The option that is NOT recursively enumerable (RE) is option b. {nL(M) = 0} where m is the binary string representing the Turing machine M.
Recursively enumerable languages are languages that can be accepted by a Turing machine. In other words, a language is recursively enumerable if there exists a Turing machine that, when given an input from that language, will eventually halt and accept that input.
Let's break down each option to understand which one is NOT RE:
a. Any NP-hard problem: NP-hard problems are a class of computational problems that are at least as hard as the hardest problems in the complexity class NP. While it is true that some NP-hard problems may not be recursively enumerable, it is not a definitive statement. Therefore, option a cannot be identified as NOT RE based solely on the fact that it is an NP-hard problem.
b. {nL(M) = 0} where m is the binary string representing the Turing machine M: This option describes the language consisting of binary strings that represent Turing machines whose language is empty. In other words, it represents the set of Turing machines that do not accept any inputs. This language is NOT recursively enumerable because there is no algorithm that can decide if a given Turing machine will never accept any inputs.
c. The CLIQUE problem: The CLIQUE problem is the problem of determining whether a graph contains a clique of a certain size. The CLIQUE problem is NP-complete, which means it is at least as hard as the hardest problems in the complexity class NP. While it is true that some NP-complete problems may not be recursively enumerable, it is not a definitive statement. Therefore, option c cannot be identified as NOT RE based solely on the fact that it is the CLIQUE problem.
d. (M, w) where M,we (0+1)* and M will stop without accepting anything if it is not a legal Turing machine: This option describes the language consisting of pairs (M, w), where M is a binary string representing a Turing machine and w is a binary string that is accepted by M. This language is recursively enumerable because we can simulate M on input w and eventually halt and accept if M accepts w. Therefore, option d is RE.
In conclusion, the option that is NOT recursively enumerable (RE) is option b. {nL(M) = 0} where m is the binary string representing the Turing machine M.