Abstract
Abstract dialectical frameworks (ADFs) are generalizations of Dung’s argumentation frameworks which allow arbitrary relationships among arguments to be expressed. In particular, arguments can not only attack each other, they also may provide support for other arguments and interact in various complex ways. The ADF approach has recently been extended in two different ways. On the one hand, GRAPPA is a framework that applies the key notions underlying ADFs – in particular their operator-based semantics – directly to arbitrary labelled graphs. This allows users to represent argumentation scenarios in their favourite graphical representations without giving up the firm ground of well-defined semantics. On the other hand, ADFs have been further generalized to the multi-valued case to enable fine-grained acceptance values. In this paper we unify these approaches and develop a multi-valued version of GRAPPA combining the advantages of both extensions.
Keywords
This research has been supported by DFG (Research Unit 1513 and project BR 1817/7-2) and FWF (project I2854).
1 Introduction
Computational models of argumentation are a highly active area of current research. The field has two main subareas, namely logic-based (also called structured) argumentation and abstract argumentation. The former studies the structure of arguments, how they can be constructed from a given formal knowledge base, and how they logically interact with each other. The latter, in contrast, assumes a given set of abstract arguments together with specific relations among them. The focus is on evaluating the arguments based on their interactions with one another. This evaluation typically uses a specific semantics, thus identifying subsets of the available arguments satisfying intended properties so that the chosen set arguably can be viewed as representing a coherent world view.
In the abstract approach, Dung’s argumentation frameworks (AFs) [18] and their associated semantics are widely used. In a nutshell, an AF is a directed graph with each vertex being an abstract argument and each directed edge corresponding to an attack from one argument to another. These attacks are then resolved using appropriate semantics. The semantics are typically based on two important concepts, namely conflict-freeness and admissibility. The former states that if there is a conflict between two arguments, i.e. one argument attacks the other, then the two cannot be jointly accepted. The latter specifies that every set of accepted arguments must defend itself against attacks. A variety of semantics has been defined, ranging from Dung’s original complete, preferred, stable, and grounded semantics to the more recent ideal and cf2 semantics. The different semantics reflect different intuitions about what “coherent world view” means in this context, see e.g. [5] for an overview.
Despite their popularity, there have been various attempts to generalize AFs as many researchers felt a need to cover additional relevant relationships among arguments (see e.g. the work of [14]). One of the most systematic and flexible outcomes of this research are abstract dialectical frameworks (ADFs) [7, 11]. ADFs allow for arbitrary relationships among arguments. In particular, arguments can not only attack each other, they also may provide support for other arguments and interact in various complex ways. This is achieved by adding explicit acceptance conditions to the arguments which are most naturally expressed in terms of a propositional formula (with atoms referring to parent arguments). This way, it is possible to specify individually for a particular argument, say, under what conditions the available supporting arguments outweigh the counterarguments. Meanwhile various applications of ADFs have been presented, for instance in legal reasoning [1,2,3,4] and text exploration [13]. Also a mobile argumentation app based on ADF techniques has been developed [19].
The operator-based semantics of ADFs can be traced back to the work of [15,16,17] on approximation fixpoint theory (AFT), an algebraic framework for studying semantics of knowledge representation formalisms. We refer to the work of [20] for a detailed analysis of the relationship between ADFs and AFT. The presentation of our approach in this paper does not assume specific background knowledge in AFT.
In the meantime, ADFs themselves have been extended in two different ways. GRAPPA [12] allows argumentation scenarios to be represented as arbitrary edge-labelled graphs. Acceptance functions now are defined in terms of the (multi-set of) active labels, that is, labels of incoming links whose source nodes are true in an interpretation. To conveniently express these functions so-called acceptance patterns are used. The definition of the different semantics then is based on similar operators as for ADFs. The GRAPPA approach has several advantages over ADFs, in particular, it is often easier to model relevant argumentation problems in terms of labelled graphs and users commonly illustrate argumentation scenarios using such graphs. GRAPPA thus allows users to stay as close as possible to their graphical representations yet turns them into full-fledged knowledge representation formalisms by providing formal semantics. Moreover, various intuitive acceptance conditions (e.g. “accept if there are more active \(+\) labels than active − labels”) have the same representation as acceptance pattern for all nodes in the graph, whereas the corresponding acceptance condition for ADFs depends on the node at hand. For a joint discussion of ADFs and GRAPPA, we refer to the recent handbook article [8].
A second extension are the recently proposed weighted ADFs (wADFs) [6, 9, 10]Footnote 1. Rather than being based on partial two-valued interpretations, the semantics of wADFs is based on partial V-valued interpretations, where V is some chosen set of acceptance degrees for arguments. This obviously allows for much more fine-grained distinctions among arguments in the semantics. The authors show that the extension to the multi-valued case is surprisingly smooth: the standard ADF operators only need mild reformulations to be able to capture arbitrary acceptance degrees.
The goal of this paper is to combine the two mentioned extensions of ADFs, thus bringing together the best of the two worlds. We are interested here in a formalism that extends GRAPPA the same way as wADFs extend ADFs. The approach we propose, called multi-valued GRAPPA (mvGRAPPA), is thus a knowledge representation formalism on the basis of simple labelled graphs equipped with a variety of multi-valued semantics rooted in argumentation and will be developed in the remainder of the paper.
