Equilibrium States in Numerical Argumentation Networks

08/28/2014
by   D. Gabbay, et al.
King's College London
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Given an argumentation network with initial values to the arguments, we look for algorithms which can yield extensions compatible with such initial values. We find that the best way of tackling this problem is to offer an iteration formula that takes the initial values and the attack relation and iterates a sequence of intermediate values that eventually converges leading to an extension. The properties surrounding the application of the iteration formula and its connection with other numerical and non-numerical techniques proposed by others are thoroughly investigated in this paper.

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1 Orientation and Background

1.1 Orientation

A finite system , with a binary relation on , can be viewed in many different ways; among them are

  1. As an abstract argumentation framework [10], and

  2. As a generator of equations [13, 14]

When viewed as an abstract argumentation framework, the basic concepts studied are those of extensions (being certain subsets of ) and different semantics (being sets of extensions). When studied as generators of equations, one can generate equations in such a way that the solutions to the equations correspond to (complete) extensions and sets of such solutions correspond to semantics.

This paper offers an iteration schema for finding specific solutions to the equations responding to initial requirements and shows what these solutions correspond to in the abstract argumentation sense.

We now explain the role iteration formulas play in general in the equational context.

When we have a system of equations designed to model an application area111For example, equations of fluid flow in hydrodynamics or equations of particle motion in mechanics, or equations modelling argumentation networks according to the equational approach (to be explained later), or equations modelling a biological system of predator-prey ecology, or some polynomial equation arising in macroeconomics. we face two problems: 1) find any solution to the system of equations, which will have a meaning in the application area giving rise to the equations; 2) given boundary conditions and/or other requirements not necessarily mathematical which are meaningful in the application area,222For example, initial conditions in the case of particle mechanics, or initial size of population in the ecology, or arguments that we would like to be accepted. we would like to find a solution to the system of equations that is compatible/respects the initial conditions/requirements.

These two problems are distinct. The first one of finding any solution is a numerical analysis problem. There are various iteration methods in numerical analysis to find solutions, of which one of the most known is Newton’s method.333This method starts with an initial guess of a possible solution and uses various iteration formulae hoping that it will converge to a solution (for an introduction on numerical analysis see [21]). The second problem is totally different. It calls for an understanding of the requirements coming from the application area and possibly the design of a specialised iteration formula which respects the type of requirements involved.

This paper provides the Gabbay-Rodrigues Iteration Schema, for the case of the equational approach to argumentation, seeking solutions (which we shall see will correspond to complete extensions) respecting as much as possible initial demands and restrictions of what arguments are in or out of the extension. We compare what our iteration schema does with Caminada and Pigozzi’s down-admissible and up-complete constructions [7]. Because we are dealing with iteration formulas (involving limits) and we are comparing with set theoretical operations (as in Caminada and Pigozzi’s paper) we have to be detailed and precise and despite it being conceptually clear and simple, the proofs turn out to be mathematically involved, and require some patience from our readers. However, once we establish the properties of our iteration schema, its use and application are straightforward and computationally simple, especially in the context of such tools as MATHEMATICA and others like it. The reader may wish to just glance at the technical proofs and concentrate on the examples and discussions. Note the iteration schema idea is very general and applies to other systems of equations possibly using other iteration formulas.

The actual technical development of the paper will start in Section 2. In Appendix A we emphasise the distinction between the above two problems with two detailed examples, the first modelling the dynamics of predator-prey interactions and the second about merging/voting in argumentation networks. We shall see that Newton’s method does not work in these scenarios and that there is the need for a new type of iteration schema. Thus this paper is not just incremental to the equational approach but constitutes a serious and necessary conceptual extension.

1.2 Background

An abstract argumentation framework is a formalism proposed by Dung [10] and defined in terms of a tuple , where is a non-empty set of arguments and is a binary attack relation. We will refer to an abstract argumentation framework simply as an argumentation network. If , we say that the argument attacks the argument . can be seen as a directed graph (see Figure 1). As informally introduced in Section 1, will be used to denote the set , i.e., the set of arguments attacking the argument . Following graph theory convention, if has no attackers (i.e., ), we say that is a source node in . Given a set , we write as a shorthand for , such that . Furthermore, following [4], we use to denote the set .

