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is made only of linear variations of θ
0
, ie every θ
ι
is of the form θ
ι
¼ a
i
. θ
0
where
a
i
∈
R; then it not necessary to test d
0
for each θ
ι
but it is enough to test
(or redesign) for the operation “multiplication of θ
ι
with a real number”. Hence
the design of generic technology takes advantage of the structure of the invariances
(here linear dependence in Θ). And it finally designs a specific relationship—hence
a structure—between the invariances and the design space.
Hence to study the design of generic technology, we need a model of knowledge
structures and their evolution during the design process. As mentioned earlier, C-K
theory can be useful in our case, because: (1) invariance is possible in C-K; (2) the
knowledge space is a “free” variable, in the sense as C-K theory is supposed to work
with many models of K. What we need to study the design of generic technology
with C-K is just to add a specific model of K.
In this paper we choose to consider that pieces of knowledge can be studied as
matroid structures. Why matroid? Because matroid is a very general language to
deal with independence in many equivalent models (graphs, linear algebra, field
extensions,
...
). Moreover, it offers ways to characterize structures and their evo-
lutions (a structure can be characterized by rank, circuits, bases, lattice of flats; the
evolution of structures can be modeled with operations of duals, minors, sums,
deletion, contraction, extension,
...
). Hence adding matroid structures to K-space
(in C-K theory) provides us with powerful analytical tools to follow the transfor-
mation of structures during a process of designing a generic technology. Note that
in matroid theory, the operations on matroids were mainly used to “analyse”
matroids, ie to identify micro-structures into more complex ones. We will use the
same tools to rather understand how, during a design process, new structures
emerge from given ones.
13.3.3 A Model of C-K with Matroids in K
Matroid structures were introduced by Whitney, in the 1930s, to capture abstractly
the essence of (linear) dependence. A matroid is a pair (E, I ) consisting of a finite
set E and a collection I of subset of E satisfying the following properties: (i) I is
non-empty; (ii) every subset of every member of I is also in I (I is hereditary); (iii) if
X and Y are in I and jXj¼jYj + 1 (the operator j
...
j designates the number of
elements in a set of elements), then there is an element x in X-Y such that Y [ {x}
is in I (independence augmentation condition). I are the independent sets of a
matroid on E, M(E). There are many forms for matroid (defined on matrices, on
algebraic extensions,
...
). In particular, it is very easy to consider the matroid given
by a graph: Given a graph G with vertice V(G), the set of vertices of the graph and
E(G) the set of edges of the graph. Then let I be the collection of subset of E that do
not contain all of the edges of any cycle closed path (or cycle) of G. Then (E, I) is a
matroid on G; it is called the cycle matroid of the graph G and is noted M(G).
We use this structure to model the design process associated to the design of
generic technologies:
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