Set containment inference and syllogisms

AbstractType hierarchies and type inclusion (isa) inference are now standard in many knowledge representation schemes. In this paper, we show how to determine consistency and inference for collections of statements of the form mammal isa vertebrate. These containment statements relate the contents of two sets (or types). The work here is new in permitting statements with negative information: disjointness of sets, or non-inclusion of sets. For example, we permit the following statements also: mammal isa non(reptile), non(vertebrate) isa non(mammal), not(reptile isa amphibian). Binary containment inference is the problem of determining the consequences of positive constraints P and negative constraints not(P) on sets, where positive constraints have the form P:X⊆Y. Negations of these constraints therefore have the form not(P):X∩non(Y)≠ > 0slash;, so two sets have a nonempty intersection.We show binary containment inference is solved by rules essentially equivalent to Aristotle's syllogisms. Necessary and sufficient conditions for consistency, as well as sound and complete sets of inference rules, are presented for binary containment. The sets of inference rules are compact, and lead to polynomial-time inference algorithms, so permitting negative constraints does not result in intractability for this problem