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It is well known that context-free languages are closed under the reversal of $L$. My answer to the question "Is the time reversal symmetry of non-deterministic computations important?" notices that this fact can be derived from the time reversal symmetry of non-deterministic computations. Hence:

This raises the question whether deterministic context-free languages are closed under the reversal of $L$, because this is the only one of the examples where the deterministic and the non-deterministic languages are not identical.

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Thomas Klimpel
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    This question has been raised before. They are not closed under reversal. The first letter can be used to distinguish two different continuations, and this "signal" is lost after reversal. – Hendrik Jan Apr 10 '15 at 10:01
  • @HendrikJan I can't help to find this state of affairs very interesting! The article http://planetmath.org/closurepropertiesonlanguages is still not fixed, and the knowledge seems to be quite scattered. – Thomas Klimpel Apr 10 '15 at 10:36
  • The planetmath article you link to is then wrong on the reversal for DCFL. As I explain in the comment on an answer in the earlier question, it is not explicitly mentioned by Hofcroft and Ullman in their book, but it can be derived from the facts presented there. – Hendrik Jan Apr 10 '15 at 11:17

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