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Problem 15

Consider the problem of determining whether a Turing machine $$M$$ on an input $$w$$ ever attempts to move its head left at any point during its computation on $$w$$. Formulate this problem as a language and show that it is decidable.

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The given problem can be formulated as a language L: $$L = \{ \langle M, w \rangle \mid$$ M is a Turing machine that tries to move its head to the left at least once during the computation on input w$$\}$$. To show that L is decidable, we constructed a Turing machine D that simulates M on input w, observing its moves. If D detects a leftward move, it accepts; if it reaches the end of the simulation without detecting a leftward move, it rejects. D satisfies both correctness and termination properties, proving that L is decidable.
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Step 1: Define the Language

Let's define a language L for the given problem: $$L = \{ \langle M, w \rangle \mid$$ M is a Turing machine that tries to move its head to the left at least once during the computation on input w$$\}$$. The goal is to show that L is decidable.

Step 2: Construct a Decider for L

To do this, we will need to construct a Turing machine D that decides L. The Turing machine D works as follows for a given input $$\langle M, w \rangle$$: 1. Simulate M on w. 2. During the simulation, observe all the moves of M's head. 3. If at any point, we find a move where M's head moves left, immediately "accept" since this is a positive instance of the language. 4. If the simulation of M on w halts and no leftward move was detected, "reject" since it is a negative instance of the language.

Step 3: Analyze the Decider's Properties

In order to prove that the constructed Turing machine D is indeed a decider for L, we need to examine the following properties: - **Correctness**: For every $$\langle M, w \rangle$$, D accepts if and only if M makes a move to the left on input w. This is true in our construction. D executes M on w and accepts once it finds the leftward move. If it finishes simulating M on w without finding any leftward move, it rejects the input. - **Termination**: For every $$\langle M, w \rangle$$, D halts. D always halts. This is because either it finds the left head move during the simulation and immediately "accepts," or it finishes simulating M on w, meaning it has exhausted all possible computations of M on w without detecting a leftward move, and "rejects." Since D satisfies both the correctness and termination properties, D is indeed a decider for L.

Step 4: Conclusion

We have successfully formulated the given problem as a language L and shown that L is decidable by constructing a Turing machine D that decides L. D accepts an input when it detects a leftward move during the computation of M on w, and rejects otherwise, after exhausting all possible states.

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