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Problem 17

# Let $B=\left\\{\left\langle M_{1}\right\rangle,\left\langle M_{2}\right\rangle, \ldots\right\\}$ be a Turing-recognizable language consisting of TM descriptions. Show that there is a decidable language C consisting of TM descriptions such that every machine described in $$B$$ has an equivalent machine in $$C$$ and vice versa.

Expert verified
We can construct a decidable language, C, containing TM descriptions, such that every machine in the Turing-recognizable language B has an equivalent machine in C, and vice versa. For every Turing Machine M_i in B, create a new Turing Machine M'_i in C that simulates M_i's behavior but ensures halting in all cases. This construction ensures that each machine in B has an equivalent machine in C and can be reversed to show the equivalence in the other direction.
See the step by step solution

## Step 1: Review the definitions

Turing-recognizable languages are languages that can be accepted by a Turing Machine (TM). Decidable languages are a subset of Turing-recognizable languages, meaning a TM not only accepts but also halts on every input, either by accepting or rejecting it.

## Step 2: Find a way to represent a decidable language, C

Let's construct TM descriptions for the decidable language C, using the Turing Machines from the recognizable language B. For every Turing Machine M_i in B, we will create a new Turing Machine M'_i in C with the required properties. We will implement M'_i with the following steps: 1. Simulate the behavior of M_i on the given input. 2. If M_i accepts the input, accept it in M'_i as well. 3. If M_i rejects the input or does not halt, we make sure M'_i halts and rejects the input.

## Step 3: Show equivalence between machines in B and C

To show that every machine in B has an equivalent in C, and vice versa, we will consider machines M_i in B and M'_i in C. We will argue in the following way: 1. For a given Turing Machine M_i in B, we have constructed the corresponding M'_i in C. In M'_i, if M_i accepts an input, M'_i will also accept it, ensuring that M'_i recognizes the same language as M_i. 2. Furthermore, if M_i does not halt or rejects an input, M'_i still halts and rejects it. This ensures that M'_i is decidable, satisfying the requirement for C. 3. Since we have constructed the Turing Machine M'_i in C for every M_i in B, it guarantees that for each machine in B, there exists an equivalent machine in C. 4. Similarly, we can reverse the construction, taking machines from C and reverting the changes made to find an equivalent machine in B. Thus, showing that for each machine in C, there exists an equivalent machine in B. In conclusion, we have shown that there is a decidable language, C, consisting of Turing Machine descriptions, such that every machine described in the Turing-recognizable language, B, has an equivalent machine in C, and vice versa.

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