Our article is organized as follows. The generalization of GRAPPA to the multi-valued case is presented in Sect. 2. The generalization is based on acceptance functions which determine the value of a node in the argument graph based on multi-sets of value/label pairs. We show that our concept of acceptance functions allows for a natural definition of different semantics that enjoy the expected relationships. The challenge how to represent acceptance functions adequately and in a user-friendly way is then addressed Sect. 3, where we propose so-called acceptance programs to specify those functions. Various examples illustrating the flexibility and expressiveness of this approach are discussed in Sect. 4. Section 5 concludes the paper.
2 Multi-valued GRAPPA
In this section we introduce syntax and semantics of multi-valued GRAPPA. We will introduce the generalized definitions right away and point out where they differ from the original definitions in [12]. Our approach allows for arbitrary sets of values in interpretations.
We assume a dedicated undefined truth value \(\mathbf {u} \).
Definition 1
A set V of values (acceptance degrees) is called \(\mathbf {u}\)-free if \(\mathbf {u} \not \in V\).
For every \(\mathbf {u} \)-free set of values used in this work we assume an underlying information ordering \(<_i \) on \(V \cup \{\mathbf {u} \}\), where \(<_i \,\,=\{\langle {\mathbf {u},v} \rangle \mid v \in V\}\), i.e., the information content of all values in V is equal and strictly greater than that of \(\mathbf {u} \). We write \(v \le _i v'\) whenever \(v <_i v'\) or \(v=v'\).
Definition 2
Let L be a set of labels and V be a \(\mathbf {u} \)-free set of values. A V-valued acceptance function over L is a function \(c:(V \times L \rightarrow \mathbb {N})\rightarrow V\). The set of V-valued acceptance functions over L is denoted by \(F^{V,L} \).
Intuitively, c determines the truth value to be assigned to a node n. This value is an element of V and depends on the truth values assigned to n’s parent nodes, but also on the labels of links connecting parent nodes to n. Since the same value/label pair may appear more than once – and since this may be relevant for the value assigned to n – the actual assignment is based on a multi-set. Intuitively, this set counts the number of edges to n with a given label from parent nodes that are assigned a given truth value.
Definition 3
Let V be a \(\mathbf {u} \)-free set of values. A V-valued labelled argument graph (V-LAG) is a tuple \(G =\langle {S,E,L,\lambda ,\alpha } \rangle \) where
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S is a set of nodes (statements),
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\(E\subseteq S\times S\) is a set of edges,
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L is a set of labels,
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\(\lambda :E\rightarrow L\) assigns labels to edges, and
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\(\alpha :S\rightarrow F^{V,L} \).
This definition is almost identical to the definition of LAGs in [12]. The single exception is \(\alpha \) which now assigns an acceptance function over an arbitrary fixed set V, whereas acceptance functions in the earlier paper were only allowed to assign \({\mathbf {t}}\) and \({\mathbf {f}}\). In other words, V was fixed to \(\{{\mathbf {t}},{\mathbf {f}}\}\) in [12].
A V-valued interpretation for S (or simply interpretation when V and S are clear from context) is a function \(v: S \rightarrow V_\mathbf {u} \) with \(V_\mathbf {u} = V\cup \left\{ \mathbf {u} \right\} \). Slightly abusing terminology we will also apply \(\le _i \) and \(<_i \) to interpretations with the standard pointwise reading. For instance, \(v \le _i v'\) stands for: \(v(s) \le _i v'(s)\) for all \(s \in S\). An interpretation v is total if it does not assign \(\mathbf {u} \) to any statement. A completion of v is any total interpretation w with \(v \le _i w\). We denote the set of completions of v by \({[v]_c} \).
Definition 4
Let V be a \(\mathbf {u} \)-free set of values, \(G =\langle {S,E,L,\lambda ,\alpha } \rangle \) a V-LAG, and v a total V-valued interpretation for S. The multi-set of valued labels of \(s\in S\) in G under v, \(m^{v}_s\), is defined as
for each \({\mathbf {x}}\in V\) and each \(l\in L\).
Example 1
Let \(G =\langle {S,E,L,\lambda ,\alpha } \rangle \) be a V-LAG over values \(V=\big \{\mathbf {yes},\mathbf {no},\mathbf {maybe} \big \}\) and with labels \(L=\left\{ +,-\right\} \). Suppose \(s\in S\) is a node in G with three parents \(p_1\), \(p_2\), and \(p_3\) such that the links from \(p_1\), \(p_2\) and \(p_2\) to s are all labelled ‘\(+\)’. G is depicted in Fig. 1.
For an interpretation v with \(v(p_1)=\mathbf {yes} \) and \(v(p_2)=v(p_3)=\mathbf {maybe} \) we have
Like for GRAPPA and ADFs, our semantics is based on a characteristic operator on interpretations. As mentioned in the introduction, it is inspired from similar operators in AFT.
Definition 5
(Characteristic Operator \(\varGamma ^{V}_G\)). Let V be a \(\mathbf {u} \)-free set of values, \(v: S \rightarrow V_\mathbf {u} \) an interpretation, and \(G =\langle {S,E,L,\lambda ,\alpha } \rangle \) a V-LAG. Applying \(\varGamma ^{V}_G\) to v yields a new interpretation (the consensus over \({[v]_c} \)) defined as
where \(\textstyle \bigsqcap _i\) denotes the greatest lower bound in \((V_\mathbf {u},\le _i)\).
