Three Variables Suffice for Real-Time Logic

  • Timos Antonopoulos
  • Paul Hunter
  • Shahab Raza
  • James Worrell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9034)


A natural framework for real-time specification is monadic first-order logic over the structure (ℝ, < , + 1)—the ordered real line with unary + 1 function. Our main result is that (ℝ, < , + 1) has the 3-variable property: every monadic first-order formula with at most 3 free variables is equivalent over this structure to one that uses 3 variables in total. As a corollary we obtain also the 3-variable property for the structure (ℝ, < ,f) for any fixed linear function f:ℝ → ℝ. On the other hand, we exhibit a countable dense linear order (E, < ) and a bijection f:E → E such that (E, < ,f) does not have the k-variable property for any k.


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  1. 1.
    Dawar, A.: How many first-order variables are needed on finite ordered structures? In: We Will Show Them! Essays in Honour of Dov Gabbay, vol. 1, pp. 489–520. College Publications (2005)Google Scholar
  2. 2.
    Gabbay, D.M.: Expressive functional completeness in tense logic. In: Mönnich, U. (ed.) Aspects of Philosophical Logic, pp. 91–117. Reidel, Dordrecht (1981)CrossRefGoogle Scholar
  3. 3.
    Gabbay, D.M., Pnueli, A., Shelah, S., Stavi, J.: On the temporal basis of fairness. In: POPL, pp. 163–173. ACM Press (1980)Google Scholar
  4. 4.
    Grohe, M., Schweikardt, N.: The succinctness of first-order logic on linear orders. Logical Methods in Computer Science 1(1) (2005)Google Scholar
  5. 5.
    Hirshfeld, Y., Rabinovich, A.: Continuous time temporal logic with counting. Inf. Comput. 214, 1–9 (2012)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Hirshfeld, Y., Rabinovich, A.M.: Timer formulas and decidable metric temporal logic. Inf. Comput. 198(2), 148–178 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hodges, W.: A Shorter Model Theory. Cambridge University Press, New York (1997)MATHGoogle Scholar
  8. 8.
    Hodkinson, I., Simon, A.: The k-variable property is stronger than H-dimension k. Journal of Philosophical Logic 26(1), 81–101 (1997)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Hunter, P.: When is metric temporal logic expressively complete? In: CSL. LIPIcs, vol. 23, pp. 380–394. Schloss Dagstuhl (2013)Google Scholar
  10. 10.
    Hunter, P., Ouaknine, J., Worrell, J.: Expressive completeness for metric temporal logic. In: LICS, pp. 349–357. IEEE Computer Society Press (2013)Google Scholar
  11. 11.
    Immerman, N.: Upper and lower bounds for first order expressibility. J. Comput. Syst. Sci. 25(1), 76–98 (1982)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Immerman, N., Kozen, D.: Definability with bounded number of bound variables. Inf. Comput. 83(2), 121–139 (1989)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kamp, H.: Tense Logic and the Theory of Linear Order. PhD thesis, University of California (1968)Google Scholar
  14. 14.
    Poizat, B.: Deux ou trois choses que je sais de L n. J. Symb. Log. 47(3), 641–658 (1982)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Rossman, B.: On the constant-depth complexity of k-clique. In: STOC, pp. 721–730. ACM (2008)Google Scholar
  16. 16.
    Venema, Y.: Expressiveness and completeness of an interval tense logic. Notre Dame Journal of Formal Logic 31(4), 529–547 (1990)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Timos Antonopoulos
    • 1
  • Paul Hunter
    • 2
  • Shahab Raza
    • 1
  • James Worrell
    • 1
  1. 1.Department of Computer ScienceOxford UniversityOxfordUK
  2. 2.Département d’InformatiqueUniversité Libre de BruxellesBruxellesBelgium

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