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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

It is a longstanding open problem whether there is an algorithm to decide the Positivity Problem for linear recurrence sequences (LRS) over the integers, namely whether given such a sequence, all its terms are non-negative. Decidability is known for LRS of order 5 or less, i.e., for those sequences in which every new term depends linearly on the previous five (or fewer) terms. For simple LRS (i.e., those sequences whose characteristic polynomials have no repeated roots), decidability of Positivity is known up to order 9.
In this paper, we focus on the important subclass of reversible LRS, i.e., those integer LRS ⟨u_n⟩_{n=0}^∞ whose bi-infinite completion ⟨u_n⟩_{n=-∞}^∞ also takes exclusively integer values; a typical example is the classical Fibonacci (bi-)sequence ⟨ … , 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, … ⟩. Our main results are that Positivity is decidable for reversible LRS of order 11 or less, and for simple reversible LRS of order 17 or less.

George Kenison, Joris Nieuwveld, Joël Ouaknine, and James Worrell. Positivity Problems for Reversible Linear Recurrence Sequences. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 130:1-130:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kenison_et_al:LIPIcs.ICALP.2023.130, author = {Kenison, George and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Positivity Problems for Reversible Linear Recurrence Sequences}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {130:1--130:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.130}, URN = {urn:nbn:de:0030-drops-181821}, doi = {10.4230/LIPIcs.ICALP.2023.130}, annote = {Keywords: The Positivity Problem, Linear Recurrence Sequences, Verification} }

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

We consider the model-checking problem for parametric probabilistic dynamical systems, formalised as Markov chains with parametric transition functions, analysed under the distribution-transformer semantics (in which a Markov chain induces a sequence of distributions over states).
We examine the problem of synthesising the set of parameter valuations of a parametric Markov chain such that the orbits of induced state distributions satisfy a prefix-independent ω-regular property.
Our main result establishes that in all non-degenerate instances, the feasible set of parameters is (up to a null set) semialgebraic, and can moreover be computed (in polynomial time assuming that the ambient dimension, corresponding to the number of states of the Markov chain, is fixed).

Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Joël Ouaknine, David Purser, Markus A. Whiteland, and James Worrell. Parameter Synthesis for Parametric Probabilistic Dynamical Systems and Prefix-Independent Specifications. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 10:1-10:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baier_et_al:LIPIcs.CONCUR.2022.10, author = {Baier, Christel and Funke, Florian and Jantsch, Simon and Karimov, Toghrul and Lefaucheux, Engel and Ouaknine, Jo\"{e}l and Purser, David and Whiteland, Markus A. and Worrell, James}, title = {{Parameter Synthesis for Parametric Probabilistic Dynamical Systems and Prefix-Independent Specifications}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {10:1--10:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.10}, URN = {urn:nbn:de:0030-drops-170732}, doi = {10.4230/LIPIcs.CONCUR.2022.10}, annote = {Keywords: Model checking, parametric Markov chains, distribution transformer semantics} }

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

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this paper is an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct - a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool Skolem point to the practical applicability of this method.

Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, David Purser, and James Worrell. Skolem Meets Schanuel. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 20:1-20:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilu_et_al:LIPIcs.MFCS.2022.20, author = {Bilu, Yuri and Luca, Florian and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Purser, David and Worrell, James}, title = {{Skolem Meets Schanuel}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {20:1--20:15}, 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.2022.20}, URN = {urn:nbn:de:0030-drops-168180}, doi = {10.4230/LIPIcs.MFCS.2022.20}, annote = {Keywords: Skolem Problem, Skolem Conjecture, Exponential Local-Global Principle, p-adic Schanuel 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 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The Skolem Problem asks to decide whether a given integer linear recurrence sequence (LRS) has a zero term. Decidability of this problem has been open for many decades, with little progress since the 1980s. Recently, a new approach was initiated via the notion of a Skolem set - a set of positive integers relative to which the Skolem Problem is decidable. More precisely, 𝒮 is a Skolem set for a class ℒ of integer LRS if there is an effective procedure that, given an LRS in ℒ, decides whether the sequence has a zero in 𝒮. A recent work exhibited a Skolem set for the class of all LRS that, while infinite, had density zero. In the present work we construct a Skolem set of positive lower density for the class of simple LRS .

Florian Luca, Joël Ouaknine, and James Worrell. A Universal Skolem Set of Positive Lower Density. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 73:1-73:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{luca_et_al:LIPIcs.MFCS.2022.73, author = {Luca, Florian and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{A Universal Skolem Set of Positive Lower Density}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {73:1--73:12}, 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.2022.73}, URN = {urn:nbn:de:0030-drops-168711}, doi = {10.4230/LIPIcs.MFCS.2022.73}, annote = {Keywords: Linear Recurrence Sequences, Skolem Problem, Exponential Diophantine Equations, Sieve Methods} }

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Invited Talk

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

Holonomic techniques have deep roots going back to Wallis, Euler, and Gauss, and have evolved in modern times as an important subfield of computer algebra, thanks in large part to the work of Zeilberger and others over the past three decades (see, e.g., [Doron Zeilberger, 1990; Petkovšek et al., 1997]). In this talk, I give an overview of the area, and in particular present a select survey of known and original results on decision problems for holonomic sequences and functions. I also discuss some surprising connections to the theory of periods and exponential periods, which are classical objects of study in algebraic geometry and number theory; in particular, I relate the decidability of certain decision problems for holonomic sequences to deep conjectures about periods and exponential periods, notably those due to Kontsevich and Zagier.
Parts of this exposition draws upon [George Kenison et al., 2021].

Joël Ouaknine. Holonomic Techniques, Periods, and Decision Problems (Invited Talk). In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, p. 3:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ouaknine:LIPIcs.MFCS.2021.3, author = {Ouaknine, Jo\"{e}l}, title = {{Holonomic Techniques, Periods, and Decision Problems}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {3:1--3:1}, 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.2021.3}, URN = {urn:nbn:de:0030-drops-144431}, doi = {10.4230/LIPIcs.MFCS.2021.3}, annote = {Keywords: Holonomic and hypergeometric sequences, Inequality problems, Continued fractions, Periods} }

Document

**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 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

An infinite sequence ⟨u_n⟩_n of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each u_n ≥ 0, and minimal if, given any other linearly independent sequence ⟨v_n⟩_n satisfying the same recurrence relation, the ratio u_n/v_n → 0 as n → ∞.
In this paper we give a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences. More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.

George Kenison, Oleksiy Klurman, Engel Lefaucheux, Florian Luca, Pieter Moree, Joël Ouaknine, Markus A. Whiteland, and James Worrell. On Positivity and Minimality for Second-Order Holonomic Sequences. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 67:1-67:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kenison_et_al:LIPIcs.MFCS.2021.67, author = {Kenison, George and Klurman, Oleksiy and Lefaucheux, Engel and Luca, Florian and Moree, Pieter and Ouaknine, Jo\"{e}l and Whiteland, Markus A. and Worrell, James}, title = {{On Positivity and Minimality for Second-Order Holonomic Sequences}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {67:1--67:15}, 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.2021.67}, URN = {urn:nbn:de:0030-drops-145071}, doi = {10.4230/LIPIcs.MFCS.2021.67}, annote = {Keywords: Holonomic sequences, Minimal solutions, Positivity Problem} }

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**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with two or more parameters.
More precisely, consider a d-dimensional square matrix M whose entries are algebraic functions in one or more real variables. Given initial and target vectors u,v ∈ ℚ^d, the parametric point-to-point orbit problem asks whether there exist values of the parameters giving rise to a concrete matrix N ∈ ℝ^{d× d}, and a positive integer n ∈ ℕ, such that N^{n} u = v.
We show decidability for the case in which M depends only upon a single parameter, and we exhibit a reduction from the well-known Skolem Problem for linear recurrence sequences, suggesting intractability in the case of two or more parameters.

Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Florian Luca, Joël Ouaknine, David Purser, Markus A. Whiteland, and James Worrell. The Orbit Problem for Parametric Linear Dynamical Systems. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 28:1-28:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{baier_et_al:LIPIcs.CONCUR.2021.28, author = {Baier, Christel and Funke, Florian and Jantsch, Simon and Karimov, Toghrul and Lefaucheux, Engel and Luca, Florian and Ouaknine, Jo\"{e}l and Purser, David and Whiteland, Markus A. and Worrell, James}, title = {{The Orbit Problem for Parametric Linear Dynamical Systems}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {28:1--28:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.28}, URN = {urn:nbn:de:0030-drops-144053}, doi = {10.4230/LIPIcs.CONCUR.2021.28}, annote = {Keywords: Orbit problem, parametric, linear dynamical systems} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We study decision problems for sequences which obey a second-order holonomic recurrence of the form f(n + 2) = P(n) f(n + 1) + Q(n) f(n) with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P. We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f(1), f(2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones.

