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Problem 14

Show that for any two languages \(A\) and \(B\), a language \(J\) exists, where $A \leq \mathrm{T} J\( and \)B \leq \mathrm{T} J$.

Expert verified

We have shown the existence of a language \(J\) for any two languages \(A\) and \(B\) such that \(A \leq_T J\) and \(B \leq_T J\). We defined \(J = \{0w : w \in A\} \cup \{1x : x \in B\}\) and showed that both \(A\) and \(B\) are Turing reducible to \(J\) by constructing Turing machines that use an oracle for \(J\) to decide if strings are in \(A\) or \(B\).

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Chapter 6

Prove that there exist two languages \(A\) and \(B\) that are Turing-incomparable- that is, where \(A \not{Z} \mathrm{~T} B\) and \(B \not \mathrm{T} A\).

Chapter 6

Show that the set of incompressible strings contains no infinite subset that is Turing-recognizable.

Chapter 6

Is the statement \(\exists x \forall y[x+y=y]\) a member of Th \((\mathcal{N},+)\) ? Why or why not? What about the statement \(\exists x \forall y[x+y=x] ?\)

Chapter 6

Give a model of the sentence $$ \begin{aligned} \phi_{e q}=& \forall x\left[R_{1}(x, x)\right] \\ & \wedge \forall x, y\left[R_{1}(x, y) \leftrightarrow R_{1}(y, x)\right] \\ & \wedge \forall x, y, z\left[\left(R_{1}(x, y) \wedge R_{1}(y, z)\right) \rightarrow R_{1}(x, z)\right] \end{aligned} $$

Chapter 6

Give an example in the spirit of the recursion theorem of a program in a real programming language (or a reasonable approximation thereof) that prints itself out.

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