SimplicityTheory

Simplicity, Complexity, Unexpectedness, Cognition, Probability, Information

by Jean-Louis Dessalles     (created 31 December 2008, updated April 2016)

Simplicity Theory (ST) is a cognitive theory based on the following observation:
human individuals are highly sensitive to any discrepancy in complexity.
Their interest is aroused by any situation which appears "too simple" to them.

This has consequences for the computation of
(narrative) Interest     Relevance     Emotional Intensity.

ST’s central result is that Unexpectedness is the difference between expected complexity and observed complexity.
One important consequence is the formula p = 2–U, which states that ex post probability p depends,
not on complexity, but on unexpectedness U, which is a difference in complexity.

Contents

Context

Finding out what elicits human interest is a fundamental scientific question which, quite surprisingly, has not been given much scientific attention in past decades. This is unfortunate, given that: The scientific study of the determining factors of human interest may have been hindered by the tacit assumption that these factors are way too complex, multiple and fuzzy to be properly modelled. The results presented in these pages were obtained thanks to the converse assumption: our minds would be much simpler than commonly believed. Ironically, simplicity plays a major role in the theory presented here. Our minds seem to possess the amazing ability to monitor the complexity of some of their own processes. Here, complexity has to be taken in its technical acceptation (size of minimal description).

These pages are an invitation to challenge the model they present. All suggestions, critiques and contributions are welcome. Please send reactions to JL at@ dessalles.fr.    

Significance for language and cognitive sciences

ST provides a formal and predictive model that impacts on the following domains:

Simplicity Theory (ST)

ST’s main claim

An event is unexpected
if it is simpler (=less complex)
to describe than to generate.


All terms are important here.
Formally:

Unexpectedness:         U = Cw – C


    U: unexpectedness
    Cw: generation complexity (= size of the minimal description of parameters values the "world" needs to generate the situation)
    C: description complexity (= size of the minimal description that makes the situation unique)

Generation complexity Cw(s) is the complexity (minimal description) of all parameters that have to be set for the situation s to exist in the "world". For example, the complexity of generating the fact that you reach me by chance, knowing that you have to cross five four-road junctions to do so, is 5 × log2 3 = 7.9 bits.

We may consider Cw(s) as the length of a minimal program that the "W-Machine" can use to generate s.
The W-machine (or World-machine) is a computing machine that represents the way the observer represents the world and its constraints.

Description complexity C(s) is the length of the shortest available description of s (that makes s unique).
This notion corresponds to the usual definition introduced in the years 1960 by Andrei Kolmogorov, Gregory Chaitin and Ray Solomonoff. However, we must instantiate the machine used in that definition: C(s) corresponds to the complexity of cognitive operations performed by an observer. We may consider C(s) as the length of a minimal program that the O-machine can use to generate s.
The O-machine (or Observation-machine) is a computing machine that can rely on all cognitive abilities and knowledge of the observer.

Caveat: The main problem when beginning with ST is that one fails to consider events, i.e. unique situations. For instance, you may think that a window is a simple object, because it is easy to describe. Indeed! But that window over there is complex, precisely because it looks like any other windows. To make it unique, I need some significant amount of information in addition to the fact that it is a window (see Conceptual Complexity).          Consider, for instance, the number 131072. There is a difference between merely saying that it is a number and producing a way of distinguishing it from all other numbers. Kolmogorov complexity corresponds to the latter action. Describing a number supposes that one eventually can reconstitute all its digits (e.g. by saying that it’s 2^17).

Remark: You’re right if you think that an abnormally complex object may be unexpected. This is consistent with the above definition, as the exceptionally complex object is abnormally simple because of its being an exception (see the "Pisa Tower" effect).

Simplicity theory has been used to explain several important phenomena concerning human interest in spontaneous communication and in news (see bibliography below). More should come. Below are a few didactic examples.

Predictions: Some phenomena explained by ST

Thanks to the above definition of unexpectedness, several quantitative predictions can be made about what human beings would regard as unexpected / improbable / interesting. Some of these predictions are illustrated in the following concrete examples. Most predictions can easily been checked by common sense, and experiments with human participants clearly confirm them (Dimulescu & Dessalles 2009). Note that no alternative theory seems available to predict these same facts.

Relevance

Probability, Simplicity and Information

A new definition of probability

One of ST’s main claims is that unexpectedness can be used to define ex post probability, thanks to the following formula:

Ex-post probability

p = 2–U


This formula indicates that human beings assess probability through complexity, not the reverse. Though the very notion of probability is dispensable in Cognitive Science, the preceding formula is important to account for the many ‘biases’ that have been noticed in human probabilistic judgment (Saillenfest & Dessalles, 2015).

Cognitive Information

ST’s formula p=2–U looks like the converse of Claude Shannon’s definition of information. We might thus be tempted to equate unexpectedness U with information I:

I = U

This means that complexity and unexpectedness are cognitively fundamental, and that probability is, at best, a derived notion. We expect human individuals to be able to convert complexity drops into (low) probability, but not the converse (hence the impressive so-called cognitive ‘biases’ that Daniel Kahneman, Amotz Tversky and others discovered) (see Saillenfest & Dessalles, 2015).

