Journal of Logic, Language and Information

, Volume 17, Issue 2, pp 217–227

Inessential Features, Ineliminable Features, and Modal Logics for Model Theoretic Syntax



While monadic second-order logic (MSO) has played a prominent role in model theoretic syntax, modal logics have been used in this context since its inception. When comparing propositional dynamic logic (PDL) to MSO over trees, Kracht (1997) noted that there are tree languages that can be defined in MSO that can only be defined in PDL by adding new features whose distribution is predictable. He named such features “inessential features”. We show that Kracht’s observation can be extended to other modal logics of trees in two ways. First, we demonstrate that for each stronger logic, there exists a tree language that can only be defined in a weaker logic with inessential features. Second, we show that any tree language that can be defined in a stronger logic, but not in some weaker logic, can be defined with inessential features. Additionally, we consider Kracht’s definition of inessential features more closely. It turns out that there are features whose distribution can be predicted, but who fail to be inessential in Kracht’s sense. We will look at ways to modify his definition.


Model theoretic syntax Modal logic Tree automata 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Afanasiev L., Blackburn P., Dimitriou I., Gaiffe B., Goris E., Marx M., de Rijke M. (2005) PDL for ordered trees. Journal of Applied Non-Classical Logic 15(2): 115–135CrossRefGoogle Scholar
  2. Blackburn P., Meyer-Viol W. (1994). Linguistics, logic and finite trees. Logic Journal of the IGPL, 2(1): 3–29CrossRefGoogle Scholar
  3. Cornell T., Rogers J. (2000). Model theoretic syntax. In: Cheng L.L.-S., Sybesma R. (eds) The GLOT international state-of-the-article book. Berlin, de Gruyter, pp. 177–198Google Scholar
  4. Courcelle, B. (1990). Graph rewriting: An algebraic and logic approach. In: J. van Leeuwen (Ed.), Handbook of theoretical computer science (Vol. B, Chapt. 5, pp. 193–242). Elsevier.Google Scholar
  5. Gécseg, F., & Steinby, M. (1997). Tree languages. In: G. Rozenberg, & A. Salomaa (Eds.), Handbook of formal languages (Vol. 3, pp. 1–68). Springer.Google Scholar
  6. Kamp, H. (1968). Tense logic and the theory of linear order. Ph.D. thesis, University of California, Los Angeles.Google Scholar
  7. Kracht M. (1997). Inessential features. In: Lecomte A., Lamarche F., Perrier G. (eds) Logical aspects of computational linguistics. Berlin, Springer, pp. 43–62CrossRefGoogle Scholar
  8. Kracht M. (1999). Tools and techniques in modal logic. Amsterdam, North-HollandCrossRefGoogle Scholar
  9. Kracht M. (2001). Logic and syntax—a personal perspective. In: Zakharyaschev M., Segerberg K., de Rijke M., Wansing H. (eds) Advances in modal logic (Vol. 2). Stanford, CSLI Publications, pp. 337–366Google Scholar
  10. Kracht M. (2003). The mathematics of language. Berlin, de GruyterGoogle Scholar
  11. Moschovakis, Y. (1974). Elementary induction on abstract structures. North-Holland Publishing Company.Google Scholar
  12. Moss L.S., Tiede H.-J. (2006). Applications of modal logic in linguistics. In: Blackburn P., van Benthem J., Wolter F. (eds) Handbook of modal logic. Amsterdam, Elsevier, pp. 1031–1076Google Scholar
  13. Palm A. (1997). Transforming tree constraints into formal grammars. Berlin, Akademische Verlagsgesellschaft Aka GmbHGoogle Scholar
  14. Potthoff A. (1994). Modulo-counting quantifiers over finite trees. Theoretical Computer Science 126(1): 97–112CrossRefGoogle Scholar
  15. Rogers J. (1997). Strict LT2: Regular :: local : recognizable. In: Retoré C. (eds) Logical aspects of computational linguistics (Nancy, 1996). Berlin, Springer, pp. 366–385CrossRefGoogle Scholar
  16. Rogers J. (1998). A descriptive approach to language-theoretic complexity. Stanford, CSLI PublicationsGoogle Scholar
  17. Schlingloff, B.-H. (1992). On the expressive power of modal logics on trees. In: A. Nerode & M. A. Taitslin (Eds.), Logical foundations of computer science—Tver ’92, Second International Symposium, Tver, Russia, July 20–24, 1992, Proceedings (pp. 441–451). Berlin: Springer-Verlag.Google Scholar
  18. Thatcher J.W. (1967). Characterizing derivation trees of context-free grammars through a generalization of finite automata theory. Journal of Computer and System Sciences 1, 317–322Google Scholar
  19. Thatcher J.W., Wright J.B. (1968). Generalized finite automata theory with an application to a decision problem of second-order logic. Mathematical Systems Theory 2, 57–81CrossRefGoogle Scholar
  20. Volger H. (1999). Principle languages and principle based parsing. In: Kolb H.-P., Mönnich U. (eds) The mathematics of syntactic structure. Berlin, de Gruyter, pp. 83–111Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceIllinois Wesleyan UniversityBloomingtonUSA

Personalised recommendations