Normal conditions for inference relations and injective models

Zhu, Zhaoda; Xian, Xiao; Zhou, Yong; Zhu, Wei

doi:10.1016/s0304-3975(03)00241-x

Cited by 5 publications

(27 citation statements)

References 16 publications

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“…We use ( ) to denote the class of all preferential models for coming from . Definition 6.1 (Zhu et al [19]). Let |∼ be an inference relation in and 0 a sublanguage of , the reduct of |∼ with respect to 0 is |∼ ∩ (Form( 0 )) 2 and denoted by |∼ ⇓ 0 .…”

Section: Some Model-theoretic Results About Preferential Modelsmentioning

confidence: 99%

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Similarity between preferential models

Zhu¹

2006

Theoretical Computer Science

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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.

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Section: Some Model-theoretic Results About Preferential Modelsmentioning

confidence: 99%

“…Let |∼ be an inference relation in and 0 a sublanguage of , the reduct of |∼ with respect to 0 is |∼ ∩ (Form( 0 )) 2 and denoted by |∼ ⇓ 0 . Definition 6.2 (Zhu et al [19]). Let W = S, l, ≺ be a preferential model for and 0 a sublanguage of .…”

Section: Some Model-theoretic Results About Preferential Modelsmentioning

confidence: 99%

Similarity between preferential models

Zhu¹

2006

Theoretical Computer Science

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“…However, we do not know a characterization (in terms of postulates) of explanatory relations representable by injective models. The analogous question for consequence relations turned out to be difficult (see [13,21,24,25]). …”

Section: Injective Representationmentioning

confidence: 99%

“…The distinction between injective and non-injective representations also occurs in the study of consequence relations. The seminal KLM paper [16] dealt with the general case and Freund [13] studied the injective case (see also [21,24,25]). …”

Section: Introductionmentioning

confidence: 99%

Representation theorems for explanatory reasoning based on cumulative models

Diaz

Uzcátegui

2008

Journal of Applied Logic

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We show several representation theorems for explanatory reasoning based on cumulative models. An explanatory process is given by a binary relation £ between formulas in a propositional language where the intended meaning of α £ γ is "γ is a preferred explanation of α". To each cumulative model E (a variation of those studied by Kraus, Lehmann and Magidor) is associated an explanatory relation £ E . We show results of the following type: An explanatory relation £ satisfies certain logical postulates iff it coincides with £ E for some cumulative model E of the appropriate type.

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“…It brings up an interesting theoretical problem: what kind of injective inference relations may be characterized by postulates of existent types? The first author and his collaborators have done some tentative work on this problem in [21]. To this end, a notion of normal condition is introduced.…”

Section: Introductionmentioning

confidence: 99%

A characterization theorem for injective model classes axiomatized by general rules

Zhu

Zhang

2006

Theoretical Computer Science

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We continue the work in Zhu et al. [Normal conditions for inference relations and injective models, Theoret. Comput. Sci. 309 (2003) 287-311]. A class of strict partial order structures (posets, for short) is said to be axiomatizable if the class of all injective preferential models from may be characterized in terms of general rules. This paper aims to obtain some characteristics of axiomatizable classes. To do this, a monadic second-order frame language is presented. The relationship between ℵ 0 -axiomatizability and second-order definability is explored. Then a notion of an admissible set is introduced. Based on this notion, we show that any preferential model, which does not contain any four-node substructure, must be a reduct of some injective model. Furthermore, we furnish a necessary and sufficient condition for the axiomatizability of classes of injective preferential models using general rules. Finally, we show that, in some sense, the class of all posets without any four-node substructure is the largest among axiomatizable classes.

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Normal conditions for inference relations and injective models

Cited by 5 publications

References 16 publications

Similarity between preferential models

Similarity between preferential models

Representation theorems for explanatory reasoning based on cumulative models

A characterization theorem for injective model classes axiomatized by general rules

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