Eike Neumann, Joël Ouaknine, and James Worrell. Decision Problems for Second-Order Holonomic Recurrences. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 99:1-99:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{neumann_et_al:LIPIcs.ICALP.2021.99, author = {Neumann, Eike and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Decision Problems for Second-Order Holonomic Recurrences}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {99:1--99:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.99}, URN = {urn:nbn:de:0030-drops-141682}, doi = {10.4230/LIPIcs.ICALP.2021.99}, annote = {Keywords: holonomic sequences, Positivity Problem, Skolem Problem} }

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Invited Talk

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Holonomic techniques have deep roots going back to Wallis, Euler, and Gauss, and have evolved in modern times as an important subfield of computer algebra, thanks in large part to the work of Zeilberger and others over the past three decades. In this talk, I will give an overview of the area, and in particular will present a select survey of known and original results on decision problems for holonomic sequences and functions. (Holonomic sequences satisfy linear recurrence relations with polynomial coefficients, and holonomic functions satisfy linear differential equations with polynomial coefficients.) I will also discuss some surprising connections to the theory of periods and exponential periods, which are classical objects of study in algebraic geometry and number theory; in particular, I will relate the decidability of certain decision problems for holonomic sequences to deep conjectures about periods and exponential periods, notably those due to Kontsevich and Zagier.

Joël Ouaknine. Holonomic Techniques, Periods, and Decision Problems (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 4:1-4:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ouaknine:LIPIcs.FSTTCS.2020.4, author = {Ouaknine, Jo\"{e}l}, title = {{Holonomic Techniques, Periods, and Decision Problems}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {4:1--4:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.4}, URN = {urn:nbn:de:0030-drops-132451}, doi = {10.4230/LIPIcs.FSTTCS.2020.4}, annote = {Keywords: holonomic techniques, decision problems, recurrence sequences, minimal solutions, Positivity Problem, continued fractions, special functions, periods, exponential periods} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix M ∈ ℚ^{d × d}, an initial vector x ∈ ℚ^{d}, a granularity g ∈ ℚ_+ and a rounding operation [⋅] projecting a vector of ℚ^{d} onto another vector whose every entry is a multiple of g, we are interested in the behaviour of the orbit 𝒪 = ⟨[x], [M[x]],[M[M[x]]],… ⟩, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability - whether a given target y ∈ ℚ^{d} belongs to 𝒪 - is PSPACE-complete for hyperbolic systems (when no eigenvalue of M has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.

Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Joël Ouaknine, Amaury Pouly, David Purser, and Markus A. Whiteland. Reachability in Dynamical Systems with Rounding. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 36:1-36:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{baier_et_al:LIPIcs.FSTTCS.2020.36, author = {Baier, Christel and Funke, Florian and Jantsch, Simon and Karimov, Toghrul and Lefaucheux, Engel and Ouaknine, Jo\"{e}l and Pouly, Amaury and Purser, David and Whiteland, Markus A.}, title = {{Reachability in Dynamical Systems with Rounding}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {36:1--36:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.36}, URN = {urn:nbn:de:0030-drops-132778}, doi = {10.4230/LIPIcs.FSTTCS.2020.36}, annote = {Keywords: dynamical systems, rounding, reachability} }

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**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We consider the problem of synthesising polynomial ranking functions for single-path loops over the reals with continuous semi-algebraic update function and compact semi-algebraic guard set. We show that a loop of this form has a polynomial ranking function if and only if it terminates. We further show that termination is decidable for such loops in the special case where the update function is affine.

Eike Neumann, Joël Ouaknine, and James Worrell. On Ranking Function Synthesis and Termination for Polynomial Programs. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 15:1-15:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{neumann_et_al:LIPIcs.CONCUR.2020.15, author = {Neumann, Eike and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On Ranking Function Synthesis and Termination for Polynomial Programs}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.15}, URN = {urn:nbn:de:0030-drops-128278}, doi = {10.4230/LIPIcs.CONCUR.2020.15}, annote = {Keywords: Semi-algebraic sets, Polynomial ranking functions, Polynomial programs} }

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**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We exhibit an algorithm to compute the strongest algebraic (or polynomial) invariants that hold at each location of a given guard-free linear hybrid automaton (i.e., a hybrid automaton having only unguarded transitions, all of whose assignments are given by affine expressions, and all of whose continuous dynamics are given by linear differential equations). Our main tool is a control-theoretic result of independent interest: given such a linear hybrid automaton, we show how to discretise the continuous dynamics in such a way that the resulting automaton has precisely the same algebraic invariants.

Rupak Majumdar, Joël Ouaknine, Amaury Pouly, and James Worrell. Algebraic Invariants for Linear Hybrid Automata. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 32:1-32:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{majumdar_et_al:LIPIcs.CONCUR.2020.32, author = {Majumdar, Rupak and Ouaknine, Jo\"{e}l and Pouly, Amaury and Worrell, James}, title = {{Algebraic Invariants for Linear Hybrid Automata}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {32:1--32:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.32}, URN = {urn:nbn:de:0030-drops-128443}, doi = {10.4230/LIPIcs.CONCUR.2020.32}, annote = {Keywords: Hybrid automata, algebraic invariants} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Consider a discrete dynamical system given by a square matrix M ∈ ℚ^{d × d} and a starting point s ∈ ℚ^d. The orbit of such a system is the infinite trajectory ⟨ s, Ms, M²s, …⟩. Given a collection T₁, T₂, …, T_m ⊆ ℝ^d of semialgebraic sets, we can associate with each T_i an atomic proposition P_i which evaluates to true at time n if, and only if, M^ns ∈ T_i. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system (M,s) and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.

Toghrul Karimov, Joël Ouaknine, and James Worrell. On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 54:1-54:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{karimov_et_al:LIPIcs.MFCS.2020.54, author = {Karimov, Toghrul and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {54:1--54:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.54}, URN = {urn:nbn:de:0030-drops-127215}, doi = {10.4230/LIPIcs.MFCS.2020.54}, annote = {Keywords: Linear dynamical systems, Orbit Problem, LTL model checking} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Continuous linear dynamical systems are used extensively in mathematics, computer science, physics, and engineering to model the evolution of a system over time. A central technique for certifying safety properties of such systems is by synthesising inductive invariants. This is the task of finding a set of states that is closed under the dynamics of the system and is disjoint from a given set of error states. In this paper we study the problem of synthesising inductive invariants that are definable in o-minimal expansions of the ordered field of real numbers. In particular, assuming Schanuel’s conjecture in transcendental number theory, we establish effective synthesis of o-minimal invariants in the case of semi-algebraic error sets. Without using Schanuel’s conjecture, we give a procedure for synthesizing o-minimal invariants that contain all but a bounded initial segment of the orbit and are disjoint from a given semi-algebraic error set. We further prove that effective synthesis of semi-algebraic invariants that contain the whole orbit, is at least as hard as a certain open problem in transcendental number theory.

Shaull Almagor, Edon Kelmendi, Joël Ouaknine, and James Worrell. Invariants for Continuous Linear Dynamical Systems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 107:1-107:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{almagor_et_al:LIPIcs.ICALP.2020.107, author = {Almagor, Shaull and Kelmendi, Edon and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Invariants for Continuous Linear Dynamical Systems}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {107:1--107:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.107}, URN = {urn:nbn:de:0030-drops-125141}, doi = {10.4230/LIPIcs.ICALP.2020.107}, annote = {Keywords: Invariants, continuous linear dynamical systems, continuous Skolem problem, safety, o-minimality} }

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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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Invited Talk

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Automated invariant generation is a fundamental challenge in program analysis and verification, going back many decades, and remains a topic of active research. In this talk I'll present a select overview and survey of work on this problem, and discuss unexpected connections to other fields including algebraic geometry, group theory, and quantum computing. (No previous knowledge of these topics will be assumed.)
This is joint work with Ehud Hrushovski, Amaury Pouly, and James Worrell.

