4.4 Impact Analysis
4.4.4 Progress in Decision Sequences
In the paragraphs above we developed two ways of measuring change in decision out- comes, one based on knowledge that was added or removed, and one based on conflict arising from a change of mind. We applied strengthDiff and argDiff to two outcomes of the same decision, in order to gauge the difference in argument strength and number between them.
Instead of analysing the difference between two outcomes of the same decision, we are now going to look at the difference between two decision outcomes Res1 and
Res2 where Res1 was followed by Res2, instead of replaced by it, resulting in a char-
acterisation of the “progress” of a decision sequence. That is, we get an impression of how the coherence (conflict-freeness) and justification of decisions evolves.
The difference between Res2replacing Res1and Res2following Res1is that in the
first case, the option represented by Res1 is not part of the eventual design anymore,
because Res1 was completely replaced by Res2. In the second case, both Res1 and
Res2 are part of the final design, and Res2 is not a revision of Res1 but a refinement
of it. When talking about decision outcomes in this section we will always use their embeddings (see Section 4.3.2 on page 118) to ensure that all implicit assumptions are included in the analysis.
Again, we characterise progress in terms of arguments, not for example in terms of how many of requirements have been met and how many are still open. This approach is a good demonstration of the unique contribution that an argumentation-based model such as ours may bring to the management of engineering design processes.
4.4. Impact Analysis 150 outcomes: monotonic, weakening and alteration. These relations can be applied to decision sequences by applying them to successive pairs of outcomes. For example, in a decision sequence (Res1, Res2, Res3, Res4) we may find that the transition from Res1
to Res2was monotonic, and the transition from Res2to Res3was an alteration. We then
express each of the three relations in terms of argDiff and strengthDiff.
4.4.4.1
Monotonic
Ideally, decision processes advance linearly towards their target, without any change of requirements, reversals of decisions or other detours. The justifications of individual decisions (i.e. the design documents) can be collated to an overall design without any inconsistencies. Therefore, this kind of progress does not introduce any attacks on previously accepted arguments, and results in conflict-free embeddings.
In our formal model, a monotonic transition from Res1 to Res2 has the property
that all of Res1’s arguments are part of the grounded extension of emb(Res2), that is,
they are still acceptable in Res2.
Definition 65 (Monotonic). A sequence of decision outcomes S = (Res1, Res2) is mono-
tonic iff
args(Res1) ⊆ Σgr(argGraph(embS(Res2)))
Example 46. (Res1, Res3) from Example 34 is monotonic.
An equivalent definition of monotonic is that the argument graph of embS(Res2)
has an empty attacks-relation, as the following result shows.
Proposition 36. A sequence of decision outcomes S = (Res1, Res2) is monotonic if and
and only if
attacks(argGraph(embS(Res2))) = /0
Proof. (⇐) Let S = (Res1, Res2) such that attacks(argGraph(embS(Res2))) = /0. Let
a∈ args(Res1). By Definition 54 Cond. 1, Res1⊆ embS(Res1), and by Definition 54
Cond. 3, embS(Res1) ⊆ embS(Res2), so a ∈ args(embS(Res2)).
Since attacks(argGraph(embS(Res2))) = /0, a ∈ Σgr(argGraph(embS(Res2))).
4.4. Impact Analysis 151 (⇒) Let S = (Res1, Res2) be a decision sequence such that S is montonic. Let G2= argGraph(embS(Res2)). Assume that attacks(G2) 6= /0 (Proof by contradiction). Then
there exist two arguments a1, a2∈ args(embS(Res2)) such that (a1, a2) ∈ attacks(G2).
Since both argGraph(Res1) and argGraph(Res2) are conflict-free, the attack (a1, a2)
must be such that
1. a1∈ args(Res1) and a2∈ args(Res2) or
2. a1∈ args(Res2) and a2∈ args(Res1) or
3. a1or a2were introduced by Definition 55 Cond. 2, reactivate.
In case (1), a1∈ args(G2) (by the assumption that S is monotonic, Definition 65). Then,
a2 is attacked by an argument in the grounded extension and therefore a2∈ Σ/ gr(G2).
