The Instructor’s Guide to Real Induction

Clark, Pete L.

doi:10.1080/0025570x.2019.1549902

Cited by 10 publications

(19 citation statements)

References 26 publications

(17 reference statements)

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“…Similar to axiom RI, the axiom RI& internalizes the real induction principle [6] but also accounts for an arbitrary domain constraint Q. Its RHS conjuncts labeled a and b correspond to 1 and 2 in Def.…”

Section: Resultsmentioning

confidence: 99%

“…Since x ∈ S, this implies that x + ϵ is a greater lower bound of S than x, contradiction. Proposition 5.4 is based on the completeness of the reals [6] and applies to compact intervals [a, b] of R. Applying it to the time axis of ODE solutions yields real induction along solutions of di erential equations. For brevity, only the real induction axiom for systems without evolution domain constraints is presented here, leaving the general version to Appendix A.1, since evolution domains are de nable in dL [30].…”

Section: Real Inductionmentioning

confidence: 99%

“…axiom follows from the real induction principle[6] and the Picard-Lindelöf theorem [46, §10.VI].Real induction axiom RI can be understood as ∀ [. .…”

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confidence: 99%

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Differential Equation Invariance Axiomatization

Platzer

Tan

2020

J. ACM

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This article proves the completeness of an axiomatization for differential equation invariants described by Noetherian functions. First, the differential equation axioms of differential dynamic logic are shown to be complete for reasoning about analytic invariants. Completeness crucially exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. Cleverly chosen differential ghosts are the proof-theoretical counterpart of dark matter. They create a new hypothetical state, whose relationship to the original state variables satisfies invariants that did not exist before. The reflection of these new invariants in the original system then enables its analysis. An extended axiomatization with existence and uniqueness axioms is complete for all local progress properties, and, with a real induction axiom, is complete for all semianalytic invariants. This parsimonious axiomatization serves as the logical foundation for reasoning about invariants of differential equations. Indeed, it is precisely this logical treatment that enables the generalization of completeness to the Noetherian case.

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Section: Resultsmentioning

confidence: 99%

Section: Real Inductionmentioning

confidence: 99%

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Differential Equation Invariance Axiomatization

Platzer

Tan

2020

J. ACM

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“…The principle of continuous induction introduced in [3] (see also [4]) is equivalent to the following statement: where ϕ is an arbitrary formula. ✧ Continuous induction in [10] (see also [9] and [4]) is equivalent to the following:…”

Section: Continuous Induction Formalized In First-order Logicmentioning

confidence: 99%

First-Order Continuous Induction and a Logical Study of Real Closed Fields

Salehi

Zarza

2019

Bull. Iran. Math. Soc.

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Over the last century, the principle of "induction on the continuum" has been studied by different authors in different formats. All of these different readings are equivalent to one of the three versions that we isolate in this paper. We also formalize those three forms (of "continuous induction") in first-order logic and prove that two of them are equivalent and sufficiently strong to completely axiomatize the first-order theory of the real closed (ordered) fields. We show that the third weaker form of continuous induction is equivalent with the Archimedean property. We study some equivalent axiomatizations for the theory of real closed fields and propose a firstorder scheme of the fundamental theorem of algebra as an alternative axiomatization for this theory (over the theory of ordered fields).

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“…Our final axiom is based on the real induction principle [3]. It internalizes the topological properties of solutions.…”

Section: Real Inductionmentioning

confidence: 99%

Differential Equation Axiomatization

Platzer

Tan

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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We prove the completeness of an axiomatization for differential equation invariants. First, we show that the differential equation axioms in differential dynamic logic are complete for all algebraic invariants. Our proof exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. Cleverly chosen differential ghosts are the proof-theoretical counterpart of dark matter. They create new hypothetical state, whose relationship to the original state variables satisfies invariants that did not exist before. The reflection of these new invariants in the original system then enables its analysis.We then show that extending the axiomatization with existence and uniqueness axioms makes it complete for all local progress properties, and further extension with a real induction axiom makes it complete for all real arithmetic invariants. This yields a parsimonious axiomatization, which serves as the logical foundation for reasoning about invariants of differential equations. Moreover, our results are purely axiomatic, and so the axiomatization is suitable for sound implementation in foundational theorem provers.

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The Instructor’s Guide to Real Induction

Cited by 10 publications

References 26 publications

Differential Equation Invariance Axiomatization

Differential Equation Invariance Axiomatization

First-Order Continuous Induction and a Logical Study of Real Closed Fields

Differential Equation Axiomatization

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