Joël Ouaknine. Program Invariants (Invited Talk). In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, p. 3:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ouaknine:LIPIcs.CONCUR.2019.3, author = {Ouaknine, Jo\"{e}l}, title = {{Program Invariants}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {3:1--3:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.3}, URN = {urn:nbn:de:0030-drops-109056}, doi = {10.4230/LIPIcs.CONCUR.2019.3}, annote = {Keywords: Automated invariant generation, program analysis and verification} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers.

Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell. On Reachability Problems for Low-Dimensional Matrix Semigroups. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 44:1-44:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{colcombet_et_al:LIPIcs.ICALP.2019.44, author = {Colcombet, Thomas and Ouaknine, Jo\"{e}l and Semukhin, Pavel and Worrell, James}, title = {{On Reachability Problems for Low-Dimensional Matrix Semigroups}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.44}, URN = {urn:nbn:de:0030-drops-106209}, doi = {10.4230/LIPIcs.ICALP.2019.44}, annote = {Keywords: membership problem, half-space reachability problem, matrix semigroups, Heisenberg group, general linear group} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the problem of deciding termination of single-path while loops with integer variables, affine updates, and affine guard conditions. The question is whether such a loop terminates on all integer initial values. This problem is known to be decidable for the subclass of loops whose update matrices are diagonalisable, but the general case has remained open since being conjectured decidable by Tiwari in 2004. In this paper we show decidability of determining termination for arbitrary update matrices, confirming Tiwari’s conjecture. For the class of loops considered in this paper, the question of deciding termination on a specific initial value is a longstanding open problem in number theory. The key to our decision procedure is in showing how to circumvent the difficulties inherent in deciding termination on a fixed initial value.

Mehran Hosseini, Joël Ouaknine, and James Worrell. Termination of Linear Loops over the Integers (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 118:1-118:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hosseini_et_al:LIPIcs.ICALP.2019.118, author = {Hosseini, Mehran and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Termination of Linear Loops over the Integers}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {118:1--118:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.118}, URN = {urn:nbn:de:0030-drops-106940}, doi = {10.4230/LIPIcs.ICALP.2019.118}, annote = {Keywords: Program Verification, Loop Termination, Linear Integer Programs, Affine While Loops} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d in N, a square matrix A in Q^{d x d}, and semialgebraic source and target sets S,T subseteq R^d. The question is whether there exists x in S and n in N such that A^nx in T.
The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d <= 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory - Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

Shaull Almagor, Joël Ouaknine, and James Worrell. The Semialgebraic Orbit Problem. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 6:1-6:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{almagor_et_al:LIPIcs.STACS.2019.6, author = {Almagor, Shaull and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{The Semialgebraic Orbit Problem}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {6:1--6:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.6}, URN = {urn:nbn:de:0030-drops-102450}, doi = {10.4230/LIPIcs.STACS.2019.6}, annote = {Keywords: linear dynamical systems, Orbit Problem, first order theory of the reals} }

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**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational linear recurrence sequence and computes effective fine-grained lower bounds on the growth rate of the sequence.

Shaull Almagor, Brynmor Chapman, Mehran Hosseini, Joël Ouaknine, and James Worrell. Effective Divergence Analysis for Linear Recurrence Sequences. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 42:1-42:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{almagor_et_al:LIPIcs.CONCUR.2018.42, author = {Almagor, Shaull and Chapman, Brynmor and Hosseini, Mehran and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Effective Divergence Analysis for Linear Recurrence Sequences}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {42:1--42:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.42}, URN = {urn:nbn:de:0030-drops-95802}, doi = {10.4230/LIPIcs.CONCUR.2018.42}, annote = {Keywords: Linear recurrence sequences, Divergence, Algebraic numbers, Positivity} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

The termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Such deceptively simple questions also relate to a number of deep open problems, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this paper, we introduce the class of o-minimal invariants, which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel's conjecture in transcendental number theory.

Shaull Almagor, Dmitry Chistikov, Joël Ouaknine, and James Worrell. O-Minimal Invariants for Linear Loops. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 114:1-114:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{almagor_et_al:LIPIcs.ICALP.2018.114, author = {Almagor, Shaull and Chistikov, Dmitry and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{O-Minimal Invariants for Linear Loops}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {114:1--114:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.114}, URN = {urn:nbn:de:0030-drops-91188}, doi = {10.4230/LIPIcs.ICALP.2018.114}, annote = {Keywords: Invariants, linear loops, linear dynamical systems, non-termination, o-minimality} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

The Orbit Problem consists of determining, given a matrix A in R^dxd and vectors x,y in R^d, whether there exists n in N such that A^n=y. This problem was shown to be decidable in a seminal work of Kannan and Lipton in the 1980s. Subsequently, Kannan and Lipton noted that the Orbit Problem becomes considerably harder when the target y is replaced with a subspace of R^d. Recently, it was shown that the problem is decidable for vector-space targets of dimension at most three, followed by another development showing that the problem is in PSPACE for polytope targets of dimension at most three.
In this work, we take a dual look at the problem, and consider the case where the initial vector x is replaced with a polytope P_1, and the target is a polytope P_2. Then, the question is whether there exists n in N such that A^n P_1 intersection P_2 does not equal the empty set. We show that the problem can be decided in PSPACE for dimension at most three. As in previous works, decidability in the case of higher dimensions is left open, as the problem is known to be hard for long-standing number-theoretic open problems.
Our proof begins by formulating the problem as the satisfiability of a parametrized family of sentences in the existential first-order theory of real-closed fields. Then, after removing quantifiers, we are left with instances of simultaneous positivity of sums of exponentials. Using techniques from transcendental number theory, and separation bounds on algebraic numbers, we are able to solve such instances in PSPACE.

Shaull Almagor, Joël Ouaknine, and James Worrell. The Polytope-Collision Problem. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 24:1-24:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{almagor_et_al:LIPIcs.ICALP.2017.24, author = {Almagor, Shaull and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{The Polytope-Collision Problem}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.24}, URN = {urn:nbn:de:0030-drops-74521}, doi = {10.4230/LIPIcs.ICALP.2017.24}, annote = {Keywords: linear dynamical systems, orbit problem, algebraic algorithms} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

The Orbit Problem consists of determining, given a linear transformation A on d-dimensional rationals 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 P which are subsets of R^d, i.e., sets that are stable under A and contain x and 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 invariants of polynomial size.
It is worth noting that the existence of semilinear invariants, on the other hand, is (to the best of our knowledge) not known to be decidable.

Nathanaël Fijalkow, Pierre Ohlmann, Joël Ouaknine, Amaury Pouly, and James Worrell. Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 29:1-29:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fijalkow_et_al:LIPIcs.STACS.2017.29, author = {Fijalkow, Nathana\"{e}l and Ohlmann, Pierre and Ouaknine, Jo\"{e}l and Pouly, Amaury and Worrell, James}, title = {{Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {29:1--29:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.29}, URN = {urn:nbn:de:0030-drops-70059}, doi = {10.4230/LIPIcs.STACS.2017.29}, annote = {Keywords: Verification,algebraic computation,Skolem Problem,Orbit Problem,invariants} }

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**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

Freeze LTL is a temporal logic with registers that is suitable for specifying properties of data words. In this paper we study the model checking problem for Freeze LTL on one-counter automata. This problem is known to be undecidable in full generality and PSPACE-complete for the special case of deterministic one-counter automata. Several years ago, Demri and Sangnier investigated the model checking problem for the flat fragment of Freeze LTL on several classes of counter automata and posed the decidability of model checking flat Freeze LTL on one-counter automata as an open problem. In this paper we resolve this problem positively, utilising a known reduction to a reachability problem on one-counter automata with parameterised equality and disequality tests. Our main technical contribution is to show decidability of the latter problem by translation to Presburger arithmetic.

Antonia Lechner, Richard Mayr, Joël Ouaknine, Amaury Pouly, and James Worrell. Model Checking Flat Freeze LTL on One-Counter Automata. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 29:1-29:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lechner_et_al:LIPIcs.CONCUR.2016.29, author = {Lechner, Antonia and Mayr, Richard and Ouaknine, Jo\"{e}l and Pouly, Amaury and Worrell, James}, title = {{Model Checking Flat Freeze LTL on One-Counter Automata}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.29}, URN = {urn:nbn:de:0030-drops-61841}, doi = {10.4230/LIPIcs.CONCUR.2016.29}, annote = {Keywords: one-counter automata, disequality tests, reachability, freeze LTL, Presburger arithmetic} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems, such as linear hybrid automata and continuoustime Markov chains. Decidability of the problem is currently open — indeed decidability is open even for the sub-problem in which a zero is sought in a bounded interval. In this paper we show decidability of the bounded problem subject to Schanuel's Conjecture, a unifying conjecture in transcendental number theory. We furthermore analyse the unbounded problem in terms of the frequencies of the differential equation, that is, the imaginary parts of the characteristic roots.
We show that the unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. We complete the picture by showing that decidability of the unbounded problem in the case of two (or more) rationally linearly independent frequencies would entail a major new effectiveness result in Diophantine approximation, namely computability of the Diophantine-approximation types of all real algebraic numbers.

