The Mathematics of Inheritance Systems

Virtually all AI and object-oriented programming languages include inheritance systems. Common examples are FRL, KRL, KLONE, NETL, Simula, Smalltalk, Flavors, and Ada. Systems that permit exceptions to inherited properties, which involves a form of nonmonotonic reasoning, remained unformalized until quite recently. The lack of a formal theory hid some defects in existing implementations. One defect was an incorrect treatment of networks with multiple grounded expansions. (A grounded expansion is the nonmonotonic equivalent of a "theory" in ordinary monotonic reasoning.) Another defect was a tendency to reason incorrectly when redundant statements are presented. We present a formal mathematical theory of inheritance and show how the defects in existing systems can be corrected. Our formalism bears some relation to default and nonmonotonic logics, but includes an important hierarchical notion the other systems lack. We prove theorems about the consistency, existence, uniqueness, and constructibility of grounded expansions, and we provide a formal semantics for inheritance in terms of constructible lattices of predicates. The formal theory of inheritance is then applied to the formal analysis of a massively parallel computer architecture known as a parallel marker propagation machine (PMPM), of which the best-known example is Fahlman's NETL Machine. This machine has been proposed as a high speed inference engine for AI. We show that a PMPM can perform correct inheritance reasoning only in certain limited cases. However, through a technique known as "conditioning," the topology of a network can be altered to force the PMPM to produce correct results. We show that conditioning is always possible, and present a simple yet efficient conditioning algorithm. An extension to the usual notion of inheritance allows us to infer, if Fred is a citizen and Dick a crook, that Fred dislikes Dick from the relation "citizens dislike crooks." An exception to such a relation might be "gullible citizens don't dislike elected crooks." We extend our earlier results by developing a formal theory of inheritable relations, including exceptions and their interaction with the IS-A hierarchy, and showing how a correct implementation on a PMPM may be achieved through conditioning. *This research was sponsored by the Defense Advanced Research Projects Agency (DOD), ARPA Order No. 3597, monitored by the Air Force Avionics Laboratory Under Contract F33615-81-K-1539. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of the Defense Advanced Research Projects Agency or the US Government.