Figure 1: A sample argumentation network.

Given an argumentation network, one usually wants to reason about the status of its arguments, i.e., whether an argument persists or is defeated by other arguments. It should be clear that arguments that have no attacks on them always persist. However, an attack from to may not in itself be sufficient to defeat , because may be defeated by some argument that attacks it, and thus one needs an evaluation process to determine the status of all arguments systematically. In Dung’s original formulation, this was done through an acceptability semantics defining conditions for the acceptability of an argument. The semantics can be defined in terms of extensions — subsets of with special properties. These subsets are based on two fundamental notions which are explained next.

A set is said to be conflict-free if for all elements , we have that . Intuitively, arguments of a conflict-free set do not attack each other. However, this does not necessarily mean that all arguments in the set are properly supported. Well supported sets satisfy special admissibility criteria. We say that an argument is acceptable with respect to , if for all , such that , there is an element , such that . A set is admissible if it is conflict-free and all of its elements are acceptable with respect to itself. An admissible set is a complete extension if and only if contains all arguments which are acceptable with respect to itself. is called a preferred extension of , if and only if is maximal with respect to set inclusion amongst all complete extensions of . Similarly, is called a stable extension of if and only if is conflict-free and for every , there is an element , such that .

Figure 2: Sample argumentation networks.

Now consider the argumentation networks (L) and (R) depicted in Figure 2. According to the semantics given above, the network (L) has three extensions , and . Both and are preferred and stable extensions. The network (R) only has only one extension, which is empty, and hence this is also its only preferred extension. This extension is however not stable.

Besides Dung’s acceptability semantics, it is also possible to give meaning to these networks through Caminada’s labelling semantics [6, 5] and through Gabbay’s equational approach [13, 14]. These are explained next.

The labelling semantics.

The labelling semantics uses labelling functions satisfying certain conditions tailored so as to obtain a complete correspondence with Dung’s semantics.

The labelling of an argument in disagreement with Dung’s semantics is said to be “illegal”. This is explained further as follows.

Definition 1.1 (Illegal labelling of an argument [7])

Let be an argumentation network and a labelling function for .

  1. An argument is illegally labelled in by if and there exists such that .

  2. An argument is illegally labelled out by if and there is no such that .

  3. An argument is illegally labelled und by if and either for all , or there exists , such that .

A legal (complete) labelling is a labelling in which no argument is illegally labelled.

It is possible to have more than one legal labelling function for the same argumentation network. Each labelling function will correspond to an extension in Dung’s semantics. For example, for network (L), we have the three functions , and below.

in out und
out in und

For the network (R), we have only the function such that . This gives the empty extension.

The equational approach.

The equational approach views an argumentation network as a mathematical graph generating equations for functions in the unit interval . Any solution to these equations conceptually corresponds to an extension. Of course, the end result depends on how the equations are generated and we can get different solutions for different equations. Once the equations are fixed, the totality of the solutions to the system of equations is viewed as the totality of extensions via an appropriate mapping. One equation schema we can possibly use for generating equations is the  below, where is the value of a node :

()

Another possibility is :

()

It is easy to see that according to  the value of any source argument will be (since they have no attackers) and the value of any argument with an attacker with value will be . The situation is more complex with nodes participating in cycles. Consider the network (L) again, with equations

If values are taken from the unit interval, this system of equations will accept any solution such that . We can divide these solutions between three classes: , ; , and , with . These again correspond to the three extensions , and given before.

In fact, Gabbay has shown that in the case of the totality of solutions to the system of equations corresponds to the totality of extensions in Dung’s sense [14]. The correspondence is best explained in terms of the labelling semantics, using the following correspondence:

in
out
und

The advantage of the equational approach is that it allows us to think of an argumentation network as a numeric system in which nodes are given certain values depending on specific rules governing their interaction with their neighbours. A rule may for instance require the value of a node to be if the value of any attacking node is . Another rule may force the value of a node to be if it has no attacking nodes. The schema and embed these rules, and they agree with Dung’s semantics. A solution to the system of equations is any combination of values of nodes satisfying the equations. Of course, since the node values are no longer discrete we have more freedom to design rules which are appropriate for a given application. Part of the objective of this paper is to explore the nature of these rules.

