0.1 Background and orientation
Our starting point is the important papers of Baroni, Giacomin and Guida on odd loops and SCC recursiveness [1, 6]. In their papers the authors offer the CF2 semantics in response to difficulties arising from the Dung semantics handling of odd and even loops. In our paper we outline our equational approach to argumentation networks and show how the CF2 semantics can be obtained from perturbations to the equations associated with the networks. This approach will offer additional methodological support for the CF2 semantics, while at the same time show the power of the equational approach. We offer our own loopbusting equational semantics LB, which includes CF2 as a special case.
The structure of this paper is as follows. Section 2 reproduces the motivating discussion from [1] for the CF2 semantics and points out its weaknesses. Section 3 introduces the equational semantics. Section 4 defines our loop busting semantics LB. Section 5 introduces our semantics LB2 and compares with CF2 on the technical level. We conclude with a general discussion in Section 6.
0.2 CF2 semantics as introduced in the SCC paper [1]
Baroni et al. devote a long discussion about the inadequacy of the traditional semantics in handling odd and even loops. They say, and I quote:
“the length of the leftmost cycle should not affect the justification states [of an argument]. More generally, it is counterintuitive that different results in conceptually similar situations depend on the length of the cycle. Symmetry reasons suggest that all cycles should be treated equally and should yield the same results.”
We now reproduce Figure 8 of [1], and discuss the problems associated with it.
The only preferred extension for Figure 1(a) is , while for Figure 1(b) we have the extensions and . These two results are conceptually different, in (a) is not prevented from being justified while in (b) it is prevented.
A more striking problem is the one outlined in Figure 9 of Baroni et al. [1], here reproduced as Figure 2.
The only extension in traditional Dung semantics is all undecided. Common sense, however, expects to be out and to be in. In [5], this is characterised as “one of the main unsolved problems in argumentationbased semantics”.
The CF2 semantics of [1] treats the loops of Figures 1(a) and 1(b) and 2 all in the same way, by taking as CF2 extensions maximal conflictfree sets. We therefore get for Figure 1(a) the CF2 extensions
and for Figure 1(b) we get
and for Figure 2 we get the extensions
Let us put forward a figure of our own, Figure 3. This is a 9 point cycle. The CF2 semantics will take all maximal conflict free subsets as extensions, including among them and its cyclic translations as well as and its cyclic translations (e.g. , etc.).
We shall see later that some of our loop busting semantics LB yield only and its cyclic translations and not , but other LB semantics does yield it.
We agree with [1] on the need for a new approach but we feel that the CF2 semantics offered as a solution requires further independent methodological justification. The notion of conflict freeness is a neutral notion and does not use the central notion of “attack” of the Dung semantics. When we get a loop like , in a real life application as in Figure 1(a), there are good reasons for the loop in the context of the application area where it arises, and we want a decisive solution to the loop in terms of {in, out}, which makes sense in the application area. We do not want just a technical, nondecisive choice of maximal conflict free sets, a sort of compromise which involves no real decision making. Imagine we have a loop with , and we go to a judge and we expect some effective decision making. We hope for something like “I think is not serious”.
Taking the maximal conflict free sets in this case, namely and means nothing. We would perceive that the judge is not doing his job properly and that he is just offering us options which are obvious and noncontroversial, given the geometry of the loop! See [3] for extensive examples of resolving loops in a practical realistic way.
Another problem, in our opinion, with the CF2 semantics is that it is an overkill as far as loopbreaking is concerned. If we look at Figures 1(a) and 1(b) and replace them by Figure 4 and 5 our loopbreaking needs are the same according to [1], but in Figure 4, we do not need the extensions from the loopbreaking point of view. In our LB semantics we do not mind if there will be less extensions than CF2, in the odd cycle case of Figure 4 but insist that there will be the same extensions as the traditional Dung semantics in the even cycle of Figure 5. CF2 gives more extensions then the traditional Dung extensions for Figure 5.
We should realise that within the context of argumentation theory alone, the maximal conflict free CF2 solution seems somewhat arbitrary, a device which is just technically successful.
It is also the only device available for the loop breaking in this context.
The next sections will discuss the equational approach of [4], and introduce the new LB semantics.
0.3 The equational approach
Let be an argumentation frame is the set of arguments and is the attack relation. The equational approach views as a bearer of equations with the elements of as the variables ranging over and with as the generator of equations. Let and let be all of its attackers. We write two types of equations and .^{1}^{1}1In [4] there are more options.
For we write


