RE (complexity)

In computability theory and computational complexity theory, RE (recursively enumerable) is the class of decision problems for which a 'yes' answer can be verified by a Turing machine in a finite amount of time.^[1] Informally, it means that if the answer to a problem instance is 'yes', then there is some procedure that takes finite time to determine this, and this procedure never falsely reports 'yes' when the true answer is 'no'. However, when the true answer is 'no', the procedure is not required to halt; it may go into an "infinite loop" for some 'no' cases. Such a procedure is sometimes called a semi-algorithm, to distinguish it from an algorithm, defined as a complete solution to a decision problem.^[2]

Similarly, co-RE is the set of all languages that are complements of a language in RE. In a sense, co-RE contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever.

^ Complexity Zoo: Class RE
^ Korfhage, Robert R. (1966). Logic and Algorithms, With Applications to the Computer and Information Sciences. Wiley. p. 89. A method of solution will be called a semi-algorithm for [a problem] P on [a device] M if the solution to P (if one exists) appears after the performance of finitely many steps. A semi-algorithm will be called an algorithm if, in addition, whenever the problem has no solution the method enables the device to determine this after a finite number of steps and halts.

[1] Complexity Zoo: Class RE

[2] Korfhage, Robert R. (1966). Logic and Algorithms, With Applications to the Computer and Information Sciences. Wiley. p. 89. A method of solution will be called a semi-algorithm for [a problem] P on [a device] M if the solution to P (if one exists) appears after the performance of finitely many steps. A semi-algorithm will be called an algorithm if, in addition, whenever the problem has no solution the method enables the device to determine this after a finite number of steps and halts.

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