Undecidability of the Equivalence Problem for Turing Machines (EQTM )
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Introduction & Overview
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Quick Overview
The **Equivalence Problem for Turing Machines (EQTM)** asks if two given Turing Machines (M1 and M2) accept the exact same language (i.e., L(M1) = L(M2)). This problem is **undecidable**. Its undecidability is proven by a many-one reduction from the Empty Language Problem (ETM). The reduction takes an instance of ETM (\) and constructs a pair (\), where M\_empty is a trivial Turing Machine that accepts the empty language. Deciding if L(M) = L(M\_empty) (an instance of EQTM) is directly equivalent to deciding if L(M) is empty (the original ETM instance). Since ETM is undecidable, EQTM must also be undecidable.