Double negation and execution model in Prolog

The caption below does tell you what this particular algorithm is about:

Figure 4.2 An abstract interpreter for logic programs

Also, its description reads:

Output: An instance of G that is a logical consequence of P, or no otherwise.

That is, the algorithm in 4.2 only shows you how to compute a logical consequence for logic programs. It only gives you an idea for how Prolog actually works. And in particular cannot explain the !. Also, the algorithm in 4.2 is only able to explain how one solution ("consequence") is found, but Prolog tries to find all of them in a systematic manner called chronological backtracking. The cut interferes with chronological backtracking in a very particular manner which cannot be explained at the level of this algorithm.

You wrote:

Additionally (page 120 of the same book), Prolog chooses goals (choose goal A) in left-to-right order, and searches clauses (choose renamed clause ...) in the order they show up in the program.

That misses one important point which you can read on page 120:

Prolog's execution mechanism is obtained from the abstract interpreter by choosing the leftmost goal ... and replacing the non-deterministic choice of a clause by sequential search for a unifiable clause and backtracking.

So it is this little addition "and backtracking" which makes things more complex. You cannot see this in the abstract algorithm.

Here is a tiny example to show that backtracking is not explicitly handled in the algorithm.

p :-
 q(X),
 r(X).

q(1).
q(2).

r(2).

We would start with p which is rewritten to q(X), r(X) (there is no other way to continue).

Then, q(X) is selected, and θ = {X = 1}. So we have r(1) as the resolvent. But now, we do not have any matching clause, so we "exit the while loop" and answer no.

But wait, there is a solution! So how do we get it? When q(X) was selected, there was also another option for θ, i.e. θ = {X = 2}. The algorithm itself is not explicit about the mechanism to perform this operation. It only says: If you make the right choice everywhere, you will find an answer. To get a real algorithm out of that abstract one, we thus need some mechanism to do this.

Your program is not a pure Prolog program, since it contains a !/0 in n/1. You may ask yourself the simpler question: With your definitions, why does the query ?- n(f(X)). fail although there clearly is a fact n(X) in your program, meaning that n(X) is true for every X, and should therefore hold in particular for f(X) as well? This is because the program's clauses can no longer be considered in isolation due to the usage of !/0, and the execution model for pure Prolog cannot be used. A more modern and pure alternative for such impure predicates are often constraints, for example dif/2, with which you can constrain a variable to be distinct from a term.

When you reach the last step:

• RESOLVENT: !, fail, !, fail

the cut ! here means, "erase everything". So the resolvent becomes empty. (this is faking it of course, but is close enough). cuts have no meaning at all here, the first fail says to flip the decision, and 2nd fail to flip it back. Now resolvent is empty - the decision was "YES", and remains so, twice flipped. (this is also faking it ... the "flipping" only makes sense in the presence of backtracking).

You can't of course place a cut ! on the list of goals in the resolvent, as it is not just one of the goals to fulfill. It has an operational meaning, it normally says "stop trying other choices" but this interpreter keeps no track of any choices (it "as if" makes all the choices at once). fail is not just a goal to fulfill too, it says "where you've succeeded say that you didn't, and vice versa".

So may I presume that the execution model in textbooks is not precisely what Prolog uses?

yes of course, the real Prologs have cut and fail unlike the abstract interpreter that you referred to. That interpreter has no explicit backtracking and instead has multiple successes by magic (its choice is inherently non-deterministic as if all the choices are made at once, in parallel - real Prologs only emulate that through sequential execution with explicit backtracking, to which the cut is referring - it simply has no meaning otherwise).