Circumscription

Circumscription is a term used in geometry and logic (and computer science) to describe a relationship between shapes and circles, or to formalize a form of nonmonotonic reasoning.

In geometry, a circumscribed circle, or circumcircle, of a polygon is a circle that passes through all the polygon’s vertices. A polygon that has a circumcircle is called cyclic. The center of the circle is the circumcenter, and the circle’s radius is the circumradius. Conversely, a polygon can be circumscribed about a circle when each of its sides is tangent to the circle; in this case the circle is the polygon’s incircle and the polygon is said to be circumscribed around the circle. These dual notions—circumscribed around a circle and circumscribed by a circle—are common in the study of triangles and other polygons. Not all polygons are cyclic, but every triangle has a circumcircle.

In logic and artificial intelligence, circumscription refers to a formalism for nonmonotonic reasoning introduced by John McCarthy in 1980. It expresses that certain predicates should be minimized (and others possibly maximized) in all models that satisfy a given theory. This captures default or common-sense reasoning, such as “normally, birds fly,” while permitting exceptions. The aim is to derive conclusions under a closed-world-like assumption by selecting models with minimal extensions for specified predicates. Circumscription has influenced related nonmonotonic logics and provides a foundational approach to reasoning with incomplete information, though it can be computationally challenging to implement.