Introduction to UndecidabilityIn the theory of computation, we often come across such problems that are answered either 'yes' or 'no'. The class of problems which can be answered as 'yes' are called solvable or decidable. Otherwise, the class of problems is said to be unsolvable or undecidable. Undecidability of Universal Languages:The universal language L_{u} is a recursively enumerable language and we have to prove that it is undecidable (nonrecursive). Theorem: L_{u} is RE but not recursive. Proof: Consider that language L_{u} is recursively enumerable language. We will assume that L_{u} is recursive. Then the complement of L_{u} that is L`u is also recursive. However, if we have a TM M to accept L`u then we can construct a TM L_{d}. But L_{d} the diagonalization language is not RE. Thus our assumption that L_{u} is recursive is wrong (not RE means not recursive). Hence we can say that L_{u} is RE but not recursive. The construction of M for L_{d} is as shown in the following diagram:
Next TopicUndecidable Problem about TM
