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A Method for Invariant Generation for Polynomial Continuous Systems

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Verification, Model Checking, and Abstract Interpretation (VMCAI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9583))

Abstract

This paper presents a method for generating semi-algebraic invariants for systems governed by non-linear polynomial ordinary differential equations under semi-algebraic evolution constraints. Based on the notion of discrete abstraction, our method eliminates unsoundness and unnecessary coarseness found in existing approaches for computing abstractions for non-linear continuous systems and is able to construct invariants with intricate boolean structure, in contrast to invariants typically generated using template-based methods. In order to tackle the state explosion problem associated with discrete abstraction, we present invariant generation algorithms that exploit sound proof rules for safety verification, such as differential cut (\({\text {DC}}\)), and a new proof rule that we call differential divide-and-conquer (\({\text {DDC}}\)), which splits the verification problem into smaller sub-problems. The resulting invariant generation method is observed to be much more scalable and efficient than the naïve approach, exhibiting orders of magnitude performance improvement on many of the problems.

This material is based upon work supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under grants EP/I010335/1 and EP/J001058/1, the National Science Foundation by NSF CAREER Award CNS-1054246, NSF EXPEDITION CNS-0926181, CNS-0931985 and DARPA FA8750-12-2-0291.

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Notes

  1. 1.

    A semi-algebraic set is a subset of \(\mathbb {R}^n\) characterized by a finite boolean combination of sets defined by polynomial equalities and inequalities.

  2. 2.

    In the sense of not having an explicit dependence on the time variable t.

  3. 3.

    Evolution constraints are often used to define operating modes in hybrid and cyber-physical systems (so-called mode, or location invariants in the parlance of hybrid automata [1, 13]).

  4. 4.

    Considering the continuous system \( \dot{\mathbf {x}} = f(\mathbf {x}) \ \& \ H\) as a program, the safety assertion \( \psi \rightarrow [ \dot{\mathbf {x}} = f(\mathbf {x}) \ \& \ H ] \ \phi \) expresses the (continuous) Hoare triple \( \{ \psi \} \ \dot{\mathbf {x}} = f(\mathbf {x}) \ \& \ H \ \{ \phi \}\).

  5. 5.

    All three regions are invariant sets in the terminology of dynamical systems [5, Chapter II].

  6. 6.

    expression simplified in Mathematica.

  7. 7.

    See http://homepages.inf.ed.ac.uk/s0805753/invgen for the problems.

  8. 8.

    The comparison was performed on an i5-3570 K CPU clocked at 3.40 GHz.

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Authors and Affiliations

  1. LFCS, School of Informatics, University of Edinburgh, Edinburgh, Scotland, UK

    Andrew Sogokon & Paul B. Jackson

  2. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

    Khalil Ghorbal & André Platzer

Authors
  1. Andrew Sogokon
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  2. Khalil Ghorbal
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  3. Paul B. Jackson
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  4. André Platzer
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Correspondence to Andrew Sogokon .

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  1. EPFL IC-DO, Lausanne, Switzerland

    Barbara Jobstmann

  2. Microsoft Research, Redmond, Washington, USA

    K. Rustan M. Leino

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Sogokon, A., Ghorbal, K., Jackson, P.B., Platzer, A. (2016). A Method for Invariant Generation for Polynomial Continuous Systems. In: Jobstmann, B., Leino, K. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2016. Lecture Notes in Computer Science(), vol 9583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49122-5_13

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