Intuitively, \(\varGamma ^{V}_G(v)\) maps statements to the greatest truth value (with respect to the information order) that is compatible with all results of evaluating G under some completion of v.
Example 2
Let \(G =\langle {S,E,L,\lambda ,\alpha } \rangle \) be defined as in Example 1 and let the acceptance function \(\alpha (s)\) of node s be given by
Hence, s is assigned \(\mathbf {yes}\) if there are no ‘\(+\)’-parents assigned \(\mathbf {no} \).Footnote 2 Furthermore, s is assigned \(\mathbf {maybe}\) if there are more or equally many ‘\(+\)’-parents assigned \(\mathbf {yes} \) than \(\mathbf {no} \). Consider the interpretation \(v'\) with \(v'(p_1)=\mathbf {yes} \) and \(v'(p_2)=\mathbf {maybe} \) and \(v'(p_3)=\mathbf {u} \). The set of completions of \(v'\) is \({[v']_c} =\{v,v_y,v_n\}\) where v is given as in Example 1, and \(v_y\) and \(v_n\) coincide with \(v'\) and v except for the assignments \(v_y(p_3)=\mathbf {yes} \) and \(v_n(p_3)=\mathbf {no} \) of \(p_3\).
The multi-sets of valued labels of these completions are given by
We have \(\alpha (s)(m^{v}_s)=\mathbf {yes} \), \(\alpha (s)(m^{v_y}_s)=\mathbf {yes} \), and \(\alpha (s)(m^{v_n}_s)=\mathbf {maybe} \). Consequently, \(\varGamma ^{V}_G(v)(s)= \textstyle \bigsqcap _i\left\{ \mathbf {yes},\mathbf {maybe} \right\} =\mathbf {u} \). Note that in the example we did not use the − label for simplicity.
We are now in the position to define the standard semantics of argumentation on top of V-LAGs in the expected way.
Definition 6
Let \(G =\langle {S,E,L,\lambda ,\alpha } \rangle \) be a V-LAG. An interpretation \(v: S \rightarrow V_\mathbf {u} \) is
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a model of G iff \(v(s) \ne \mathbf {u} \) for all \(s \in S\) and \(\varGamma ^{V}_G(v) = v\).
Intuition: the value of a node s in v is exactly the one required by the acceptance function of S.
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grounded for G iff \(v= lfp (\varGamma ^{V}_G)\), i. e., v is the least fixpoint of \(\varGamma ^{V}_G\) w.r.t \(\le _i \).
Intuition: v collects all the information which is beyond any doubt.
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admissible for G iff \(v \le _i \varGamma ^{V}_G(v)\).
Intuition: v does not contain unjustifiable information.
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preferred for G iff it is \(\le _i\)-maximal admissible for G.
Intuition: v has maximal information content without giving up admissibility.
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complete for G iff \(v=\varGamma ^{V}_G(v)\).
Intuition: v contains exactly the justifiable information.
For \(\sigma \in \{\textit{adm},\textit{com},\textit{prf}\}\), \(\sigma (G)\) denotes the set of all admissible (resp. complete, preferred) interpretations with respect to G. Moreover, we use \(\textit{mod}(G)\) to denote the models of G.
We still need to show existence of the least fixpoint of \(\varGamma ^{V}_G\). This is a consequence of the monotonicity of the operator \(\varGamma ^{V}_G\). The pair \((\left\{ v:S\rightarrow V_\mathbf {u} \right\} ,\le _i)\) forms a complete partial order in which the characteristic operator \(\varGamma _D\) of wADFs is monotone.
Proposition 1
The operator \(\varGamma ^{V}_G\) is \(\le _i \)-monotone, that is, \(v \le _i w\) implies \(\varGamma _D(v) \le _i \varGamma _D(w)\) for all interpretations \(v,w:S\rightarrow V_\mathbf {u} \).
Existence of the least fixpoint of \(\varGamma ^{V}_G\) then follows via the fixpoint theorem for monotone operators in complete partial orders.
Next, we show that the well-known relationships between Dung semantics carry over to our generalizations.
Proposition 2
Let G be a V-LAG. It holds that
We now show how stable semantics can be generalized following the approach in [10]. Stable semantics treats truth values asymmetrically. For standard ADFs \({\mathbf {f}}\) (false) can be assumed to hold (by default), whereas \({\mathbf {t}}\) (true) needs to be justified by a derivation. This is achieved by building the reduct of an ADF and then checking whether the grounded interpretation of the reduct coincides with the original model on the nodes which “survive” in the reduct. Moving from the two-valued to the multi-valued case allows us to choose what the assumed, respectively derived truth values are. Stable semantics thus becomes parametrized by a subset W of the set of values V.
