j})$ for no distinct $1\leq i,j\leq 4$. Assume $(a^{\small\mbox{{BBA}}},0),(a_{0},0.0),(a_{1},0.1),(a_{2},0.2),(a_{3},0.3),(a_{4},0.4),\linebreak(a_{3},0.0.1),(a_{4},0.0.2)\in\varpi(a^{\small\mbox{{BBA}}})$. The following are all the standard complete labelling distinct for $a^{\small\mbox{{BBA}}}$.

• •

$\lambda_{1}((a_{i},0.i))=+$ for $i\in\{0,2,4\}$, $\lambda_{1}((a_{i},0.i))=-$ for $i\in\{1,3\}$,

$\lambda_{1}((a_{3},0.0.1))=+$, $\lambda_{1}((a_{4},0.0.2))=-$.

• •

$\lambda_{2}((a_{i},0.i))=+$ for $i\in\{0,3\}$, $\lambda_{2}((a_{i},0.i))=-$ for $i\in\{1,2,4\}$.

$\lambda_{2}((a_{3},0.0.1))=+$, $\lambda_{2}((a_{4},0.0.2))=-$.

• •

$\lambda_{3}((a_{0},0.0))=+$, $\lambda_{3}((a_{1},0.1))=-$, $\lambda_{3}((a_{i},0.i))=\ ?$ for $i\in\{2,3,4\}$.

$\lambda_{3}((a_{3},0.0.1))=+$, $\lambda_{3}((a_{4},0.0.2))=-$.

$\lambda_{1}$ and $((a_{3},0.0.1))$ do not satisfy S. $\lambda_{2}$ and $((a_{3},0.0.1))$ do. Now, while $\lambda_{3}((a_{3},0.3))=?\not=+=\lambda_{3}((a_{3},0.0.1))$ and $\lambda_{3}((a_{4},0.4))=\ ?\not=-=\lambda_{3}((a_{4},0.0.2))$, S is respected, since $?\succeq+,-$.

Denote the argumentation in F by $a^{\small\mbox{{BBA}}}$. Assume $(a^{\small\mbox{{BBA}}},0),(a_{i},0.i)\in\varpi(a^{\small\mbox{{BBA}}})$ for $5\leq i\leq 10$. Assume $\textsf{Eq}(a_{5},a_{7})$ and $\textsf{Eq}(a_{9},a_{8})$. There is a standard complete labelling $\lambda$ of $a^{\small\mbox{{BBA}}}$ such that $\lambda((a_{5},0.5))=\lambda((0.7,0.7))=\lambda((a_{9},0.9))=+$, $\lambda((a_{6},0.6))=\lambda((a_{8},0.8))=\lambda((a_{10},0.10))=-$. Then it could be understood that the same argument ($a_{8}$ and $a_{9}$) in 0 is assigned both + and $-$ simultaneously, which on some occasions outside of rhetoric argumentation is not convenient. $\star$ is the condition against such labelling.

4.5 Generalisation of standard complete labelling

Let us refine our earlier definition of standard complete labelling with those constraints. We define $\mathcal{C}$ to be $\{\textbf{G},\textbf{S},\star\}$, and refer to its subset by $C$ with or without a subscript.

Definition 12 (Complete labelling)

For any $a^{\small\mbox{{BBA}}}$, any $\lambda\in\Lambda$ and any $C$, we say that $\lambda$ is a complete labelling of $a^{\small\mbox{{BBA}}}$ under $C$ iff every $(\{a_{1},\ldots,a_{n}\},\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ satisfies all the following conditions.

• •

$\lambda((a_{i},\langle m\rangle.n_{i}))=+$, $1\leq i\leq n$, iff both:

• –

every $a_{j}$, $\leq j\leq n$, attacking $a_{i}$ in $\langle m\rangle$ satisfies $\lambda((a_{j},\langle m\rangle.n_{j}))=-$.

• –

if $\{\textbf{S},\star\}\in C$, then assigning + to $(a_{i},\langle m\rangle.n_{i})$ does not lead to non-satisfaction of a semantic constraint $c\in C$ for any $(a,\langle m_{1}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$.