Ventsislav Chonev, Joël Ouaknine, and James Worrell. On the Skolem Problem for Continuous Linear Dynamical Systems. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 100:1-100:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chonev_et_al:LIPIcs.ICALP.2016.100, author = {Chonev, Ventsislav and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On the Skolem Problem for Continuous Linear Dynamical Systems}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {100:1--100:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.100}, URN = {urn:nbn:de:0030-drops-62357}, doi = {10.4230/LIPIcs.ICALP.2016.100}, annote = {Keywords: differential equations, reachability, Baker’s Theorem, Schanuel’s Conjecture, semi-algebraic sets} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Herman's self-stabilization algorithm, introduced 25 years ago, is a well-studied synchronous randomized protocol for enabling a ring of N processes collectively holding any odd number of tokens to reach a stable state in which a single token remains. Determining the worst-case expected time to stabilization is the central outstanding open problem about this protocol. It is known that there is a constant h such that any initial configuration has expected stabilization time at most hN2. Ten years ago, McIver and Morgan established a lower bound of 4/27 ~ 0.148 for h, achieved with three equally-spaced tokens, and conjectured this to be the optimal value of h. A series of papers over the last decade gradually reduced the upper bound on h, with the present record (achieved in 2014) standing at approximately 0.156. In this paper, we prove McIver and Morgan's conjecture and establish that h = 4/27 is indeed optimal.

Maria Bruna, Radu Grigore, Stefan Kiefer, Joël Ouaknine, and James Worrell. Proving the Herman-Protocol Conjecture. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 104:1-104:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bruna_et_al:LIPIcs.ICALP.2016.104, author = {Bruna, Maria and Grigore, Radu and Kiefer, Stefan and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Proving the Herman-Protocol Conjecture}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {104:1--104:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.104}, URN = {urn:nbn:de:0030-drops-62393}, doi = {10.4230/LIPIcs.ICALP.2016.104}, annote = {Keywords: randomized protocols, self-stabilization, Lyapunov function, expected time} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We investigate the Matrix Powering Positivity Problem, PosMatPow: given an m X m square integer matrix M, a linear function f: Z^{m X m} -> Z with integer coefficients, and a positive integer n (encoded in binary), determine whether f(M^n) \geq 0. We show that for fixed dimensions m of 2 and 3, this problem is decidable in polynomial time.

Esther Galby, Joël Ouaknine, and James Worrell. On Matrix Powering in Low Dimensions. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 329-340, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{galby_et_al:LIPIcs.STACS.2015.329, author = {Galby, Esther and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On Matrix Powering in Low Dimensions}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {329--340}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.329}, URN = {urn:nbn:de:0030-drops-49240}, doi = {10.4230/LIPIcs.STACS.2015.329}, annote = {Keywords: matrix powering, complexity, Baker's theorem} }

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**Published in:** Dagstuhl Reports, Volume 4, Issue 3 (2014)

This report documents the program and the outcomes of Dagstuhl Seminar 14141 "Reachability Problems for Infinite-State Systems", held from March 30th until April 4th, 2014. The seminar gathered 44 participants and the program consisted of 34 presentations. Participants were asked to contribute open questions prior and
during the seminar. A list of these open questions appears in a separate section of the present report. This list generated collaborations among participants and gave rise to research publications solving (partially), for example, question 5.13, namely "what functions are computable by VASS?"

Javier Esparza, Alain Finkel, Pierre McKenzie, and Joel Ouaknine. Reachability Problems for Infinite-State Systems (Dagstuhl Seminar 14141). In Dagstuhl Reports, Volume 4, Issue 3, pp. 153-180, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@Article{esparza_et_al:DagRep.4.3.153, author = {Esparza, Javier and Finkel, Alain and McKenzie, Pierre and Ouaknine, Joel}, title = {{Reachability Problems for Infinite-State Systems (Dagstuhl Seminar 14141)}}, pages = {153--180}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2014}, volume = {4}, number = {3}, editor = {Esparza, Javier and Finkel, Alain and McKenzie, Pierre and Ouaknine, Joel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.3.153}, URN = {urn:nbn:de:0030-drops-46121}, doi = {10.4230/DagRep.4.3.153}, annote = {Keywords: Infinite-State Systems, Reachability Problems, Formal Verification, Well-Structured Transition Systems, Counter Machines, Vector Addition Systems, Timed Systems} }

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**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Church's Problem asks for the construction of a procedure which, given a logical specification S(I,O) between input strings I and output strings O, determines whether there exists an operator F that implements the specification in the sense that S(I,F(I)) holds for all inputs I. Buechi and Landweber gave a procedure to solve Church's problem for MSO specifications and operators computable by finite-state automata.
We consider extensions of Church's problem in two orthogonal directions: (i) we address the problem in a more general logical setting, where not only the specifications but also the solutions are presented in a logical system; (ii) we consider not only the canonical discrete time domain of the natural numbers, but also the continuous domain of reals.
We show that for every fixed bounded length interval of the reals, Church's problem is decidable when specifications and implementations are described in the monadic second-order logics over the reals with order and the +1 function.

Mark Jenkins, Joël Ouaknine, Alexander Rabinovich, and James Worrell. The Church Synthesis Problem with Metric. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 307-321, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{jenkins_et_al:LIPIcs.CSL.2011.307, author = {Jenkins, Mark and Ouaknine, Jo\"{e}l and Rabinovich, Alexander and Worrell, James}, title = {{The Church Synthesis Problem with Metric}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {307--321}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.307}, URN = {urn:nbn:de:0030-drops-32390}, doi = {10.4230/LIPIcs.CSL.2011.307}, annote = {Keywords: Church's Problem, monadic logic, games, uniformization} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

A fundamental problem in numerical computation and computational geometry is to determine the sign of arithmetic expressions in radicals. Here we consider the simpler problem of deciding whether $\sum_{i=1}^m C_i A_i^{X_i}$ is zero for given rational numbers $A_i$, $C_i$, $X_i$. It has been known for almost twenty years that this can be decided in polynomial time. In this paper we improve this result by showing membership in uniform TC0. This requires several significant departures from Blömer's polynomial-time algorithm as the latter crucially relies on primitives, such as gcd computation and binary search, that are not known to be in TC0.

Paul Hunter, Patricia Bouyer, Nicolas Markey, Joël Ouaknine, and James Worrell. Computing Rational Radical Sums in Uniform TC^0. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 308-316, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{hunter_et_al:LIPIcs.FSTTCS.2010.308, author = {Hunter, Paul and Bouyer, Patricia and Markey, Nicolas and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Computing Rational Radical Sums in Uniform TC^0}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {308--316}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.308}, URN = {urn:nbn:de:0030-drops-28739}, doi = {10.4230/LIPIcs.FSTTCS.2010.308}, annote = {Keywords: Sum of square roots, Threshold circuits, Complexity} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

A channel machine consists of a finite controller together with
several fifo channels; the controller can read messages from the
head of a channel and write messages to the tail of a channel. In
this paper, we focus on channel machines with insertion errors,
i.e., machines in whose channels messages can spontaneously appear.
Such devices have been previously introduced in the study of Metric
Temporal Logic. We consider the termination problem: are all the
computations of a given insertion channel machine finite? We show
that this problem has non-elementary, yet primitive recursive
complexity.

