# The number of words of length $n$ in a context-sensitive language

Let $$L$$ be a context-sensitive language, $$s_{L}(n)$$ is denoted by the number of words of length $$n$$ in $$L$$.

What is known about $$s_{L}(n)$$?

Note that it is known that $$s_{L}(n)$$ is either polynomial, or grows exponentially if $$L$$ is a context-free language.

• Possible duplicate of cstheory.stackexchange.com/questions/42190/…. – domotorp Nov 23 '19 at 7:01
• A simple note: for every (total) function $f(n)$ that can be computed by a LBA with unary input, you can make $s_L(n) = f(n)$ (just check $x < f(n)$) – Marzio De Biasi Nov 25 '19 at 7:47