Downward-directed transitive frames with universal relations

In this paper we identify modal logics of some bimodal Kripke frames corresponding to geometrical structures. Each of these frames is a set of 'geometrical' objects with some natural accessibility relation plus the universal relation. For these logics we present nite axiom systems and prove completeness. We also show that all these logics have the nite model property and are PSPACE-complete. To prove this, we show that under certain restrictions, adding the universal modality preserves 'good' properties of a monomodal logic.

[1]  Carsten Lutz,et al.  Modal Logics of Topological Relations , 2006, Log. Methods Comput. Sci..

[2]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[3]  Valentin B. Shehtman,et al.  Chronological Future Modality in Minkowski Spacetime , 2002, Advances in Modal Logic.

[4]  D. Holdstock.  Past, present--and future? , 2005, Medicine, conflict, and survival.

[5]  Frank Wolter,et al.  Solution to a Problem of Goranko and Passy , 1994, J. Log. Comput..

[6]  Edith Hemaspaandra,et al.  The Price of Universality , 1996, Notre Dame J. Formal Log..

[7]  Valentin B. Shehtman,et al.  Modal logics of domains on the real plane , 1983 .

[8]  Valentin Goranko,et al.  Using the Universal Modality: Gains and Questions , 1992, J. Log. Comput..

[9]  Robert Goldblatt.  Diodorean modality in Minkowski spacetime , 1980 .

[10]  Valentin B. Shehtman,et al.  "Everywhere" and "Here" , 1999, J. Appl. Non Class. Logics.

[11]  Ilya Shapirovsky,et al.  Modal logics of closed domains on Minkowski plane ★ , 2007, J. Appl. Non Class. Logics.

[12]  Valentin B. Shehtman,et al.  Modal Logics of Regions and Minkowski Spacetime , 2005, J. Log. Comput..

[13]  Ilya Shapirovsky.  On PSPACE-decidability in Transitive Modal Logic , 2004, Advances in Modal Logic.

[14]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .

[15]  A. Chagrov,et al.  Modal Logic (Oxford Logic Guides, vol. 35) , 1997 .

[16]  Yoav Shoham,et al.  A propositional modal logic of time intervals , 1991, JACM.

[17]  Y HalpernJoseph,et al.  A propositional modal logic of time intervals , 1991 .

[18]  Michael Zakharyaschev,et al.  Modal Logic , 1997, Oxford logic guides.