User:IssaRice/Computability and logic/Rogers fixed point theorem using Sipser's notation

Sipser's textbook presents the recursion theorem, but not the fixed point theorem. Here is one way to state the fixed point theorem using Sipser's notation.

Theorem (Fixed point theorem). Let $F$ be a Turing machine which computes a function $f:{\mathrm{\Sigma }}^{*}\to {\mathrm{\Sigma }}^{*}$. Then there exists a Turing machine $R$ which computes a function $r:{\mathrm{\Sigma }}^{*}\to {\mathrm{\Sigma }}^{*}$, and a Turing machine $T$ which computes a function $t:{\mathrm{\Sigma }}^{*}\to {\mathrm{\Sigma }}^{*}$, such that $f(⟨R⟩)=⟨T⟩$ and $r(w)=t(w)$ for all $w\in {\mathrm{\Sigma }}^{*}$.

Remark. This is basically a "curried" version of the recursion theorem. Instead of $r(w)=f(⟨R⟩,w)$, we have (with abuse of notation) $r(w)=f(⟨R⟩)(w)$.

The proof is pretty similar to the one in Sipser's book. We build $T$ in four parts, $A,B,F,E$, where:

• $A$ is $P_{⟨BFE⟩}$
• $B$ is the Turing machine that, on input $w,⟨M⟩$:
  • computes $q(⟨M⟩)$
  • combines the result with $⟨M⟩$ to make a complete Turing machine
  • prints $w$ along with the description of this Turing machine and halts, i.e. outputs $w,q(⟨M⟩)⟨M⟩$
• $F$ is as given
• $E$ is an "eval" Turing machine that, on input $w,⟨M⟩$:
  • changes the tape contents to $w$
  • runs $M$

See also

• https://machinelearning.subwiki.org/wiki/User:IssaRice/Computability_and_logic/Diagonalization_lemma#Rogers.27s_fixed_point_theorem