Patricia Bouyer, Nicolas Markey, Joël Ouaknine, Philippe Schnoebelen, and James Worrell. On Termination for Faulty Channel Machines. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 121-132, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{bouyer_et_al:LIPIcs.STACS.2008.1339, author = {Bouyer, Patricia and Markey, Nicolas and Ouaknine, Jo\"{e}l and Schnoebelen, Philippe and Worrell, James}, title = {{On Termination for Faulty Channel Machines}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {121--132}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1339}, URN = {urn:nbn:de:0030-drops-13390}, doi = {10.4230/LIPIcs.STACS.2008.1339}, annote = {Keywords: Automated Verification, Computational Complexity} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

It is a longstanding open problem whether there is an algorithm to decide the Positivity Problem for linear recurrence sequences (LRS) over the integers, namely whether given such a sequence, all its terms are non-negative. Decidability is known for LRS of order 5 or less, i.e., for those sequences in which every new term depends linearly on the previous five (or fewer) terms. For simple LRS (i.e., those sequences whose characteristic polynomials have no repeated roots), decidability of Positivity is known up to order 9.
In this paper, we focus on the important subclass of reversible LRS, i.e., those integer LRS ⟨u_n⟩_{n=0}^∞ whose bi-infinite completion ⟨u_n⟩_{n=-∞}^∞ also takes exclusively integer values; a typical example is the classical Fibonacci (bi-)sequence ⟨ … , 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, … ⟩. Our main results are that Positivity is decidable for reversible LRS of order 11 or less, and for simple reversible LRS of order 17 or less.

George Kenison, Joris Nieuwveld, Joël Ouaknine, and James Worrell. Positivity Problems for Reversible Linear Recurrence Sequences. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 130:1-130:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kenison_et_al:LIPIcs.ICALP.2023.130, author = {Kenison, George and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Positivity Problems for Reversible Linear Recurrence Sequences}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {130:1--130:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.130}, URN = {urn:nbn:de:0030-drops-181821}, doi = {10.4230/LIPIcs.ICALP.2023.130}, annote = {Keywords: The Positivity Problem, Linear Recurrence Sequences, Verification} }

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

We consider the model-checking problem for parametric probabilistic dynamical systems, formalised as Markov chains with parametric transition functions, analysed under the distribution-transformer semantics (in which a Markov chain induces a sequence of distributions over states).
We examine the problem of synthesising the set of parameter valuations of a parametric Markov chain such that the orbits of induced state distributions satisfy a prefix-independent ω-regular property.
Our main result establishes that in all non-degenerate instances, the feasible set of parameters is (up to a null set) semialgebraic, and can moreover be computed (in polynomial time assuming that the ambient dimension, corresponding to the number of states of the Markov chain, is fixed).

Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Joël Ouaknine, David Purser, Markus A. Whiteland, and James Worrell. Parameter Synthesis for Parametric Probabilistic Dynamical Systems and Prefix-Independent Specifications. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 10:1-10:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baier_et_al:LIPIcs.CONCUR.2022.10, author = {Baier, Christel and Funke, Florian and Jantsch, Simon and Karimov, Toghrul and Lefaucheux, Engel and Ouaknine, Jo\"{e}l and Purser, David and Whiteland, Markus A. and Worrell, James}, title = {{Parameter Synthesis for Parametric Probabilistic Dynamical Systems and Prefix-Independent Specifications}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {10:1--10:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.10}, URN = {urn:nbn:de:0030-drops-170732}, doi = {10.4230/LIPIcs.CONCUR.2022.10}, annote = {Keywords: Model checking, parametric Markov chains, distribution transformer semantics} }

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

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this paper is an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct - a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool Skolem point to the practical applicability of this method.

Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, David Purser, and James Worrell. Skolem Meets Schanuel. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 20:1-20:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilu_et_al:LIPIcs.MFCS.2022.20, author = {Bilu, Yuri and Luca, Florian and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Purser, David and Worrell, James}, title = {{Skolem Meets Schanuel}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {20:1--20:15}, 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.2022.20}, URN = {urn:nbn:de:0030-drops-168180}, doi = {10.4230/LIPIcs.MFCS.2022.20}, annote = {Keywords: Skolem Problem, Skolem Conjecture, Exponential Local-Global Principle, p-adic Schanuel 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 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The Skolem Problem asks to decide whether a given integer linear recurrence sequence (LRS) has a zero term. Decidability of this problem has been open for many decades, with little progress since the 1980s. Recently, a new approach was initiated via the notion of a Skolem set - a set of positive integers relative to which the Skolem Problem is decidable. More precisely, 𝒮 is a Skolem set for a class ℒ of integer LRS if there is an effective procedure that, given an LRS in ℒ, decides whether the sequence has a zero in 𝒮. A recent work exhibited a Skolem set for the class of all LRS that, while infinite, had density zero. In the present work we construct a Skolem set of positive lower density for the class of simple LRS .

Florian Luca, Joël Ouaknine, and James Worrell. A Universal Skolem Set of Positive Lower Density. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 73:1-73:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{luca_et_al:LIPIcs.MFCS.2022.73, author = {Luca, Florian and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{A Universal Skolem Set of Positive Lower Density}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {73:1--73:12}, 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.2022.73}, URN = {urn:nbn:de:0030-drops-168711}, doi = {10.4230/LIPIcs.MFCS.2022.73}, annote = {Keywords: Linear Recurrence Sequences, Skolem Problem, Exponential Diophantine Equations, Sieve Methods} }

Document

Invited Talk

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

Holonomic techniques have deep roots going back to Wallis, Euler, and Gauss, and have evolved in modern times as an important subfield of computer algebra, thanks in large part to the work of Zeilberger and others over the past three decades (see, e.g., [Doron Zeilberger, 1990; Petkovšek et al., 1997]). In this talk, I give an overview of the area, and in particular present a select survey of known and original results on decision problems for holonomic sequences and functions. I also discuss some surprising connections to the theory of periods and exponential periods, which are classical objects of study in algebraic geometry and number theory; in particular, I relate the decidability of certain decision problems for holonomic sequences to deep conjectures about periods and exponential periods, notably those due to Kontsevich and Zagier.
Parts of this exposition draws upon [George Kenison et al., 2021].

Joël Ouaknine. Holonomic Techniques, Periods, and Decision Problems (Invited Talk). In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, p. 3:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ouaknine:LIPIcs.MFCS.2021.3, author = {Ouaknine, Jo\"{e}l}, title = {{Holonomic Techniques, Periods, and Decision Problems}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {3:1--3:1}, 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.2021.3}, URN = {urn:nbn:de:0030-drops-144431}, doi = {10.4230/LIPIcs.MFCS.2021.3}, annote = {Keywords: Holonomic and hypergeometric sequences, Inequality problems, Continued fractions, Periods} }

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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 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

An infinite sequence ⟨u_n⟩_n of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each u_n ≥ 0, and minimal if, given any other linearly independent sequence ⟨v_n⟩_n satisfying the same recurrence relation, the ratio u_n/v_n → 0 as n → ∞.
In this paper we give a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences. More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.

George Kenison, Oleksiy Klurman, Engel Lefaucheux, Florian Luca, Pieter Moree, Joël Ouaknine, Markus A. Whiteland, and James Worrell. On Positivity and Minimality for Second-Order Holonomic Sequences. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 67:1-67:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kenison_et_al:LIPIcs.MFCS.2021.67, author = {Kenison, George and Klurman, Oleksiy and Lefaucheux, Engel and Luca, Florian and Moree, Pieter and Ouaknine, Jo\"{e}l and Whiteland, Markus A. and Worrell, James}, title = {{On Positivity and Minimality for Second-Order Holonomic Sequences}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {67:1--67:15}, 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.2021.67}, URN = {urn:nbn:de:0030-drops-145071}, doi = {10.4230/LIPIcs.MFCS.2021.67}, annote = {Keywords: Holonomic sequences, Minimal solutions, Positivity Problem} }

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**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with two or more parameters.
More precisely, consider a d-dimensional square matrix M whose entries are algebraic functions in one or more real variables. Given initial and target vectors u,v ∈ ℚ^d, the parametric point-to-point orbit problem asks whether there exist values of the parameters giving rise to a concrete matrix N ∈ ℝ^{d× d}, and a positive integer n ∈ ℕ, such that N^{n} u = v.
We show decidability for the case in which M depends only upon a single parameter, and we exhibit a reduction from the well-known Skolem Problem for linear recurrence sequences, suggesting intractability in the case of two or more parameters.

Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Florian Luca, Joël Ouaknine, David Purser, Markus A. Whiteland, and James Worrell. The Orbit Problem for Parametric Linear Dynamical Systems. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 28:1-28:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{baier_et_al:LIPIcs.CONCUR.2021.28, author = {Baier, Christel and Funke, Florian and Jantsch, Simon and Karimov, Toghrul and Lefaucheux, Engel and Luca, Florian and Ouaknine, Jo\"{e}l and Purser, David and Whiteland, Markus A. and Worrell, James}, title = {{The Orbit Problem for Parametric Linear Dynamical Systems}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {28:1--28:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.28}, URN = {urn:nbn:de:0030-drops-144053}, doi = {10.4230/LIPIcs.CONCUR.2021.28}, annote = {Keywords: Orbit problem, parametric, linear dynamical systems} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We study decision problems for sequences which obey a second-order holonomic recurrence of the form f(n + 2) = P(n) f(n + 1) + Q(n) f(n) with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P. We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f(1), f(2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones.

