Topological perspective on the hybrid proof rules (bibtex)

by Balder ten Cate and Tadeusz Litak

Abstract:

We consider the non-orthodox proof rules of hybrid logic from the viewpoint of topological semantics. Topological semantics is more general than Kripke semantics. We show that the hybrid proof rule BG is topologically not sound. Indeed, among all topological spaces the BG rule characterizes those that can be represented as a Kripke frame (i.e., the Alexandroff spaces). We also demonstrate that, when the BG rule is dropped and only the Name rule is kept, one can prove a general topological completeness result for hybrid logics axiomatized by pure formulas. Finally, we indicate some limitations of the topological expressive power of pure formulas. All results generalize to neighborhood frames.

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Reference:

Balder ten Cate and Tadeusz Litak: Topological perspective on the hybrid proof rules, In Electronic Notes in Theoretical Computer Science, 174(6), pp. 79–94, 2007. Proceedings of the International Workshop on Hybrid Logic HyLo 2006

Bibtex Entry:

@article{tCateL07:entcs,
  title={Topological perspective on the hybrid proof rules},
  journal={Electronic Notes in Theoretical Computer Science},
  volume={174},
  number={6},
  pages={79--94},
  year={2007},
  publisher={Elsevier},
  keywords={conf},
  note = "Proceedings of the International Workshop on Hybrid Logic HyLo 2006",
issn = "1571-0661",
doi = "http://dx.doi.org/10.1016/j.entcs.2006.11.027",
url = "http://www.sciencedirect.com/science/article/pii/S1571066107002447",
author = "Balder ten Cate and Tadeusz Litak",
keywords = "hybrid logic",
keywords = "proof rules",
keywords = "topological spaces ",
abstract = "We consider the non-orthodox proof rules of hybrid logic from the viewpoint of topological semantics. Topological semantics is more general than Kripke semantics. We show that the hybrid proof rule BG is topologically not sound. Indeed, among all topological spaces the BG rule characterizes those that can be represented as a Kripke frame (i.e., the Alexandroff spaces). We also demonstrate that, when the BG rule is dropped and only the Name rule is kept, one can prove a general topological completeness result for hybrid logics axiomatized by pure formulas. Finally, we indicate some limitations of the topological expressive power of pure formulas. All results generalize to neighborhood frames."
}