Understanding Circumscription in Commonsense AI Reasoning

This article explores circumscription, a foundational concept introduced by computer scientist John McCarthy in 1980 to formalize commonsense reasoning in artificial intelligence. It covers why traditional logic fails at human-like assumptions, how circumscription solves the qualification problem through non-monotonic logic, and the practical mechanics of minimizing abnormality to help AI systems jump to reasonable conclusions based on incomplete information.

Traditional mathematical logic is monotonic. This means that once a conclusion is proven true, adding new information can never make that conclusion false. However, human commonsense reasoning is non-monotonic. Humans constantly make assumptions based on incomplete information and later revise those conclusions when new facts emerge. To build artificial intelligence capable of commonsense reasoning, John McCarthy developed circumscription as a formal mathematical rule of conjecture.

McCarthy introduced circumscription primarily to solve the qualification problem. The qualification problem states that it is impossible to list all the preconditions required for a real-world action to have its intended effect. In his famous missionaries and cannibals puzzle example, McCarthy noted that to cross a river in a rowboat, one assumes the boat has oars, does not have a hole in it, and has no sea monsters blocking the way. Programming an AI to check an infinite list of potential obstacles is impossible. Circumscription allows an AI to assume that no unspecified obstacles exist unless explicitly stated otherwise.

At its core, circumscription is a method of minimizing the extension of a specific predicate—essentially assuming that as few things as possible possess a certain property. In commonsense reasoning, the property being minimized is usually "abnormality." McCarthy formalized this by introducing an abnormality predicate, often written as Ab(x).

When an AI is given a general rule, it is framed with this abnormality predicate. Consider the classic AI example regarding birds and flight:

1. Rule: If x is a bird, and x is not abnormal, then x can fly.
2. Fact: Tweety is a bird.

Using standard logic, an AI cannot conclude that Tweety flies because it does not know if Tweety is abnormal or not. By applying circumscription to the abnormality predicate, the AI minimizes the set of abnormal things. Since there is no information forcing Tweety to be abnormal, the AI assumes the set of abnormal things is empty. Therefore, Tweety is not abnormal, and the AI concludes that Tweety can fly.

The non-monotonic nature of circumscription becomes clear when new information is introduced. Suppose a third statement is added to the knowledge base:

3. New Fact: Tweety is a penguin, and penguins are abnormal birds regarding flight.

When the AI applies circumscription now, it must minimize the abnormality predicate while still respecting the known facts. The smallest possible set of abnormal things now includes Tweety. Consequently, the AI retracts its previous conclusion and determines that Tweety cannot fly.

Mechanically, McCarthy achieved circumscription by using second-order logic, which allows quantification over predicates rather than just individual objects. The circumscription formula creates a mathematical boundary around the known facts. It states that the predicate being circumscribed is true only for those objects where the known facts absolutely require it to be true.

Circumscription differs from the Closed World Assumption (CWA) used in standard databases. While CWA blindly assumes that any statement not currently known to be true is false, circumscription is much more flexible. It allows the AI system to minimize specific predicates while leaving others open, and it properly handles disjunctive information (such as knowing that either Tweety or Opus is a penguin, without knowing which one).

By formalizing circumscription, John McCarthy provided artificial intelligence with a rigorous mathematical way to apply Occam's Razor. It allows AI systems to operate effectively in the real world by assuming that the situation they are dealing with is as normal as possible, bridging the gap between rigid mathematical logic and fluid human common sense.