Question

I need to write a program that constructs a minimal dominating vector for a set of N-Dimensional vectors. A dominating vector for a set S of vectors is defined as a vector whose ith component is greater than or equal to the ith component of every vector in S, with i ranging over all dimensions of the vectors. The dimension N must be taken as input from the user.

Answer 1

Prolog doesn’t have vectors. You can use lists, or structs of arity N.
In second case of course N must be less than the max arity allowed (SWI-Prolog has unlimited length…)

If you use lists, this code should do. I assume N is implicit from list length.

'minimal dominating vector'([V|Vs], Min) :-
  'minimal dominating vector'(Vs, V, Min).

'minimal dominating vector'([], M, M).
'minimal dominating vector'([V|Vs], MinSoFar, Min) :-
   maplist(min, V, MinSoFar, Updated),
   'minimal dominating vector'(Vs, Updated, Min).

min(X, Y, M) :- X < Y -> M = X ; M = Y.

test:

?- 'minimal dominating vector'(["abc","aab","uaa"],M),format('~s',[M]).
aaa
M = [97, 97, 97].

If your Prolog doesn’t have maplist/4 (I can’t remember this detail of P#) then replace maplist(min, V, MinSoFar, Updated), with minvect(V, MinSoFar, Updated),, and add this definition

minvect([], [], []).
minvect([N|Ns], [M|Ms], [R|Rs]) :- min(N,M,R), !, minvect(Ns,Ms,Rs).

note, OT when I tried P# some year ago, I found it very slow, and memory ungry. If you have big arrays, then you’re better to use LINQ