if it has no attackers.
For we write


, if it has no attackers.
We seek solutions f for the above equations. In [4] we prove the following:

There is always at least one solution in to any system of continuous equations .

If we use then the solutions f correspond exactly to the Dung extensions of A. Namely

corresponds to in

corresponds to out

corresponds to undecided.
The actual value in reflects the degree of odd looping involving .


If we use , we give more sensitivity to loops. For example the more undecided elements attack , the closer to 0 (out) its value gets.
In the context of equations, a very natural step to take is to look at Perturbations. If the equations describe a physical or economic system in equilibrium, we want to change the solution a bit (perturb the variables) and see how it affects the system. For example, when we go to the bank to negotiate a mortgage, we start with the amount we want to borrow and indicate for how many years we want the loan and then solve equations that tell us what the monthly payment is going to be. We then might change the amount or the number of years or even negotiate the interest rate if we find the monthly payments too high.
In the equational system arising from an argumentation network we can try and fix the value of some arguments and see what happens. In the equational context, this move is quite natural. We shall see later, that fixing some values to 0 in the equations of , amounts to adopting the CF2 semantics, when done in a certain way. When done in other ways it gives the new loopbusting semantics LB.
Consider Figure 2. The equations for this figure are (we use )
The solution here is
Let us perturb the equation by adding an external force which makes a node equal zero. The best analogy I can think of is in electrical networks where you make the voltage of a node 0 by connecting it to earth.
Let be the “earth” connection for node . We now do several perturbations as examples

Let’s choose to make .
We replace equation 3 by


.^{2}^{2}2We use and write and , rather than just writing because of algebraic considerations. The current equations can be manipulated algebraically to to prove . By adding a fourth variable we prevent that.
The equations now solve to
This gives us the extension

If we try to make , we replace equation (1) by


We solve the equations and get
This corresponds to the extension .

Now let us make . We replace equation (1) by


Solving the new equations gives us
This gives us the extension .
If we compare these extensions with the CF2 extensions, we see that they are the same.
Let us see what happens with Figure 1(b). Here we have a well behaved even loop. Let us write the equations
Let us do some perturbations:

Let us make . We change equation 4 to


We solve the new equations and get
The extension is .

Let us try . we replace equation 1 by


We solve the new equations and get
The extension we get is .

Let us make . We replace equation 2 by


The solution is
This gives the extension

Let us make . the new equations for are


We solve the new set of equations and get
The extension is .

Let us make . We change equation 5 to


We solve the new equations.
From (3) and (4) we get

From (1) and (2) we get

.
Let . Then .
If we want extensions, i.e. , then we get the extensions
, case
, case . 
Let us make . The new equations are







.
The solution is
The extension we get is .

Let us summarise in Table 1.
Case Set of points made 0 Corresponding extensions (a) (b) (c) (d) (e) (f) Table 1:
0.4 The equational loopbusting semantics LB for complete loops
We now introduce our loop busting semantics, the LB semantics for complete loops. We need a series of concepts leading up to it.
[Loops] Let be an argumentation network.

A subset is a loop cycle, (or a loop set, or a loop) if we have
is said to be a complete loop if every element of is an element of some loop cycle.^{3}^{3}3Comparing with the terminology of [1], a complete loop is a union of disjoint strongly connected sets.

