Is there a relationship between predicate logic and computers? If so, what is the relationship?

I don't think that there is a special relationship between predicate logic, as opposed to any other kind of logic (propositional, modal, etc.), and computers. But there is often a special relationship between formal logic and computers. It is often possible to set up physical relations inside a computer to mirror (formal) logical relations. The computer can then take over a lot of hard work for us: it can be used to deduce logical consequences, and to check whether particular logical relations obtain. All one has to do is look at the subsequent physical states of the computer, and read off the logical relations that they embody.