Theory of Computing Systems

, Volume 63, Issue 5, pp 1027–1048 | Cite as

Complete Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem

  • Nathanaël FijalkowEmail author
  • Pierre Ohlmann
  • Joël Ouaknine
  • Amaury Pouly
  • James Worrell
Part of the following topical collections:
  1. Special Issue on Theoretical Aspects of Computer Science (STACS 2017)


The Orbit Problem consists of determining, given a matrix A on \(\mathbb {Q}^{d}\), together with vectors x and y, whether the orbit of x under repeated applications of A can ever reach y. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s. In this paper, we are concerned with the problem of synthesising suitable invariants\(\mathcal {P} \subseteq \mathbb {R}^{d}\), i.e., sets that are stable under A and contain x but not y, thereby providing compact and versatile certificates of non-reachability. We show that whether a given instance of the Orbit Problem admits a semialgebraic invariant is decidable, and moreover in positive instances we provide an algorithm to synthesise suitable succinct invariants of polynomial size. Our results imply that the class of closed semialgebraic invariants is closure-complete: there exists a closed semialgebraic invariant if and only if y is not in the topological closure of the orbit of x under A.


Verification Algebraic computation Skolem Problem Orbit problem Invariants 



We would like to thank the reviewers for their very detailed and helpful comments, in particular pointing out a flaw in the complexity analysis in the first version of the paper.


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Authors and Affiliations

  1. 1.CNRSLaBRIBordeauxFrance
  2. 2.The Alan Turing Institute of data science and artificial intelligenceLondonUK
  3. 3.IRIFUniversité Paris Diderot - Paris 7ParisFrance
  4. 4.Max Planck Institute for Software Systems (MPI-SWS)Saarland Informatics CampusSaarbrückenGermany
  5. 5.Department of Computer ScienceOxford UniversityOxfordUK

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