The Pseudo-Skolem Problem is Decidable
We study fundamental decision problems on linear dynamical systems in discrete time. We focus on pseudo-orbits, the collection of trajectories of the dynamical system for which there is an arbitrarily small perturbation at each step. Pseudo-orbits are generalizations of orbits in the topological theory of dynamical systems. We study the pseudo-orbit problem, whether a state belongs to the pseudo-orbit of another state, and the pseudo-Skolem problem, whether a hyperplane is reachable by an ε-pseudo-orbit for every ε. These problems are analogous to the well-studied orbit problem and Skolem problem on unperturbed dynamical systems. Our main results show that the pseudo-orbit problem is decidable in polynomial time and the Skolem problem on pseudo-orbits is decidable. The former extends the seminal result of Kannan and Lipton from orbits to pseudo-orbits. The latter is in contrast to the Skolem problem for linear dynamical systems, which remains open for proper orbits.
Pseudo-orbits
Orbit problem
Skolem problem
linear dynamical systems
Theory of computation~Design and analysis of algorithms
34:1-34:21
Regular Paper
Julian
D'Costa
Julian D'Costa
Department of Computer Science, University of Oxford, UK
https://orcid.org/0000-0003-2610-5241
emmy.network foundation under the aegis of the Fondation de Luxembourg.
Toghrul
Karimov
Toghrul Karimov
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
https://orcid.org/0000-0002-9405-2332
Rupak
Majumdar
Rupak Majumdar
Max Planck Institute for Software Systems, Kaiserslautern, Germany
https://orcid.org/0000-0003-2136-0542
DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
Joël
Ouaknine
Joël Ouaknine
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
https://orcid.org/0000-0003-0031-9356
ERC grant AVS-ISS (648701), and DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science). Joël Ouaknine is also affiliated with Keble College, Oxford as http://emmy.network/ Fellow.
Mahmoud
Salamati
Mahmoud Salamati
Max Planck Institute for Software Systems, Kaiserslautern, Germany
https://orcid.org/0000-0003-3790-3935
Sadegh
Soudjani
Sadegh Soudjani
Newcastle University, Newcastle upon Tyne, UK
https://orcid.org/0000-0003-1922-6678
James
Worrell
James Worrell
Department of Computer Science, University of Oxford, UK
https://orcid.org/0000-0001-8151-2443
EPSRC Fellowship EP/N008197/1.
10.4230/LIPIcs.MFCS.2021.34
Dmitri V. Anosov. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math., 90, 1967.
Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in Real Algebraic Geometry. Springer, 2006.
Rufus Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, volume 470 of Lecture Notes in Mathematics. Springer-Verlag, 1975.
Jin-Yi Cai. Computing Jordan normal forms exactly for commuting matrices in polynomial time. Int. J. Found. Comput. Sci., 5(3/4):293-302, 1994.
Jin-Yi Cai, Richard J. Lipton, and Yechezkel Zalcstein. The complexity of the A B C problem. SIAM J. Comput., 29(6), 2000.
Henri Cohen. A Course in Computational Algebraic Number Theory. Springer, 1993.
George E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In H. Brakhage, editor, Automata Theory and Formal Languages, pages 134-183, Berlin, Heidelberg, 1975. Springer Berlin Heidelberg.
Charles C. Conley. Isolated invariant sets and the Morse index, volume 25 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, 1978.
Guoqiang Ge. Algorithms Related to Multiplicative Representations of Algebraic Numbers. PhD thesis, U.C. Berkeley, 1993.
Godfrey H. Hardy and Edward M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 1999.
Michael A. Harrison. Lectures on linear sequential machines. Technical report, DTIC Document, 1969.
Ravindran Kannan and Richard J. Lipton. Polynomial-time algorithm for the orbit problem. J. ACM, 33(4):808-821, 1986.
David W. Masser. Linear relations on algebraic groups. In New Advances in Transcendence Theory. Camb. Univ. Press, 1988.
Maurice Mignotte, Tarlok N. Shorey, and Robert Tijdeman. The distance between terms of an algebraic recurrence sequence. J. für die reine und angewandte Math., 349, 1984.
Joël Ouaknine and James Worrell. Positivity problems for low-order linear recurrence sequences. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 366-379, 2014.
Joël Ouaknine and James Worrell. On linear recurrence sequences and loop termination. ACM SIGLOG News, 2(2):4-13, 2015.
Christos H. Papadimitriou and Georgios Piliouras. From nash equilibria to chain recurrent sets: An algorithmic solution concept for game theory. Entropy, 20(10):782, 2018.
James Renegar. On the computational complexity and geometry of the first-order theory of the reals. J. Symb. Comp., 1992.
Eduardo D. Sontag. Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer-Verlag, Berlin, Heidelberg, 1998.
Terence Tao. Structure and randomness: pages from year one of a mathematical blog. American Mathematical Society, 2008.
Nikolai K. Vereshchagin. The problem of appearance of a zero in a linear recurrence sequence (in russian). Mat. Zametki, 38(2), 1985.
Julian D'Costa, Toghrul Karimov, Rupak Majumdar, Joël Ouaknine, Mahmoud Salamati, Sadegh Soudjani, and James Worrell
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