A set is a loopbuster if for every loop set we have

Let be a loopbuster and let be a metapredicate describing properties of . We can talk about the semantics LB, where, (when we define it later), we use only loopbusters such that holds. Criteria for adequacy for LB are

It busts all odd numbered loops

It busts all even numbered loops and yields all allowable Dung extensions for such loops.


Our first two proposals for conditions on loopbusters is minimality. The idea is the smaller is, the more options we have.
Therefore, we define: A loopbuster set is minimal absolute if there is no loopbuster set with a smaller number of elements (we do not require !).

A loopbuster set is minimal relative if there does not exist a which is a loopbuster set.
[Loopbuster 2]
Consider Figure 8.
The loops in this figure are many. For example, we list some
Consider the loopbuster
This is not a minimal absolute set but if we delete one of its elements we get a minimal absolute set. No one element is a loopbuster.
[The loopbusting semantics LB for complete loops] Let be an argumentation network. Assume that is a complete loop, namely that each of its elements belongs to some loop cycle, as defined in item 1 of Definition 0.4. We define the LB extensions for as follows.

Let be a loopbuster for satisfying .

Let be the system of equations generated by . These have the form
where , and are all the attackers of . If has no attackers then .

For each replace the equation by the two new equations
where is a new variable syntactically depending on alone.


Solve the equations in (3) and let be any solution.
Then the set
is an LB extension.

Thus the set of all LB extensions for is the set
Note that our definition of extension for a general network will be given in the next section.
Before we prove soundness of LB relative to the traditional Dung semantics and compare LB with CF2 semantics, let us do some examples. We use Figures 2 and 1(b).
Consider Figure 2. The only loop here is . There are three minimal absolute loopbusting sets, and .
For each one of these sets we need to modify the equations of Figure 2 and solve them and see what extensions we get. This has already been done in Example 0.3, parts (a), (b) and (c).
In (a) we made , i.e. we used the loopbusting set . We solved the modified equations and got the extension . In (b) we made , i.e. we used the set , solved the modified equations and got the extension .
In (c) we made , i.e. we used the set , solved the modified equations and got the extension .
Let us now compare with the CF2 extensions for the figure (Figure 2). The maximal conflict free sets of the first loop are and . They are the same as our loopbusting sets, but they are used differently. They are supposed to be in (i.e. value 1) not out (value 0). We use to calculate the CF2 extensions and get and , indeed the same as the LB extensions.
We now consider Figure 1(b). The only minimal absolute loopbuster set here is . We have three more minimal relative sets, and .
We refer the reader to Example 0.3, where some equational calculations for this figure are carried out.

In (a) of Example 0.3, we make , we solve the modified equation and get the extension .
This takes care of the case .

Let us address the case of . We use (b) of Example 0.3, where we make . We modify the equation for and get a solution and .
We needed to also make for the loopbuster set , but as it turns out, making also makes . We thus get the extension .

Let us address the case of . This corresponds to case (d) of Example 0.3. We modify the equations and solve them and get and .
The extension is .
Again, although we did not explicitly make the requirement , the equations obtained from the requirement did the job for us.

We now check the case of . Here we get a discrepancy with case (c) of Example 0.3.
There, in case (c), we only require , solve the equations and get the extension . This is not what we want, as we also require . So let us do the calculation in detail here.
The modified equation system for is the following: xxxx



.




. We solve the equations and get .
The extension is .
[CF2 and the LB minimal absolute semantics] The LB minimal absolute semantics does not give all the CF2 extensions in the case of even loops. Consider Figure 5. The set yields the extension . is minimal absolute. Consider now being .
This yields
However, is not minimal absolute. is a CF2 extension. is a minimal relative set.
What happens here is that the minimal absolute semantics gives the same extensions for even loops as the traditional Dung extensions, but the CF2 semantics gives more. This is a weakness of CF2.
[CF2 and the minimal relative extensions] Let us discuss the results of Example 0.4 calculated for Figure 1(b) and compare them with the CF2 extensions of Figure 1(b). This will give us an idea about the relation of CF2 to the minimal relative semantics. We use Example 0.3, where all the extensions were calculated and especially refer to Table 1, given in item (g) of Example 0.3, which summarises these calculations.