Definition 7
Let \(G =\langle {S,E,L,\lambda ,\alpha } \rangle \) be a V-LAG. Let \(v: S \rightarrow V\) be a model of G (that is, v is total). Let \(W \subseteq V\) be the set of assumed truth values. The v, W-reduct of G is the V-LAG \(G^v_W = (S^v_W, E^v_W, L, \lambda ^v_W, \alpha ^v_W)\) where
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\(S^v_W = \{s \in S \mid v(s) \notin W\}\),
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\(E^v_W = E \cap (S^v_W \times S^v_W)\),
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\(\lambda ^v_W\) is \(\lambda \) restricted to \(E^v_W\),
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\(\alpha ^v_W\) is obtained from \(\alpha \) as follows: \(\alpha ^v_W(n)(m) = \alpha (n)(m')\) where, for each value/label pair (x, l), \(m'(x,l) = m(x,l) + \mid \{s \in S\setminus S^v_W \mid v(s)= x, \lambda (s,n) = l \}\mid \).
The v, W-reduct can be viewed as the partial evaluation of the original graph which takes values in W for granted. Now stable models can be defined as usual:
Definition 8
Let \(G =\langle {S,E,L,\lambda ,\alpha } \rangle \) be a V-LAG and let \(v: S \rightarrow V\) be a model of G. Let \(v_g\) be the grounded interpretation of the v, W-reduct of G. v is a W-stable model of G iff \(v(s) = v_g(s)\) for each \(s \in S^v_W\).
We conclude this section with the following result:
Proposition 3
Multi-valued GRAPPA generalizes both GRAPPA [12] and weighted ADFs with flat information ordering [10].
Proof
Sketch: (a) Weighted ADFs can be modelled by labelling each link with the source node. Acceptance functions of weighted ADFs are functions from value assignments of the parents of a node to values for that node. With nodes as labels these value assignments can be reconstructed from the multi-sets used in multi-valued GRAPPA via the corresponding value/label pairs. This allows us to model the acceptance functions of weighted ADFs. (b) GRAPPA is just the special case of multi-valued GRAPPA with \(V = \{{\mathbf {f}}, {\mathbf {t}}\}\) and acceptance functions which only depend on the number of parent nodes with value \({\mathbf {t}}\).
From an abstract, mathematical point of view our generalization may seem straightforward. However, it brings with it an important practical issue which needs to be addressed: how to conveniently represent acceptance functions? In original GRAPPA the issue was dealt with by so-called acceptance patterns. An acceptance pattern is basically a condition that evaluates to \({\mathbf {t}}\) or \({\mathbf {f}}\) and directly determines the value of a node via this evaluation. In our new multi-valued setting this simple approach obviously does not work. To solve this issue we propose a rule-based approach: we use a set of rules; each rule consists of a condition and a value from V; the value assigned is computed from the values of all rules whose conditions evaluate to true.
3 Acceptance Programs
In this section we develop a method to represent multi-valued acceptance functions via so-called acceptance programs. An acceptance program consists of a collection of acceptance rules R and an aggregation function \({AG} \). Each rule \(r \in R\) is of the form
where, intuitively, \({\mathsf {v}}\) describes some value in V and \({\mathsf {b}}\) is an expression that evaluates to true or false for any given multi-set m of value/label pairs. This way the rules determine potential values taken from V. The role of the aggregation function \({AG}: 2^V \rightarrow V\) is to determine a unique value to be assigned to the node at hand based on the potential values. For convenience we also consider programs based on non-ground rules. We now formally define syntax and semantics of acceptance programs.
3.1 Syntax
We define the syntax in a bottom up fashion. Acceptance rules are built from three basic types of expressions: label expressions, value expressions, and numeric expressions. On top of these we define Boolean expressions.
Definition 9
A GRAPPA signature \(\varSigma = (V,L,Var_V,Var_L,F,Rel)\) consists of a set of values V, a set of labels L, sets of value and label variables \(Var_V\) and \(Var_L\), respectively, a set of function symbols F and a set of binary relation symbols Rel.
A label expression (over \(\varSigma \))Footnote 3 is
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a label \(l\in L\) or
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a label variable \({X_{L}} \in Var_L\).
A value expression is recursively defined as
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a value \({\mathbf {x}}\in V\),
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a value variable \(\mathbf {X} \in Var_V\),
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a value function term \(f(\overrightarrow{{\mathsf {e}}})\) where \(f\in F\) \(\overrightarrow{{\mathsf {e}}} =\langle {{\mathsf {e}}_1,\dots ,{\mathsf {e}}_k} \rangle \) is a vector of label, value, or numeric expressions,
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\(min_{W,\preccurlyeq }({\mathsf {l}})\), or
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\(max_{W,\preccurlyeq }({\mathsf {l}})\) where W is a set of value expressions, \(\preccurlyeq \in Rel\) a binary relation symbol, and \({\mathsf {l}}\) a label expression.
A numeric expression is of the form
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\(c\in \mathbb {R} \),
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\(\#_{W}({\mathsf {l}})\), where \({\mathsf {l}}\) is a label expression,
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\(sum_{W}\),
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\(count_{W}\),
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\(min_{W}\), or
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\(max_{W}\) where W is a set of value expressions, or
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\(({\mathsf {n}}\oplus {\mathsf {n}}')\) where \({\mathsf {n}}\) and \({\mathsf {n}}'\) are numeric expressions and \(\oplus \in \left\{ +,*\right\} \).
