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# 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.

**Example**

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 −**

DSPACE(S(n))=DSPACE(O(S(n)))

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

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

LBA=NSPACE(n)---------------------(1)

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**

LBA=NSPACE(n)............by(1)

⊆DSPACE(n^{2})............by(2)

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

⊆PSPACE

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

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