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How does one define the concept of correlation between languages?

Is there any 'standard' measure of 'correlation' between two (possibly inf) sets of strings / an analogue of the concept in this setting?

Was thinking could use the algorithmic mutual information between their descriptions--maybe this works for languages with finite descriptions?

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  • $\begingroup$ Various reasonable definitions are possible; I suggested one such in my answer. What is your ultimate motivation in asking this? $\endgroup$
    – Aryeh
    Nov 28, 2020 at 15:39

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Your idea only works on languages with finite descriptions; here is an approach that's well defined for all languages, but it requires choosing some probability distribution $\mu$ on the set of all finite words, $\Sigma^*$. Assuming that you've chosen one, let $A$ and $B$ be your two languages. Let $X$ be the random variable corresponding to the event that a $\mu$-random word belongs to $A$, and define $Y$ analogously for $B$. Now you can define correlation between $A$ and $B$ as the correlation between $X$ and $Y$ in the classical sense: $$ cov(X,Y) = E[XY]-E[X]E[Y] = \sum_{x\in\Sigma^*}1[x\in A\cap B]\mu(x) - \sum_{x,y\in\Sigma^*}1[x\in A]1[y\in B]\mu(x)\mu(y) , $$ and $cor(X,Y)=\frac{cov(X,Y)}{\sigma_X\sigma_Y}$, where $$ \sigma_X^2 = \sum_{x\in\Sigma^*}1[x\in A]\mu(x) - \left( \sum_{x\in\Sigma^*}1[x\in A]\mu(x) \right)^2. $$

A reasonable choice of $\mu$ would be to let word length be distributed, say, geometrically (or any other distribution on the integers), and within each fixed word length, $\mu$ is uniform. But of course the "reasonableness" or usefulness of this definition will depend entirely on your purposes.

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