How to prove a language about futuristic events (uncertain facts) is decidable or not?

Question

I get some weird questions about the decidability of a language that focuses on futuristic events (uncertain facts). It is framed like this:

We define the following language:

L = {d, t| The temperature of the dth day in 2022 is t degree Celcius}.

How to prove or disprove this language is decidable? My intuition is that it is undecidable, but I find it hard to reduce $A_{TM}$ to it or find a direct proof.

Answer 1

An important thing to keep in mind is that we may be able to prove that something is decidable even if we can't figure out precisely how to decide it. For example, each of the following sets is decidable:

• $X=\{0\}$ iff alien life exists and $X=\{1\}$ otherwise.

  • Why? Well, there are two possibilities for what $X$ could be, and each is decidable. We don't (yet) know which one is right, but we can say for sure that $X$ is decidable.

• $Y=$ The set of $n$ such that there is a block of at least $n$ many consecutive $7$s in the decimal expansion of $\pi$.

  • Why? Well, note that $Y$ is closed downwards: if $n\in Y$ and $m<n$ then $m\in Y$. So $Y$ is an initial segment of $\mathbb{N}$ (possibly all of $\mathbb{N}$ itself - indeed, this is conjectured to be the case), and all of those are decidable.

Now let's look at your set. Certainly it's true that we can't be sure right now what algorithm decides it, even if it is decidable. However, it's also conceivable that it is decidable - maybe by sheer coincidence the pattern of temperatures in $2022$ happens to be really simple.

This suggests that there might be a "silly" reason that your set is decidable after all. Can you find something that indicates that that set is actually much simpler than it may first appear? (HINT: how many days will there be in $2022$?)