This violates condition 2 of Definition 54, so embS is not an embedding of S, which
contradicts Theorem 5.
In case (2), by Definition 54 Cond. 2, a1∈ Σgr(G2) and therefore a2∈ Σ/ pgrr(G2).
This contradicts the assumption that S is monotonic (Definition 65).
In case of (3), there is an “underlying” attack (a2, a02) by. Definition 48 which can
be reduced to case (1) or (2).
Proposition 37. If a decision sequence S = (Res1, Res2) is monotonic then
strengthDiff(Res1, EmbS(Res2)) = 0.
Proof. From Propage 36 we know that attacks(argGraph(embS(Res2))) = /0, and we
can apply Propage 32 to get strengthDiff(Res1, EmbS(Res2)) = 0.
The opposite direction of Propage 37 does not hold, because Propage 32 also only holds for the “if-then” case (see the discussion on page 136 for a counterexample).
However, if Res1 and Res2 are embedded in some larger decision process
(. . . , Res1, Res2, . . .) then their counterparts Res01 and Res02 are not necessarily conflict-
free anymore, because attacking arguments may have been introduced by earlier deci- sions.
4.4.4.2
Weakening
The support for a decision Res1 is weakened in embS(Res2) if the option of Res2sub-
4.4. Impact Analysis 152 Res1. In this case, the design that was agreed on in Res1 has not been altered (only
specialised) in Res2, but its support is weaker.
Definition 66 (Weakening). In a sequence of decision outcomes S = (Res1, Res2), Res1
isweakened by Res2if
1. option(Res1) ⊆ option(Res2) and
2. ∃a ∈ args(Res1) such that a /∈ Σgr(argGraph(embS(Res2)))
Example 47. In Example 34, the transition (Res1, Res2) is a weakening one, because
argument a3= [a1; aluminium ⇒ non corrosive; non corrosive] is not part of the grounded extension of Res2’s embedding. It is attacked by argument a6.
In a practical application of the theory, occurrences of weakening should be flagged to the user, because they imply that some of the arguments used to justify a decision were attacked (invalidated) later on, even though the decision itself has not been changed. Weakening may be an accidental side effect of decision making.
4.4.4.3
Alteration
As a generalisation of weakening (Definition 66), it is possible that an argument of Res1
is defeated in embS(Res2), without option(Res1) ⊆ option(Res2).
Definition 67 (Alteration). A sequence of decision outcomes S = (Res1, Res2) is an
alteration iff
∃a ∈ args(Res1) such that a /∈ Σgr(argGraph(embS(Res2)))
Example 48. In Example 34, the transition Res2, Res3 is an alteration, because the
arguments a5= [bolts] and a7= [shim] are part of Res2but not of Res3.
The following result shows that any change in the option of a decision outcome results in either an alteration or a monotonic transition.
Proposition 38. Let Res1, Res2 be two decision outcomes with option(Res1) 6=
4.4. Impact Analysis 153 Proof. Let Res1= (O1, K1) and Res2= (O2, K2) be two decision outcomes with O16=
O2.
Monotonic or alteration... Either O1⊆ O2 or not. If O1⊆ O2, then either (a)
args(Res1) ⊆ Σpr(argGraph(embS(Res2))) or (b) args(Res1) * Σpr(argGraph(embS(Res2))).
In case (a), (Res1, Res2) is monotonic by Definition 65. In case (b), there exists an ar-
gument a ∈ args(Res1) such that a /∈ Σgr(argGraph(embS(Res2))). Then, (Res1, Res2)
is an alteration by Definition 67.
If O1* O2 then there exists a literal l ∈ O1 such that l /∈ O2. Hence there is
an argument [l] ∈ args(Res1) with [l] /∈ args(Res2) and (Res1, Res2) is an alteration by
Definition 67.
... but not bothFollows from the logical form of the two defitions.
In practical applications, alterations may be a sign of a healthy decision process, because they occur if a previous error has been corrected (as shown in Example 48). However, it is clear from the definition that every instance of weakening is also an alteration, so any alterations should be analysed further.