Let $f : \{0, 1\}^* \to \{0, 1\}^*$ be a computable function. Given any encoding $\left<M\right>$ of Turing machines over binary (i.e., a function from the set of Turing machines to the set of strings over $\{0, 1\}^*$), consider the following language:

$$ L_f = \{\left<M\right> \mid M \mathrm{\ accepts\ } f(\left<M\right>)\} $$

Question: Does there exist a function $f$ for which the decidability/recognizability of this language depends upon the particular encoding used?


Claim: for any function $f:\{0,1\}^*\to\{0,1\}^*$ (not necessarily computable) and any admissible (see comments below) encoding, the language $$ L_f = \{\left<M\right> \mid M \mathrm{\ accepts\ } f(\left<M\right>)\} $$ is not decidable. Proof. Suppose, for a contradiction, that $L_f$ is decidable -- say, by a TM $M_f$. Now we construct the following TM $M'$. First, $M'$ computes its encoding $\left<M'\right>$ -- which it can, using the Recursion Theorem. Then, $M'$ runs $M_f$ with input $\left<M'\right>$, which tells it whether or not $M'$ is supposed to accept $f(\left<M'\right>)$. If the answer is YES, then $M'$ rejects every input, otherwise, $M'$ accepts every input. In any case, the behavior of $M'$ contradicts the prescription furnished by $M_f$ -- a contradiction.

Update. Of course, if $f$ is computable, then $L_f$ is in RE (i.e., is Turing-recognizable) --- trivially so.

  • $\begingroup$ Thanks, this is interesting. Didn't know the Recursion theorem before. $\endgroup$ – RandomStudent Sep 21 '20 at 8:24
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    $\begingroup$ This does not work for arbitrary encodings, but only for those that satisfy the recursion theorem. $\endgroup$ – Emil Jeřábek Sep 21 '20 at 12:56
  • $\begingroup$ What are the conditions for an encoding to satisfy the recursion theorem? $\endgroup$ – Aryeh Sep 21 '20 at 12:58
  • $\begingroup$ Admissible numberings of Turing machines satisfy it, but in general, you are not going to get necessary and sufficient conditions simpler than “those that satisfy the recursion theorem”. $\endgroup$ – Emil Jeřábek Sep 21 '20 at 13:52
  • $\begingroup$ I modified my answer, but I'd like to get to the bottom of this sometime. Sipser certainly glosses over it, defining an encoding as a mapping from an object (algorithm, TM) to a string. Everything seems to work; where is he cheating? $\endgroup$ – Aryeh Sep 21 '20 at 14:05

For a given computable $f$, the decidability of $L_f$ is independent of the encoding of Turing machines if and only $f$ is eventually injective (i.e., there exists a finite $X\subseteq\def\N{\mathbb N}\N$ such that $f\restriction(\N\smallsetminus X)$ is injective, or equivalently, $\{\def\<#1>{\langle#1\rangle}\<n,m>:n\ne m,f(n)=f(m)\}$ is finite).

  1. If $f$ is computable and eventually injective, then $L_f$ is undecidable for any encoding.

Assume for contradiction that $L_f$ is decidable. Then $$\{m\in\N:\exists n\in\N\:(f(n)=m\land n\notin L_f)\}$$ is r.e., hence it is accepted by infinitely many Turing machines; in particular, using the assumption, it is accepted by a TM $M$ such that $f(n)\ne f(\<M>)$ for all $n\ne\<M>$. Then $$\<M>\in L_f\iff M\text{ accepts }f(\<M>)\iff\<M>\notin L_f,$$ a contradiction.

  1. If $f$ is not eventually injective, then the decidability of $L_f$ depends on the encoding.

As shown in Aryeh’s answer, $L_f$ is undecidable under the standard encoding, hence it suffices to exhibit an encoding that makes $L_f$ is decidable. Let $\{M_m:m\in\N\}$ be any enumeration of TMs, and define $X\subseteq\N$ by $$n\in X\iff|\{m<n:f(n)=f(m)\}|\text{ is even.}$$ We define a new enumeration $\{M'_n:n\in\N\}$ by $M'_n=M_{m(n)}$, where $m(n)$ is defined by induction on $n$: $$m(n)=\begin{cases} \min\bigl\{m\in\N\smallsetminus\{m(i):i<n\}:M_m\text{ accepts }f(n)\bigr\},&\text{if }n\in X,\\ \min\bigl\{m\in\N\smallsetminus\{m(i):i<n\}:M_m\text{ does not accept }f(n)\bigr\},&\text{if }n\notin X. \end{cases}$$ To see that $\{M'_n:n\in\N\}$ indeed enumerates all TMs, assume for contradiction that $m$ is least such that $m\ne m(n)$ for all $n\in\N$. Since $f$ is not eventually injective, there are $n<n'$ such that $f(n)=f(n')$ and $\{0,\dots,m-1\}\subseteq\{m(i):i<n\}$. We may assume $n'>n$ is smallest such that $f(n')=f(n)$, which implies $$n\in X\iff n'\notin X.$$ Thus, either $n''=n$ or $n''=n'$ satisfies $$M_m\text{ accepts }f(n'')\iff n''\in X,$$ which implies $m(n'')=m$ by the definition of $m(n'')$, a contradiction.

Thus, we may define an encoding of TM by $\<M'_n>=n$, and then it follows from the definition that $$L_f=X,$$ which is decidable.

The answer above assumes that an encoding of TM has to be injective; this is not specified in the OP, but I suppose this is just an omission, as I fail to see how something that assigns the same code to two different TM can be considered an “encoding” of TM.

Anyway, if we allow non-injective encodings, the answer is that the decidability of $L_f$ depends on the encoding for every $f$: it is undecidable under the standard encoding by Aryeh’s answer, while it is trivially decidable (because finite) for the constant encoding $\<M>=0$.

  • $\begingroup$ Nice! Thanks for the illuminating answer! =) $\endgroup$ – user21820 Sep 28 '20 at 17:06

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