Undecidable Problem about Turing MachineIn this section, we will discuss all the undecidable problems regarding turing machine. The reduction is used to prove whether given language is desirable or not. In this section, we will understand the concept of reduction first and then we will see an important theorem in this regard. ReductionReduction is a technique in which if a problem P1 is reduced to a problem P2 then any solution of P2 solves P1. In general, if we have an algorithm to convert an instance of a problem P1 to an instance of a problem P2 that have the same answer then it is called as P1 reduced P2. Hence if P1 is not recursive then P2 is also not recursive. Similarly, if P1 is not recursively enumerable then P2 also is not recursively enumerable. Theorem: if P1 is reduced to P2 then
Proof:
Empty and non empty languages:There are two types of languages empty and non empty language. L_{e}t L_{e} denotes an empty language, and L_{ne} denotes non empty language. L_{e}t w be a binary string, and Mi be a TM. If L(M_{j}) = Ф then Mi does not accept input then w is in L_{e}. Similarly, if L(M_{j}) is not the empty language, then w is in L_{ne}. Thus we can say that L_{e} = {M  L(M) = Ф} Both L_{e} and L_{ne} are the complement of one another.
Next TopicPost Correspondence Problem
