A big black dog chases a big brown dog with a green object in his mouth while another big brown dog tags behind.
There are dogs chasing each other outside.
If a black dog chasing a brown dog has another dog that tags behind, one could say that the dogs are chasing each other.  Multiple running dogs are usually found outside.
exists e1 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 x1 x2 x3 x4 x5 x6 x7.(a_behind(s10) & Manner(e1,s10) & while(e1,x6) & n_tag(x6) & of(x6,x7) & n_dog(x7) & a_brown(s9) & Theme(s9,x6) & a_big(s8) & Theme(s8,x6) & -(x6 = x5) & n_tag(x5) & of(x5,x1) & n_dog(x1) & n_dog(x1) & a_brown(s7) & Theme(s7,x5) & a_big(s6) & Theme(s6,x5) & Theme(e1,x2) & Actor(e1,x1) & v_chase(e1) & with(x2,x3) & in(x3,x4) & n_mouth(x4) & of(x4,x1) & n_male(x1) & n_object(x3) & a_green(s5) & Theme(s5,x3) & n_dog(x2) & a_brown(s4) & Theme(s4,x2) & a_big(s3) & Theme(s3,x2) & a_black(s2) & Theme(s2,x1) & a_big(s1) & Theme(s1,x1))
exists p1 x1 x5.(n_outside(x5) & exists x2 x3.((x1 = x2) & all x4.((-(x4 = x5) & n_outside(x4)) -> exists e1.(Theme(e1,x4) & Actor(e1,x3) & v_chase(e1))) & (x2 = x3) & n_dog(x2)))