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Undecidability of the Empty Language Problem (ETM )
Undecidability of the Empty Language Problem (ETM )
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Introduction &
Overview
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Quick Overview
The **Empty Language Problem (ETM)** asks if the language accepted by a given Turing Machine (TM) is empty. This problem is **undecidable**. Its undecidability is proven by a many-one reduction from the Halting Problem (HALTTM). A reduction function takes an instance of HALTTM (\) and constructs a new TM, M', such that M' accepts no strings (i.e., L(M') is empty) if and only if M *does not* halt on w. Conversely, if M halts on w, M' accepts *all* strings. Since deciding if L(M') is empty effectively decides the Halting Problem, and the Halting Problem is undecidable, ETM must also be undecidable.