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This is accomplished by the nondeterministic space hierarchy theory, given that CSL is the same as $\mathsf{NSPACE}(n)$.


Since nobody answered my question yet, I have my own try; I would be happy to have comments about it. Since context-sensitive languages are accepted by linear bounded automata, finding minimalist toy examples of languages which are not context sensitive is not easy, because such automata are already quite powerful. However, starting from the previous fact, ...

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