Simplicity Theory
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Simplicity, Complexity, Unexpectedness, Cognition,
Probability, Information
by Jean-Louis Dessalles
(created
2008.12.31)
(updated 2010.03.30)
Generation complexity and
W-Machine
Some events require complex combinations of circumstances
Generation complexity measures the minimal amount of
instructions that should be given to the W-machine
("world machine") for it to generate the event. The W-machine
represents the "world" as the observer knows or imagines it.
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In pachinko, small balls
are fired by the player to the top of the machine. They fall down through an
array of pins that alter their course. Most balls end up in a collector and
are lost for the player, but some may reach winning pockets. |
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The
following example (see figure below) is inspired by the pachinko game. Imagine that a
ball falls down along a binary tree such as the one depicted below. At each branching
node, it must “decide” whether it should turn right or left. It eventually
reaches a leaf of the tree after k
such decisions. The generation complexity of the event “the ball reached leaf x” is Cw = k.
Note that most events of
the type “the ball reached leaf x”
are not unexpected. By definition, unexpectedness
is the difference between generation complexity (as defined here) and description complexity: Cw – C. As there are 2k leaves, one generally needs
C = k bits to single out one of them. Therefore, unexpectedness U = 0 for most leaves. However, if the observer can use a
simple feature to single out the winning leaf (e.g. the only coloured leaf),
then C = C(f) and unexpectedness U =
k – C(f) may be large. Note that C(f) is small, as f is the only perceptual feature among
leaves.
It seems that individuals
are able two combine several basic machines to assess Cw.
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Lottery: when a given situation can be considered as
one among N equivalent alternatives,
the W-machine requires a minimum of log2 N bits to produce it. This rule can
be used when an object, a location or a moment is picked out of a range of
possibilities.
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Knowledge: When an event contradicts the
observer’s beliefs (see the running nuns
example), Cw
is derived from the W-complexity of the least complex causal hypothesis that
must be revised for the contradiction to be cancelled (see Dessalles, 2008b,
chap. 6.2).
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Causal story: individuals are able to combine
the W-complexities of successive events, considered as a computation sequence. For independent
successive decision ci
the “world” has to take: Cw(c1*c2*c3*s) = Cw(c1) + Cw(c2) + Cw(c3) + Cw(s|c1&c2&c3).
See the rabid bat and the running
nuns for illustration.
W-complexity is a crucial
component of unexpectedness. The computation of unexpectedness
U requires some knowledge of the
causal functioning of the “world” and of its constraints. In a world in which
everything is possible, nothing is unexpected. Only a constrained world can
surprise us.
The W-machine is a computing machine
that respects the constraints of the "world" and that is given the
current state of the "world" as input. The W-complexity is the
additional input that must be given to the W-machine for it to generate the
situation. An object b that is
considered to belong to the known "world" requires zero complexity to
be generated: Cw(b) = 0.
Caveat: the
world taken as reference has no objective character; it is the world as it is
known by the observer (or, in the case of fiction, imagined).
Bibliography
Dessalles, J-L.
(2008a). Coincidences
and the encounter problem: A formal account. In B. C. Love, K. McRae &
V. M. Sloutsky (Eds.), Proceedings of the 30th Annual Conference of the Cognitive Science
Society, 2134-2139. Austin, TX: Cognitive Science Society.
Dessalles,
J-L. (2008b). La pertinence et ses origines cognitives -
Nouvelles théories.
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