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I read in a book (Hromkovic, Communication Complexity and Parallel Computating) that there is an infinite number of non recursively - enumerable (non-RE) languages that are composed of only 1 element? But is that possible? I though that for a language to be non-RE (or even undecidable), the language has to be infinite.

Link L
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No, all finite languages are regular because they can be accepted by finite automata. There are at least three possible explanations for what you read:

  1. the author(s) wrote something different and you are paraphrasing incorrectly;
  2. the author(s) wrote something other than what they intended, possibly using sloppy terminology;
  3. the author(s) wrote something incorrect due to a misunderstanding of subject matter not belonging to their specialty.

If you quote the relevant passage, it might be possible to explore which of these options is most likely. Note that everybody makes mistakes - people who read books and people who write books alike. Note also that I use author(s) and not authors because it's possible this passage was written in isolation by one author and was not properly reviewed.

Patrick87
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  • After re-reading the text, I think case (1) is more likely: here are more precise words from the proof for Theorem 2.3.5.5, page 81 of the book: "There are infinitely many languages L in LRE with the property L(n)=1 for every n in the natural number set". (LRE are recursively enumerable languages) – Link L Mar 23 '19 at 05:59
  • and L(n) refers to a string belonging to language L with length 1 – Link L Mar 23 '19 at 06:19
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    @LinkL Maybe what they're saying is that there are infinitely many RE languages which contain strings of all natural-number lengths. This is true. It's also true that there are infinitely many RE languages containing strings of length 1. So yes, probably something like that is meant. – Patrick87 Mar 24 '19 at 00:07