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Problem 1

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.

Expert verified

A Python program that prints its own source code can be demonstrated as follows:
```python
def main():
with open(__file__, 'r') as source_code_file:
source_code = source_code_file.read()
print(source_code)
if __name__ == "__main__":
main()
```
This program utilizes the recursion theorem concept by reading its own source code file using `__file__` attribute and printing its content.

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Chapter 6

Show that for any language \(A\), a language \(B\) exists, where $A \leq \mathrm{T} B\( and \)B \not \mathrm{T} A$.

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

Let \(S=\\{\langle M\rangle \mid M\) is a TM and $L(M)=\\{\langle M\rangle\\}\\} .\( Show that neither \)S\( nor \)\bar{S}$ is Turing-recognizable.

Chapter 6

Show that for any \(c\), some strings \(x\) and \(y\) exist, where $\mathrm{K}(x y)>\mathrm{K}(x)+\mathrm{K}(y)+c .$

Chapter 6

Show that \(\overline{E Q_{\mathrm{TM}}}\) is recognizable by a Turing machine with an oracle for \(A_{\mathrm{TM}}\).

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