The CF2 extensions are all the conflict free subsets. These are .
Comparing with the semantics of Table 1, we get the following: the LB minimal absolute extensions are one only, namely . The LB minimal relative extensions are and .
We see that LB minimal absolute gives less extensions (but breaks loops) while LB minimal relative gives one more extension. Obviously we need to identify a policy which will yield exactly the CF2 extensions.

Let us examine case (4) of Example 0.4 more closely. This is the case of of Figure 1(b). The loopbuster set was introduced to bust two loops. The loop and the loop . was included to bust the first loop and was included to bust the second loop. Our equational computations show in case (c) of Example 0.3 that if we start with we get that it follows that . But belongs also to the second loop . So on its own is a loopbuster for both loops and we do not need to include in the loopbuster. So is not minimal relative because can do the job. The above considerations show that the definition of minimal relative loopbusting sets needs to be adjusted. This needs to be done in a methodologically correct manner and will be addressed in the next section.
Note that if we accept that is the minimal relative loopbusting set, then the calculated extension for this case is , in complete agreement with the CF2 semantics!
We now need to demonstrate the soundness of the LB semantics. The perceptive reader will ask himself, how do the LB extensions relate to the extensions of traditional Dung semantics? After all, we start with the standard equational semantics, which for the case of is identical with the Dung semantics, but then using a loopbusting set of one kind or another, we get a new set of equations and call the solutions LB extensions. What are these solutions and what meaning can we give them?
Obviously, we need some sort of soundness result. This is the job of the next theorem.
[Representation theorem for LB semantics] Let be an argumentation net being a complete loop as in Definition 8 and let be a loopbusting subset of (of some sort ). Let be the family of LB extensions obtained from and by following the procedures of Definition 8. Then can be obtained also following the procedure below

For each , let be a new point not in . Let be all different for different s.

Define as follows:

The network is an ordinary Dung network and has traditional Dung extensions. We have (for ):
The new equations for each in are
where are all the attackers of in .
Since , we get that
provided .
Of course means .
So we get the same modified equations as required by the LB semantics in Definition 8.
0.5 The equational semantics LB and its connection with CF2
We now define the family of LB semantics and identify the loopbusting counterpart of CF2. We need to develop some concepts first. We begin with a high school example.
[High school example]

Solve the following equations in the unknowns .
The point I want to make is that we solve the equations directionally. We first find the values of and from equations (a) and (b) to be and and then substitute in equation (c) and solve it. We get


Let us change the problem a bit. We have the equations
Here we may again consider equations (a) and (b) first but also use the approximation . We find and solve the third to get .


A third possibility is to look at equations (a) and (b) and decide to ignore them altogether,^{4}^{4}4Of course, ignoring (a) and (b) needs to be justified. and substitute . We get


Another example is the equation
To solve this equation we decide on the perturbation which ignores on account of it being relatively small. We solve
we get .
We present a perturbation protocol for solving equations of the form

Let be a set of variables and be a set of equations of the form , where are the variables appearing in , and ranges over . We seek solutions to the system with values hopefully in . If are all continuous functions in , then we know that there are solutions with values in , but are there solutions with values in ?
Even if we are looking for and happy with any kind of solution, we may wish to shorten the computation by starting with some good guesses, or some approximation or follow any kind of protocol which will enable us to perturb the equations and get some results which we would find satisfactory from the point of view of our application area.
In the case of equations arising from argumentation networks, we would like perturbations which help us overcome oddnumbered loops.
Note that in numerical analysis such equations are well known. If are variables in and are continuous functions in , we want to solve the equations
One well known method is that of successive approximations. We guess a starting value
and continue by substituting
Under certain conditions on the functions (Lipschitz condition), the values converge to a limit and that would be a solution. What we are going to do in this paper is in the same spirit.

