A Uniform Substitution Calculus for Differential Dynamic Logic

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9195)

Abstract

This paper introduces a new proof calculus for differential dynamic logic (\(\mathsf {d}\mathcal {L}\)) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform substitutions make it possible to rely on axioms rather than axiom schemata, substantially simplifying implementations. Instead of subtle schema variables and soundness-critical side conditions on the occurrence patterns of variables, the resulting calculus adopts only a finite number of ordinary \(\mathsf {d}\mathcal {L}\) formulas as axioms. The static semantics of differential dynamic logic is captured exclusively in uniform substitutions and bound variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this paper introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivations as first-class axioms in \(\mathsf {d}\mathcal {L}\).

References

  1. 1.
    Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5(2), 56–68 (1940)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Church, A.: Introduction to Mathematical Logic, vol. I. Princeton University Press, Princeton (1956)MATHGoogle Scholar
  3. 3.
    Henkin, L.: Banishing the rule of substitution for functional variables. J. Symb. Log. 18(3), 201–208 (1953)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reas. 41(2), 143–189 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Platzer, A.: The complete proof theory of hybrid systems. In: LICS, pp. 541–550. IEEE (2012)Google Scholar
  7. 7.
    Platzer, A.: The structure of differential invariants and differential cut elimination. Log. Meth. Comput. Sci. 8(4), 1–38 (2012)MATHGoogle Scholar
  8. 8.
    Platzer, A.: Differential game logic. CoRR abs/1408.1980 (2014)Google Scholar
  9. 9.
    Platzer, A.: A uniform substitution calculus for differential dynamic logic. CoRR abs/1503.01981 (2015)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

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