• •

$\lambda((a_{i},\langle m\rangle.n_{i}))=-$, $1\leq i\leq n$, iff both:

• –

there exists some $1\leq j\leq n$ such that $\lambda((a_{j},\langle m\rangle.n_{j}))=+$ and that $a_{j}$ attacks $a_{i}$ in $\langle m\rangle$ and there is no $1\leq k\leq n$ such that $\lambda((a_{k},\langle m\rangle.n_{k}))=+$ and that $a_{k}$ supports $a_{i}$ in $\langle m\rangle$.

• –

if $\{\textbf{S},\star\}\in C$, then assigning $-$ to $(a_{i},\langle m\rangle.n_{i})$ does not lead to non-satisfaction of a semantic constraint $c\in C$ for any $(a,\langle m_{1}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$.

• •

$\lambda((a_{i},\langle m\rangle.n_{i}))\not=+$, $1\leq i\leq n$, if $\textbf{G}\in C$ and $(a_{i},\langle m\rangle.n_{i})$ does not satisfy G.

Any argument that violates the graphic constraint will not be assigned + if $\textbf{G}\in C$. For both + and $-$, the first condition matches exactly the condition given for a standard complete labelling. The second condition ensures conditions for a standard complete labelling to be maximally respected, in the sense that they normally apply unless by applying them there will be a non-satisfaction of a semantic constraint $c\in C$.

Broader intuition is as follows. If a standard labelling of $a^{\small\mbox{{BBA}}}$ satisfies all $c\in C$, then the standard complete labelling should itself be a complete labelling under $C$. Thus, when a standard complete labelling is not a complete labelling under $C$, it is because the labelling either induces violation of some semantic constraint $c\in C$ for some arguments occurring in $a^{\small\mbox{{BBA}}}$ or + assignment to an argument that violates the graphic constraint if $\textbf{G}\in C$. In such cases, therefore, it will be required to make minimal change to the standard complete labelling so the resulting labelling satisfies semantic constraints (if they are in $C$) and does not assign $+$ to an argument that violates G (if it is in $C$). The second conditions for $+$ and $-$ ensure that the change be indeed minimal with a complete labelling.

Example 6 (Complete labelling)

(Continued) For the argumentation in D with the same assumptions and $\lambda_{1,2,3}$ as in Example 5, $\lambda_{1}$ is not a complete labelling under $C$ if $\textbf{S}\in C$: $(a_{0},0.0)$ is assigned +, however, it does not satisfy S, so it should be assigned ?. $\lambda_{2}$ and $\lambda_{3}$ are a complete labelling under any $C$. In addition, $\lambda_{4}$ such that $\lambda_{4}((a_{i},0.i))=?,i\in\{0,1\}$, $\lambda_{4}((a_{2},0.2))=\lambda_{4}((a_{4},0.4))=+$, $\lambda_{4}((a_{3},0.3))=-$ is a complete labelling. $\lambda_{4}$ is not a standard complete labelling.

Theorem 4.2 (Conservation)

For any $a^{\small\mbox{{BBA}}}$ and any $\lambda$, if $C=\emptyset$, then $\lambda$ being a complete labelling of $a^{\small\mbox{{BBA}}}$ under $C$ is equivalent to $\lambda$ being a standard complete labelling of $a^{\small\mbox{{BBA}}}$.

Theorem 4.3 (No inclusion)

For any $C\not=\emptyset$, there exists $a^{\small\mbox{{BBA}}}$ such that some standard complete labelling of $a^{\small\mbox{{BBA}}}$ is not a complete labelling under $C$ and that some complete labelling of $a^{\small\mbox{{BBA}}}$ under $C$ is not a standard complete labelling.

This result holds even for $a^{\small\mbox{{BBA}}}$ that does not involve any support (see Example 6).

4.6 Acceptability Semantics

Definition 13 (Types of complete sets and acceptability semantics)

For any $a^{\small\mbox{{BBA}}}$, we say that $A_{1}\subseteq\textsf{Arg}(a^{\small\mbox{{BBA}}},0)$ is: complete under $C$ iff there exists a complete labelling $\lambda\in\Lambda$ of $a^{\small\mbox{{BBA}}}$ under $C$ such that $A_{1}=\{a\in\mathcal{A}\ |\ \exists n\in\mathbb{N}\ \exists(a,0.n)\in\varpi(a^{\small\mbox{{BBA}}}).\lambda((a,0.n))=+\}$; grounded under $C$ iff it is the set intersection of all complete sets under $C$; preferred under $C$ iff it is a maximal complete set under $C$; and semi-grounded iff it is a minimal complete set.