We start by generalising some concepts a bit further. Consider the network in Figure 3 in which . To agree with Dung’s semantics, if the value of any attacker of is , we want the value of to be . If all of the attackers of have value , we want the value of to be . For any other combination of values of the attackers we want the value of to be anything other than or . So within the traditional semantics but taking the extended set of values of the unit interval, we can think of a single attack by a node with value as the order-reversing operation which returns the value . This is a kind of negation.444If we make und equals , then an attack by a single undecided node will have value . Since a node can have multiple attacks, we also need an operation to combine the values of the attackers. We can think of this as a type of conjunction, which numerically can be obtained through several operations. For instance, in fuzzy logic, the standard semantics of (weak) conjunction is given by the operation .

Figure 3: Multiple attacks on a node.

Therefore, the value of a node can be defined as

which is equivalent to

obtained by our now familiar schema . Note that the conjunction operation in the schema  is product. The operations and product are two examples of t-norms. They are two instances of functions that are particularly suitable for argumentation semantics. The following definition elaborates on this further.

Definition 1.2

A function with domain being the family of all finite sequences of elements from and range is argumentation-friendly if satisfies the following conditions.

  • 555The values of for any sequence containing the value is the same as the value of for the subsequence without the .

  • if and only if

  • if and only if for every

  • is continuous as a multi-variable function666In fact, this condition is only needed to guarantee the existence of solutions to the equations.

Example 1.1

Below are some examples of argumentation-friendly functions:

  1. , for any satisfying (T1)–(T6).

Later on, we will see that argumentation-friendly functions will be used both to calculate aggregation of attacks as well as for combining the value of attacks with initial values. However, as we mentioned attack is a type of negation and hence when operating on the attack of a node with value , we will consider the complement of to , i.e., .

Notice that t-norms satisfy conditions (T1)–(T4) above.

Definition 1.3

For any assignment of values define the sets and .

Theorem 1.1

Let be a network, an argumentation-friendly function, and a system of equations written for , where for each node , . Take any solution to , it follows that is a complete extension.

Proof. Suppose that is not conflict-free. Then there are , such that . Since , then . But and , and hence . It then follows by (T4) that and hence , a contradiction.

Now suppose that . We show that for all there exists , such that . If , then and then by (T5) it follows that , for all and hence for all . Take any such . Since , we have by (T4) that for some , . It then follows that .

Theorem 1.2

Let be a network, an argumentation-friendly function, and a system of equations written for , where for each node , . Then for every preferred extension of , there exists a solution to such that

  • If , then

  • If , then

  • If , then

Proof. Let us start by partitioning the set using into three sets , , and . Note that the elements of are the undecided elements in with respect to . Each element of is not attacked by any element of and its attackers cannot all come from , i.e., at least one attacker comes from itself. Consider the argumentation network . Write a system of equations using for . For each , the equation is

By Brouwer’s theorem, the above equations have a solution .777The Euclidean version of the theorem states that if is a real-valued function, defined and continuous on a bounded closed interval of the real line where , for all , then has a fixed-point. In our case, there are variables in the network

, which we can associate with the vector

. We can then see each equation as , where is a continuous function on the -dimensional space (see Theorem 1.2 in [21]). To be clear is defined on , giving values , such that for every ,

We are seeking however a solution defined for all of , which satisfies the system of equations for :

Furthermore, we want to be such that for , , for and for . We now define such a solution . Let

, for all
, for all
, for all

We have to show now that indeed solves the system of equations for . Take :

Case 1: . We defined . We need to show that . Since , then all of its attackers are in , and then (by definition), for all . Therefore, , by (T5).

Case 2: . We defined . We need to show that . Since , then there exists , such that . By definition, , and then , by (T4).

case 3: . We defined . We need to show that . We noted above, that implies that none of its attackers belong to and therefore any remaining attackers not in must be in . By definition, , therefore and by (T2), such values can be safely deleted in the calculation of . Therefore, deleting all such values will show that .

Having shown that above solves the system of equations , we can use Theorem 1.1 to show that is a complete extension. We now ask whether any of the values , for can be or . The answer is no, for if for any , then and then , which is impossible, since is a preferred extension. Analogously, we can only get for some , if for some of its attackers , , which as we mentioned is impossible. This completed the proof.

The condition of preferred extension of the Theorem 1.2 is necessary, as shown in the example below.

