SimplicityTheory |
Simplicity, Complexity, Unexpectedness, Cognition, Probability, Information
Simple structures seem unexpected. |
By definition, unexpectedness U is the difference between generation complexity and description complexity: C_{w} – C. Let’s compute both terms.
Combinations |
Complexity | Probability |
1 2 3 4 5 6 34 35 36 37 38 39 10 11 12 44 45 46 7 8 9 37 38 39 8 9 26 27 28 29 10 20 30 31 32 33 1 2 5 6 15 49 . . . 14 24 36 38 42 44 |
3 |
p/8 × 10^6 p/10^6 p/32768 p/16384 p/16384 p/16384 p/4096 . . . p |
The description complexity depends on the structure the subject is able to detect in the combination.
Similarly, when a child sitting in a car sees 66666 on the clock, she can’t help drawing the driver’s attention to the event. Generating such a number requires five instantiations (complexity = 5 × log_{2} 10) whereas its description requires only one instantiation and one iterated copy (complexity = log_{2} 10 + C_{copy}). If we neglect the complexity of copying, formula p=2^{–U} gives a subjective probability of 10^{–4}, which explains the child’s behaviour.
Note that the definition of probability p=2^{–U} conflicts with traditional definitions to be found in complexity theory (Solomonoff 1997), for which simple objects are more likely. Solomonoff’s definition, however, pays attention only to generation complexity and not to description complexity. This is why it makes wrong predictions in the case of Lottery draws.
Saillenfest, A. & Dessalles, J.-L. (2015). Some probability judgments may rely on complexity assessments. Proceedings of the 37th Annual Conference of the Cognitive Science Society, to appear. Austin, TX: Cognitive Science Society.
Savoie, D. & Ladouceur, R. (1995). Évaluation et modification de conceptions erronées au sujet des loteries. Canadian Journal of Behavioural Science, 27 (2), 199-213.
Solomonoff, R. J. (1997). The discovery of algorithmic probability. Journal of Computer and System Sciences, 55 (1), 73-88.