A Boolean expression is of the form
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\(m \sim n\) where m and n are numeric expressions and \(\sim \,\,\in \left\{ <,\le ,=,\ge ,>,\ne \right\} \),
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\(\mathbf {v} \sim \mathbf {w} \) where \(\mathbf {v} \) and \(\mathbf {w} \) are value expressions and \(\sim \,\, \in Rel\),
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\(\bot \),
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\(\top \),
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\({\mathsf {b}}\otimes {\mathsf {b}}'\) where \({\mathsf {b}}\) and \({\mathsf {b}}'\) are Boolean expressions and \(\otimes \in \left\{ \wedge ,\vee ,\rightarrow \right\} \), or
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\(\lnot a\) where a is a Boolean expression.
We are now in a position to define rules. As mentioned above rules derive potential values based on some Boolean condition:
Definition 10
An acceptance rule is of the form
where \({\mathsf {v}}\) is a value expression and \({\mathsf {b}}\) a Boolean expression. An acceptance rule is ground if it contains no (label nor value) variables.
Since rules may identify multiple potential values (or none), acceptance programs need an additional component which computes (or simply picks) one specific value out of the set of candidate values. This is the role of the aggregation function.
Definition 11
An acceptance program (over \(\varSigma \)) is a pair \(\langle {{AG},R} \rangle \), where \({AG}:2^V\rightarrow V\) is a value aggregation function and R is a set of acceptance rules. is ground if all of its acceptance rules are ground. \(\varPi _{\varSigma }\) is the set of acceptance programs over \(\varSigma \).
One obvious choice for aggregation functions are functions which, given some total order on V, pick the maximal, respectively the minimal element among the candidates. We will see specific examples in Sect. 4.
Definition 12
Let V be a \(\mathbf {u} \)-free set of values. A V-valued GRAPPA instance over \(\varSigma \) is a tuple \(G =\langle {S,E,L,\lambda ,\pi } \rangle \) where S, E, L and \(\lambda \) are defined as for a V-LAG and \(\pi \) is a function \(\pi :S\rightarrow \varPi _{\varSigma }\) that assigns to each statement \(s\in S\) an acceptance program over \(\varSigma \). G is ground if \(\pi (s)\) is ground for all \(s\in S\).
In order to handle variables, we use variable substitutions as defined next.
Definition 13
Let \(G =\langle {S,E,L,\lambda ,\pi } \rangle \) be a V-valued GRAPPA instance. A variable substitution for G is a mapping \(\theta \) that assigns every label variable \({X_{L}}\), respectively every value variable \(\mathbf {X} \) that appears in some acceptance program \(\pi (s)\) (\(s\in S\)), a label from L, respectively, a value from V. For an acceptance rule r, \(r \theta \) denotes the acceptance rule obtained by replacing every variable v in r by \(\theta (v)\). The set of variable substitutions for G is denoted by \(\varTheta _G\).
The grounding of G is given by \(\textit{gr}(G) =\langle {S,E,L,\lambda ,\pi '} \rangle \) such that for every \(s\in S\), \(\pi '(s)=\langle {{AG},\{r \theta \mid r\in R, \theta \in \varTheta _G\}} \rangle \), where \(\pi (s)=\langle {{AG},R} \rangle \).
3.2 Semantics
In the following, we assume a GRAPPA signature \(\varSigma = (V,L,Var_V,Var_L,F,Rel)\) where each \(f \in F\) with arity k comes with an associated function \(\hat{f}:(L\cup V\cup \mathbb {R})^k\rightarrow V\). Similarly, each relation symbol \(r \in Rel\) has an associated binary relation \(\hat{r}\) on V.
As a first step we need to define the value assigned to a given multi-set by an acceptance program. Since the semantics of non-ground GRAPPA instances is defined in terms of their groundings, we can restrict the following definitions of valuation to the ground case.
Definition 14
Let m be a multi-set \(m:V \times L \rightarrow \mathbb {N} \). The valuation of a ground value expression (over \(\varSigma \)) is given by
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\(val^m({\mathbf {x}})={\mathbf {x}}\) for \({\mathbf {x}}\in V\)
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\(val^m(f({\mathsf {e}}_1,\dots ,{\mathsf {e}}_n)) = \hat{f}(val^m({\mathsf {e}}_1),\dots ,val^m({\mathsf {e}}_n))\)
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\(val^m(min_{W,\preccurlyeq }(l)) = min_{\hat{\preccurlyeq }}\{ val^m({\mathsf {v}}) \mid {\mathsf {v}}\in W, m(\langle {val^m({\mathsf {v}}),l} \rangle ) > 0, l\in L \}\)
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\(val^m(max_{W,\preccurlyeq }(l)) = max_{\hat{\preccurlyeq }}\{ val^m({\mathsf {v}}) \mid {\mathsf {v}}\in W, m(\langle {val^m({\mathsf {v}}),l} \rangle ) > 0, l\in L \}\) Footnote 4
Naturally, value constants are interpreted by themselves and function symbols by the application of their associated functions on their evaluated arguments. Intuitively, \(min_{W,\preccurlyeq }(l)\) extracts the \(\hat{\preccurlyeq }\)-minimal value of all l-parents that evaluates to a value from W. The intuition of the max-case is analogous.
