Set containment inference
Type hierarchies and type inclusion 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)
Among the various types of containment, we consider "binary containment inference", 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\(\subseteq\)Y, where X, Y, are types or their complements. A negative constraint is equivalent to a statement of the form X∩non(Y) ≠ 0. Positive constraints assert containment relations among sets, while negative constraints assert that two sets have a non-empty intersection. We show binary containment inference is solved by rules essentially equivalent to Aristotle's Syllogisms. The containment inference problem can also be formulated and solved in predicate logic. When only positive constraints P are specified, binary containment inference is equivalent to Propositional 2-CNF Unsatisfiability (unsatisfiability of conjunctive propositional formulas limited to at most two literals per conjunct). In either situation, necessary and sufficient conditions for consistency, as well as sound and complete sets of inference rules are presented. Polynomial-time inference algorithms are consequences, showing that adding negation does not result in intractability for this problem.
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- 1.Atzeni, P. and D. Stott Parker, Formal Properties of Net-based Knowledge Representation Schemes, Proc. 2nd International Conference on Data Engineering (Los Angeles, CA, February 5–7, 1986), IEEE Computer Society, Washington, D.C., 1986.Google Scholar
- 2.Boole, G., An Investigation of The Laws of Thought, Dover, 1958.Google Scholar
- 3.Gardner, M., Logic Machines and Diagrams, University of Chicago Press, 1982.Google Scholar
- 4.Garey, M.R. and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Co., San Francisco, 1979.Google Scholar
- 5.Loveland, D., Automated Theorem Proving, North-Holland, 1976.Google Scholar
- 6.Maier, D., The Theory of Relational Data Bases, Computer Science Press, Rockville, MD, 1983.Google Scholar
- 7.Ullman, J.D., Principles of Data Base Systems, 2nd Ed., Computer Science Press, Rockville, MD, 1982.Google Scholar