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Complete Algorithms for Algebraic Strongest Postconditions and Weakest Preconditions in Polynomial ODE’S

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SOFSEM 2018: Theory and Practice of Computer Science (SOFSEM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10706))

Abstract

A system of polynomial ordinary differential equations (ode’s) is specified via a vector of multivariate polynomials, or vector field, F. A safety assertion \(\psi \longrightarrow [F]\,\phi \) means that the system’s trajectory will lie in a subset \(\phi \) (the postcondition) of the state-space, whenever the initial state belongs to a subset \(\psi \) (the precondition). We consider the case when \(\phi \) and \(\psi \) are algebraic varieties, that is, zero sets of polynomials. In particular, polynomials specifying the postcondition can be seen as conservation laws implied by \(\psi \). Checking the validity of algebraic safety assertions is a fundamental problem in, for instance, hybrid systems. We consider generalized versions of this problem, and offer algorithms to: (1) given a user specified polynomial set P and a precondition \(\psi \), find the smallest algebraic postcondition \(\phi \) including the variety determined by the valid conservation laws in P (relativized strongest postcondition); (2) given a user specified postcondition \(\phi \), find the largest algebraic precondition \(\psi \) (weakest precondition). The first algorithm can also be used to find the weakest algebraic invariant of the system implying all conservation laws in P valid under \(\psi \). The effectiveness of these algorithms is demonstrated on a challenging case study from the literature.

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Notes

  1. 1.

    Provided the involved coefficients can be finitely represented, e.g. are rational.

  2. 2.

    Any elimination ordering [8] for the parameters \(a_i\) could as well be considered.

  3. 3.

    Note that linear expressions with a constant term, such as \(2+5a_1 +42a_2-3a_3\) are not allowed.

  4. 4.

    For instance, if \(\pi =(a_1+a_2)x_1+a_3x_2\) then \(\pi [v]=0\) corresponds to the constraints \(a_1=-a_2\) and \(a_3=0\).

  5. 5.

    Code and examples available at http://local.disia.unifi.it/boreale/papers/PrePost.py.

  6. 6.

    We could dispense with them by considering a complete template of degree 3.

  7. 7.

    For instance, one should compare the polynomial \(\psi _3=q^2-2 {\theta }(M/{I_{yy}})\), which is part of the invariant cluster in [10], with our polynomial \(- ( 1/2) q^2 + \theta (M/{I_{yy}}) + ( 1/2 )q_0^2\) in the second summand of \(\pi '\) above.

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Authors and Affiliations

  1. Dipartimento di Statistica, Informatica, Applicazioni (DiSIA) “G. Parenti”, Università di Firenze, Viale Morgagni 65, I-50134, Firenze, Italy

    Michele Boreale

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  1. Michele Boreale
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Correspondence to Michele Boreale .

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Editors and Affiliations

  1. Vienna University of Technology , Vienna, Austria

    A Min Tjoa

  2. ISAE-ENSMA, Chasseneuil-du-Poitou, France

    Ladjel Bellatreche

  3. Vienna University of Technology, Vienna, Austria

    Stefan Biffl

  4. Utrecht University, Utrecht, The Netherlands

    Jan van Leeuwen

  5. Academy of Sciences, Prague, Czech Republic

    Jiří Wiedermann

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Boreale, M. (2018). Complete Algorithms for Algebraic Strongest Postconditions and Weakest Preconditions in Polynomial ODE’S. In: Tjoa, A., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds) SOFSEM 2018: Theory and Practice of Computer Science. SOFSEM 2018. Lecture Notes in Computer Science(), vol 10706. Edizioni della Normale, Cham. https://doi.org/10.1007/978-3-319-73117-9_31

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  • DOI: https://doi.org/10.1007/978-3-319-73117-9_31

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  • Publisher Name: Edizioni della Normale, Cham

  • Print ISBN: 978-3-319-73116-2

  • Online ISBN: 978-3-319-73117-9

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