Let us proceed formally adopting a purely equational point of view and take a subset of the variables and decide for our own reasons to substitute the value for all the variables in in the equations .
How we choose is not said here, we assume that we have some protocols for finding such a . In the application area of argumentation, these protocols will be different loopbusting protocols LB.
For the moment, formally from the equational point of view, we have a set of equations
with variables and a , which we want to make 0. How do we proceed?This has to be done carefully and so we replace for each , the equation
by the pair of equations
We now propagate these values through the new set of equations, solve what we can solve and end up with new equations of the form
for , where is the new equation for and are its variables. We have
The variables of get all value 0 and maybe more variables solve to some numerical values. Note that we can allow also for the case of .
We always have a solution because the functions involved are all continuous.
Let . is the set of which get a definite numerical value, for which , the set of variables they depend on, is empty. We have .
Let be a function collecting these values on , i.e. , for .

We refer to as the set of all elements instantiated to numerical values at step 1. We declare all variables of as having rank 1.

Let be the system of equations for the variables in .
We now have a new system of variables and we can repeat the procedure by using a new set chosen to make .
We can carry this procedure repeatedly until we get numerical values for all variables. Say that at step we have that the union of all sets equals . Then also each element of has a clear rank , the step at which was instantiated. Call this procedure Protocol . Note that we did not say why and how we choose the sets . In the case of equations arising from argumentation networks, these sets will be loopbusting sets.

Note that the equations initially give variables either 0 or 1 and our loop busters also give variables , and and are such that they keep the variables in , then all the functions f involved are functions
We now explain why we use in our perturbation. Consider the equations



.
These equations correspond to a 3element argumentation loop.
We take the and want to execute a perturbation. If we do just substitute , we get a contradiction because the equations prove algebraically through manipulation that
So we need to change the equation governing . We write xxxx
Algebraically we now have 4 equations in 4 variables
The solution is
We cannot any more execute an algebraic manipulation to get !
Let us recall Example 0.3, manipulating the equations arising from Figure 2. This is an illustration of our procedure. We used the loopbusting sets and , and followed the procedure as described in Remark 0.5.
Let us now proceed with more concepts leading the way to the full definition of our loopbusting LB semantics.
We saw how to get a set of equations from any argumentation network . Now we want to show how to get an argumentation network from any set of equations .
Furthermore, once we have a set of equations , we can perturb it to get a new set of equations using some perturbation set and then from the equations get a new argumentation network . The net result of all these steps is that we start with a network and a perturbation set of nodes and we end up with a new network which we can denote by . If is a loopbusting set, then is the loopbusted result of applying to .

Let be a set of variables and let be a system of equations , where is the set of variables actually appearing in . We now define the associated argumentation network as follows:

Let

Let hold iff .^{5}^{5}5The definition of as is a very special definition, making essential use of the fact that the equation


Let be a network and let be its system of equations.
is a system of equations as in (a) above. Let be some of the variables in . Let f be a function giving numerical values 0 to the variables in .
Let be the system of equations obtained from by substituting the values in the equations for the variables of . The variables of are . Consider now the argumentation network
We say that was derived from using f. We can also use the notation or .
Let us use the network of Figure 14 to illustrate the process outlined in Remark 0.5. This figure is used extensively in [2] and also quoted in [3].
The variables of this figure are
The equations are, using as follows:
Let us take and let f be the function making (i.e. ). (This is a loopbusting move, breaking the loop ).
The new equations for are xxxx
or we can simply write xxxx
Substituting this value in the equations and solving we get the new system of equations for the unknown variable as follows

, known value

, known value

, known value




We get the solution function giving the known values to the variables (these are the variables of rank 2) and the new system of equations (6), (7), (8), (9). Using item (1) of Definition 0.5, we get the derived network in Figure 15.
We can continue now with this loop and choose a loopbusting variable say
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