We call the set of all complete / grounded / semi-grounded / preferred sets under some $C$ complete / grounded / semi-grounded / preferred semantics under $C$.

Note that ultimately we need to tell which subsets of $\textsf{Arg}(a^{\small\mbox{{BBA}}},0)$ are acceptable: this explains why we only look at $\langle m\rangle=0$ for the semantics.

Example 7 (Acceptability semantics)

(Continued) For the argumentation in D with the same assumptions as in Example 5, if $\textbf{S}\not\in C$, then we have the following semantics.

• complete: $\{\{a_{0}\},\{a_{0},a_{2},a_{4}\},\{a_{0},a_{3}\}\}$.             grounded: $\{\{a_{0}\}\}$.
• semi-grounded: $\{\{a_{0}\}\}$.                                 preferred: $\{\{a_{0},a_{2},a_{4}\},\{a_{0},a_{3}\}\}$.

If $\textbf{S}\in C$, then we have the following semantics.

• complete: $\{\{a_{0}\},\{a_{3}\},\{a_{0},a_{2},a_{4}\}\}$.                 grounded: $\{\emptyset\}$.
• semi-grounded: $\{\{a_{0}\},\{a_{3}\}\}$                          preferred: $\{\{a_{3}\},\{a_{0},a_{2},a_{4}\}\}$.

For the argumentation in F with the same assumptions as in Example 5, if $\star\in C$, then we have $\{\{a_{5}\},\{a_{9}\},\{a_{5},a_{9}\}\}$ (complete), $\{\emptyset\}$ (grounded), $\{\{a_{5}\},\{a_{9}\}\}$ (semi-grounded) and $\{\{a_{5},a_{9}\}\}$ (preferred).

Theorem 4.4 (Existence)

For any $a^{\small\mbox{{BBA}}}$ and any $C$, we have all the following:

There exists at least one complete set under $C$. 2. 2.

The grounded set may not be a complete set under $C$. 3. 3.

A semi-grounded set is a complete set under $C$. 4. 4.

If there is only one semi-grounded set, then it is the grounded set under $C$.

Additionally, if $\textbf{S},\star\not\in C$, but not necessarily otherwise, there is only one semi-grounded set under $C$.

5 Related Work and Conclusion

Structured Argumentation. Structured argumentation [11, 16] lets arguments associated in premise-conclusion relation. Consider ABA-style structured argumentation [11]:$(\mathcal{H},\mathcal{I},\mathcal{J},\mathcal{K})$ for: some finite set $\mathcal{H}$ of entities; $\mathcal{I}:\mathcal{H}\times 2^{\mathcal{H}}$; $\emptyset\not\subseteq\mathcal{J}\subseteq\mathcal{H}$; and $\mathcal{K}:\mathcal{J}\rightarrow\mathcal{H}$. $H_{1}\subseteq\mathcal{H}$ is said to support $h_{1}\in\mathcal{H}$ iff $(h_{1},H_{1})\in\mathcal{I}$. For any $(h,H)\in\mathcal{I}$, if $H\not=\emptyset$, then $h\in\mathcal{H}\backslash\mathcal{J}$. This tuple $(\mathcal{H},\mathcal{I},\mathcal{J},\mathcal{K})$ ought to be such that the set $\mathcal{T}$ below exists.

Let $\mathcal{T}:\mathcal{H}\times 2^{\mathcal{T}}$ be the least set satisfying both (1) $(h,\emptyset)\in\mathcal{T}$ if $h\in\mathcal{J}$, and (2) if $(h_{1},t_{1}),\ldots,(h_{n},t_{n})\in\mathcal{T}$ and if $(h_{0},\{h_{1},\ldots,h_{n}\})\in\mathcal{I}$, then $(h_{0},\{(h_{1},t_{1}),\ldots,(h_{n},t_{n})\})\linebreak\in\mathcal{T}$. Every $t\in\mathcal{T}$ is an argument in ABA. $(h_{1},t^{\prime}_{1})\in\mathcal{T}$ attacks $t_{2}\in\mathcal{T}$ iff $\mathcal{K}(h)=h_{1}$ for some $h\in\mathcal{J}$ occurring in $t_{2}$. $T_{1}\subseteq\mathcal{T}$ defends $t\in\mathcal{T}$ just when every $t_{p}\in\mathcal{T}$ attacking $t$ is attacked by at least some $t_{q}\in T_{1}$. It is complete just when: (1) there is no attack among the members of $T_{1}$; and (2) it includes all $t\in\mathcal{T}$ it defends.