Example 3
Let \(V=[\mathbf {0};\mathbf {1} ]\), \(L=\{+,-\}\), and \(m=[\langle {0.1,+} \rangle ,\langle {0.3,+} \rangle ,\langle {0.4,-} \rangle ,\langle {0.7,+} \rangle )]\). Then, the valuation of the the value expression \(max_{[\mathbf {0};\mathbf {0.5} ],\le }(+)\), where \(\le \) is the relation symbol for the natural order, is given by \(val^m(max_{[\mathbf {0};\mathbf {0.5} ],\le }(+))= max\{\mathbf {0.1},\mathbf {0.3} \}=\mathbf {0.3} \).
Definition 15
Let m be a multi-set \(m:V \times L \rightarrow \mathbb {N} \). The valuation of a ground numeric expression (over \(\varSigma \)) is given by
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\(val^m(c)=c\) for \(c\in \mathbb {R} \)
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\(val^m(\#l_W) = \sum _{{\mathsf {v}}\in W} m(\langle {val^m({\mathsf {v}}),l} \rangle )\)
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\(val^m(sum_{W}) = \sum _{{\mathsf {v}}\in W, l \in L} m(\langle {val^m({\mathsf {v}}),l} \rangle )\)
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\(val^m(count_{W}) = | \{ l \mid {\mathsf {v}}\in W, m(\langle {val^m({\mathsf {v}}),l} \rangle )>0\}|\)
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\(val^m(min_{W}) = min\{ l \in L \mid m(\langle {val^m({\mathsf {v}}),l} \rangle ) > 0, {\mathsf {v}}\in W \}\)
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\(val^m(max_{W}) = max\{ l \in L \mid m(\langle {val^m({\mathsf {v}}),l} \rangle ) > 0, {\mathsf {v}}\in W \}\) Footnote 5
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\(val^m(o \oplus p) = val^m(o)\oplus val^m(p)\) where \(\oplus \in \left\{ +,*\right\} \)
Notice the semantic differences of the expressions \(\#l_W\), \(sum_{W}\), and \(count_{W}\): \(\#l_W\) counts the number of l-parents that evaluate to values from W. In contrast, \(sum_{W}\) and \(count_{W}\) are not dependent on a label. The expression \(sum_{W}\) returns the number of all parents that evaluate to values from W and \(count_{W}\) counts the number of different labels of parents that evaluate to a value from W. The expressions \(min_{W}\) and \(max_{W}\) give the minimal, respectively maximal, label to a parent that evaluates to a value from W.
Definition 16
Let m be a multi-set \(m:V \times L \rightarrow \mathbb {N} \), where n and o numeric expressions, \(\mathbf {v} \) and \(\mathbf {w} \) value expressions, and a and b Boolean expressions. Function \(val^m\) maps Boolean expressions to \({\mathbf {t}}\) or \({\mathbf {f}}\). In particular, the valuation of a ground Boolean expression (over V and L) is given by
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\(val^m(n \sim o)= {\mathbf {t}}\) iff \(val^m(n) \sim val^m(o)\), where \(\sim \,\,\in \left\{ <,\le ,=,\ge ,>,\ne \right\} \)
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\(val^m(\mathbf {v} \sim \mathbf {w})={\mathbf {t}}\) iff \(val^m(\mathbf {v}) \hat{\sim } val^m(\mathbf {w})\) with \(\sim \,\, \in Rel\)
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\(val^m(\bot )={\mathbf {f}}\)
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\(val^m(\top )={\mathbf {t}}\)
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\(val^m(a \wedge b)={\mathbf {t}}\) iff \(val^m(a)={\mathbf {t}}\) and \(val^m(b)={\mathbf {t}}\)
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\(val^m(a \vee b)={\mathbf {t}}\) iff \(val^m(a)={\mathbf {t}}\) or \(val^m(b)={\mathbf {t}}\)
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\(val^m(a \rightarrow b)={\mathbf {t}}\) iff \(val^m(a)={\mathbf {f}}\) or \(val^m(b)={\mathbf {t}}\)
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\(val^m(\lnot a)={\mathbf {t}}\) iff \(val^m(a)={\mathbf {f}}\)
Definition 17
Let m be a multi-set \(m:V \times L \rightarrow \mathbb {N} \). For a ground acceptance program \(\pi =\langle {{AG},R} \rangle \in \varPi _{\varSigma }\), its valuation is defined as
With these definitions we can reformulate the definition of the characteristic operator taking into account that acceptance functions are represented by acceptance programs:
Definition 18
(Characteristic Operator \(\varGamma ^{V}_G\)). Let V be a \(\mathbf {u} \)-free set of values, \(v: S \rightarrow V_\mathbf {u} \) an interpretation, and G a V-valued GRAPPA instance with \(\textit{gr}(G) =\langle {S,E,L,\lambda ,\pi } \rangle \). Applying \(\varGamma ^{V}_G\) to v yields a new interpretation (the consensus over \({[v]_c} \)) defined as
where \(\textstyle \bigsqcap _i\) denotes the greatest lower bound in \((V_\mathbf {u},\le _i)\).
The semantics of a GRAPPA instance is then defined analogously to that of a V-LAG (see Definition 6). For the case of stable semantics, also a v, W-reduct of a GRAPPA instance can be defined in a similar fashion as for a V-LAG, using the same construction of multi-set \(m'\) as in Definition 7.
