Prove that Linear bounded automata LBA ⊂ PSPACE in TOC?

Linear Bounded Automaton (LBA) is a restricted form of Turing Machine in which input tape is finite.


Prove that LBA ⊂ PSPACE

PSPACE is a superset of the set of context-sensitive languages.

Now to prove LBA=PSPACE,

We use theorem of Space compression with tape reduction which states that,

For every k-tape S(n) space-bounded off-line Turing Machine M and constant c>0,there exists a one-tape cS(n) space-bounded off-line turing machine N such that L(M)=L(N).

Following identity holds for −


and NSPACE(S(n))=NSPACE(O(S(n)))

Since LBA is one-tape n space-bounded Turing Machine it follows −


Now by Savitch theorem, if S is fully space constructible and S(n)>log(n) then

NSPACE(S(n)) ⊆DSPACE(S^{2}(n)) -------------(2)

Final proof



⊂DSPACE(n^{3}) Space Hierarchy Theorem


Space Hierarchy then requires S(n) which is fully space-constructible.