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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.

Expert verified

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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Chapter 5

$$ \text { Show that } A \text { is decidable iff } A \leq_{\mathrm{m}} 0^{*} 1^{*} \text {. } $$

Chapter 5

Say that a variable \(A\) in CFG \(G\) is necessary if it appears in every derivation of some string \(w \in G\). Let $N E C E S S A R Y_{\mathrm{CFG}}=\\{\langle G, A\rangle \mid A$ is a necessary variable in \(G\\}\). a. Show that \(N E C E S S A R Y_{\mathrm{CFG}}\) is Turing-recognizable. b. Show that \(N E C E S S A R Y_{\mathrm{CFG}}\) is undecidable.

Chapter 5

Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape during the course of its computation on any input string. Formulate this problem as a language and show that it is undecidable.

Chapter 5

Let \(\Gamma=\\{0,1, \sqcup\\}\) be the tape alphabet for all TMs in this problem. Define the busy beaver function $B B: \mathcal{N} \longrightarrow \mathcal{N}\( as follows. For each value of \)k\(, consider all \)k$-state TMs that halt when started with a blank tape. Let \(B B(k)\) be the maximum number of 1s that remain on the tape among all of these machines. Show that \(B B\) is not a computable function.

Chapter 5

A useless state in a Turing machine is one that is never entered on any input string. Consider the problem of determining whether a Turing machine has any useless states. Formulate this problem as a language and show that it is undecidable.

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