premise: Dog running with pet toy being chased by another dog.
hypothesis: dog is running and being chased by another dog
explanation_1: Dog running indicates that a dog is running; being basically involves running by definition, but 'and' is necessary because the fact that the dog is running was already repeated and the first sentence was incomplete anyway.  Oh, wait, so is the second one.
explanation_2: A dog with a pet toy is being chased by another dog.
explanation_3: Running is a paraphrase of is running.
premise_logic: exists e2 e1 s1 x1 x2 x3 x4 x5.(n_dog(x4) & ne_nam_dog(x1) & with(e1,x2) & Actor(e2,x5) & n_dog(x5) & -(x5 = x4) & Theme(e2,x3) & v_chase(e2) & (x2 = x3) & n_toy(x2) & a_pet(s1) & Theme(s1,x2) & Actor(e1,x1) & v_run(e1))
hypothesis_logic: exists e2 e1 x1 x2 x3.(Actor(e2,x3) & n_dog(x3) & -(x3 = x1) & n_dog(x1) & n_dog(x1) & Theme(e2,x2) & v_chase(e2) & n_dog(x2) & Actor(e1,x1) & v_run(e1))
premise_sim: exists e2 e1 s1 x1 x2 x3 x4 x5.(with(e1,x2) & v_run(e1) & v_chase(e2) & ne_nam_dog(x1) & n_toy(x2) & n_dog(x5) & n_dog(x4) & a_pet(s1) & Theme(s1,x2) & Theme(e2,x3) & Actor(e2,x5) & Actor(e1,x1) & -(x5 = x4) & (x2 = x3))
hypothesis_sim: exists e2 e1 x1 x2 x3.(v_run(e1) & v_chase(e2) & n_dog(x3) & n_dog(x2) & n_dog(x1) & Theme(e2,x2) & Actor(e2,x3) & Actor(e1,x1) & -(x3 = x1))