Example 1.2

Consider the complete extension of the network below. is not preferred, since is a proper subset of .

The network generates the following equations.

(1)
(2)
(3)
(4)

Since , we get that and these values satisfy equations (1) and (2) above. However, replacing (3) in (4) gives us

If is product, this gives us , and hence , and hence , and therefore no solution corresponding to using exists. Note that the two preferred extensions and include . No extension can include .

However, with as , we have that (4) becomes

and for this set of equations, the values , , form a solution corresponding to .

The loop in the example above is quite elucidating. Let us analyse it in some more detail.

Example 1.3

Consider the network with a single self-referencing loop below.

The network generates the equation:

Notice that and hence we have that , whatever the function is, as long as it satisfies (T1)–(T5).

Note that satisfies (T1)–(T4). As a result, we have that:

Corollary 1.1

Let be a network and a system of equations written for , where for each node , . Take any solution to . It follows that is a complete extension.

This follows from Theorem 1.1. What it means is that any solution to the system of equations defined in terms of  can be translated into a complete extension simply by defining that extension as the set containing the nodes whose solution values are . Obviously, different solutions will give rise to different extensions.

Proposition 1.1

Let be a network and a system of equations written for , where for each node , . Then for every complete extension of , there exists a solution to satisfying:

  • If , then .

  • If , then .

  • If , then .

Proof. Let be a complete extension. Consider the following assignment of values to the nodes in :

  • if , then

  • if , then

  • , otherwise

We now show that the values above form a solution to the system of equations . As in Theorem 1.2, replacing the above values in the original system of equations will reduce them to the following types.

We have seen that and since , (1) is satisfied. Similarly, and since , so is (2). Notice that the image of is . All values in are greater than , but at least one of them is , therefore , and hence the above assignment solves the equations.

So far, we have shown the basics of the equational numerical approach to abstract argumentation frameworks. In the next section we consider two additional developments that follow naturally. Firstly, we know that solutions do exist to the system of equations, but can we find them using some numerical method? For example, by applying iterations given some initial guess?888As can be done to find the square root of numbers using Newton’s method. Secondly, we would like to apply our methodology to questions of merging, voting, or any other application where a set of initial values emerges and needs to be transformed to the “closest” extension. How can we do that? The following section provides a method to answer these questions.

2 The Gabbay-Rodrigues Iteration Schema

Suppose we are given initial values which do not correspond to any extension in the way that we presented them in the previous section. These values may come attached to the nodes for different reasons. For instance, the arguments themselves may be expressed as some proof in a fuzzy logic and the initial values can represent the values of the conclusions of the proofs, or they can be obtained as the result of the merging of some networks, or they may come from some voting mechanism, etc. Whatever the reason, the initial values may or may not correspond to a complete extension in Dung’s sense and we seek a mechanism that would allow us to find the “best” possible extension corresponding to them.

Consider the equation :

()

 is satisfied when the value of the node is legal (in Caminada and Pigozzi’s terminology [7]). That is, if the value of is and the value of all of ’s attackers are ; or if the value of is and at least of one ’s attackers has value ; or if the value of and at least one of ’s attackers has value in and no attacker of has value . If we aim to correct the values of the nodes in a network iteratively, we need a mechanism that leaves legal in, out and und node values intact, changing illegal in or out values into und.999We will come to the correction of illegal und nodes later. To make a distinction between these classes of values, we will call the values in crisp and the values in undecided.

Now consider the following averaging function:

For legal assignments of values, we have three cases to consider:

  • is legally in. In this case and all of its attackers have value . We want the value of to remain . We have that:

  • is legally out. In this case and at least one of its attackers has value . We want the value of to remain . We have that:

  • is legally und. In this case , none of its attackers has value and at least one of its attackers has value greater than . This means that and therefore . Let and . It follows that and . We want the value of to remain undecided, although we are prepared to accept changes to its initial value as long as its final value remains in the interval . We have that:

    Notice that and , therefore . If , then and hence . If , then and . Therefore, . It then follows that and therefore the value of remains in .

