A brown dog is running through the snow.
A brown dog is running after a stick.
Just because the dog is running does not mean it has to be running after a stick or anything else.
exists e1 s1 x1 x2.(n_snow(x2) & through(e1,x2) & Actor(e1,x1) & v_run(e1) & n_dog(x1) & a_brown(s1) & Theme(s1,x1))
exists e1 s1 x1 x2.(after(e1,x2) & n_stick(x2) & Actor(e1,x1) & v_run(e1) & n_dog(x1) & a_brown(s1) & Theme(s1,x1))