The term of unexpectedness U allows to make several non-trivial predictions about what constitutes valuable information. These predictions include logarithmic variations with distance (see The "next door" effect), the role of prominent places or individuals (see The "Eiffel Tower" effect), habituation effects, the importance of coincidences (see The Lincoln-Kennedy effect), recency effects, transitions, violations of norms (see The running nuns), and records (see The "Robert Wadlow" effect). These theory-based predictions apply to personalized information and to newsworthiness in the media as well.

The other fundamental dimension of information is however missing: emotional intensity E. One should rather write:

I = U + Eh

Emotional intensity, Moral judgments, Responsibility

A crucial observation is that unexpectedness U controls Emotional Intensity in narratives (and beyond).

This means that emotional events elicit more intense emotions when they appear more unexpected.
This observation has far-reaching consequences. Here are a few of them.

Logic, independence, causality

Frequently asked questions

  1. Complexity can’t be computed. How would people have access to it?
        ST considers a resource-bounded version of Kolmogorov complexity, which is computable.
        See the Observation machine.
        Note also that probability theory relies on Set theory. But most sets considered in probabilistic reasoning are not computable
        (e.g. the set of all people who got a flu).
  2. Some improbable objects are improbable because they are complex, not simple.
        Answer in the Pisa Tower effect.
  3. Simple situations are boring, not interesting.
        Such "situations" are frequent. Each instance is complex, as you need information to discriminate it among all instances to make it an event.
        See Caveat above.    See also the Conceptual Complexity page.
  4. Expected situations can still be interesting.
        ’Unexpected’ does not mean ‘unpredicted’. See the Lunar Eclipse example.
  5. It is not always clear what belongs to generation and what belongs to description.
        Depending on the predicates involved, part of the computation may migrate between
        the generation side and the description side (leaving the difference constant).
        See the Inverted Stamp example.
  6. What’s the link with entropy?
        When statistical distributions are available, unexpectedness equals cross-entropy.
  7. Generation complexity and Description Complexity are not based on prefix-free codes. Why?
        Prefix-free codes are used to communicate information.
        Here, we need to use the standard definition of complexity, in which code ‘words’ are externally delimited.
  8. With the definition of probability given by p = 2–U, probabilities sum up to more than 1!.
        Yes, indeed. This of course not a problem, as p corresponds to ex-post probability.
        Only strictly ex-ante probabilities are supposed to sum to 1.
        For instance, two unremarkable lottery draws may both have ex-post probability close to 1.

Links

Further reading

Chater, N. (1999). The search for simplicity: A fundamental cognitive principle?. The Quarterly Journal of Experimental Psychology, 52 (A), 273-302.

Chater, N. & Brown, G. D. A. (2008). From universal laws of cognition to specific cognitive models. Cognitive Science, 32, 36-67.

Chater, N. & Vitányi, P. (2003). Simplicity: a unifying principle in cognitive science?. Trends in cognitive sciences, 7 (1), 19-22.

Dessalles, J-L. (2006). A structural model of intuitive probability. In D. Fum, F. Del Missier & A. Stocco (Eds.), Proceedings of the seventh International Conference on Cognitive Modeling, 86-91. Trieste, IT: Edizioni Goliardiche.

Dessalles, J-L. (2008). 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. (2008). La pertinence et ses origines cognitives - Nouvelles théories. Paris: Hermes-Science Publications.

Dessalles, J-L. (2010). Emotion in good luck and bad luck: predictions from simplicity theory. In S. Ohlsson & R. Catrambone (Eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society, 1928-1933. Austin, TX: Cognitive Science Society.

Dessalles, J-L. (2011). Simplicity Effects in the Experience of Near-Miss. In L. Carlson, C. Hoelscher & T. F. Shipley (Eds.), Proceedings of the 33rd Annual Conference of the Cognitive Science Society, 408-413. Austin, TX: Cognitive Science Society.

Dessalles, J-L. (2013). Algorithmic simplicity and relevance. In D. L. Dowe (Ed.), Algorithmic probability and friends - LNAI 7070, 119-130. Berlin, D: Springer Verlag.

Dimulescu, A. & Dessalles, J-L. (2009). Understanding narrative interest: Some evidence on the role of unexpectedness. In N. A. Taatgen & H. van Rijn (Eds.), Proceedings of the 31st Annual Conference of the Cognitive Science Society, 1734-1739. Amsterdam, NL: Cognitive Science Society.

Leyton, M. (2001). A generative theory of shape. New York: Springer Verlag, 2145.

Saillenfest, A. & Dessalles, J-L. (2012). Role of kolmogorov complexity on interest in moral dilemma stories. In N. Miyake, D. Peebles & R. Cooper (Eds.), Proceedings of the 34th Annual Conference of the Cognitive Science Society, 947-952. Austin, TX: Cognitive Science Society.

Saillenfest, A. & Dessalles, J-L. (2014). Can Believable Characters Act Unexpectedly?. Literary & Linguistic Computing, 29 (4), 606-620.

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, 2069-2074. Austin, TX: Cognitive Science Society.