Open-universe Probability Models Show Merit in Unifying Efforts

P E R H A P S T H E M O S T enduring idea from the early days of AI is that of a declarative system reasoning over explicitly represented knowledge with a general inference engine. Such systems require a formal language to describe the real world; and the real world has things in it. For this rea son, classical AI adopted first-order logic—the mathe matics of objects and relations—as its foundation. The key benefit of first-order logic is its expressive power, which leads to concise—and hence learnable—models. For example, the rules of chess occupy 10 pages in first-order logic, 105 pages in propositional logic, and 1038 pages in the language of finite automata. The power comes from separating predicates from their arguments and quantifying over those arguments: so one can write rules about On(p, c, x, y, t) (piece p of color c is on square x, y at move t) without filling in each specific value for c, p, x, y, and t. Modern AI research has addressed another important property of the real world—pervasive uncertainty about both its state and its dynamics—using probability theory. A key step was Pearl’s devel opment of Bayesian networks, which provided the beginnings of a formal language for probability models and enabled rapid progress in reasoning, learning, vision, and language understanding. The expressive power of Bayes nets is, however, limited. They assume a fixed set of variables, each taking a value from a fixed range; thus, they are a propositional formalism, like Boolean circuits. The rules of chess and of many other domains are beyond them. What happened next, of course, is that classical AI researchers noticed the pervasive uncertainty, while modern AI researchers noticed, or remembered, that the world has things in it. Both traditions arrived at the same place: the world is uncertain and it has things in it. To deal with this, we have to unify logic and probability. But how? Even the meaning of such a goal is unclear. Early attempts by Leibniz, Bernoulli, De Morgan, Boole, Peirce, Keynes, and Carnap (surveyed by Hailperin and Howson14) involved attaching probabilities to logical sentences. This line of work influenced AI Unifying Logic and Probability