Time complexity of a language whose alphabet has a single symbol Consider a language $L$ such that $L \subseteq \Sigma^*$, where the cardinality of $\Sigma$ is $1$ (i.e. the alphabet has only one symbol). E.g. $L \subseteq \{a\}^*$.

Can anything be said about the time complexity of $L$? Is it the case that $L \in P$? 

I can't come up with any example of a language with a single symbol that is not in $P$.
The language $\{a^n : \text{ the $n$th Turing machine halts on the empty string} \}$ is undecidable.

In fact, a random unary language is undecidable almost surely, for the simple reason that there are only countably many computable languages.