Three Variables Suffice for Real-Time Logic

  • Timos AntonopoulosEmail author
  • 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.


Induction Hypothesis Linear Order Temporal Logic Atomic Formula Linear Temporal Logic 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Timos Antonopoulos
    • 1
    Email author
  • 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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