Advertisement

A Uniform Substitution Calculus for Differential Dynamic Logic

  • André Platzer
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 \(\mathsf {d}\mathcal {L}\).

Keywords

Free Variable Function Symbol Predicate Symbol Static Semantic Axiom Schema 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgment

I thank the anonymous reviewers for their helpful feedback.

References

  1. 1.
    Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5(2), 56–68 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Church, A.: Introduction to Mathematical Logic, vol. I. Princeton University Press, Princeton (1956)zbMATHGoogle Scholar
  3. 3.
    Henkin, L.: Banishing the rule of substitution for functional variables. J. Symb. Log. 18(3), 201–208 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reas. 41(2), 143–189 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle 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

Personalised recommendations