Similarity between preferential models

Abstract

The notion of bisimulation is an important concept in process algebra and modern modal logic. This paper explores the notion of B-similarity, which is a kind of bisimulation between preferential models. We characterize the equivalence of preferential models in terms of B-similarity. However, this result is applicable only for preferential models of finite depth. To overcome this defect, we introduce a weak notion of similarity called M-similarity, and obtain a result corresponding to Hennessy-Milner Theorem and Keisler-Shelah's Isomorphism Theorem in modal logic and first-order logic, respectively. As its application, we investigate the expressive power of Boolean combinations of conditional assertions (BCA, for short), and prove that BCAs are the fragments of first-order language preserved under M-similarity. Moreover, we obtain a characterization for elementary classes defined by BCAs. A notion of first-order translation originating from modal logic plays an important role in this paper. In order to illustrate that first-order translation is a powerful tool in the study of nonmonotonic logic, some model-theoretic results about preferential models are proved based on this translation.

A characterization theorem for injective model classes axiomatized by general rules

“…Incidentally, a nice model-theoretical characterization for this fragment is obtained in [17]. On the other hand, the smoothness can also be expressed in first-order language.…”

“…We obtain a number of useful and interesting results concerning the above translation [17]. Here we list only ones related to this paper.…”

“…This subsection recalls some related definitions and results in [17]. To investigate the relationship between modal logic and first-order logic, modal logicians introduce a technology called standard translation [2], which plays an important role in establishing the correspondence theory [2].…”

“…To investigate the relationship between modal logic and first-order logic, modal logicians introduce a technology called standard translation [2], which plays an important role in establishing the correspondence theory [2]. In [17], in order to explore the notion of bisimulation for preferential models, we borrow this technology and introduce a first-order translation function described as follows.…”

“…However, another problem raises: how to use the above theorem in the context of preferential models? Fortunately, the notion of first-order translation is a bridge between preferential logic and first-order logic, and we can use this bridge to import results and ideas from first-order model theory to preferential logic [17]. Two related results are listed below.…”