"is-not-provable" vs. "is-not-true" -- not(Goal)

If the difference between the Prolog "is-not-provable" operator (\+) and the standard negation operator of logic is not taken into account, you may find that some of your programs will behave in an unexpected manner. Here is an example:

     married(adam, eve).
     married(X) :-
             married(X, _).
     married(X) :-
             married(_, X).
     human(X) :-
     human(X) :-

The question

     | ?- \+ married(Who).

is a perfectly good one, to which one might at first glance expect the response to be john or sue. However, the meaning of this clause to Prolog is

     is it the case that the term married(Who) has
     no provable instances?

to which the answer is no, as married(adam) is an instance of married(Who) and is provable by Prolog. The question

     | ?- \+ dead(X).

is also a perfectly good one, and after perhaps complaining that dead/1 is undefined, Prolog will report the answer yes, because dead(X) has no instance that can be proven from this database. In effect, this means "for all X, dead(X) is not provable". Even though "dead(adam) is not provable" is a true consequence of this statement, Prolog will not report X = adam as a solution of this question. This is not the function of \+/1.

Note also that "dead(adam) is false" is not a valid consequence of this database, even though "dead(adam) is not provable" is true. To deduce one from the other requires the use of the "Closed World Assumption", which can be paraphrased as "anything that I do not know and cannot deduce is not true". See any good book on logic programming (such as Foundations of Logic Programming by John Lloyd, Springer-Verlag 1984) for a fuller explanation of this.

We would very often like an operation that corresponds to logical negation. That is, we would like to ask

     | ?- not married(X).

and have Prolog tell us the names of some people who are not married, or to ask

     | ?- not dead(X).

and have Prolog name some people who are not dead. The unprovability operator will not do this for us. However, we can use \+/1 as if it were negation, but only for certain tasks under some conditions that are not very restrictive.

The first condition is that if you want to simulate not(p) with \+(p), you must first ensure that you have complete information about p. That is, your program must be such that every true ground instance of p can be proved in finite time, and that every false ground instance of p will fail in finite time. Database programs often have this property.

Even then, given a non-ground instance of p, not(p) would be expected to bind some of the variables of p. But by design, \+(p) never binds any variables of p. Therefore the second condition is that when you call \+(p), p should be ground, or \+(p) will not simulate not(p) properly.

Checking the first condition requires an analysis of the entire program. Checking that p is ground is relatively simple. Therefore, the library file library(not) defines an operation

     not Goal

which checks whether its Goal argument is ground, and if it is, attempts to prove \+ Goal. Actually, the check done is somewhat subtler than that. The simulation can be sound even when some variables remain; for example, if left_in_stock(Part, Count) has at most one solution for any value of Part, then

     \+ (left_in_stock(Part,Count), Count < 10)

is perfectly sound provided you do not use Count elsewhere in the clause. You can tell not/1 that you take responsibility for a variable's being safe by existentially quantifying it (see the description of setof/3), so

     not Count^(left_in_stock(Part,Count), Count < 10)

demands only that Part must be ground. Even so, this is not particularly good style, and you would be better off adding a predicate

     fewer_in_stock_than(Part, Limit) :-
             left_in_stock(Part, Count),
             Count < Limit.

and asking the question

     not fewer_in_stock_than(Part, 10)

If you want to find instances that do not satisfy a certain test, you can generally enumerate likely candidates and check each one. For example,

     | ?- human(H), not married(H).
     H = john ;
     H = sue
     | ?- man(M), not dead(M).
     M = john ;
     M = adam

The present library definition of not/1 warns you when you would get the wrong answer, and offers you the choice of aborting the computation or accepting possible incorrect results.