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2025-12-09 15:43:08 +01:00
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@ -24,7 +24,7 @@ The Turing machine is the one described by the following:
- States: $S = \{ s_0, s_1 \}$
- Initial state: $s_0$
- Input alphabet: $\{c_1,c_2\}$
- Tape alphabet: $\{\text{\textvisiblespace}},c_1,c_2\}$
- Tape alphabet: $\{\text{\textvisiblespace},c_1,c_2\}$
- Transition function:
+ $\delta (s_0, c_1) = (s_1,c_1,R)$
+ $\delta (s_0, c_2) = (s_1,c_2,R)$
@ -62,3 +62,8 @@ See =Turing.hs=
Prove that every Turing-computable partial function in $\mathbb{N} \rightharpoonup \mathbb{N}$ is also $\chi$ computable. You can assume that the definition of “Turing-computable” uses Turing machines of the kind used in the previous exercise.
Hint: Use the interpreter from the previous exercise. Do not forget to convert the input and output to the right formats.
** Answer
We know that we can construct an interpreter for Turing machines, and a translation of $\mathbb{N}$ to the language a Turing machine.
We also know that one can construct an interpreter in the form of a Turing machine of $\chi$, and the opposite.
This means that a partial function can be translated from $\chi$ to a Turing machine.
So, given a partial function, one can translate it to a Turing machine, and translate the input, and run the implemented Turing interpreter from the last task, and then translate the result back into $\chi$.