What (L1)–(L3) above give us is that legal labellings are preserved.101010Legal undecided values may change, although they remain in the undecided range (by (L3)). Later on, we shall see that our iteration schema also eventually corrects all illegal values. It does so in two stages. In the first stage, all illegal crisp values are turned into undecided (this is done in iterations). In the second stage, all remaining illegal undecided values converge to whatever legal crisp values they should be, so that in the limit, all of the values in the sequence are legal. Therefore, the Gabbay-Rodrigues Iteration Schema introduced below provides a numerical iterative method to turn any initial illegal assignment of values to arguments into its closest legal assignment.111111The precise definition of “closest” will be made clear in Theorem 2.9.

Definition 2.1

Let be an argumentation network and be an assignment of values to the nodes in . The Gabbay-Rodrigues Iteration Schema is defined by the following system of equations , where for each node , the value is defined in terms of the values of the nodes in as follows:

We call the system of equations for using the above iteration schema its GR system of equations.

We ask whether we can regard the iteration schema above as an equation schema as in the previous section, i.e.,

(GR)

To further clarify this point, let us take an equation written with an argumentation-friendly function for a node in terms of its attackers. The equation would be

It is clear that if one of the attackers of is , the value of solves to , and if all the attackers of are , the value of will solve to . This follows from the properties (T1)–(T5) of an argumentation-friendly function. Now let us compare and see what happens when we use the formula above. If the value of one of the attackers of is , the first component of the sum will be , whereas the second component will be , because the equation is implicit, we have the equation

which solves to , which is correct. If the values of all attackers of are , then we get the equation

which solves to , which again gives a correct result. Otherwise, assume that the values of all attackers are either or , with at least one of them being . We get the equation

which again solves to the correct value of . By correct we mean that the results are exactly compatible with the Caminada labelling mentioned in Section 1, where means is in, means is out and means is und.

Therefore, the Gabbay-Rodrigues schema remains faithful to the spirit of Dung’s semantics captured through the legal Caminada labellings just as  does. Its advantage over  is that it can be used iteratively as we will show in the rest of this section. 121212As an equation, we can regard the expression  (GR) just as another type of , a special .

We start by showing some properties of the schema. The first one ensures that the values of all nodes remain in the unit interval in all iterations.

Proposition 2.1

Let be an argumentation network and an assignment of initial values to the nodes in . Let each assignment , , be calculated by the Gabbay-Rodrigues Iteration Schema for . It follows that , for all and all .

Proof. The base of the induction is the initial value assignment that holds trivially. The induction step is proven by looking at the maximum and minimum values that the nodes can take and showing that the sum in the iterated schema is always a number in . Now, suppose that indeed for all nodes , , for a given iteration . Pick any node . It follows that

So we have that , where , , , and .

The lowest value for is obtained with the lowest values for and , when we get that . If , then . If , then we get . The highest value for is obtained with the highest values for and , when we get that . If , then . If , then we get . In all cases, .

We now show that a given “legal” set of initial values for the nodes in satisfies the equations and hence the values remain unchanged.

Proposition 2.2

Let be a network and its GR system of equations. Then for every complete extension of and all , if is defined using by the clauses (C1)–(C3) below, we have that .

  • If , then

  • If , then

  • If and , then

Proof. Let be a complete extension and suppose . Then and hence, i) either , or ii) for all , (since is admissible). As a result, , and hence we have that

If on the other hand, , then . Therefore, there exists some , such that and hence . It follows that

Finally, if , then and . We must have that for all , (otherwise, we would have that ). We must also have that for some , , otherwise would defend and since it is complete , but then . Therefore, , and hence we have that

Obviously, if for all nodes , as above, then for all nodes , , for all .

Furthermore, crisp values do not “swap” between each other and undecided values do not become crisp:

Theorem 2.1

Let be an argumentation network, a system of equations for using the Gabbay-Rodrigues Iteration Schema, and an assignment of initial values to the nodes in . Let , , , … be a sequence of value assignments where each , , is generated by . Then the following properties hold for all and for all

  1. If , then .

  2. If , then .

  3. If , then .

Proof.

  1. Suppose , then .

  2. Suppose , then .

  3. Suppose . We first show that . Note that . Therefore, we have that

    It is easy to see that the first component of the above sum is greater than or equal to , whereas the second is strictly greater than , and hence .

    Since we start with values in , Proposition 2.1, gives us that , for all . We therefore only need to show that . Again we have that