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From: Jan Ziak <0xe2.0x9a.0x9b@gmail.com>
Newsgroups: comp.compilers
Subject: Re: At what point is a language so abstract that it simply cannot be compiled?
Followup-To: comp.theory
Date: Tue, 14 May 2019 08:52:41 -0700 (PDT)
Organization: Compilers Central
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On Monday, May 13, 2019 at 9:31:05 PM UTC+2, Derek M. Jones wrote:
> John,
>
> > [...
> > There are plenty of problems that are known to be undecidable.  The
> > best known is the halting problem.  You cannot write a program that
> > can inspect any arbitrary program and always tell you whether the
> > program it's inspecting will eventually halt or it will run forever.
> ....
> > -John]
>
> This wording of the halting problem tends to lead to all kinds of
> misunderstandings.
>
> I prefer to phrase it as:
> It has been provided that there exist programs for which it is not
> possible to deduce whether it will eventually halt or not.
>
>
> For many programs it is possible to deduce whether they will
> halt, or not.

Hello.

My preference is to phrase it like this:

For any program P that is able to decide in finite time whether a subset S of
its Turing-complete inputs will halt it is possible to construct an input X
not in S which will cause P to either loop forever or return the result
"unable to decide whether X halts".

From practical viewpoint (which does not allow infinite runtimes), P must
always terminate in finite time and return one of these three results:

1: X halts
2: X loops forever
3: unable to decide whether X halts or loops forever

(3) can be further extended in precision by allowing conditional expressions
in the result, such as:

3.A: If <condition> then [X halts] else [X loops forever]
3.B: If <condition> then [X halts] else [unable to decide whether X halts or
loops forever]

Precision of (3) is maximized when <condition> has the form of a universal
expression which evaluates to true or false in finite time.

Sincerely
Jan
[This thread has gotten far afield from compilers.  Please note followup-to. -John]