Replacement Rules Research Articles

On the implementation of AMBIT, a language for symbol manipulation

A brief description is given of the implementation technique for the replacement rule of the AMBIT programming language. The algorithm for the “AMBIT scan” and an example of its application are given. The algorithm is applicable to other members of the family of string transformation languages of which AMBIT is a member, and it provides a rationale for the design of the AMBIT language.

Minimum Cost Replacement Under Markovian Deterioration

Conditions are derived under which optimal replacement rules for systems subject to Markovian deterioration and failure are given by control limit replacement rules. An efficient method of deriving such rules is given which is based on the simplex algorithm.

Replacement of periodically inspected equipment. (An optimal optional stopping rule)

AbstractThe problem of choosing an optimal replacement rule is considered for deteriorating equipment where the amount of deterioration is observed periodically. A “near” optimal rule is obtained. The main tool is an optimal stopping lemma related to semimartingales.

Extensions of the Lewis system S5

Dugundji has proved that none of the Lewis systems of modal logic, S1 through S5, has a finite characteristic matrix. The question arises whether there exist proper extensions of S5 which have no finite characteristic matrix. By an extension of a sentential calculus S, we usually refer to any system S′ such that every formula provable in S is provable in S′. An extension S′ of S is called proper if it is not identical with S. The answer to the question is trivially affirmative in case we make no additional restrictions on the class of extensions. Thus the extension of S5 obtained by adding to the provable formulas the additional formula p has no finite characteristic matrix (indeed, it has no characteristic matrix at all), but this extension is not closed under substitution—the formula q is not provable in it. McKinsey and Tarski have defined normal extensions of S4* by imposing three conditions. Normal extensions must be closed under substitution, must preserve the rule of detachment under material implication, and must also preserve the rule that if α is provable then ~◊~α is provable. McKinsey and Tarski also gave an example of an extension of S4 which satisfies the first two of these conditions but not the third. One of the results of this paper is that every extension of S5 which satisfies the first two of these conditions also satisfies the third, and hence the above definition of normal extension is redundant for S5. We shall therefore limit the extensions discussed in this paper to those which are closed under substitution and which preserve the rule of detachment under material implication. These extensions we shall call quasi-normal. The class of quasi-normal extensions of S5 is a very broad class and actually includes all extensions which are likely to prove interesting. It is easily shown that quasi-normal extensions of S5 preserve the rules of replacement, adjunction, and detachment under strict implication. It is the purpose of this paper to prove that every proper quasi-normal extension of S5 has a finite characteristic matrix and that every quasi-normal extension of S5 is a normal extension of S5 and to describe a simple class of characteristic matrices for S5.