4 Examples
In this section we illustrate our approach with a number of examples. We will make use of aggregation functions of the form provided next. Note that in case the rules provide no candidates at all, a certain default value needs to be picked. For some total order \(\trianglelefteq \) on V, we use
-
\(max_\trianglelefteq (V')\), where \(max_\trianglelefteq (V')={\mathbf {v}}\) when \({\mathbf {v}}\) is the \(\trianglelefteq \)-maximal element of \(V'\) if \(V'\ne \emptyset \), and the \(\trianglelefteq \)-minimal element of V, otherwise,
-
\(min_\trianglelefteq (V')\), where \(max_\trianglelefteq (V')={\mathbf {v}}\) where \({\mathbf {v}}\) is the \(\trianglelefteq \)-minimal element of \(V'\) if \(V'\ne \emptyset \), and the \(\trianglelefteq \)-maximal element of V, otherwise, and
-
\(\varSigma (V')\), where V is numeric and \(\varSigma (V')=\varSigma _{v\in V'}v\) returns the sum of elements of \(V'\).
We start with examples where acceptance programs evaluate to statically chosen values. Recall that a parent \(n'\) of a node n is an l-parent if the link \((n',n)\) is labelled with l.
Example 4
Let \(L=\left\{ +,-\right\} , V=\left\{ \mathbf {yes},\mathbf {no},\mathbf {maybe} \right\} \) and assume we want to assign \(\mathbf {yes} \) to a node if (a) more ‘\(+\)’-parents have value \(\mathbf {yes} \) than ‘−’-parents or (b) the same number of ‘\(+\)’- and ‘−’-parents have value \(\mathbf {yes} \) but more ‘\(+\)’-parents have value \(\mathbf {maybe} \) than ‘−’-parents. In addition, we assign \(\mathbf {maybe} \) to the node if (a) the same number of ‘\(+\)’- and ‘−’-parents have value \(\mathbf {yes} \) and the same number of ‘\(+\)’- and ‘−’-parents have value \(\mathbf {maybe} \). We assign \(\mathbf {no} \) to the node in all other cases.
Using the aggregation function \({AG} =min_{(\mathbf {yes} \trianglelefteq \mathbf {maybe} \trianglelefteq \mathbf {no})}\) the following set of rules R produces the intended values:
Note that \(\mathbf {no} \) is always among the values derived from these rules. The aggregation function guarantees that nevertheless the right value is chosen, in case one of the other rules fire.
Consider the V-valued GRAPPA instance \(\langle S,E,L,\lambda ,\pi \rangle \), given in Fig. 2, with \(S=\{s_1,s_2\}\), \(E=\{(s_2,s_1)\}\), \(L=\{+,-\}\), \(\lambda ((s_2,s_1))=\,+\), and \(\pi \) assigning above acceptance program \(({AG},R)\) to both \(s_1\) and \(s_2\).
Admissible semantics yield interpretations \(\{ s_1\mapsto \mathbf {u},s_2\mapsto \mathbf {u} \}\), \(\{s_1\mapsto \mathbf {u},s_2\mapsto \mathbf {maybe} \}\), \(\{ s_1\mapsto \mathbf {yes},s_2\mapsto \mathbf {u} \}\), and \(\{ s_1\mapsto \mathbf {yes},s_2\mapsto \mathbf {maybe} \}\). For the complete, grounded, preferred, and model semantics, we obtain a single interpretation \(\{ s_1\mapsto \mathbf {yes},s_2\mapsto \mathbf {maybe} \}\): intuitively, \(s_2\) maps to \(\mathbf {maybe} \) because it has no parents and thus the second rule of the program applies. \(s_1\) maps to \(\mathbf {yes} \) because there is neither a ‘\(+\)’-parent nor a ‘−’-parent with value \(\mathbf {yes} \), but one ‘\(+\)’-parent (namely \(s_2\)) with value \(\mathbf {maybe} \) but no such ‘−’-parent.
Example 5
In the next example we use the unit interval as set of values \(V=[\mathbf {0};\mathbf {1} ]\) in order to exemplarily illustrate terms from the value domain of form \(max_{W,\le }(l)\): the term selects the maximal truth value from W (with respect to the natural ordering \(\le \) over reals) appearing in the multi-set for label l.
Moreover, let the labels be \(L=\left\{ -,+,++\right\} \). We want to express the following conditions (1) Parent nodes with value at least 0.5 and connected via edges with labels \(++\) can veto for full acceptance (i. e., assigned value \(\mathbf {1} \)). (2) Nodes with some greater ‘\(+\)’-parent value than ‘−’-parent values get \(\mathbf {0.75} \). (3) Nodes with equal maximal ‘\(+\)’ and ‘−’-parent values get \(\mathbf {0.6} \) if there are more nodes with \(+\) than − that have this maximal value (here we use \(max_{W,\le }(l)\) for indexing a \(\#\)-based term). (4) Nodes with equal maximal ‘\(+\)’ and ‘−’-parent values get \(\mathbf {0.5} \) if there are equally many nodes with \(+\) and − that have this maximal value. (5) Nodes with equal maximal ‘\(+\)’ and ‘−’-parent values get \(\mathbf {0.4} \) if there are less nodes with \(+\) than − that have this maximal value. (6) Otherwise, we assign value \(\mathbf {0} \). For the aggregation function \({AG} \) we use \(max_{\le }\) and specify the conditions via the following acceptance rules
So far, we have provided rules that statically assign truth values (i. e., the rule heads have been given via concrete values). However, our language allows arbitrary value expressions in rule heads. The following example gives a simple application of that.
