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Show that $\mathit{E}{\mathit{Q}}_{\mathbf{C}\mathbf{F}\mathbf{G}}$ is undecidable.

Show that $\mathit{E}{\mathit{Q}}_{\mathbf{C}\mathbf{F}\mathbf{G}}$ is co-Turing-recognizable.

Find a match in the following instance of the Post Correspondence Problem.$\mathbf{\{}\mathbf{\left[}\frac{\mathbf{a}\mathbf{b}}{\mathbf{a}\mathbf{b}\mathbf{a}\mathbf{b}}\mathbf{\right]}\mathbf{,}\mathbf{}\mathbf{\left[}\frac{\mathbf{b}}{\mathbf{a}}\mathbf{\right]}\mathbf{,}\mathbf{}\mathbf{\left[}\frac{\mathbf{a}\mathbf{b}\mathbf{a}}{\mathbf{b}}\mathbf{\right]}\mathbf{,}\mathbf{}\mathbf{\left[}\frac{\mathbf{a}\mathbf{a}}{\mathbf{a}}\mathbf{\right]}\mathbf{\}}$

Show that if A is Turing-recognizable and $A{â\copyright ½}_m\stackrel{¯}{A}$, then A is decidable.

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

Question: 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.

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

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

Show that the Post Correspondence Problem is decidable over the unary alphabet$\underset{}{∑}=\left\{1\right\}$.

Define a two-headed finite automaton (2DFA) to be a deterministic finite automaton that has two read-only, bidirectional heads that start at the left-hand end of the input tape and can be independently controlled to move in either direction. The tape of a 2DFA is finite and is just large enough to contain the input plus two additional blank tape cells, one on the left-hand end and one on the right-hand end, that serve as delimiters. A 2DFA accepts its input by entering a special accept state. For example, a 2DFA can recognize the language$\left\{a^n{b}^n{c}^n|nâ‰\yen 0\right\}$ .

• a. Let ${\mathbf{A}}_{\mathbf{2}\mathbf{DFA}}=\{<\mathbf{M},\mathbf{x}\mathbf{>}|\mathbf{M}$is a 2DFA and M accepts$x\overline{)\}}$ . Show that A_2DFA is decidable.
• b. Let ${\mathbf{E}}_{\mathbf{2}\mathbf{DFA}}=\{<\mathbf{M}\mathbf{>}|\mathbf{M}$is a 2DFA and $\mathbf{L}\left(\mathbf{M}\right)=âˆ\dots \}$. Show that E_2DFA is not decidable.