I read multiple answers that state if a language is not context free, then its complement is context free (correct me if I am wrong). Is this true for the opposite? the complement of a context free language is context free?
Asked
Active
Viewed 1.9k times
1
abedzantout
- 802
- 4
- 15
- 28
-
2I don't believe this is really a programming question; it would have been better asked on http://cs.stackexchange.com. But I answered it anyway. I couldn't find a single answer which states that the complement of a non-context-free language is context-free, but not all answers on SO are correct. Perhaps you could include one or two examples of such answers. – rici Dec 13 '15 at 01:34
1 Answers
8
Neither statement is true. The complement of a context-free language can be context-free or not; the complement of a non-context free language can be context-free or not.
Every regular language is context-free. Regular languages are closed under complement, so the complement of a regular language is regular. Consequently, any regular language and its complement are a pair of complementary context-free languages.
The classic example of a non-context-free language whose complement is context-free is {ww|w∈{0,1}*}. (The proof that its complement is context-free is in the answer to this question.)
For a non-context-free language whose complement is also not context-free, a simple example is the language whose valid strings are all pairs {(i, x) i halts on input x} (where i is the description of a Turing machine). That language is not context-free, but it is recursively enumerable. Its complement is not even recursively-enumerable. (See Wikipedia)
