A brown dog is walking on a wooded path with white flowers.
A brown dog is walking to its owner.
The brown dog on the wooded path could be walking alone.  Nothing in the sentence suggests that a person or it's owner is there.
exists e1 s1 s2 s3 x1 x2 x3.(with(e1,x3) & n_flower(x3) & a_white(s3) & Theme(s3,x3) & on(e1,x2) & n_path(x2) & a_wooded(s2) & Theme(s2,x2) & Actor(e1,x1) & v_walk(e1) & n_dog(x1) & a_brown(s1) & Theme(s1,x1))
exists e1 s1 x1 x2.(to(e1,x2) & n_owner(x2) & of(x2,x1) & n_thing(x1) & Actor(e1,x1) & v_walk(e1) & n_dog(x1) & a_brown(s1) & Theme(s1,x1))