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Hyper arrow structures. Arrow logics. III. (English) Zbl 0906.03017

Kracht, Marcus (ed.) et al., Advances in modal logic. Vol. 1. Selected papers of the 1st AiML conference, Free University of Berlin, Germany, October 1996. Stanford, CA: Center for the Study of Language and Information (CSLI). CSLI Lect. Notes. 87, 269-290 (1998).
Summary: [For Part II see M. Marx et al. (eds.), Arrow logic and multi-modal logic, 141-187 (1996; Zbl 0874.03027).]
The notion of hyper arrow structure of dimension \(n\) is introduced. This is a two-sorted system with a set of points, a set of arrows and \(n\) projection functions associating with each arrow a corresponding set of points. For \(n=1\) examples are Property Systems [the author, “logical analysis of positive and negative similarity relations in property systems”, in M. De Glas et al. (eds.), WOCFAI’91, First World Conf. on the Fundamentals of AI, 491-499 (1991)] and hypergraphs. For \(n=2\) this is a generalization of the notion of directed multi-graph such that each arrow has a set of beginnings and a set of ends. Information systems in the sense of Z. Pawlak [Rough sets. Theoretical aspects of reasoning about data (1991; Zbl 0758.68054)] are also examples of hyper arrow structures.
To each hyper arrow structure of dimension \(n\) a relational system over the set of arrows is associated, called a hyper arrow frame of dimension \(n\). Using a characterization theorem, an abstract definition of hyper arrow frames of dimension \(n\) is given, based on a special theory of filters and ideals, generalizing the Stone representation theory for distributive lattices. The corresponding modal logic, called Basic Hyper Arrow Logic of dimension \(n\), \(\text{BHAL}^n\), is introduced. A completeness theorem of \(\text{BHAL}^n\) with respect to hyper arrow frames of dimension \(n\) is proved which uses an intermediate nonstandard semantics. It is shown that \(\text{BHAL}^n\) is decidable, and that it enjoys the finite model property with respect to its nonstandard models.
For the entire collection see [Zbl 0897.00022].

MSC:

03B45 Modal logic (including the logic of norms)
03G25 Other algebras related to logic
03B70 Logic in computer science
68T27 Logic in artificial intelligence
05C99 Graph theory
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