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ROOFS AND BOXES
Computer Science Department
Stanford, CA 94305
1998 Sep 9, 12:59 p.m.
This note presents an example, (roofs-and-boxes), to refute the idea
that sequence extrapolation is a paradigmatic problem for AI. This
plausible idea was that intelligence predicted the sequence of future
sensations from the past sequence of sensations. The roofs-and-boxes
example illustrates that intelligence requires knowing about objects in
the world and not just about one’s history of sensations—even if one’s
goal is to predict future sensations.
The justiﬁcation for writing this up many years after I discussed
it in lectures is that almost all machine learning research does not
undertake to infer structures in the world and not just classify the
data. I’ll explain this point after giving the example.
This note presents an example, roofs-nd-oxes, to refute the idea that sequence
extrapolation is the paradigmatic problem for AI. This plausible idea was
that intelligence predicted the sequence of future sensations from the past
sequence of sensations. This idea led to programs for sequence extrapolation.
The ﬁrst programs predicted sequences of integers generated by polynomials,
and later programs dealt with sequences generated by programs that included
Programs for sequence extrapolation were written by Edward Fredkin,
Donald Michie, Jan Mycielski and others. I don’t have the references yet.
My objection to taking this as a paradigm is that the prediction of the
future in real life involves many other kinds of learning than that involved
in direct sequence extrapolation. Speciﬁcally, human learning often involves
the discovery of objects in the environment and their eﬀects on experience.
The roofs-and-boxes example illustrates that intelligence requires knowing
about objects in the world and not just about one’s history of sensations—
even if one’s goal is to predict future sensations.
2 The Roofs and Boxes example
Consider the problem of extrapolating a sequence formed in the following
A billiard ball rolls frictionlessly in a rectangular arena and is friction-
lessly reﬂected when it hits the wall with angle of reﬂection equal to angle of
incidence. Inside the arena there are also some rectangular boxes that reﬂect
the ball when it hits their sides. There are also some rectangular roofs that
have no eﬀect on the ball but hide it from observation.
A sequence of zeroes and ones is generated by a mechanism that observes
the arena from above once per second (or nanosecond if you are impatient).
If the ball is under a roof, it is invisible and a zero is generated. Otherwise
a one is generated.
Now consider extrapolating this sequence from an initial segment not
knowing about the roofs and boxes. None of the techniques of sequence
extrapolation studied by the above-mentioned authors is applicable.
If you know that the sequence is generated by roofs and boxes you can
try to ﬁt models of the locations of the roofs and boxes. With enough data
and computation, you will succeed.
If you don’t have the idea of roofs and boxes, you will have to invent
it. Donald Michie opined that a good cryptanalyst might come up with the
idea. Looking for and analyzing repeated subsequences might help.
3 Making experiments
Here’s a variant system that might be easier to analyze, because it lends itself
to experiment. Suppose the observer, still seeing only zeroes and ones, has a
button he can press. The eﬀect of the button, perhaps unbeknownst to him,
is to deﬂect the ball through an angle of 0.01 degrees. Pressing the button
once will aﬀect the sequence but usually only after some time. For a while
the ball will be bouncing oﬀ the same surfaces.
An observer who has been told or has formed the hypothesis that he is
facing a roofs-and-boxes problem can locate the roofs and boxes simply but
tediously in the case when the sides of the roofs an boxes are parallel to edges
of the arena. He presses his button and waits a long time to see if the zeroes
and ones form an approximately periodic pattern. If so he has a measure of
the distance of a box from the edge and the amount of overhang. If not he
moves on until he has such a pattern. After analyzing one such pattern he
moves on till he ﬁnds another. Eventually he will get the pattern of roofs
and boxes and can predict the future.
How clever must one be to hypothesize that it is a roofs-and-boxes prob-
lem? It is a problem of scientiﬁc creativity. A scientist might ﬁddle for a
long time before coming up with the hypothesis. Donald Michie said that a
good cryptanalyst would probably succeed.
How hard would it be to write a program that could honestly discover
the roofs-and-boxes theory?
Roofs-and-boxes illustrates the idea that even to extrapolate experience a
robot must know or learn about phenomena in the world. Learning programs
have to discover phenomena in the world and not just patterns in the data.
To put the matter in another way, important patterns in the data usually
take the form of observations of phenomena in the world.
/@steam.stanford.edu:/u/jmc/e98/roofs.tex: begun Sun Aug 2 14:08:24 1998, latexed September 9, 1998 at 12:59 p.m.