Eike Neumann, Joël Ouaknine, and James Worrell. Decision Problems for Second-Order Holonomic Recurrences. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 99:1-99:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{neumann_et_al:LIPIcs.ICALP.2021.99, author = {Neumann, Eike and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Decision Problems for Second-Order Holonomic Recurrences}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {99:1--99:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.99}, URN = {urn:nbn:de:0030-drops-141682}, doi = {10.4230/LIPIcs.ICALP.2021.99}, annote = {Keywords: holonomic sequences, Positivity Problem, Skolem Problem} }

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Invited Talk

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Holonomic techniques have deep roots going back to Wallis, Euler, and Gauss, and have evolved in modern times as an important subfield of computer algebra, thanks in large part to the work of Zeilberger and others over the past three decades. In this talk, I will give an overview of the area, and in particular will present a select survey of known and original results on decision problems for holonomic sequences and functions. (Holonomic sequences satisfy linear recurrence relations with polynomial coefficients, and holonomic functions satisfy linear differential equations with polynomial coefficients.) I will also discuss some surprising connections to the theory of periods and exponential periods, which are classical objects of study in algebraic geometry and number theory; in particular, I will relate the decidability of certain decision problems for holonomic sequences to deep conjectures about periods and exponential periods, notably those due to Kontsevich and Zagier.

Joël Ouaknine. Holonomic Techniques, Periods, and Decision Problems (Invited Talk). In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 4:1-4:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ouaknine:LIPIcs.FSTTCS.2020.4, author = {Ouaknine, Jo\"{e}l}, title = {{Holonomic Techniques, Periods, and Decision Problems}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {4:1--4:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.4}, URN = {urn:nbn:de:0030-drops-132451}, doi = {10.4230/LIPIcs.FSTTCS.2020.4}, annote = {Keywords: holonomic techniques, decision problems, recurrence sequences, minimal solutions, Positivity Problem, continued fractions, special functions, periods, exponential periods} }

Document

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix M ∈ ℚ^{d × d}, an initial vector x ∈ ℚ^{d}, a granularity g ∈ ℚ_+ and a rounding operation [⋅] projecting a vector of ℚ^{d} onto another vector whose every entry is a multiple of g, we are interested in the behaviour of the orbit 𝒪 = ⟨[x], [M[x]],[M[M[x]]],… ⟩, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability - whether a given target y ∈ ℚ^{d} belongs to 𝒪 - is PSPACE-complete for hyperbolic systems (when no eigenvalue of M has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.

Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Joël Ouaknine, Amaury Pouly, David Purser, and Markus A. Whiteland. Reachability in Dynamical Systems with Rounding. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 36:1-36:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{baier_et_al:LIPIcs.FSTTCS.2020.36, author = {Baier, Christel and Funke, Florian and Jantsch, Simon and Karimov, Toghrul and Lefaucheux, Engel and Ouaknine, Jo\"{e}l and Pouly, Amaury and Purser, David and Whiteland, Markus A.}, title = {{Reachability in Dynamical Systems with Rounding}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {36:1--36:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.36}, URN = {urn:nbn:de:0030-drops-132778}, doi = {10.4230/LIPIcs.FSTTCS.2020.36}, annote = {Keywords: dynamical systems, rounding, reachability} }

Document

**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We consider the problem of synthesising polynomial ranking functions for single-path loops over the reals with continuous semi-algebraic update function and compact semi-algebraic guard set. We show that a loop of this form has a polynomial ranking function if and only if it terminates. We further show that termination is decidable for such loops in the special case where the update function is affine.

Eike Neumann, Joël Ouaknine, and James Worrell. On Ranking Function Synthesis and Termination for Polynomial Programs. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 15:1-15:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{neumann_et_al:LIPIcs.CONCUR.2020.15, author = {Neumann, Eike and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On Ranking Function Synthesis and Termination for Polynomial Programs}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.15}, URN = {urn:nbn:de:0030-drops-128278}, doi = {10.4230/LIPIcs.CONCUR.2020.15}, annote = {Keywords: Semi-algebraic sets, Polynomial ranking functions, Polynomial programs} }

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**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We exhibit an algorithm to compute the strongest algebraic (or polynomial) invariants that hold at each location of a given guard-free linear hybrid automaton (i.e., a hybrid automaton having only unguarded transitions, all of whose assignments are given by affine expressions, and all of whose continuous dynamics are given by linear differential equations). Our main tool is a control-theoretic result of independent interest: given such a linear hybrid automaton, we show how to discretise the continuous dynamics in such a way that the resulting automaton has precisely the same algebraic invariants.

Rupak Majumdar, Joël Ouaknine, Amaury Pouly, and James Worrell. Algebraic Invariants for Linear Hybrid Automata. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 32:1-32:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{majumdar_et_al:LIPIcs.CONCUR.2020.32, author = {Majumdar, Rupak and Ouaknine, Jo\"{e}l and Pouly, Amaury and Worrell, James}, title = {{Algebraic Invariants for Linear Hybrid Automata}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {32:1--32:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.32}, URN = {urn:nbn:de:0030-drops-128443}, doi = {10.4230/LIPIcs.CONCUR.2020.32}, annote = {Keywords: Hybrid automata, algebraic invariants} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Consider a discrete dynamical system given by a square matrix M ∈ ℚ^{d × d} and a starting point s ∈ ℚ^d. The orbit of such a system is the infinite trajectory ⟨ s, Ms, M²s, …⟩. Given a collection T₁, T₂, …, T_m ⊆ ℝ^d of semialgebraic sets, we can associate with each T_i an atomic proposition P_i which evaluates to true at time n if, and only if, M^ns ∈ T_i. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system (M,s) and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.

Toghrul Karimov, Joël Ouaknine, and James Worrell. On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 54:1-54:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{karimov_et_al:LIPIcs.MFCS.2020.54, author = {Karimov, Toghrul and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {54:1--54:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.54}, URN = {urn:nbn:de:0030-drops-127215}, doi = {10.4230/LIPIcs.MFCS.2020.54}, annote = {Keywords: Linear dynamical systems, Orbit Problem, LTL model checking} }

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Continuous linear dynamical systems are used extensively in mathematics, computer science, physics, and engineering to model the evolution of a system over time. A central technique for certifying safety properties of such systems is by synthesising inductive invariants. This is the task of finding a set of states that is closed under the dynamics of the system and is disjoint from a given set of error states. In this paper we study the problem of synthesising inductive invariants that are definable in o-minimal expansions of the ordered field of real numbers. In particular, assuming Schanuel’s conjecture in transcendental number theory, we establish effective synthesis of o-minimal invariants in the case of semi-algebraic error sets. Without using Schanuel’s conjecture, we give a procedure for synthesizing o-minimal invariants that contain all but a bounded initial segment of the orbit and are disjoint from a given semi-algebraic error set. We further prove that effective synthesis of semi-algebraic invariants that contain the whole orbit, is at least as hard as a certain open problem in transcendental number theory.

Shaull Almagor, Edon Kelmendi, Joël Ouaknine, and James Worrell. Invariants for Continuous Linear Dynamical Systems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 107:1-107:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{almagor_et_al:LIPIcs.ICALP.2020.107, author = {Almagor, Shaull and Kelmendi, Edon and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Invariants for Continuous Linear Dynamical Systems}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {107:1--107:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.107}, URN = {urn:nbn:de:0030-drops-125141}, doi = {10.4230/LIPIcs.ICALP.2020.107}, annote = {Keywords: Invariants, continuous linear dynamical systems, continuous Skolem problem, safety, o-minimality} }

Document

**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} }

Document

Invited Talk

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Automated invariant generation is a fundamental challenge in program analysis and verification, going back many decades, and remains a topic of active research. In this talk I'll present a select overview and survey of work on this problem, and discuss unexpected connections to other fields including algebraic geometry, group theory, and quantum computing. (No previous knowledge of these topics will be assumed.)
This is joint work with Ehud Hrushovski, Amaury Pouly, and James Worrell.

