Given a set
$$S=\{x∣ \text{there is an x-block of 5's in the decimal expansion of π}\}$$
(Note: x-block is a maximal block of x successive 5's).
Identify class of language?
Somewhere it explained : S is regular. It might not be possible to find what is x, but once it is found, it will be an integer and hence can be represented using an automaton.
My question is :
Since,there is infinite digits in expansion of $\pi$, so , how we recognize $x$?
This is pretty likely to be open, if $\pi$ is proved normal the set would be all positive integers.
We know the language is recursively enumerable but there aren't known techniques to show it is recursive.
If you change the definition of "x-block" from "a maximal block of consecutive 5's" to "a block of consecutive 5's", then the language is regular. Proof: either the length of blocks is unbounded (then $L$ contains all positive integers), or the length of blocks is bounded by a number $N$ (then $L$ is finite).