The Model Of Description Logic And It's Modal Expansion

In this paper, we mainly discuss the model of description logic and its modal extension.We achive these gools in two ways, one is discussing the condition which make cyclic ALCN—Tbox have models, and pointing out the mistake of proposition 2.9 in reference [1],furthermore , we also give some modification of proposition 2.9; the other one is modal expanding the description logicALC , we call the modal expansion of ALC as MALC ,then we define the syntax and semantic of MALC, at last,we discuss different ways to construct the model of MALC.The proposition 2.9 of reference [1] is as following(in this paper ,we use proposition 2.9 to note the proposition 2.9 of reference [1] ): LetT be a terminology such that each cycle in GT contains an even number of negative arcs , thenT is monotone.The main contributions of this paper are divided into the following three parts:(1) we have pointed out the mistake of proposition 2.9 and give some modification of that proposition, and got some condition which make the conclusion of proposition 2.9 hold.(2) The modal expansion of description logic are defined in a general sense, then we define the syntax and semantics of MALC.(3) We define the disjoint unions model and generated submodels of MALC,at the end ,some properties of disjoint unions model and generated submodels of MALC are proved.The key conclusions of this paper are following:Theorem 5.2.6 If all name symbol in definition about concept of a TboxT are occurs in the scope of an even number of negations.Then, T is monotone, what's follow is T exist greatest fixpont model and least fixpoint model.Theorem 5.2.7 If T is a Tbox,and the condition of proposition 2.9 is hold, we useΛto represent the set of all models aboutT .Then, the mapping which are defined onΛis monotone,it is followed that there exist greatest foxpoint model and least fixpoint model of TboxT inΛ.Theorem 6.1.2 Let M ibe a model of modal description logic's models.Then,for each formula , for each element w of M i, we have M i,w , iff ∪i∈n Mi,w .Theorem 6.2.5 Let M =(W , ,D,I )and M 1=( W1 , 1,D1, I1 )are two models of modal description logic MALC, M 1is a generated submodel of M . Then, for each formula , for each element w of M 1,we have M 1,w iff M ,w .Theorem 6.3.2 Let M =(W , ,D,I )and M 1=( W1 , 1,D1, I1 )are two models of modal description logicMALC, M i= ( Wi , 1,D1, I i) ( i∈n)are models of MALC.Then:(1) for each i∈n, for each element wof M i,we have M ,w∪i∈n Mi,w;(2) if M 1is a generated submodel of M .Then, for each element wof M 1,we have M 1,w M ,w.