Collective Discovery Events: Web-Based Mathematical Problem-Solving with Codelets - Computational Creativity Research: Towards Creative Machines - page 373

codelet is the overall supervisor of the task PT , with A being the “supervisor agent”
who will decide at some future point whether the task was handled satisfactorily.
It would be a normal occurrence in this setting that agents at this high level in
the hierarchy of codelets have ideas of “strategies” that might work to obtain a proof
of the theorem. For example, agent A might decide to ask for the solution of four
different and independent approaches to PT , whichwe denote as PT i , )

and will explain soon. Alternatively, it could be that A introduces only one or two
of the PT i , whereas other agents, after being informed of the availability of PT in
the system, add the rest of the PT i (up to i

( i = 1 ,..., 4

4 for our purposes of illustration).
Whenever an agent sees one of the PT i and deems it interesting—reserving it for
solution—a corresponding codelet

=

, who undertakes the role of a prover, is
introduced in the systempool. Thus, assuming all four PT i are reserved, we now have
four more codelets in the pool:

A i , PT i

,
the status of all of which is monitored by the system software. We assume that each
of the four agents A i has different background and skills and is thereby motivated
by different ideas about how to tackle the PT i task at hand. Each PT i might lead to
the creation of new subtasks, and every time a member of the community accepts
working on a subtask a new codelet is created. But let's see what each of the tasks
PT i might look like:

A 1 , PT 1 , A 2 , PT 2 , A 3 , PT 3 , and A 4 , PT 4

1. might be dealing with PT as a purely visual problem, i.e., as one of

dividing the two smaller squares into pieces that then add up to cover exactly the
largest square. This decision means that a choice was made to use some “dia-
grammatic” language to express the argumentation in favor of the validity of the
problem. Thus, A 1 's proof would be formulated in a “diagrammatic” style. This
choice might have been encouraged by A 1 's background and expertise, or by A 1 's
belief that such a proof would be intuitive and convincing. Such a “proof” would
consist of not only showing that such a division is visually possible, but also
that it can be carried out in the “classic” manner, by compass and ruler alone—a
condition imposed upon the solution. An example of such a division is shown in
Fig. 18.1 (proof #28 in [ 1 ]):
Here, the diagram might be termed “self-explanatory”, and agent A 1 might stop
at this point thinking that this is sufficient as a proof. In other words, the agent,
acting as a prover might assume that the item (i.e., the diagram) that A 1 communi-
cates to other codelets—which may undertake the role of interpreters—contains
sufficient information, and thereby might succeed in convincing a potential inter-
preter. However, the overall supervisor agent A (acting as interpreter) might ask
for a further elaboration. Specifically, A might ask to be convinced that the above
division is possible in all cases of shapes of the triangle ABC . i.e., A might
evaluate the item of the prover as insufficient and unconvincing in establishing
the general validity of PT 1 . The overall supervisor A may be suspicious of the
language chosen by the prover to present the argument, believing that the prover
was misled by the language. This would create a new codelet,

A 1 , PT 1

, where
A 1 . 1 is either the same agent as A 1 , or someone else (possibly one having more
advanced expertise in logical reasoning), who accepted the challenge. To show

A 1 . 1 , PT 1