@InProceedings{PattinsonSchroder08,
  author = {Dirk Pattinson and Lutz Schr{\"o}der},
  title = {Beyond Rank 1: Algebraic Semantics and Finite Models for Coalgebraic Logics},
  year = {2008},
  editor = {Roberto Amadio},
  booktitle = {Foundations of Software Science and Computation Structures (FOSSACS 2008)},
  publisher = {Springer},
  series = {Lecture Notes in Computer Science},
  volume = {4962},
  pages = {66-80},
  keywords = {Coalgebra modal logic frame conditions algebraic semantics quantitative uncertainty coalition logic},
  url = {http://dx.doi.org/10.1007/978-3-540-78499-9_6},
  comment = { <a href = "http://www8.informatik.uni-erlangen.de/~schroeder/papers/fmp-fap.pdf"> [preprint] </a>},
  abstract = {  Coalgebras provide a uniform framework for the semantics of a large
  class of (mostly non-normal) modal logics, including e.g. monotone
  modal logic, probabilistic and graded modal logic, and coalition
  logic, as well as the usual Kripke semantics of modal logic.  In
  earlier work, the finite model property for coalgebraic logics has
  been established w.r.t. the class of emph{all} structures
  appropriate for a given logic at hand; the corresponding modal
  logics are characterised by being axiomatised in rank 1, i.e.
  without nested modalities. Here, we extend the range of coalgebraic
  techniques to cover logics that impose global properties on their
  models, formulated as frame conditions with possibly nested
  modalities on the logical side (in generalisation of frame
  conditions such as symmetry or transitivity in the context of Kripke
  frames).
  We show that the finite model property for such logics follows from
  the finite algebra property of the associated class of complex
  algebras, and then investigate sufficient conditions for the finite
  algebra property to hold.  Example applications include extensions
  of coalition logic and logics of uncertainty and knowledge.
},
}