NU-Prolog's and Gödel's logical and sound `if-then-else` extension

问题

A paper about mercury says the following:

The if-then-else and negation constructs in most variants of Prolog are non-logical and unsound: they can cause the system to compute answers which are inconsistent with the program viewed as a logical theory. Some existing logic programming systems such as NU-Prolog and Gödel provide logical and sound replacements for these Prolog constructs. Unfortunately, these systems enforce safety via runtime groundness checks. This effect can increase the runtime of a program by an arbitrarily large factor; if the goals checked for groundness include large terms, the checks can be prohibitively expensive.

NU-Prolog and Gödel look rather dead and non-free, but I still wonder:

• What are these logical and sound if-then-else replacements?
• Do they have analogs in SWI or GNU Prologs?
• How do they work? How could they work? Adding logical negation to Prolog turns it into general FOL, right? You would basically need a general FOL theorem prover to work with it?
• Are they different from if_/3? if_/3 has to be extended to be used with new conditions. Would one have to do this in NU-Prolog and Gödel also?

回答1:

A break through in if-then-else could be a new annotation. By annotation I understand things like mode declarations, determinancy declarations, etc.. For an if then else, a complete declaration would be nice. Lets assume we could declare a predicate or built-in p/n complete. This would mean it has the property for ground arguments t1,..,tn:

   T |- p(t1,..,tn)

- or -

   T |- ~p(t1,..,tn)

Or in short it would be a decidable predicate if the theory T is recursively enumerable. If we then recall that if-then-else is logically:

    ITE(A,B,C) == (A => B) & (~A => C)

We could then use the complete annotation as follows. Lets assume A = p(t1,..,tn). Because of the annotation the Prolog system would try to prove A. If it doesn't succeed, because of the complete annotation, the Prolog system can infer that ~A would succeed. And therefore it can use the else branch without a proof attempt of ~A.

But interestingly this is already what the ISO core standard if-then-else does, (A->B;C) does also only prove A once. So whats the problem? I guess the problem is that A might be more complex and not necessarily ground. Or even that a predicate p/n might be incomplete, or we even don't know whether it is complete. And in all these cases the ISO core standard nevertheless allows us to use the (A->B;C).

The groundness problem can sometimes be solved by using a runtime groundness checks. This is probably what Mercury refers to:

    when(ground(A), (A->B;C))

SWI-Prolog even applies a trick to make the groundness check cheaper, see also some further discussion on Discourse:

%!  trigger_ground(@Term, :Goal)
%
%   Trigger Goal when Term becomes   ground.  The current implementation
%   uses nonground/2, waiting for an   atribtrary  variable and re-check
%   Term  when  this  variable   is    bound.   Previous   version  used
%   term_variables and suspended  on  a   term  constructed  from  these
%   variables. It is clear  that  either   approach  performs  better on
%   certain types of terms. The term_variables/2  based approach wins on
%   large terms that are almost  ground.   Possibly  we need a nonground
%   that also returns the number of tests   performed  and switch to the
%   term_variables/2 based approach if this becomes large.

trigger_ground(X, Goal) :-
    (   nonground(X, V)
    ->  '$suspend'(V, when, trigger_ground(X, Goal))
    ;   call(Goal)
).

回答2:

I found this review of MU- and NU-Prologs here, if anyone's interested:

来源：https://stackoverflow.com/questions/65386644/nu-prologs-and-g%c3%b6dels-logical-and-sound-if-then-else-extension