Uniform Substitution for Differential Game Logic

  • André PlatzerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10900)


This paper presents a uniform substitution calculus for differential game logic ( Open image in new window ). Church’s uniform substitutions substitute a term or formula for a function or predicate symbol everywhere. After generalizing them to differential game logic and allowing for the substitution of hybrid games for game symbols, uniform substitutions make it possible to only use axioms instead of axiom schemata, thereby substantially simplifying implementations. Instead of subtle schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many axiom schema instances to sound ones, the resulting axiomatization adopts only a finite number of ordinary Open image in new window formulas as axioms, which uniform substitutions instantiate soundly. This paper proves soundness and completeness of uniform substitutions for the monotone modal logic Open image in new window . The resulting axiomatization admits a straightforward modular implementation of Open image in new window in theorem provers.


  1. 1.
    Bohrer, B., Rahli, V., Vukotic, I., Völp, M., Platzer, A.: Formally verified differential dynamic logic. In: Bertot, Y., Vafeiadis, V. (eds.) CPP. ACM (2017).
  2. 2.
    Church, A.: Introduction to Mathematical Logic. Princeton University Press, Princeton (1956)zbMATHGoogle Scholar
  3. 3.
    Fulton, N., Mitsch, S., Quesel, J.D., Völp, M., Platzer, A.: KeYmaera X: an axiomatic tactical theorem prover for hybrid systems. In: Felty, A., Middeldorp, A. (eds.) CADE. LNCS, vol. 9195, pp. 527–538. Springer, Berlin (2015). Scholar
  4. 4.
    Parikh, R.: Propositional game logic. In: FOCS, pp. 195–200. IEEE (1983).
  5. 5.
    Platzer, A.: Differential game logic. ACM Trans. Comput. Log. 17(1), 1:1–1:52 (2015). Scholar
  6. 6.
    Platzer, A.: A complete uniform substitution calculus for differential dynamic logic. J. Autom. Reas. 59(2), 219–265 (2017). Scholar
  7. 7.
    Platzer, A.: Differential hybrid games. ACM Trans. Comput. Log. 18(3), 19:1–19:44 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Platzer, A.: Uniform substitution for differential game logic. CoRR abs/1804.05880 (2018)Google Scholar
  9. 9.
    Platzer, A., Quesel, J.D.: KeYmaera: a hybrid theorem prover for hybrid systems. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR. LNCS, vol. 5195, pp. 171–178. Springer, Berlin (2008). Scholar
  10. 10.
    Quesel, J.-D., Platzer, A.: Playing hybrid games with KeYmaera. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 439–453. Springer, Heidelberg (2012). Scholar
  11. 11.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

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