The Countable Core Model Of RCC

Spatial Logic is an important area in the Computer Science and Artificial Intelligence. It rises rapidly and inevitably. In the last few years, many strong research groups have been formed and governments in different countries have put huge amount of grants to support their own researches in this direction. Very few branches of science and technology can be in comparison with it in the sense of the scale and the speed of the development. The outcoming book: Handbook of Spatial Logic [1] that has been published in the year of 2007 (the book contains more than 1000 pages) is just an example to see its development.Among all existing formal systems of spatial logic, RCC(Region Connection Calculus) formal system solely has wide and deep impact. It is regarded as the most important system. To its semantics, mainly, the countable models, researchers both oversea and in China had paid a lot of attention and efforts to the study on these models. Among them: Li/ying, Düntsch, Wang and Mccloskey, Stell, etc... Their works are significant and valuable. But on the other hand, mathematically speaking, in the study on countable models, we still need a general theoretical framework, in this theoretical framework, one can reveal more new models and treat those models in a unified way.In this paper, we take a new angle to investigate all common countable models, among them, the best-known three models: Bω,Bx and B?. We first consider the connection relation C(x, y). As a subset of A×A, C(x, y) ? A×A. It is possible that some connection is redundant in the sense that if one deletes some connections from C, the remaining < B, C > is still a model of RCC. This idea leads us to go further, our effort has produced some interesting results. Our main contribution in this direction can classified as following:(ⅰ) A new type of models has been defined and its inductive construction has been found. We call this new type models as "Core-Models". Denoted as Bc. We have proved that for the other best-known three models Bω, Bx and B?, each of them can be obtained by extending a core-model inside themselves. The discovery of core model enables us to treat three models in a general framework. Our work have cleared some ambiguity existed in some literatures. Also, the finding of core models has opened a more space for the semantic study on the formal system RCC.(ⅱ) We have introduced the notion of formal contactness. The notion is definable by the first-order language of Boolean Contact Algebra. By using this notion, we also have proved that there is a core model Bc, Bc is not isomorphic with any one of the three existing models: Bω, Bx and B. It shows the notion of core models is truly extension of the known models.(ⅲ) The compactness property of models of RCC has been studied. We also give the details of the construction of a class of core models of linear ordered type: Bl. This type of models has significance in the study of applications of RCC.As we all know that only better semantics can produce better syntax. We think, our works have contributed to the foundation of better semantics study on RCC as well as to other formal systems of spatial logic.