Joël Ouaknine. Program Invariants (Invited Talk). In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, p. 3:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ouaknine:LIPIcs.CONCUR.2019.3, author = {Ouaknine, Jo\"{e}l}, title = {{Program Invariants}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {3:1--3:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.3}, URN = {urn:nbn:de:0030-drops-109056}, doi = {10.4230/LIPIcs.CONCUR.2019.3}, annote = {Keywords: Automated invariant generation, program analysis and verification} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers.

Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell. On Reachability Problems for Low-Dimensional Matrix Semigroups. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 44:1-44:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{colcombet_et_al:LIPIcs.ICALP.2019.44, author = {Colcombet, Thomas and Ouaknine, Jo\"{e}l and Semukhin, Pavel and Worrell, James}, title = {{On Reachability Problems for Low-Dimensional Matrix Semigroups}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.44}, URN = {urn:nbn:de:0030-drops-106209}, doi = {10.4230/LIPIcs.ICALP.2019.44}, annote = {Keywords: membership problem, half-space reachability problem, matrix semigroups, Heisenberg group, general linear group} }

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the problem of deciding termination of single-path while loops with integer variables, affine updates, and affine guard conditions. The question is whether such a loop terminates on all integer initial values. This problem is known to be decidable for the subclass of loops whose update matrices are diagonalisable, but the general case has remained open since being conjectured decidable by Tiwari in 2004. In this paper we show decidability of determining termination for arbitrary update matrices, confirming Tiwari’s conjecture. For the class of loops considered in this paper, the question of deciding termination on a specific initial value is a longstanding open problem in number theory. The key to our decision procedure is in showing how to circumvent the difficulties inherent in deciding termination on a fixed initial value.

Mehran Hosseini, Joël Ouaknine, and James Worrell. Termination of Linear Loops over the Integers (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 118:1-118:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hosseini_et_al:LIPIcs.ICALP.2019.118, author = {Hosseini, Mehran and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Termination of Linear Loops over the Integers}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {118:1--118:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.118}, URN = {urn:nbn:de:0030-drops-106940}, doi = {10.4230/LIPIcs.ICALP.2019.118}, annote = {Keywords: Program Verification, Loop Termination, Linear Integer Programs, Affine While Loops} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d in N, a square matrix A in Q^{d x d}, and semialgebraic source and target sets S,T subseteq R^d. The question is whether there exists x in S and n in N such that A^nx in T.
The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d <= 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory - Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

Shaull Almagor, Joël Ouaknine, and James Worrell. The Semialgebraic Orbit Problem. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 6:1-6:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{almagor_et_al:LIPIcs.STACS.2019.6, author = {Almagor, Shaull and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{The Semialgebraic Orbit Problem}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {6:1--6:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.6}, URN = {urn:nbn:de:0030-drops-102450}, doi = {10.4230/LIPIcs.STACS.2019.6}, annote = {Keywords: linear dynamical systems, Orbit Problem, first order theory of the reals} }

Document

**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational linear recurrence sequence and computes effective fine-grained lower bounds on the growth rate of the sequence.

Shaull Almagor, Brynmor Chapman, Mehran Hosseini, Joël Ouaknine, and James Worrell. Effective Divergence Analysis for Linear Recurrence Sequences. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 42:1-42:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{almagor_et_al:LIPIcs.CONCUR.2018.42, author = {Almagor, Shaull and Chapman, Brynmor and Hosseini, Mehran and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Effective Divergence Analysis for Linear Recurrence Sequences}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {42:1--42:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.42}, URN = {urn:nbn:de:0030-drops-95802}, doi = {10.4230/LIPIcs.CONCUR.2018.42}, annote = {Keywords: Linear recurrence sequences, Divergence, Algebraic numbers, Positivity} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

The termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Such deceptively simple questions also relate to a number of deep open problems, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this paper, we introduce the class of o-minimal invariants, which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel's conjecture in transcendental number theory.

Shaull Almagor, Dmitry Chistikov, Joël Ouaknine, and James Worrell. O-Minimal Invariants for Linear Loops. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 114:1-114:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{almagor_et_al:LIPIcs.ICALP.2018.114, author = {Almagor, Shaull and Chistikov, Dmitry and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{O-Minimal Invariants for Linear Loops}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {114:1--114:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.114}, URN = {urn:nbn:de:0030-drops-91188}, doi = {10.4230/LIPIcs.ICALP.2018.114}, annote = {Keywords: Invariants, linear loops, linear dynamical systems, non-termination, o-minimality} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

The Orbit Problem consists of determining, given a matrix A in R^dxd and vectors x,y in R^d, whether there exists n in N such that A^n=y. This problem was shown to be decidable in a seminal work of Kannan and Lipton in the 1980s. Subsequently, Kannan and Lipton noted that the Orbit Problem becomes considerably harder when the target y is replaced with a subspace of R^d. Recently, it was shown that the problem is decidable for vector-space targets of dimension at most three, followed by another development showing that the problem is in PSPACE for polytope targets of dimension at most three.
In this work, we take a dual look at the problem, and consider the case where the initial vector x is replaced with a polytope P_1, and the target is a polytope P_2. Then, the question is whether there exists n in N such that A^n P_1 intersection P_2 does not equal the empty set. We show that the problem can be decided in PSPACE for dimension at most three. As in previous works, decidability in the case of higher dimensions is left open, as the problem is known to be hard for long-standing number-theoretic open problems.
Our proof begins by formulating the problem as the satisfiability of a parametrized family of sentences in the existential first-order theory of real-closed fields. Then, after removing quantifiers, we are left with instances of simultaneous positivity of sums of exponentials. Using techniques from transcendental number theory, and separation bounds on algebraic numbers, we are able to solve such instances in PSPACE.

Shaull Almagor, Joël Ouaknine, and James Worrell. The Polytope-Collision Problem. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 24:1-24:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{almagor_et_al:LIPIcs.ICALP.2017.24, author = {Almagor, Shaull and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{The Polytope-Collision Problem}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.24}, URN = {urn:nbn:de:0030-drops-74521}, doi = {10.4230/LIPIcs.ICALP.2017.24}, annote = {Keywords: linear dynamical systems, orbit problem, algebraic algorithms} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

The Orbit Problem consists of determining, given a linear transformation A on d-dimensional rationals 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 P which are subsets of R^d, i.e., sets that are stable under A and contain x and 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 invariants of polynomial size.
It is worth noting that the existence of semilinear invariants, on the other hand, is (to the best of our knowledge) not known to be decidable.

Nathanaël Fijalkow, Pierre Ohlmann, Joël Ouaknine, Amaury Pouly, and James Worrell. Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 29:1-29:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fijalkow_et_al:LIPIcs.STACS.2017.29, author = {Fijalkow, Nathana\"{e}l and Ohlmann, Pierre and Ouaknine, Jo\"{e}l and Pouly, Amaury and Worrell, James}, title = {{Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {29:1--29:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.29}, URN = {urn:nbn:de:0030-drops-70059}, doi = {10.4230/LIPIcs.STACS.2017.29}, annote = {Keywords: Verification,algebraic computation,Skolem Problem,Orbit Problem,invariants} }

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**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

Freeze LTL is a temporal logic with registers that is suitable for specifying properties of data words. In this paper we study the model checking problem for Freeze LTL on one-counter automata. This problem is known to be undecidable in full generality and PSPACE-complete for the special case of deterministic one-counter automata. Several years ago, Demri and Sangnier investigated the model checking problem for the flat fragment of Freeze LTL on several classes of counter automata and posed the decidability of model checking flat Freeze LTL on one-counter automata as an open problem. In this paper we resolve this problem positively, utilising a known reduction to a reachability problem on one-counter automata with parameterised equality and disequality tests. Our main technical contribution is to show decidability of the latter problem by translation to Presburger arithmetic.

Antonia Lechner, Richard Mayr, Joël Ouaknine, Amaury Pouly, and James Worrell. Model Checking Flat Freeze LTL on One-Counter Automata. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 29:1-29:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lechner_et_al:LIPIcs.CONCUR.2016.29, author = {Lechner, Antonia and Mayr, Richard and Ouaknine, Jo\"{e}l and Pouly, Amaury and Worrell, James}, title = {{Model Checking Flat Freeze LTL on One-Counter Automata}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.29}, URN = {urn:nbn:de:0030-drops-61841}, doi = {10.4230/LIPIcs.CONCUR.2016.29}, annote = {Keywords: one-counter automata, disequality tests, reachability, freeze LTL, Presburger arithmetic} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems, such as linear hybrid automata and continuoustime Markov chains. Decidability of the problem is currently open — indeed decidability is open even for the sub-problem in which a zero is sought in a bounded interval. In this paper we show decidability of the bounded problem subject to Schanuel's Conjecture, a unifying conjecture in transcendental number theory. We furthermore analyse the unbounded problem in terms of the frequencies of the differential equation, that is, the imaginary parts of the characteristic roots.
We show that the unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. We complete the picture by showing that decidability of the unbounded problem in the case of two (or more) rationally linearly independent frequencies would entail a major new effectiveness result in Diophantine approximation, namely computability of the Diophantine-approximation types of all real algebraic numbers.

