4.4 Impact Analysis
4.4.2 Impact Analysis Based on Knowledge Added or Removed
If we have a decision outcome Res1 and add knowledge to it, the result will be a de-
cision outcome Res2 with args(Res1) ⊆ args(Res2). When using abstract argument
graphs (A, Att), the only statements we can make about added or removed knowledge (as opposed to conflict) are about the “argument” component A. In Dung’s theory, the arguments in A are atomic, so any distance measure based on A must be a general distance measure for sets.
We could take into account that the argument graphs considered here are pro- duced from decision outcomes, essentially ASPIC+ knowledge bases. However, this would create an asymmetry in our distance measures, because strengthDiff (Definition 61) does not take the underlying ASPIC+ knowledge base into account, and relies only on the resulting argument graph. For this reason, we will use a simple set-based dis- tance measure. The distance measure argDiff(Res1, Res2) essentially counts how many
arguments are only found in one of Res1, Res2.
Definition 62 (Argument-Based Difference of Decision Outcomes). Let Res1, Res2be
two decision outcomes. Theargument-based distance of Res1, Res2is defined as
argDiff(Res1, Res2) = |args(Res1)∆args(Res2)|
Example 41. For the decision outcomes Res1and Res2from Example 37, we get
argDiff(Res1, Res2) = |args(Res1)∆args(Res2)|
= |{a1, a4, a5, a7, a8, a9}|
4.4. Impact Analysis 138 The measure argDiff stays true to Dung’s formalism by not assuming any- thing about the arguments other than the fact that they form a set. It is clear that argDiff(Res1, Res2) cannot be zero if strengthDiff(Res1, Res2) is non-zero, because each of Res1, Res2is conflict-free.
Proposition 33. For any two decision outcomes Res1, Res2, ifstrengthDiff(Res1, Res2) >
0 then argDiff(Res1, Res2) > 0.
Proof. Let Res1, Res2 be two decision outcomes such that strengthDiff(Res1, Res2) >
0. Let G = (A, Att) = argGraph(Res1∪ Res2) and let A1= args(Res1) and let A2 =
args(Res2). Since strengthDiff(Res1, Res2) > 0, there must be at least one attack in Att,
so Att 6= /0. Let (a, b) ∈ Att and assume a ∈ A1 and b ∈ A2(without loss of generality,
by Propage 25). The argument graph (A1, Att1) = argGraph(Res1) is conflict-free (by
Propage 24), so b /∈ A1. Therefore, b ∈ A1∆A2, so A1∆A26= /0 and argDiff(Res1, Res2) >
0.
Example 42. In terms of the running example from Section 4.3, we get the following argument differences for the decision outcomes defined in Example 34.
argDiff(Res1, Res2) = |{a2, a3, a4, a5, a6, a7, a8}| = 7
argDiff(Res2, Res3) = |{a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12}| = 11
argDiff(Res1, Res3) = |{a9, a10, a11, a12}| = 4
While thestrengthDiff values for Res1, Res2 and Res3 are relatively small (see page
136, Example 40), the difference in arguments between the outcomes varies consider- ably. The design and its justification have evolved, while conflicting justifications have been kept to a minimum, which can be interpreted as sign of a healthy decision process. In the following section we will take a closer look at the interplay of strengthDiff and argDiff.
4.4.2.1
Relationship between argDiff and strengthDiff
The two measures of impact correspond to the conflict-based (strengthDiff, Definition 61) and knowledge-based (argDiff, Definition 62) distance measures. The possible results of comparing decision outcomes Res and Res0 with strengthDiff and argDiff, can be grouped into four categories, forming an imaginary square:
4.4. Impact Analysis 139 1. strengthDiff(Res, Res0) = 0 and argDiff(Res, Res0) = 0
2. strengthDiff(Res, Res0) 6= 0 and argDiff(Res, Res0) = 0 3. strengthDiff(Res, Res0) = 0 and argDiff(Res, Res0) > 0 4a. strengthDiff(Res, Res0) < 0 and argDiff(Res, Res0) > 0 4b. strengthDiff(Res, Res0) > 0 and argDiff(Res, Res0) > 0
The first case only occurs when Res = Res0. The second case is impossible to achieve (see Propage 34 below). This leaves us with three interesting cases: Knowledge has been changed, but there are no changes in argument strength (case 3), knowledge changed, argument strength decreased (case 4a) and knowledge changed, argument strength increased (case 4a).
Proposition 34. For any two decision outcomes Res, Res0, ifargDiff(Res, Res0) = 0 then strengthDiff(Res, Res0) = 0.
Proof. Let G = (A, Att) = argGraph(Res ∪ Res0) and assume argDiff(Res, Res0) = 0. Then args(Res) = args(Res0), so Res = Res0(by the assumption that there are no unused rules, see page 33). Since args(Res) is conflict-free, attacks(Res ∪ Res0) = /0, and we can apply Propage 32 to get strengthDiff(Res, Res0) = 0.
We will now analyse each of the two remaining cases.
First, if strengthDiff(Res, Res0) = 0 and argDiff(Res, Res0) > 0 then either there are no conflicts in argGraph(Res ∪ Res0), or the strength of Res’s arguments is exactly the same as that of the arguments of Res0(cf. the discussion of Propage 32 on page 136). In either case, the added knowledge did not result in a strengthening of the decision’s justification.
If however strengthDiff(Res, Res0) > 0, then the average strength of the ar- guments of Res0 is higher (in Res ∪ Res0) than that of the arguments of Res. If strengthDiff(Res, Res0) < 0, then the average strength of Res is higher, and switch- ing from Res to Res0resulted in a relative weakening of the reasons used to justify the decision.
4.4. Impact Analysis 140 Example 43. For decision outcomes Res1 and Res3 from Example 4.3 we get
strengthDiff(Res1, Res3) = 0 and argDiff(Res1, Res3) = 4, implying that their justi-
fications overlap (albeit not completely) but are not inconsistent. In our running example, Res1 is followed by Res3, but this result shows that we could replace Res1 with Res3without having to take a different stance on previously accepted arguments.
On the other hand, if strengthDiff(r, r0) > 0, the two decision outcomes are incom- patible, and replacing r with r0requires us to reject some assumptions that were made originally.
Example 44. The two outcomes in Example 37 are not compatible, because strengthDiff(Res1, Res2) = 2. If we replaced Res1 with Res2, we would have to re-
ject arguments a1, a5 and a8. This has implications for subsequent decisions that were based on Res1: Any arguments which have a1, a5or a8as sub-arguments will be incompatible with Res1’s replacement (Res2) and therefore have to be adjusted.
With strengthDiff and argDiff, we can get an idea of how different the justifications of two decisions are. They bring us closer – as alluded to in the previous two examples – to achieving our larger goal: To assess the impact of adjusting past decisions on the overall decision process, not just on the immediately affected decision outcome.