Two Variable vs. Linear Temporal Logic in Model Checking and Games

  • Michael Benedikt
  • Rastislav Lenhardt
  • James Worrell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6901)


Verification tasks have non-elementary complexity for properties of linear traces specified in first-order logic, and thus various limited logical languages are employed. In this paper we consider two restricted specification logics, linear temporal logic (LTL) and two-variable first-order logic (FO2). LTL is more expressive, but FO2 is often more succinct, and hence it is not clear which should be easier to verify. In this paper we take a comprehensive look at the issue, giving a comparison of verification problems for FO2, LTL, and the subset of LTL expressively equivalent to FO2, unary temporal logic (UTL). We give two logic-to-automata translations which can be used to give upper bounds for FO2 and UTL; we apply these to get new bounds for both non-deterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). We couple this with lower-bound arguments for FO2 and UTL. Our results give both a unified approach to understanding the behavior of FO2 and UTL, along with a nearly comprehensive picture of the complexity of verification for these logics.


Markov Chain Model Check Temporal Logic Markov Decision Process Linear Temporal Logic 
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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michael Benedikt
    • 1
  • Rastislav Lenhardt
    • 1
  • James Worrell
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordUK

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