Formula size games for modal logic and mu-calculus

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We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic FO and (basic) modal logic ML⁠. We also present a version of the game for the modal μ-calculus Lμ and show that FO is also non-elementarily more succinct than Lμ⁠.
Original languageEnglish
Article numberexz025
Pages (from-to)1311-1344
Issue number8
Publication statusPublished - 2019
Publication typeA1 Journal article-refereed


  • Succinctness
  • bisimulation invariant first-order logic
  • formula size game
  • modal logic
  • modal mu-calculus
  • succinctness

Publication forum classification

  • Publication forum level 2


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