Encoding of ABA into $a^{\small\mbox{{BBA}}}=(A,R,R^{\mathbf{s}})$ is rather smooth. We define $\mathcal{X}$ to be such that $x_{h}\in\mathcal{X}$ iff $h\in\mathcal{J}$. We define $A$ to be the least set that satisfies both: (1) $a_{h}=(\{x_{h}\},\emptyset,\emptyset)$ is in $A$ if $(h,\emptyset)\in\mathcal{T}$ for some $h$; and (2) $a_{t}=(\{a_{T_{1}},a_{h}\},\emptyset,\{(a_{T_{1}},a_{h})\})$ such that $a_{T_{1}}=(\{a_{t_{1}},\ldots,a_{t_{n}}\},\emptyset,\emptyset)$ is in $A$ if $T_{1}=\{t_{1},\ldots,t_{n}\}$ for some $T_{1}\subseteq\mathcal{T}$ and some $h$ such that $(h,T_{1})\in\mathcal{T}$. We define $R$ to be such that $(a_{1},a_{2})\in R$ for some $a_{1},a_{2}\in A$ iff, if $a_{1}$ ($a_{2}$) is the encoding of $t_{1}\in\mathcal{T}$ ($t_{2}\in\mathcal{T}$), then $t_{1}$ attacks $t_{2}$. Finally, we define $R^{\mathbf{s}}$ to be empty. Then $A_{1}\subseteq A$ is a complete set ($\langle m\rangle=0$) under $C=\emptyset$ iff $T_{1}\subseteq\mathcal{T}$ such that $A_{1}$ is the encoding of $T_{1}$ is complete. Trivially, ABA is an instance of BBA.

More than just the result of subsumption, the process of encoding highlights the degree of structuralisation allowed in BBA and ABA, which is similarly true of ASPIC [16]: (1) support (premise-conclusion) in ABA is accessible for a child (premise) of a subtree of $t\in\mathcal{T}$, but no support, if it should occur in the parent (conclusion) of a subtree of $\mathcal{T}$, is accessible. That is, for the argument: [something is a support for [that something is a support for something]], the outer support is accessible in ABA formalisation, but the inner support, within the conclusion, must be encapsulated. BBA relaxes this restriction. Similarly, (2) any construct of the sort: [[that something attacks something] supports something] is expressed in ABA by encapsulating the inner attack. In BBA, the inner attack is also accessible. The concept of block argumentation thus allows for a greater reach to the internal structure of arguments. It is our hope, then, that the study in this paper will be of interest also to the scientific community working on structured argumentation.

Dependency among Arguments. Dung abstract argumentation does not specify the nature of an abstract entity. It is possible two arguments in an argumentation graph are interpreted identical. For rhetoric argumentation, acceptance of an argument may be considered with respect to the position in the graph in which it occurs; thus such a scenario incurs comparatively small an issue. Outside rhetoric argumentation, however, acceptability statuses of arguments are often preferred to be regarded indicative of acceptance of the arguments and not their acceptance with respect to their positions in the graph. It is then that multiple occurrences of the same argument in a graph are less desirable. An extreme case is “$a_{1}$ attacks $a_{2}$”. In case they are interpreted the same, classic semantics predict the argument to be both acceptable and not acceptable, and yet it sceptically accepts it, i.e. $a_{1}$. Such issue from dependencies seems to have been seldom reflected back to abstract argumentation semantics. In abstract dialectical frameworks [7], acceptability statuses of arguments are given based on those of their attackers/supporters. In this work, we showed that practically any arguments in a graph, whether or not they are connected, may have dependencies. We proposed use of both graphical and semantic constraints.