Example 6
Let \(L=\left\{ +\right\} \), \(V=[\mathbf {0};\mathbf {1} ]\) and consider an acceptance rule of the form
which simply states that, if there is some parent-node, assign the maximal value of all parents to the current node. Note that in this case, any aggregation function \({AG} \) that satisfies \({AG} (\{x\})=x\) can be chosen; however, the default-value of \({AG} \) is crucial here for nodes with no parents. Consider a V-valued GRAPPA instance with two nodes \(S=\{s_1,s_2\}\) and two edges \(E=\{(s_1,s_2),(s_2,s_2)\}\), as illustrated in Fig. 3, including a self-loop for \(s_2\). Both nodes use the acceptance rule shown above.
First assume both nodes use aggregation function \({AG} _1=max_\le \). As \(s_1\) has no parent, the rule body will never fire in the evaluation of \(s_1\). As a consequence \({AG} _1\) will be applied on the empty set, yielding value \(\mathbf {0} \) for \(s_1\) under every interpretation. Node \(s_2\) has two parents, \(s_1\) and itself, therefore the rule will fire and \(s_2\) can take any value from \([\mathbf {0};\mathbf {1} ]\).
Now assume both nodes use aggregation function \({AG} _2=min_\le \). Also here, \({AG} _2\) will be applied on the empty set for node \(s_1\), this time yielding value \(\mathbf {1} \) for \(s_1\) under every interpretation. The rule will fire for \(s_2\) and its head will evaluate to the maximal value of each ‘\(+\)’-parent, i.e., to \(\mathbf {1} \) as \(s_1\) is assigned \(\mathbf {1} \).
Example 7
Our final example shows that acceptance programs can also be used to specify propagation of values throughout a network. To this end, we assume that labels and values are both numbers, i.e. \(L=V=\mathbb {N} \). Suppose we want nodes to be assigned the weighted sum of the values of parent nodes, where labels are used as weight factors. We provide an acceptance program \(\langle AG,\{r\}\rangle \) with a single non-ground rule
and AG being the summation function \(\varSigma \). Note that multiplication \(*\) here is formally a value function, \(\mathbf {X} \) a value variable and \({X_{L}}\) a label variable. Let us consider the graph given in Fig. 4 and apply \(\langle AG,\{r\}\rangle \) to node s.
Consider interpretation v with \(v(p_1)=v(p_2)=\mathbf {1} \), \(v(p_3)=\mathbf {2} \), and \(v(p_4)=\mathbf {0} \). The relevant parts of the the grounding of r are provided by substituting \(\mathbf {X} \) by 1, 2, and resp. 0 and \({X_{L}}\) by 3 and 5.
The evaluation of the \(\#_W(\cdot )\) value expression under v is obtained via multi-set
Thus, the value assigned to s is thus determined by the first and the third rule of the grounding rules shown above (all others have factor 0) which evaluate to the sum of \(\mathbf {12} \).
5 Conclusions
In this paper we have successfully combined two recent extensions of ADFs, namely the extension to the multi-valued case where the user can pick the set of truth values which is best-suited for a particular application, and the GRAPPA approach which defines operator-based argumentation semantics directly on top of arbitrary labelled argument graphs. We believe this combination is highly useful for the following reasons. First of all, it is important to come up with knowledge representation formalisms that provide means to express the relevant information in a way that is as user friendly as possible - without giving up precisely defined formal semantics. We believe graphical representation are particularly well-suited to play this role, and our operator-based semantics turns graphs into full-fledged knowledge representation formalisms. On the other hand, it is important to provide enough flexibility for more fine-grained distinctions than possible with only two truth values. Multi-valued GRAPPA combines these features and thus, as we believe, is a useful tool for argumentation.
The challenge posed by the combination aimed for in this paper was not so much the generalization of the relevant definitions underlying the operator-based semantics. It was the identification of an adequate representation of the acceptance functions. We believe acceptance programs are a sufficiently flexible yet manageable means to this end. Acceptance programs may seem at odds with the requirement of user-friendliness discussed above. This is to a large extent due to the generality of our approach which allows us to handle arbitrary sets of values. We expect that for specific sets of values simpler representations - and maybe a small number of useful predefined functions - can be identified. This is a topic of future research.
Notes
- 1.
- 2.
We call a parent \(n'\) of a node n l-parent if the link \((n',n)\) is labelled with l.
- 3.
We will often leave \(\varSigma \) implicit in definitions from now on.
- 4.
The values \(val^m(max_{W,\preccurlyeq }(l))\) and \(val^m(min_{W,\preccurlyeq }(l))\) are only defined when \(\hat{\preccurlyeq }\) is an order that has a maximal, respectively, minimal, element for every subset of V. The expressions \(max_{W,\preccurlyeq }(l)\) and \(min_{W,\preccurlyeq }(l)\) may only be used when this is the case.
- 5.
The use of \(min_{W}\) and \(max_{W}\) is restricted to settings where the label domain L is numeric.
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Brewka, G., Pührer, J., Woltran, S. (2019). Multi-valued GRAPPA. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_6
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