34. Cryptography 1

You have an alphabet of $m$ letters, $\alpha$. Two languages, $\Upsilon$ and $\Phi$ , use $\alpha$ and have probability distributions

$\Upsilon=(\upsilon_1,\upsilon_2,\ldots,\upsilon_m), \quad \Phi=(\phi_1,\phi_2,\ldots,\phi_m)$

where $\upsilon_i > 0$ and $\phi_i > 0$ and $\sum_{i=1}^m \upsilon_i = \sum_{i=1}^m \phi_i = 1$

You are provided a text that you know is in one of the two languages, each language having the same probability of having generated it, although you do not understand either language. You observe the text, one letter at a time.

Design an algorithm which, after observing $n$ letters, decides whether the text was generated by $\Upsilon$ or $\Phi$. The algorithm should maximise the probability of correct identification.