Higher-Level and Meta Argumentation. Meta argumentation [15, 22] facilitates layers of argumentation, to discuss attack relation among arguments and so on in a preceding layer. From a layer to a layer, there is a clear boundary (and the separation does not disappear if one is to flatten them). Our motivating example in Section 2 shows, however, that [something attacks something] may itself be an argument, which BBA can handle uniformly as with unitary ones. Temporal / modal argumentation networks [3], as far as we are aware, is the first abstract argumentation study that hinted at the possibility that an unattacked argument may not be outright accepted. A temporal argument $\diamond a$ asks effectively if $a$ is accepted in the argumentation at a connected possible world for the argument to be accepted-able. The temporal argument allows an argument in another possible world to be referred to. In this work, we have generalised such semantic dependency with the semantic constraints. Higher-level argumentation [12] considers substitution of an argumentation framework into an argument to identify which arguments in the substituted argumentation are the actual interactors to the outside. By contrast, in block argumentation, the key is to express the dual roles of an argument: as an argument and as an argumentation (similar emphasis given for coalition formation [2]) and how they influence acceptability semantics. In block argumentation, it is possible that an argument as an argumentation, and not necessarily some arguments in the argumentation, attacks or supports other arguments.

Conclusion. We presented block argumentation, with comparisons to structured argumentation for which we have demonstrated a prospect of further generalisation. Block argumentation reveals multiple occurrences of the same arguments within an argumentation, and, as we have identified, both graphical and semantic coherency pose challenges to the classic principle of always accepting unattacked arguments. We proposed to tackle this issue with constraints. For future work, we plan to study their granularity to cater for specific applications, as well as to incorporate probabilistic or dynamic approaches.

Acknowledgement

The first author thanks Jiraporn Pooksook for a tutorial on ABA.

Appendix (Proofs)

Theorem 5.1 (Correspondence between extension-based complete sets and complete labellings)

For any $a^{\small\mbox{{BBA}}}$ and any $\langle m\rangle$, $A_{1}\subseteq\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ is standard complete in $\langle m\rangle$ only if there is some standard complete labelling $\lambda$ of $a^{\small\mbox{{BBA}}}$ such that $\lambda((a_{p},\langle m\rangle.n))=+$ is equivalent to $a_{p}\in A_{1}$ for any $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$. Conversely, $\lambda$ is a standard complete labelling of $a^{\small\mbox{{BBA}}}$ only if, for every $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$, $\{a_{p}\in\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)\ |\ \exists n\in\mathbb{N}.\lambda((a_{p},\langle m\rangle.n))=+\}$ is a standard complete set in $\langle m\rangle$.

Proof

Assume $A_{1}\subseteq\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ is standard complete in $\langle m\rangle$. Assume some $\lambda\in\Lambda$ such that $\lambda((a_{p},\langle m\rangle.n))=+$ for any $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$ with $a_{p}\in A_{1}$, that $\lambda((a_{p},\langle m\rangle.n))=-$ for any $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$ with $a_{p}\in\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ if $a_{p}$ is attacked by $A_{1}$, and that $\lambda((a_{p},\langle m\rangle.n))=\ ?$ for any other $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$ with $a_{p}\in\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$. We show that $\lambda$ is a standard complete labelling. We show firstly that every $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$ such that $\lambda((a_{p},\langle m\rangle.n))=+$ satisfies the corresponding condition (Definition 6). Suppose, by way of showing contradiction, that there is some $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$ such that $\lambda((a_{p},\langle m\rangle.n))=+$ does not satisfy the corresponding condition. Then, there must exist some $(a_{1},\langle m\rangle.n_{1})\in\varpi(a^{\small\mbox{{BBA}}}))$ such that $\lambda((a_{1},\langle m\rangle.n_{1}))=+$ and that $a_{1}$ attacks $a_{p}$ in $\langle m\rangle$. However, it cannot be $+$, since $A_{1}$ is conflict-free, contradiction. Second, we show that every $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$ such that $\lambda((a_{p},\langle m\rangle.n))=-$ satisfies the corresponding condition (Definition 6). By construction of $\lambda$, however, $a_{p}$ is attacked by a member of $A_{1}$. Since $A_{1}$ is defended, there exists no member of $A_{1}$ that supports $a_{p}$. Finally, we show that every $(a_{p},\langle m\rangle.n)\in\varpi(a^{\small\mbox{{BBA}}})$ such that $\lambda((a_{p},\langle m\rangle.n))=\ ?$ does not satisfy the conditions for $+$ or $-$ (Definition 6). If it satisfied the condition for $+$, then every $a_{1}$ attacking it would get $-$. By construction of $\lambda$, however, every such $a_{1}$ would be attacked by a member of $A_{1}$ in $\langle m\rangle$. Since $A_{1}$ includes all arguments it defends in $\langle m\rangle$, it follows that $a_{p}\in A_{1}$, contradiction. If it satisfied the condition for $-$, then it would be attacked by a member of $A_{1}$ in $\langle m\rangle$. By construction of $\lambda$, it is not possible that $a_{p}$ gets $?$.

