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What is defined as a free or connected variable. I did a little research, but I could not understand very well the mathematical concept and differentiate them.
If possible, explain with examples defined in Prolog showing when a variable is said to be free and when turned on.
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A free variable is a variable that does not yet have a value. A linked variable is one that has (and in the absence of backtracking after a variable receives a value it is linked to it forever, it is not possible to assign another value to it).
Example:
?- member(X, Y).
Y = [X|_G3168] .
By calling member/2, so much X how much Y were free variables. After the call, Y is now linked to a list (started with X and finished with anything). X continues free, so that you can assign it the value you want:
?- member(X, Y), X = 42.
X = 42,
Y = [42|_G3116] .
While Y is no longer free, if you try to unify it with something other than what it is (a list with at least one element) the execution will fail (triggering a backtrack):
?- member(X, Y), !, Y = 42.
false.
(the cut is to avoid an infinite loop because member/2 would continue to generate new Ys, failing and returning)
Note that like the contents of Y contains free variables, you can unify it with another compatible list. You wouldn’t be changing your value - the list would remain the same. Only its elements (head and tail) that before were free and will cease to be:
?- member(X, Y), Y = [1,2,3,4,5].
X = 1,
Y = [1, 2, 3, 4, 5] .
(at that point you could ask for "more solutions" to get the other values of X, but in the end it would also fall into an infinite loop).
If you want to know whether a variable has a value or not, it is no use comparing it with something else - since comparing it will end up unifying it with that thing and therefore making it turned on. Instead, use the predicates var/1 and nonvar/1 which are successful if the variable is free/on respectively:
?- member(X, Y), var(X), nonvar(Y).
Y = [X|_G4611629] .
?- member(X, Y), !, var(Y).
false.
?- member(X, Y), !, nonvar(X).
false.
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