Ventsislav Chonev, Joël Ouaknine, and James Worrell. On the Skolem Problem for Continuous Linear Dynamical Systems. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 100:1-100:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chonev_et_al:LIPIcs.ICALP.2016.100, author = {Chonev, Ventsislav and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On the Skolem Problem for Continuous Linear Dynamical Systems}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {100:1--100:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.100}, URN = {urn:nbn:de:0030-drops-62357}, doi = {10.4230/LIPIcs.ICALP.2016.100}, annote = {Keywords: differential equations, reachability, Baker’s Theorem, Schanuel’s Conjecture, semi-algebraic sets} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Herman's self-stabilization algorithm, introduced 25 years ago, is a well-studied synchronous randomized protocol for enabling a ring of N processes collectively holding any odd number of tokens to reach a stable state in which a single token remains. Determining the worst-case expected time to stabilization is the central outstanding open problem about this protocol. It is known that there is a constant h such that any initial configuration has expected stabilization time at most hN2. Ten years ago, McIver and Morgan established a lower bound of 4/27 ~ 0.148 for h, achieved with three equally-spaced tokens, and conjectured this to be the optimal value of h. A series of papers over the last decade gradually reduced the upper bound on h, with the present record (achieved in 2014) standing at approximately 0.156. In this paper, we prove McIver and Morgan's conjecture and establish that h = 4/27 is indeed optimal.

Maria Bruna, Radu Grigore, Stefan Kiefer, Joël Ouaknine, and James Worrell. Proving the Herman-Protocol Conjecture. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 104:1-104:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bruna_et_al:LIPIcs.ICALP.2016.104, author = {Bruna, Maria and Grigore, Radu and Kiefer, Stefan and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Proving the Herman-Protocol Conjecture}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {104:1--104:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.104}, URN = {urn:nbn:de:0030-drops-62393}, doi = {10.4230/LIPIcs.ICALP.2016.104}, annote = {Keywords: randomized protocols, self-stabilization, Lyapunov function, expected time} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We investigate the Matrix Powering Positivity Problem, PosMatPow: given an m X m square integer matrix M, a linear function f: Z^{m X m} -> Z with integer coefficients, and a positive integer n (encoded in binary), determine whether f(M^n) \geq 0. We show that for fixed dimensions m of 2 and 3, this problem is decidable in polynomial time.

Esther Galby, Joël Ouaknine, and James Worrell. On Matrix Powering in Low Dimensions. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 329-340, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{galby_et_al:LIPIcs.STACS.2015.329, author = {Galby, Esther and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On Matrix Powering in Low Dimensions}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {329--340}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.329}, URN = {urn:nbn:de:0030-drops-49240}, doi = {10.4230/LIPIcs.STACS.2015.329}, annote = {Keywords: matrix powering, complexity, Baker's theorem} }

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**Published in:** Dagstuhl Reports, Volume 4, Issue 3 (2014)

This report documents the program and the outcomes of Dagstuhl Seminar 14141 "Reachability Problems for Infinite-State Systems", held from March 30th until April 4th, 2014. The seminar gathered 44 participants and the program consisted of 34 presentations. Participants were asked to contribute open questions prior and
during the seminar. A list of these open questions appears in a separate section of the present report. This list generated collaborations among participants and gave rise to research publications solving (partially), for example, question 5.13, namely "what functions are computable by VASS?"

Javier Esparza, Alain Finkel, Pierre McKenzie, and Joel Ouaknine. Reachability Problems for Infinite-State Systems (Dagstuhl Seminar 14141). In Dagstuhl Reports, Volume 4, Issue 3, pp. 153-180, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@Article{esparza_et_al:DagRep.4.3.153, author = {Esparza, Javier and Finkel, Alain and McKenzie, Pierre and Ouaknine, Joel}, title = {{Reachability Problems for Infinite-State Systems (Dagstuhl Seminar 14141)}}, pages = {153--180}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2014}, volume = {4}, number = {3}, editor = {Esparza, Javier and Finkel, Alain and McKenzie, Pierre and Ouaknine, Joel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.3.153}, URN = {urn:nbn:de:0030-drops-46121}, doi = {10.4230/DagRep.4.3.153}, annote = {Keywords: Infinite-State Systems, Reachability Problems, Formal Verification, Well-Structured Transition Systems, Counter Machines, Vector Addition Systems, Timed Systems} }

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**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Church's Problem asks for the construction of a procedure which, given a logical specification S(I,O) between input strings I and output strings O, determines whether there exists an operator F that implements the specification in the sense that S(I,F(I)) holds for all inputs I. Buechi and Landweber gave a procedure to solve Church's problem for MSO specifications and operators computable by finite-state automata.
We consider extensions of Church's problem in two orthogonal directions: (i) we address the problem in a more general logical setting, where not only the specifications but also the solutions are presented in a logical system; (ii) we consider not only the canonical discrete time domain of the natural numbers, but also the continuous domain of reals.
We show that for every fixed bounded length interval of the reals, Church's problem is decidable when specifications and implementations are described in the monadic second-order logics over the reals with order and the +1 function.

Mark Jenkins, Joël Ouaknine, Alexander Rabinovich, and James Worrell. The Church Synthesis Problem with Metric. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 307-321, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{jenkins_et_al:LIPIcs.CSL.2011.307, author = {Jenkins, Mark and Ouaknine, Jo\"{e}l and Rabinovich, Alexander and Worrell, James}, title = {{The Church Synthesis Problem with Metric}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {307--321}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.307}, URN = {urn:nbn:de:0030-drops-32390}, doi = {10.4230/LIPIcs.CSL.2011.307}, annote = {Keywords: Church's Problem, monadic logic, games, uniformization} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

A fundamental problem in numerical computation and computational geometry is to determine the sign of arithmetic expressions in radicals. Here we consider the simpler problem of deciding whether $\sum_{i=1}^m C_i A_i^{X_i}$ is zero for given rational numbers $A_i$, $C_i$, $X_i$. It has been known for almost twenty years that this can be decided in polynomial time. In this paper we improve this result by showing membership in uniform TC0. This requires several significant departures from Blömer's polynomial-time algorithm as the latter crucially relies on primitives, such as gcd computation and binary search, that are not known to be in TC0.

Paul Hunter, Patricia Bouyer, Nicolas Markey, Joël Ouaknine, and James Worrell. Computing Rational Radical Sums in Uniform TC^0. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 308-316, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{hunter_et_al:LIPIcs.FSTTCS.2010.308, author = {Hunter, Paul and Bouyer, Patricia and Markey, Nicolas and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Computing Rational Radical Sums in Uniform TC^0}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {308--316}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.308}, URN = {urn:nbn:de:0030-drops-28739}, doi = {10.4230/LIPIcs.FSTTCS.2010.308}, annote = {Keywords: Sum of square roots, Threshold circuits, Complexity} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

A channel machine consists of a finite controller together with
several fifo channels; the controller can read messages from the
head of a channel and write messages to the tail of a channel. In
this paper, we focus on channel machines with insertion errors,
i.e., machines in whose channels messages can spontaneously appear.
Such devices have been previously introduced in the study of Metric
Temporal Logic. We consider the termination problem: are all the
computations of a given insertion channel machine finite? We show
that this problem has non-elementary, yet primitive recursive
complexity.

Patricia Bouyer, Nicolas Markey, Joël Ouaknine, Philippe Schnoebelen, and James Worrell. On Termination for Faulty Channel Machines. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 121-132, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{bouyer_et_al:LIPIcs.STACS.2008.1339, author = {Bouyer, Patricia and Markey, Nicolas and Ouaknine, Jo\"{e}l and Schnoebelen, Philippe and Worrell, James}, title = {{On Termination for Faulty Channel Machines}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {121--132}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1339}, URN = {urn:nbn:de:0030-drops-13390}, doi = {10.4230/LIPIcs.STACS.2008.1339}, annote = {Keywords: Automated Verification, Computational Complexity} }

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