For the second part, assume $\lambda$ is a standard complete labelling. Assume also some $\langle m\rangle$ such that $\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ is non-empty. Assume $A_{1}=\{a_{p}\in\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)\ |\ \exists n\in\mathbb{N}.\lambda((a_{p},\langle m\rangle.n))=+\}$. We show that $A_{1}$ is standard complete. $A_{1}$ is trivially conflict-free. Now, suppose some $a_{1}\in\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ attacking a member of $A_{1}$. Since $\lambda$ is a standard complete labelling, it must be that $a_{1}$ gets $-$ in $\langle m\rangle$, which means that $A_{1}$ attacks but does not support $a_{1}$ in $\langle m\rangle$. Thus $A_{1}$ defends its members attacked by $a_{1}$. By the definition of $\lambda$, it is straightforward to see that $A_{1}$ includes all arguments it defends. ∎

Theorem 5.2 (Conservation)

For any $a^{\small\mbox{{BBA}}}$ and any $\lambda$, if $C=\emptyset$, then $\lambda$ being a complete labelling of $a^{\small\mbox{{BBA}}}$ under $C$ is equivalent to $\lambda$ being a standard complete labelling of $a^{\small\mbox{{BBA}}}$.

Proof

Trivial. ∎

Theorem 5.3 (No inclusion)

For any $C\not=\emptyset$, there exists $a^{\small\mbox{{BBA}}}$ such that some standard complete labelling of $a^{\small\mbox{{BBA}}}$ is not a complete labelling under $C$ and that some complete labelling of $a^{\small\mbox{{BBA}}}$ under $C$ is not a standard complete labelling.

Proof

See Example 6 when $C$ contains S or $\star$. For $C=\{\textbf{G}\}$, consider the argumentation in B as $a^{\small\mbox{{BBA}}}$. ∎

For the last theorem, it helps to define order on labelling that judges one labelling strictly more informative than another when the former preserves

• and $-$ assignments by the latter but assigns a smaller number of ?.

Definition 14 (Order in labelling)

Let $\rhd:\mathcal{A}^{\small\mbox{{BBA}}}\times\Lambda\times\Lambda$ be such that $(a^{\small\mbox{{BBA}}},\lambda_{1},\lambda_{2})\in\rhd$, written alternatively as $\lambda_{1}\rhd^{a^{\small\mbox{{BBA}}}}\lambda_{2}$ iff $\lambda_{2}((a_{p},\langle m_{p}\rangle))\succeq\lambda_{1}((a_{p},\langle m_{p}\rangle))$ for every $(a_{p},\langle m_{p}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$. For any $\lambda_{1},\lambda_{2}\in\Lambda$ and for any $a^{\small\mbox{{BBA}}}$, we say that $\lambda_{1}$ is more informative than $\lambda_{2}$ iff $\lambda_{2}\rhd^{a^{\small\mbox{{BBA}}}}\lambda_{1}$. In particular, $\lambda_{1}$ is strictly more informative than $\lambda_{2}$ iff $\lambda_{2}\rhd^{a^{\small\mbox{{BBA}}}}\lambda_{1}$ and $\lambda_{1}\not\rhd^{a^{\small\mbox{{BBA}}}}\lambda_{2}$.

We say, further, that there is a path from $a_{1}$ to $a_{2}$ in $\langle m\rangle$ iff $a_{1},a_{2}\in\textsf{Arg}(a^{\small\mbox{{BBA}}},\langle m\rangle)$ and either: $a_{1}$ attacks $a_{2}$; or else there is a path from $a_{1}$ to some $a_{3}$ such that $a_{3}$ attacks $a_{2}$.

