About the Complete Axiomatization of Dynamic Extensions of Arrow Logic

  • Philippe BalbianiEmail author
Part of the Outstanding Contributions to Logic book series (OCTR, volume 17)


This paper is devoted to the proof of the completeness of deductive systems for dynamic extensions of arrow logic. These extensions are based on the relational constructs of composition and intersection. The proof of the completeness of our deductive systems uses the canonical model construction and the subordination model construction.


Information logics Arrow logics Dynamic logics Axiomatization and Completeness 



Special acknowledgement is heartly granted to Ewa Orłowska. Her research on rough set analysis, her use of modal logic as a general tool for the formalization of reasoning about incomplete information, the multifarious papers that she has written on that subject, her papers introducing modal logics such as DAL and NIL have exerted a profound influence on my research and a great deal of it was directly motivated and influenced by her ideas.


  1. Arsov, A. (1993). Completeness Theorems for Some Extensions of Arrow Logic (Master’s Dissertation, Sofia University).Google Scholar
  2. Arsov, A. & Marx, M. (1993). Basic arrow logic with relation algebraic operators. In P. de Dekker & M. Stokhof (Eds.), Proceedings of the 9th Amsterdam Colloquium (pp. 93–112). Amsterdam University, Institute for Logic, Language and Computation.Google Scholar
  3. Balbiani, P. (2003). Eliminating unorthodox derivation rules in an axiom system for iteration-free PDL with intersection. Fundamenta Informaticae, 56(3), 211–242.Google Scholar
  4. Balbiani, P. & Orłowska, E. (1999). A hierarchy of modal logics with relative accessibility relations. Journal of Applied Non-classical Logics, 9(2–3), 303–328.CrossRefGoogle Scholar
  5. Balbiani, P. & Vakarelov, D. (2001). Iteration-free PDL with intersection: A complete axiomatization. Fundamenta Informaticae45(3), 173–194.Google Scholar
  6. Balbiani, P. & Vakarelov, D. (2004). Dynamic extensions of arrow logic. Annals of Pure and Applied Logic, 127(1–3), 1–15.CrossRefGoogle Scholar
  7. Fariñas del Cerro, L. & Orłowska, E. (1985). DAL–A logic for data analysis. Theoretical Computer Science, 36, 251–264.CrossRefGoogle Scholar
  8. Demri, S. (2000). The nondeterministic information logic NIL is PSpace-complete. Fundamenta Informaticae, 42(3–4), 211–234.Google Scholar
  9. Demri, S. & Gabbay, D. M. (2000a). On modal logics characterized by models with relative accessibility relations: Part I. Studia Logica, 65(3), 323–353.CrossRefGoogle Scholar
  10. Demri, S. & Gabbay, D. M. (2000b). On modal logics characterized by models with relative accessibility relations: Part II. Studia Logica, 66(3), 349–384.CrossRefGoogle Scholar
  11. Demri, S. & Orłowska, E. (2002). Incomplete Information: Structure, Inference, Complexity. Monographs in Theoretical Computer Science. An EATCS series Berlin: Springer.CrossRefGoogle Scholar
  12. Hughes, G. & Cresswell, M. (1984). A Companion to Modal Logic. Methuen and Co.Google Scholar
  13. Marx, M. (1995). Algebraic Relativization and Arrow Logic (Doctoral Dissertation, Amsterdam University).Google Scholar
  14. Orłowska, E. (1984). Modal logics in the theory of information systems. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 42(1/2), 213–222.CrossRefGoogle Scholar
  15. Orłowska, E. (1985a). Logic of indiscernibility relations. In A. Skowron (Ed.), Proceedings of Computation Theory with 5th Symposium, 1984 (Vol. 208, pp. 177–186). Lecture Notes in Computer Science. Zaborów, Poland: Springer.Google Scholar
  16. Orłowska, E. (1985b). Logic of nondeterministic information. Studia Logica, 44(1), 91–100.CrossRefGoogle Scholar
  17. Orłowska, E. (1988). Kripke models with relative accessibility and their application to inferences from incomplete information. In G. Mirkowska & H. Rasiowa (Eds.), Mathematical Problems in Computation Theory (Vol. 21, pp. 329–339). Banach Centre Publications.Google Scholar
  18. Orłowska, E. (1990). Kripke semantics for knowledge representation logics. Studia Logica, 49(2), 255–272.CrossRefGoogle Scholar
  19. Orłowska, E. & Pawlak, Z. (1984). Representation of nondeterministic information. Theoretical Computer Science, 29, 27–39.CrossRefGoogle Scholar
  20. Pawlak, Z. (1981). Information systems theoretical foundations. Information Systems, 6(3), 205–218.CrossRefGoogle Scholar
  21. Vakarelov, D. (1992). A modal theory of arrows: Arrow logics I. In D. Pearce & G. Wagner (Eds.), Proceedings of Logics in AI, European Workshop, JELIA ’92 (Vol. 633, pp. 1–24). Lecture Notes in Computer Science. Berlin, Germany: Springer.Google Scholar
  22. Vakarelov, D. (1995). A duality between Pawlak’s knowledge representation systems and bi-consequence systems. Studia Logica, 55(1), 205–228.CrossRefGoogle Scholar
  23. Vakarelov, D. (1996). Many-dimensional arrow structures Arrow logics II. In M. Marx, L. Pólos, & M. Masuch (Eds.), Arrow Logic and Multi-modal Logic (pp. 141–187). Studies in Logic, Language and Information. Amsterdam: Center for the Study of Language and Information.Google Scholar
  24. Vakarelov, D. (1998). Information systems, similarity relations and modal logics. In E. Orłowska (Ed.), Incomplete Information: Rough Set Analysis (Vol. 13, pp. 492–550). Studies in Fuzziness and Soft Computing. Heidelberg: Springer-Physica Verlag.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institut de recherche en informatique de Toulouse, CNRS — Toulouse UniversityToulouse Cedex 9France

Personalised recommendations