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

Sat Oct 11 16:25:10 CEST 2014

Hi Francisco,

Thanks for the reference! I’m afraid the argument doesn’t work in this
particular situation…
The problem is that it is assumed there that the subset we are looking for
consists _entirely_ of terminating programs. But what I’m asking about is a
set that might also contain some non-terminating ones. Instead I want a
termination checker to exist for this subset. What this means for that
proof is that `eval` is not total and neither is `evil`.

To give an idea of what I’m talking about consider the following “coding
style”. (By the way, the term “programming language” usually means “Gödel
numbering” in computability theory, that is, there must be a translator
from any other language; that’s why I’m using the term “coding style”–it
will be a numbering, but it won’t be a Gödel one.) So, take any programming
language (say, C++) and add a requirement that each function definition is
annotated with a boolean value saying whether the function is total or not.
Clearly, any C++ program can be written this way but the translation is not
computable (there is no way to know what to put in the annotation for a
given function). Now the termination checker is trivial, it just takes a
function and outputs its annotation. So, as I mentioned in my first post,
such coding styles exist.

But, well, this stuff is totally useless, because, as already pointed out
by Thomas, membership in the set of programs correctly written in this
style can’t be decidable.

So what I learned so far:
1. No total language is “perfect.”
2. There are “perfect” “coding styles”.
3. If a “coding style” is perfect then it is not decidable (and I keep
saying “coding style” because it is somehow a more general concept since
“programming languages” implicitly include a decision procedure).
4. Undecidable “coding styles” are useless (this is not a formally proved
proposition, but things just work this way).

The remaining questions are: how close to “perfect” a programming language
or a useful (decidable) coding style can be? How good can we (or computers)
become in translating programs into such a close-to-perfect coding style
(note that this question is redundant for programming languages as the
definition of Gödel sets actually requires existence of a computable
translator)?

On Thu, Sep 4, 2014 at 6:30 PM, Francisco Mota <fmota91 at gmail.com> wrote:

> I meant to say "decidable subset of *terminating* programs".
>
>
> On Thu, Sep 4, 2014 at 10:29 AM, Francisco Mota <fmota91 at gmail.com> wrote:
>
>> 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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>>>
>>>
>>>
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>>
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