Why do dogs chase cats?
They work out the hunting instinct - they can "break", but they do not eat. Cats too, but on "game" smaller and usually eat prey. Both are natural predators that rarely intersect in wildlife, where the situation with the relative sizes of "felines" and "dogs" is rather the opposite. And in artificial "domesticated" conditions, dogs do not adequately relate to the forced neighborhood with domesticated cats, perceiving them as an opportunity for rarely dropping out hunting, but without practical nutritional benefit, as, incidentally, dogs do with many other animals, which is what man uses in his hunting. The larger one chases after the smaller one, like Sharik from Prostokvashino - "with the terrible force of the hunting instinct." With joint education - there is no enmity - only games or "neutral enmity", like Matroskin and Sharik in the same "cartoon".
My dog and my cats are not at all hostile, and even on the contrary - they are very good friends. With one of the cats, my Kid even has some kind of special love))) he does not even let us near her, always protects.
I think that the law of predator number one works here - the strong will drive the weak, this is how, at the genetic level, they chase cats and want to eat, but they probably will not - since they are fed, and the genes drive forward, after the prey.
so I know a cat (she lives at my house), which itself rushes to dogs (honestly). But at the expense of your question I have no idea.
Dogs chase cats and cats because of such heredity from their older predator ancestors: wolves, hyenas, coyotes, lynxes, leopards, cheetahs, lions and tigers, which have been at war with each other for millions of years.
Predators hunt, what else is there to explain? I think big cats would hunt dogs if they were tamed.