Context sensitive language with non deterministic turing machine

Question

how can i show a language is context sensitive with a non deterministic turing machine?

i know that a language that is accepted by a Linear bound automaton (LBA ) is a context -sensitive language. And a LBA is a non-deterministic turing machine.

Any idea how can i relate all these and show that a language is context sensitive?

Answer 1

As templatetypedef's answer has some flaws (which I will point out in a second in a comment), I give a quick answer to your question:

The language is context sensitive if (and only if) you can give a nondeterministic turing machine using linear space that defines L.

Let L = { a^n b^n a^n } for an arbitrary integer n; a^n here means n concatenations of the symbol a. This is a typical context sensitive language. Instead of giving a CSG, you can give a LBA to show that L is context sensitive:

The turing machine M 'guesses' (thanks to nondeterminism) n [in other words you may say 'every branch of the nondeterministic search tree tries out another n], and then checks whether the input matches a^n b^n a^n. You need log n cells to store n, the matching might need (if implemented trivially) another log n cells. As n + 2log n < 2n, this machine needs only linear space, and is therefore an LBA, hence L is context sensitive.

Answer 2

This is not an exact answer, but since the context-sensitive languages are precisely those accepted by a linear-bounded automaton (a TM with O(n) space on its tape), the context-sensitive languages are precisely those in DSPACE(n). Moreover, we know that NTIME(n) = DSPACE(n). This means that if you can find a linear-time NTM that decides membership in some language L, that language must be context-sensitive. However, there still might be a context-sensitive language that does not have a linear-time NTM (I don't know whether there is a definitive answer to this or whether this is an open problem), so this is not an exact characterization.

Hope this helps!