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Imagine that complexity is measured by a positive number $f_n$.

If one has no prior knowledge about the positive number $f_n$ then from Bayesian theory one can assign $\log f_n$ an "improper" uniform prior over the range $-\infty$ to $\infty$.

A random mutation might increase or decrease the complexity of the offspring $f_{n+1}$. If we assume the offspring's complexity is likely to be close to the parent then we could simply model the random change in complexity by:

$$log f_{n+1} = log f_n + \Delta\ \ \ \ \ \hbox{with probability 0.5}$$

and

$$log f_{n+1} = log f_n - \Delta\ \ \ \ \ \hbox{with probability 0.5}$$

where $\Delta$ is some small number.

Such a random evolution will lead to the log of the complexity performing a random walk.

But now add sex.

Assume the complexity of a sexual organism is the average of the complexities of its parents.

$$f_{child} = \frac{f_{male} + f_{female}}{2}.$$

If one evolves the simple model now one finds that the complexity of the population grows exponentially.

Is this idea interesting or not? :)

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What is complexity? I think you need to very explicitly define what this concept is, with regards to your model. –  kmm Jul 24 '13 at 18:00
    
True - I could have modeled simplicity instead but that seemed a bit perverse! –  John Eastmond Jul 24 '13 at 18:05
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3 Answers 3

Replace the word "complexity" with any other word..."height", "weight", "resting metabolic rate", etc. and the model would still be solvable in a mathematical sense and it would read the same way. So, it's hard for me to accept the model as relevant to the evolution of complexity.

I think you need to make the model have a more robust definition of complexity (as mentioned in the comments) but you also need to consider the costs and benefits of complexity in your model, which will greatly affect the dynamics and actually carry biological relevance.

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Is it interesting? Perhaps, but "complexity" is a vague notion. If you want to simplify and just say "variation" then sure, sex increases variation. But so does that random mutation you brought in. Really, all you need for increased variability is some difference between generations and genetic Drift will take care of the rest. Mutation is enough, which is clear from asexual organisms. Sexual reproduction tends to increase diversity, but that has nothing to do with "complexity." The environment, in particular interaction with other organisms, drives complexity more than anything.

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A concrete theoretical version of my model could be a population of random computer programs or Turing machines. I would define the complexity of a computer program as its run time not the size of its code.

Mutating programs will have varying run times whose logs just perform a random walk.

But if you allow pairs of those programs to occasionally combine in such a way that they generate a child program with a run time that is the average of their own then the run times of the population will grow.

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