[Agda] Theoretical limits of termination checking (reference request)

[Francisco Mota]

Hi Kirill Elagin,

>From what I understand, you are asking whether there can theoretically be a
decidable subset of programs such that all total recursive functions have a
corresponding program in this subset, even if we don't know how to
translate them. The answer to that question is no, and the proof follows by
diagonalization.

Here's a reference:
http://www.haskell.org/pipermail/haskell-cafe/2003-May/004343.html

Best,
Francisco Mota


On Thu, Sep 4, 2014 at 10:06 AM, Frédéric Blanqui <frederic.blanqui at inria.fr
> wrote:

>
> Le 29/07/2014 23:26, Kirill Elagin a écrit :
>
>  Hello,
>
> I was going through the great articles by Andreas Abel and I suddenly
> started asking myself very basic questions about theoretical limits of
> termination checking. The halting problem is often cited as an explanation
> for impossibility of sound&complete termination checker. The termination
> checking problem is not exactly the halting problem, but indeed it is quite
> easy to derive impossibility of general termination checking from
> impossibility of solving the halting problem.
>
>  But then, another question arises. When we encode proofs, say, in Agda,
> we often have a terminating program in mind, but we have to write it down
> in some specific way, so that the termination checker “sees” that the
> program is fine. So, is it possible to develop a “programming style” such
> that there is a sound&complete termination checker for programs “written in
> this style”,
>
> Hi.
>
> You may be interested in the works of Bellantoni, Marion, Avanzini &
> Moser, ... See for instance
> - http://dx.doi.org/10.2168/LMCS-9(4:9)2013
> - http://dx.doi.org/10.4230/LIPIcs.RTA.2011.123
> to cite just very few papers on this topic.
>
> Frédéric.
>
>  _and_ any program can be “written in this style” (the “translation”
> function is obviously not computable)? Formally: is there a subset of
> programs, such that there is an algorithm correctly checking termination of
> programs from this subset _and_ for any program there is an equivalent in
> this subset (by “equivalent” I mean “extensionally equal”)?
> I used to think that it is impossible, but I recently realized that my
> “proof” was wrong. Turns out that when we are working with the whole
> universe of programs, undecidability of termination checking follows from
> undecidability of the halting problem. But if we restrict ourselves to a
> subset, it is no longer necessarily true, and sound&complete termination
> checking (for programs from this subset) _is_ possible for some subsets.
>
> Then, there are more questions. How good can we become at translating
> arbitrary programs to equivalents from some of those good subsets? As I
> said, the translation function is clearly not computable. But can it be
> that it is not computable only for programs we don't care about? Or maybe
> it is not computable by algorithms, but computable by human brains? Are any
> of the existing means of checking termination already “perfect” in this
> sense, that is can I write _any_ terminating function, say, in MiniAgda, so
> that it passes the termination check?
> I haven't come up with any answers to those ones yet.
>
> For some reason I couldn’t find any information on this topic. I guess
> that either those questions are so trivial and the answers to them are so
> obvious that no one even bothers to write them down, or everything about
> this was published long ago in 70s, so it’s somewhat difficult to find
> those papers now.
> I feel that negative results are most likely to come from the
> computability theory, while positive ones—from more specific fields.
>
> Is there an ultimate source of this kind of funny, useless, purely
> theoretical facts?
>
>
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