Is there a Turing Machine that can decide whether almost all other Turing Machines halt?

Suppose we have some enumeration $\mathbb{N} \rightarrow \{M_i\}$ of Turing machines, and some notion of "size" of a set of natural numbers $\| \cdot \|$, and we define:

$$f(i) = \|\{n: M_i \text{ can't decide whether }M_n \text{ halts} \}\|.$$

What characterizations of the minimum value of $f$ exist for different $\| \cdot \|$? For instance, suppose $\| S \|$ is the limsup of the proportion of numbers up to $k$ that are in $S$. Is there an $i$ for which $f(i) = 0$?


This isn't a "nice" property, because whether it's true or false depends upon the encoding.

See David et al's Asymptotically almost all $\lambda$-terms are strongly normalizing, which proves what it says in the title. However, this paper also shows that the opposite holds for SKI-combinators (into which lambda-terms can be compositionally embedded).

In the lambda calculus, a reduction is the equivalent of a step of a Turing machine, and strong normalization is the property that every reduction sequence eventually reaches a normal form -- ie, no further reductions are possible. (Since a given lambda-term may have many valid reductions, strong normalization is a bit like saying a given nondeterministic Turing machine always halts.) So the fact that asymptotically almost all $\lambda$-terms are strongly normalizing means that with probability approaching 1, reducing a large lambda terms will always reach a normal form.

However, lambda-terms can be translated in a meaning-preserving way into a combinatory calculus such as the SKI combinators (and vice-versa), and in combinator calculi asymptotically all terms loop.

  • 2
    $\begingroup$ I observe that a future visitor, not necessarily knowing the relationship between strong normalization and halting detection, may not be able to determine what position (if any) your answer takes. $\endgroup$ May 15 '18 at 4:55
  • $\begingroup$ @EricTowers Done! $\endgroup$ May 15 '18 at 10:36

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