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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

The matrix semigroup membership problem asks, given square matrices M,M₁,…,M_k of the same dimension, whether M lies in the semigroup generated by M₁,…,M_k. It is classical that this problem is undecidable in general, but decidable in case M₁,…,M_k commute. In this paper we consider the problem of whether, given M₁,…,M_k, the semigroup generated by M₁,…,M_k contains a non-negative matrix. We show that in case M₁,…,M_k commute, this problem is decidable subject to Schanuel’s Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability proof is a procedure to determine, given a matrix M, whether the sequence of matrices (Mⁿ)_{n = 0}^∞ is ultimately nonnegative. This answers a problem posed by S. Akshay [S. Akshay et al., 2022]. The latter result is in stark contrast to the notorious fact that it is not known how to determine, for any specific matrix index (i,j), whether the sequence (Mⁿ)_{i,j} is ultimately nonnegative. Indeed the latter is equivalent to the Ultimate Positivity Problem for linear recurrence sequences, a longstanding open problem.

Julian D'Costa, Joël Ouaknine, and James Worrell. Nonnegativity Problems for Matrix Semigroups. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dcosta_et_al:LIPIcs.STACS.2024.27, author = {D'Costa, Julian and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Nonnegativity Problems for Matrix Semigroups}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.27}, URN = {urn:nbn:de:0030-drops-197371}, doi = {10.4230/LIPIcs.STACS.2024.27}, annote = {Keywords: Decidability, Linear Recurrence Sequences, Schanuel’s Conjecture} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We study the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets. We establish a uniform upper bound on the number of iterations it takes for every orbit of a rational matrix to escape a compact semialgebraic set defined over rational data. Our bound is doubly exponential in the ambient dimension, singly exponential in the degrees of the polynomials used to define the semialgebraic set, and singly exponential in the bitsize of the coefficients of these polynomials and the bitsize of the matrix entries. We show that our bound is tight by providing a matching lower bound.

Julian D'Costa, Engel Lefaucheux, Eike Neumann, Joël Ouaknine, and James Worrell. Bounding the Escape Time of a Linear Dynamical System over a Compact Semialgebraic Set. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dcosta_et_al:LIPIcs.MFCS.2022.39, author = {D'Costa, Julian and Lefaucheux, Engel and Neumann, Eike and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Bounding the Escape Time of a Linear Dynamical System over a Compact Semialgebraic Set}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.39}, URN = {urn:nbn:de:0030-drops-168374}, doi = {10.4230/LIPIcs.MFCS.2022.39}, annote = {Keywords: Discrete linear dynamical systems, Program termination, Compact semialgebraic sets, Uniform termination bounds} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We study fundamental reachability problems on pseudo-orbits of linear dynamical systems. Pseudo-orbits can be viewed as a model of computation with limited precision and pseudo-reachability can be thought of as a robust version of classical reachability. Using an approach based on o-minimality of ℝ_exp we prove decidability of the discrete-time pseudo-reachability problem with arbitrary semialgebraic targets for diagonalisable linear dynamical systems. We also show that our method can be used to reduce the continuous-time pseudo-reachability problem to the (classical) time-bounded reachability problem, which is known to be conditionally decidable.

Julian D'Costa, Toghrul Karimov, Rupak Majumdar, Joël Ouaknine, Mahmoud Salamati, and James Worrell. The Pseudo-Reachability Problem for Diagonalisable Linear Dynamical Systems. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dcosta_et_al:LIPIcs.MFCS.2022.40, author = {D'Costa, Julian and Karimov, Toghrul and Majumdar, Rupak and Ouaknine, Jo\"{e}l and Salamati, Mahmoud and Worrell, James}, title = {{The Pseudo-Reachability Problem for Diagonalisable Linear Dynamical Systems}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {40:1--40:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.40}, URN = {urn:nbn:de:0030-drops-168380}, doi = {10.4230/LIPIcs.MFCS.2022.40}, annote = {Keywords: pseudo-orbits, Orbit problem, Skolem problem, linear dynamical systems, reachability} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We study the computational complexity of the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets, or equivalently the Termination Problem for affine loops with compact semialgebraic guard sets. Consider the fragment of the theory of the reals consisting of negation-free ∃ ∀-sentences without strict inequalities. We derive several equivalent characterisations of the associated complexity class which demonstrate its robustness and illustrate its expressive power. We show that the Compact Escape Problem is complete for this class.

Julian D'Costa, Engel Lefaucheux, Eike Neumann, Joël Ouaknine, and James Worrell. On the Complexity of the Escape Problem for Linear Dynamical Systems over Compact Semialgebraic Sets. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 33:1-33:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dcosta_et_al:LIPIcs.MFCS.2021.33, author = {D'Costa, Julian and Lefaucheux, Engel and Neumann, Eike and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On the Complexity of the Escape Problem for Linear Dynamical Systems over Compact Semialgebraic Sets}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {33:1--33:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.33}, URN = {urn:nbn:de:0030-drops-144734}, doi = {10.4230/LIPIcs.MFCS.2021.33}, annote = {Keywords: Discrete linear dynamical systems, Program termination, Compact semialgebraic sets, Theory of the reals} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

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.

Julian D'Costa, Toghrul Karimov, Rupak Majumdar, Joël Ouaknine, Mahmoud Salamati, Sadegh Soudjani, and James Worrell. The Pseudo-Skolem Problem is Decidable. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dcosta_et_al:LIPIcs.MFCS.2021.34, author = {D'Costa, Julian and Karimov, Toghrul and Majumdar, Rupak and Ouaknine, Jo\"{e}l and Salamati, Mahmoud and Soudjani, Sadegh and Worrell, James}, title = {{The Pseudo-Skolem Problem is Decidable}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {34:1--34:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.34}, URN = {urn:nbn:de:0030-drops-144742}, doi = {10.4230/LIPIcs.MFCS.2021.34}, annote = {Keywords: Pseudo-orbits, Orbit problem, Skolem problem, linear dynamical systems} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

The Continuous Polytope Escape Problem (CPEP) asks whether every trajectory of a linear differential equation initialised within a convex polytope eventually escapes the polytope. We provide a polynomial-time algorithm to decide CPEP for compact polytopes. We also establish a quantitative uniform upper bound on the time required for every trajectory to escape the given polytope. In addition, we establish iteration bounds for termination of discrete linear loops via reduction to the continuous case.

Julian D'Costa, Engel Lefaucheux, Joël Ouaknine, and James Worrell. How Fast Can You Escape a Compact Polytope?. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 49:1-49:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dcosta_et_al:LIPIcs.STACS.2020.49, author = {D'Costa, Julian and Lefaucheux, Engel and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{How Fast Can You Escape a Compact Polytope?}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {49:1--49:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.49}, URN = {urn:nbn:de:0030-drops-119105}, doi = {10.4230/LIPIcs.STACS.2020.49}, annote = {Keywords: Continuous linear dynamical systems, termination} }

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