META- Bongard Problems
Beyond normal Bongard Problems (BP), Harry Foundalis and
Joseph Insana discovered entire classes of Meta- Bongard Problems (MBP).
I.e. nested problems out of the normal domain, whose solution
implies a property of groups of BPs.
Every box/picture in a MBP contains an
independent BP. Some property of all the BPs in the left class distinguishes
them from those in the right class. This property has to be found.
The human or computer solver has to jump one level up to solve this kind of
problem, since the pictures inside the classes are BPs themselves.
The solution of a MBP is not about a property of the visual image,
but about a property of the solution of each individual box/BP.
The first MBP created was Harry's FP #44.
Its solution:
The left class contains BPs with solution based on difference in values
of a feature (e.g. big|small, thin lines|thick lines) | The right class
contains BPs with solution based on numerosity (e.g. one group|two groups,
double branching|triple branching).
Nearly every BP could be transformed in a MBP in the form:
PROPERTY is part of the solution on the left class while it is absent
in the solution on the right; where PROPERTY is taken from a BP, e.g.
(from BP #02) "size".
The resulting MBP would be one with solution: BPs where size matters |
BPs where size doesn't matter
But it's more interesting to think to entire classes of MBPs (examples of
MBPs for each class will be added as soon as they are created).
The following first 2 classes would group all those kind of derived MBPs:
- "BP +property | BP -property" (presence
vs absence of a property)
e.g. the one "size matters" just described
or the above mentioned FP #44
- "BP +property1 | BP +property2" (property1
vs property2)
e.g. "size vs wiggliness")
- "BP | no BP" (real BP vs fake one)
i.e. pictures distinguishable in two classes | pictures cannot be separated
(it's not a Bongard Problem)
- "noisy BP | not noisy BP"
just consider the difference between
BP #36 and BP #37,
to immediately grasp the concept of "noisy" problem (and there are problems
that are REALLY noisy, much more than BP #37; filtering the noise to keep the
signal (the information needed to reach the solution) would produce
not noisy BPs to be placed in this kind of MBP).
- "deceiving BP | not deceiving BP"
we talk of deception when an apparent solution is offered but negated by
just one box so if that particular is missed, the fake solution is given
instead.
E.g.: FP #05
(Box 2d [i.e. the one in position
center-right in the right (2nd) class] is the only one that negates the
apparent solution "parallel lines | not parallel lines") or
IP #23 (in which the number of the modifications
to the big black figure is not relevant to the solution)
- "correct BP | incorrect BP"
Correctness means obeying to the
rules concerning BP creation, described in Foundalis' pages. For example
using only geometrical features, avoiding reference to the human culture.
Another incorrect BP would be one that relies on the relative position
(order) of the boxes.
- "short description of solution | long description
of solution" (the length of the description of the solution of each
BP is the key to this kind of MBPs)
i.e. the boxes on the left would hold BPs whose solution can be phrased
with few words, as opposed to those on the right, whose BPs would present
longer phrased solutions
e.g. BP #02 has a very short possible phrasing
for its solution: "big objects | small objects", while
FP #93 requires at least: "the circle that can be
considered the shared intersection of two threads of circles is white" | "that
same circle is black"
- "non-ambiguous BP | ambiguous BP"
(1 unique solution vs 2 possible solutions)
e.g. an alternative solution to FP #01 could
be "higher total number of white pixels| higher total number of black pixels".
This is hence a problem with two possible solutions.
- "positional BP | not positional BP"
(whether the absolute location of the images within the box matters)
Think to BP #08 or
HP #09, where the absolute position within the
box is required for the solution
Suggestions for more classes of MBPs are welcome. Send email.
META-META- (META^2)?
A meta-meta-problem's solution should be about a property of the
solution to each individual MBP.
(Hence a property of the property of the solution of each BP; i.e.
a property of the property of the property of the visual images that constitute
the BPs....)
This implies that in each box of this MMBP there should be a whole MBP,
such as FP #44 (complete with its 12 sub-boxes, each of which is a BP).
Then it's just a matter of deciding how to group the MBP classes
proposed above in two categories.
E.g. on the left we can have MBPs of class I
and on the right MBPs of class VII.
Achieving this with just 100x100 pixels is a bit problematic though....
Another kind, maybe just META^(1.5) instead than META^2 is the following:
We choose the MBP class IV (noisy vs not noisy) for this example.
Each box includes a BP.
On the left the noisy BPs are marked with a straight separator line
and the not noisy ones are marked with a wiggling separator line.
On the right the separator line code is inverted (wiggling for noisy and
straight for not noisy).
I.e. both the inner BP and the outer form are important, a cross-link
between the plain BP and the MBP.
This is the reason why we prefer to think to this as a META^(1.5) BP.
