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Automatic Pre- and Postconditions for Partial Differential Equations

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Quantitative Evaluation of Systems (QEST 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12289))

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Abstract

Based on a simple automata-theoretic and algebraic framework, we study equational reasoning for Initial Value Problems (IVPs) of polynomial Partial Differential Equations (PDEs). In order to represent IVPs in their full generality, we introduce stratified systems, where function definitions can be decomposed into distinct subsystems, focusing on different subsets of independent variables. Under a certain coherence condition, for such stratified systems we prove existence and uniqueness of formal power series solutions, which conservatively extend the classical analytic ones. We then give a—in a precise sense, complete—algorithm to compute weakest preconditions and strongest postconditions for such systems. To some extent, this result reduces equational reasoning on PDE initial value (and boundary) problems to algebraic reasoning. We illustrate some experiments conducted with a proof-of-concept implementation of the method.

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Notes

  1. 1.

    In general, we shall adopt for monomials the same notation we use for strings, as the context is sufficient to disambiguate. In particular, we overload the symbol \(\epsilon \) to denote both the empty string and the unit monomial. When \(X=\emptyset \), \(X^\otimes {\mathop {=}\limits ^{\triangle }}\{\epsilon \}\).

  2. 2.

    Real arithmetic expressions will be used as a meta-notation for polynomials: e.g. \((u+u_x+1)\cdot (x+u_y)\) denotes the polynomial \(xu+uu_y+xu_x+u_xu_y+x+u_y\).

  3. 3.

    Specifically, \(\psi _0(E)(\epsilon )=E(u_0)\) for each \(E\in \mathrm{I\!R}[u]\).

  4. 4.

    In fact, more is true: \(\rightarrow _H\) is terminating and confluent, so there is a unique H-normal form F s.t. \(E\rightarrow ^*_H F\). See [13, App. A]. Therefore the arbitrariness in Definition 7 is only apparent.

  5. 5.

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

  6. 6.

    Additional examples, concerning conservation laws and boundary problems, are reported in [13]. Code and examples are available at https://github.com/micheleatunifi/PDEPY/blob/master/PDE.py. Execution times reported here are for a Python Anaconda distribution running under Windows 10 on a Surface Pro laptop.

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Boreale, M. (2020). Automatic Pre- and Postconditions for Partial Differential Equations. In: Gribaudo, M., Jansen, D.N., Remke, A. (eds) Quantitative Evaluation of Systems. QEST 2020. Lecture Notes in Computer Science(), vol 12289. Springer, Cham. https://doi.org/10.1007/978-3-030-59854-9_15

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  • DOI: https://doi.org/10.1007/978-3-030-59854-9_15

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