N. Lygeros
The Douglas Hofstadter
sequence is, to some extent, a deformation of Fibonacci's. It represents a
generic case of the existence of an abstracted relation between the immediate
future and the remote past. With the opposition to the mentality generated by
the theory of differential equations, we find in this sequence a fractal aspect
whose complexity is interpreted a priori as indeterminism because this
recursive process seems to have a chaotic behavior. Our goal is to show that
this process, ultimately deterministic and understandable, constitutes a
paradigm for non-uniform reasoning.
One of the characteristics of the reasoning described as
intelligent is the synchronic synthesis of knowledge to solve a given problem.
It seems that for relatively elementary problems - for example, exercises or
fast tests - this characteristic is amply sufficient for their resolution. The
really difficult problems, however, require the use of diachronic synthesis.
This method, although very expensive in terms of memory, is essential. Indeed,
its power not only makes it possible to overcome the difficulties encountered,
but also to completely understand the complexity of the problems.
Within this framework, let us attempt to analyze the surprising
character of the fast resolution of a complex problem. It is obvious that this
type of resolution can come from a preliminary knowledge of a problem and an
analogous resolution. Let us therefore exclude this case from our study, a
choice which all the more highlights the surprising character of the
resolution. Let us propose, then, a possible explanation of this phenomenon:
fast resolution appears surprising for one observing the solver because the
observer first carries out an implicit inference, knowing the continuity of the
reasoning in cognitive space. This inference implies for the observer that
there is no essential phase shift in the reasoning of the solver. Thus, for the
observer, the immediate outcome of the intellectual advance could even depend
only on the present. Nevertheless, we now consider a type of problem whose
heuristic model corresponds to the Douglas Hofstadter sequence. It is clear
that the value sought for a given row does not depend on those of immediately
close rows. In this type of problem, a local knowledge proves to be
insufficient and only a diachronic synthesis, and thus, in a certain total way,
allows the determination of the required value. And it is precisely this method
of the solver that surprises the observer: the solver was not locally fast but
different in an essential way.
Thus our paradigm of non-uniform reasoning explicitly shows that
the difference between a reasoning based on a diachronic synthesis and another
is qualitative rather than quantitative. Moreover, when the solver belongs to
one of the fundamental categories (cf our article: M-classification) this
qualitative difference leads to a concept incomparable in cognitive space.
source: lygeros.org (lemma 153)
source: lygeros.org (lemma 153)
