An object-algebra is presented

O N E OF THE GREATEST DIFFIculties in developing an expert system is knowledge acquisition, the process of building the knowledge base. A knowledge base might be incomplete or inconsistent from the start, since human experts are not prepared to provide all the knowledge needed in one complete and consistent chunk. There might be cases that the expert has not considered, or items (rules, for example) that need to be rephrased. Even after the knowledge base has been built, its maintenance usually requires a complete validation if any item is modified, removed, or added. Therefore, validation is considered a mandatory step in developing knowledge-based systems.'An expert system cannot be tested, even on simple cases, until much of its knowledge base is encoded. Regardless of how an expert system is developed, its builders can profit from a systematic check of the knowledge base without having to gather extensive data for test runs, even before the full reasoning mechanism is functioning. This verification can be achieved by developing a program to check the knowledge base for consistency and completeness.2Our proposed method for verifying knowledge bases is based on the unification of rules.' One characteristic that T H i s METHOD CHECKS MVOWLEDGE BASE SYSTEMS FOR COMPLETENESS AND CONSISTENCY BY GENERATlNG INFERRED R U U S AND THEN CONSlDERlNG THEIR EFFECTS IN Adistinguishes our approach from other verification tools is that it infers some of the rules that are not explicitly given in the rule base and considers their effect in the verification process. Our method can determine conflicting, redundant, subsumed, circular, and dead-end rules; redundant If conditions in rules; and cycles and contradictions within rules. We have implemented our method in acomputer program called UVT (for unification-based verification tool) and tested it on sample knowledge bases. Rules for knowledge representationWhen planning an expert system, developers must decide on a knowledge VERlFlCATlON PROCESS. "w representation scheme that is most suitable to the application. We chose a rule-based representation scheme because of the modularity it provides and the simple, uniform interpretive procedure that is often sufficient in rule-based systems. Rule-based representation is also easy to learn and use. Our method assumes that the knowledge base rules have only one literal in their consequents and aconjunctionofliterals in their antecedents. A literal is either a predicate or the negation of a predicate. A predicate has a name and a finite number of arguments, which can be variables or constants. Although most expert systems use certainty factors associated with rules to handle uncertainty, our method does not depend on certainty factors. Our verification method is also independent of the inference mechanism to be used with the

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Badora 2002 proved the following stability result. Let ε and δ be nonnegative real numbers, then for every mapping f of a ring R onto a Banach algebra B satisfying ||f x y − f x − f y || ≤ ε and ||f x • y − f x f y || ≤ δ for all x, y ∈ R, there exists a unique ring homomorphism h : R → B such that ||f x − h x || ≤ ε, x ∈ R. Moreover, b • f x − h x 0, f x − h x • b 0, for all x ∈ R and all b from the algebra generated by h R. In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.