Recently, in order to mix algebraic and logic styles of specification in a uniform framework, the notion of a logic labelled transition system (Logic LTS or LLTS for short) has been introduced and explored. A variety of constructors over LLTS, including usual process-algebraic operators, logic connectives (conjunction and disjunction) and standard temporal operators (always and unless), have been given. However, no attempt has made so far to develop general theory concerning (nested) recursive operations over LLTS and a few fundamental problems are still open. This paper intends to study this issue in pure process-algebraic style. A few fundamental properties, including precongruence and the uniqueness of consistent solutions of equations, will be established.

This paper introduces valuation structures associated with preferential models. Based on KLM valuation structures, we present a canonical approach to obtain injective preferential models for any preferential relation satisfying the property INJ, and give uniform proofs of representation theorems for injective preferential relations appeared in the literature. In particular, we show that, in any propositional language (finite or infinite), a preferential inference relation satisfies INJ if and only if it can be represented by a standard preferential model. This conclusion generalizes the result obtained by Freund. In addition, we prove that, when the language is finite, our framework is sufficient to establish a representation theorem for any injective relation.

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.

Recently, alternating transition systems are adopted to describe control systems with disturbances and their finite abstract systems. In order to capture the equivalence relation between these systems, a notion of alternating approximate bisimilarity is introduced. This paper aims to establish a modal characterization for alternating approximate bisimilarity. Moreover, based on this result, we provide a link between specifications satisfied by the samples of control systems with disturbances and their finite abstractions.

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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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