Lemma 1

Let $a^{\small\mbox{{BBA}}}$ be a member of $\mathcal{A}^{\small\mbox{{BBA}}}$, and let $\lambda_{1}$ be a standard complete labelling of $a^{\small\mbox{{BBA}}}$. Let $\Gamma$ be a set of some $(a_{1},\langle m_{1}\rangle),\ldots,(a_{n},\langle m_{n}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ such that $\lambda_{1}((a_{i},\langle m_{i}\rangle))\not=\ ?$. Let $\Lambda_{1}$ be a subclass of $\Lambda$ that contains all $\lambda$ such that $\lambda((a_{i},\langle m_{i}\rangle))=\ ?$ for any $(a_{i},\langle m_{i}\rangle)\in\Gamma$ but, for any other $(a_{p},\langle m_{p}\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$, that $\lambda$ respects the conditions given for a standard complete labelling. Then, there exists $\lambda\in\Lambda_{1}$ such that $\lambda_{1}$ is strictly more informative than $\lambda$.

Theorem 5.4 (Existence)

For any $a^{\small\mbox{{BBA}}}$ and any $C$, we have all the following:

There exists at least one complete set under $C$. 2. 2.

The grounded set may not be a complete set under $C$. 3. 3.

A semi-grounded set is a complete set under $C$. 4. 4.

If there is only one unique semi-grounded set, then it is the grounded set under $C$.

Additionally, if $\textbf{S},\star\not\in C$, but not necessarily otherwise, there is only one semi-grounded set under $C$.

Proof

2. • 4. follow immediately from the preceding discussion, on the assumption that 1. holds to be the case. Suppose, by way of showing contradiction, that there is no complete labelling. Then, it must be that there exist some $a^{\small\mbox{{BBA}}}$ and some $C$ such that every $\lambda\in\Lambda$ of $a^{\small\mbox{{BBA}}}$ contradicts at least one condition given in Definition 12 for some $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$. Consider some standard complete labelling $\lambda$ which always exists. By the supposition, $\lambda$ is not a complete labelling. Therefore, there exists some $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ that either does not satisfy some semantic constraint $c\in C$ with $\lambda$, or violates G and yet is assigned + with $\lambda$. Let us define the following function.

Function down $(a^{\small\mbox{{BBA}}}:\mathcal{A}^{\small\mbox{{BBA}}},\lambda:\Lambda,C:\mathcal{C})$ returns $(a^{\small\mbox{{BBA}}}:\mathcal{A}^{\small\mbox{{BBA}}},\lambda_{1}:\Lambda,C:\mathcal{C})$.

Assume $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ for which $\lambda\in\Lambda$ of $a^{\small\mbox{{BBA}}}$ contradicts at least one condition given in Definition 12.

return $(a^{\small\mbox{{BBA}}},\lambda,C)$ if there is no such $(a,\langle m\rangle)$. 2. 2.

If $\lambda((a,\langle m_{p}\rangle.n))\in\{+,-\}$ ($\langle m\rangle=\langle m_{p}\rangle.n$), $\star\in C$, and $(a,\langle m_{p}\rangle.n)$ does not satisfy $\star$, then there exist $(a_{1},\langle m_{p}\rangle.n_{1}),\ldots,(a_{k},\langle m_{p}\rangle.n_{k})\in\varpi(a^{\small\mbox{{BBA}}})$ such that $\bigwedge_{1\leq i\leq k}\textsf{Eq}(a,a_{i})$ and that $|\bigcup_{1\leq i\leq k}\lambda((a_{i},\langle m_{p}\rangle.n_{i}))|>1$. Consider a subclass $\Lambda_{1}$ of $\Lambda$ that contains all $\lambda_{1}$ such that $\lambda_{1}((a,\langle m_{p}\rangle.n))=\lambda_{1}((a_{1},\langle m_{p}\rangle.n_{1}))=\cdots=\lambda_{1}((a_{k},\langle m_{p}\rangle.n_{k}))=\ ?$, but which otherwise respects the conditions given for a standard complete labelling. By Lemma 1, and by the fact that there is at least one $(a_{i},\langle m_{p}\rangle.n_{i})$, $1\leq i\leq k$, that is not assigned ?, there exists some $\lambda_{1}\in\Lambda_{1}$ such that $\lambda$ is strictly more informative than $\lambda_{1}$. By construction, such $\lambda_{1}$ always exists. return $(a^{\small\mbox{{BBA}}},\lambda_{1},C)$. 3. 3.

