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Example of language accepted by Turing Machine but not by Linear Bounded Automata ? Is there any EXPSPACE language?

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Consider the language of pairs $(n,T)$ where $n$ is an integer and $T$ is a description of a Turing machine that, when started on an empty tape, eventually moves farther than $2^n$ squares away from the origin.

The empty language is in EXPSPACE. Wikipedia's EXPSPACE article mentions a known EXPSPACE-complete problem.

See also the space hierarchy theorem which gives a simple diagonal construction of languages tailored to have specific space complexities.

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    I'd like to emphasize the space hierarchy theorem: it implies that exponential space is unnecessary for TM to be more powerful than a LBA, there are languages recognizable in $O(n \log n)$ space but not $O(n)$.2012-01-24