# Is there a DCSL that cannot be recognized in O(n^2) steps by a deterministic LBA?

Is there a context sensitive language $L$ so that $L$ cannot be recognized by a deterministic linear bounded turing machine in $O(n^2)$ steps, but still can be recognized by a deterministic LBA?

The existence of $L$ would imply that linear-space-bounded cellular automata are more powerful than linear-time cellular automata: All deterministic CSL's can be recognized by linear-space-bounded cellular automata and linear-time cellular automata can be simulated by LBA's in $O(n^2)$ steps.

By using crossing-sequences, it is shown that unbounded LBA's are more powerful than linear-time-bounded LBA's. In particular the context free language $\{ ww^R\: | \: w \in \Sigma^* \}$ cannot be recognized by a turing machine in $O(n)$ steps, but can be recognized in $O(n^2)$ steps.

• A stronger question is: Is there an universal deterministic LBA, i.e. a deterministic LBA that can decide whether a given (encoded) deterministic LBA accepts a given input? This would allow diagonalization. – Henning Jul 15 '17 at 14:01