Abstract
This article introduces a relatively complete proof calculus for differential dynamic logic (dL) 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 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 calculus adopts only a finite number of ordinary dLformulas as axioms, which uniform substitutions instantiate soundly. The static semantics of differential dynamic logic and the soundness-critical restrictions it imposes on proof steps is captured exclusively in uniform substitutions and variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this article introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivatives as first-class axioms to reason about differential equations axiomatically. The resulting axiomatization of differential dynamic logic is proved to be sound and relatively complete.
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References
Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5(2), 56–68 (1940)
Church, A.: Introduction to Mathematical Logic, vol. I. Princeton University Press, Princeton, NJ (1956)
Cimatti, A., Roveri, M., Tonetta, S.: HRELTL: a temporal logic for hybrid systems. Inf. Comput. 245, 54–71 (2015). doi:10.1016/j.ic.2015.06.006
Davoren, J.M., Nerode, A.: Logics for hybrid systems. IEEE 88(7), 985–1010 (2000)
Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. J. Autom. Reas. 31(1), 33–72 (2003)
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). doi:10.1007/978-3-319-21401-6_36
Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press, Cambridge, MA (2000)
Henkin, L.: Banishing the rule of substitution for functional variables. J. Symb. Log. 18(3), 201–208 (1953)
Hughes, G.E., Cresswell, M.J.: A New Introduction to Modal Logic. Routledge, London (1996)
Liu, J., Lv, J., Quan, Z., Zhan, N., Zhao, H., Zhou, C., Zou, L.: A calculus for hybrid CSP. In: Ueda, K. (ed.) APLAS, LNCS, vol. 6461, pp. 1–15. Springer, Berlin (2010). doi:10.1007/978-3-642-17164-2_1
Pfenning, F.: Logical frameworks. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 1063–1147. MIT Press, Cambridge, MA (2001)
Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reas. 41(2), 143–189 (2008). doi:10.1007/s10817-008-9103-8
Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010). doi:10.1093/logcom/exn070
Platzer, A.: The complete proof theory of hybrid systems. In: LICS, pp. 541–550. IEEE (2012). doi:10.1109/LICS.2012.64
Platzer, A.: The structure of differential invariants and differential cut elimination. Log. Meth. Comput. Sci. 8(4), 1–38 (2012). doi:10.2168/LMCS-8(4:16)2012
Platzer, A.: Differential game logic. ACM Trans. Comput. Log. 17(1), 1:1–1:51 (2015). doi:10.1145/2817824
Platzer, A.: Differential Hybrid Games. CoRR arXiv:1507.04943 (2015)
Platzer, A.: A uniform substitution calculus for differential dynamic logic. In: Felty, A., Middeldorp, A. (eds.) CADE, LNCS, vol. 9195, pp. 467–481. Springer, Berlin (2015). doi:10.1007/978-3-319-21401-6_32
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). doi:10.1007/978-3-540-71070-7_15
Rice, H.G.: Classes of recursively enumerable sets and their decision problems. Trans. AMS 89, 25–59 (1953)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)
Walter, W.: Analysis 1. Springer, Berlin (1985)
Walter, W.: Analysis 2, 4th edn. Springer, Berlin (1995)
Walter, W.: Ordinary Differential Equations. Springer, Berlin (1998)
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An extended abstract has appeared at CADE [18].
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Platzer, A. A Complete Uniform Substitution Calculus for Differential Dynamic Logic. J Autom Reasoning 59, 219–265 (2017). https://doi.org/10.1007/s10817-016-9385-1
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DOI: https://doi.org/10.1007/s10817-016-9385-1