quick Prolog example because I'm not as familiar with Curry:
% This generates or recognizes any palindrome:
pal --> [_].
pal --> X,pal,X.
% Here we try it out and press ; to generate more answers.
?- phrase(pal,P).
P = [A];
P = [B,A,B];
...
% Here we plug in a value and it fails with [A], fails with [B,A,B], etc. until it gets to [D,C,B,A,B,C,D], which can be unified with "racecar."
?- phrase(pal, "racecar")
true.
Another example is just (X=a;X=b),(Y=b;Y=a),X=Y. This has two answers: X=a, Y=a, and X=b,Y=b. What happens is that it first tries X=a, then moves onto the second clause and tries Y=b, then moves onto the third clause and fails, because a≠b! So we backtrack to the last choicepoint, and try Y=a, which succeeds. If we tell Prolog we want more answers (by typing ;) we have exhausted both options of Y, so we'll go back to the first clause and try X=b, then start afresh with Y again (Y=b), and we get the second solution.
Prolog goes in order, and goes deep. This is notoriously problematic, because it's incomplete. Curry only evaluates choicepoints that a function's output depends on, and only when that output is needed. Curry does have disjunctions (using ? rather than Prolog's ;), unification (by =:= rather than =), and pattern guards rather than clause heads, and the evaluation strategy is different because laziness, but in terms of the fundamentals this is what "non-determinism" means in logic programming. it doesn't mean random, it means decisions are left to the machine to satisfy your constraints.
