Answer
See the explanation
Work Step by Step
To prove that if a language \( \mathrm{L} \) is recognized by a Turing machine with a two-way infinite tape, it can be recognized by a Turing machine with a one-way infinite tape, we need to show that there exists a Turing machine \( M' \) with a one-way infinite tape that recognizes \( \mathrm{L} \).
Let's denote the original Turing machine with a two-way infinite tape as \( M \). We can simulate the behavior of \( M \) on a one-way infinite tape Turing machine \( M' \) as follows:
1. **Initialization**: Start by simulating the behavior of \( M \) on the leftmost cell of the tape. Move the head of \( M' \) to the right as needed to accommodate the simulation.
2. **Simulation**: At each step, \( M' \) will simulate the transition function of \( M \) based on its current state and the symbol under its tape head. Since \( M \) is a Turing machine with a two-way infinite tape, \( M' \) can simulate the behavior of \( M \) by moving left or right as needed.
3. **Acceptance**: If \( M \) accepts the input, \( M' \) will also accept the input. Similarly, if \( M \) rejects the input, \( M' \) will also reject the input.
By simulating the behavior of \( M \) on a one-way infinite tape Turing machine \( M' \), we have shown that any language recognized by a Turing machine with a two-way infinite tape can also be recognized by a Turing machine with a one-way infinite tape. Therefore, the statement is proved.