If $\lambda((a,\langle m\rangle))\in\{+,-\}$, $\textbf{S}\in C$, and $(a,\langle m\rangle)$ does not satisfy S: Consider a subclass $\Lambda_{1}$ of $\Lambda$ that contains all $\lambda_{1}$ such that $\lambda_{1}((a,\langle m\rangle))=\ ?$, but which otherwise respects the conditions given for a standard complete labelling. By Lemma 1, and by the fact that $\lambda((a,\langle m\rangle))\not=\ ?$, there exists some $\lambda_{1}\in\Lambda_{1}$ such that $\lambda$ is strictly more informative than $\lambda_{1}$. By construction, such $\lambda_{1}$ always exists. return $(a^{\small\mbox{{BBA}}},\lambda_{1},C)$. 4. 4.

If $\lambda((a,\langle m\rangle))=+$, $\textbf{G}\in C$, and $(a,\langle m\rangle)$ does not satisfy G: Consider a subclass $\Lambda_{1}$ of $\Lambda$ that contains all $\lambda_{1}$ such that $\lambda_{1}((a,\langle m\rangle))=\ ?$, but which otherwise respects the conditions given for a standard complete labelling. By Lemma 1, and by the fact that $\lambda((a,\langle m\rangle))\not=\ ?$, there exists some $\lambda_{1}\in\Lambda_{1}$ such that $\lambda$ is strictly more informative than $\lambda_{1}$. By construction, such $\lambda_{1}$ always exists. return $(a^{\small\mbox{{BBA}}},\lambda_{1},C)$.

End Function

Let $(a^{\small\mbox{{BBA}}},\lambda_{1},C)$ be: $(a^{\small\mbox{{BBA}}},\lambda_{1},C)=\textsf{down}^{n+1}(a^{\small\mbox{{BBA}}},\lambda,C)=\textsf{down}^{n}(a^{\small\mbox{{BBA}}},\lambda,C)$ for some $n$ and for some standard complete labelling $\lambda$. For every $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$, it holds that $(a,\langle m\rangle)$ and $\lambda_{1}$ satisfy semantic constraints in $C$, and that $\lambda((a,\langle m\rangle))\not=+$ if $(a,\langle m\rangle)$ does not satisfy G if $\textbf{G}\in C$.

Now, consider $\Lambda_{1}\subseteq\Lambda$ to be the set of all labelling $\lambda_{2}$ such that $\lambda_{2}$ is more informative than $\lambda_{1}$ but not more informative than $\lambda$, and moreover that, for every $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$, it holds that $(a,\langle m\rangle)$ and $\lambda_{1}$ satisfy semantic constraints in $C$, and that $\lambda((a,\langle m\rangle))\not=+$ if $(a,\langle m\rangle)$ does not satisfy G if $\textbf{G}\in C$. Since both $\lambda_{1}$ and $\lambda$ exist, $\Lambda_{1}\not=\emptyset$. It is easy to see that a maximal element in $\Lambda_{1}$ is a complete labelling of $a^{\small\mbox{{BBA}}}$.

If $C=\textbf{G}$ or else $C=\emptyset$, then, for any $(a,\langle m\rangle)\in\varpi(a^{\small\mbox{{BBA}}})$ if there is some $\lambda$ that is strictly less informative than a standard complete labelling $\lambda_{1}$ of $a^{\small\mbox{{BBA}}}$ and if $\lambda((a,\langle m\rangle))\succeq\lambda_{1}((a,\langle m\rangle))$, then for any complete labelling $\lambda_{2}$ of $a^{\small\mbox{{BBA}}}$ under $C$, $\lambda_{2}((a,\langle m\rangle))=\lambda((a,\langle m\rangle))$. Straightforward. Example 6 for $\textbf{S}\in C$ and $\star\in C$. ∎