6 Search Results for "Salamati, Mahmoud"

Document
DOI: 10.4230/LIPIcs.MFCS.2025.79
Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space

Authors: Eike Neumann

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We study the problem of deciding whether a point escapes a closed subset of ℝ^d under the iteration of a continuous map f : ℝ^d → ℝ^d in the bit-model of real computation. We give a sound partial decision method for this problem which is complete in the sense that its halting set contains the halting set of all sound partial decision methods for the problem. Equivalently, our decision method terminates on all problem instances whose answer is robust under all sufficiently small perturbations of the function. We further show that the halting set of our algorithm is dense in the set of all problem instances. While our algorithm applies to general continuous functions, we demonstrate that it also yields complete decision methods for much more rigid function families: affine linear systems and quadratic complex polynomials. In the latter case, completeness is subject to the density of hyperbolicity conjecture in complex dynamics. This in particular yields an alternative proof of Hertling’s (2004) conditional answer to a question raised by Penrose (1989) regarding the computability of the Mandelbrot set.
Cite as

Eike Neumann. Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{neumann:LIPIcs.MFCS.2025.79,
  author =	{Neumann, Eike},
  title =	{{Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{79:1--79:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.79},
  URN =		{urn:nbn:de:0030-drops-241866},
  doi =		{10.4230/LIPIcs.MFCS.2025.79},
  annote =	{Keywords: Dynamical Systems, Computability in Analysis, Non-Linear Functions}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.80
On Expansions of Monadic Second-Order Logic with Dynamical Predicates

Authors: Joris Nieuwveld and Joël Ouaknine

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Expansions of the monadic second-order (MSO) theory of the structure ⟨ℕ;<⟩ have been a fertile and active area of research ever since the publication of the seminal papers of Büchi and Elgot & Rabin on the subject in the 1960s. In the present paper, we establish decidability of the MSO theory of ⟨ℕ;<,P⟩, where P ranges over a large class of unary "dynamical" predicates, i.e., sets of non-negative values assumed by certain integer linear recurrence sequences. One of our key technical tools is the novel concept of (effective) prodisjunctivity, which we expect may also find independent applications further afield.
Cite as

Joris Nieuwveld and Joël Ouaknine. On Expansions of Monadic Second-Order Logic with Dynamical Predicates. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 80:1-80:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{nieuwveld_et_al:LIPIcs.MFCS.2025.80,
  author =	{Nieuwveld, Joris and Ouaknine, Jo\"{e}l},
  title =	{{On Expansions of Monadic Second-Order Logic with Dynamical Predicates}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{80:1--80:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.80},
  URN =		{urn:nbn:de:0030-drops-241879},
  doi =		{10.4230/LIPIcs.MFCS.2025.80},
  annote =	{Keywords: Monadic second-order logic, linear recurrence sequences, decidability, Baker’s theorem}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.163
Verification of Linear Dynamical Systems via O-Minimality of the Real Numbers

Authors: Toghrul Karimov

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
A discrete-time linear dynamical system (LDS) is given by an update matrix M ∈ ℝ^{d× d}, and has the trajectories ⟨s, Ms, M²s, …⟩ for s ∈ ℝ^d. Reachability-type decision problems of linear dynamical systems, most notably the Skolem Problem, lie at the forefront of decidability: typically, sound and complete algorithms are known only in low dimensions, and these rely on sophisticated tools from number theory and Diophantine approximation. Recently, however, o-minimality has emerged as a counterpoint to these number-theoretic tools that allows us to decide certain modifications of the classical problems of LDS without any dimension restrictions. In this paper, we first introduce the Decomposition Method, a framework that captures all applications of o-minimality to decision problems of LDS that are currently known to us. We then use the Decomposition Method to show decidability of the Robust Safety Problem (restricted to bounded initial sets) in arbitrary dimension: given a matrix M, a bounded semialgebraic set S of initial points, and a semialgebraic set T of unsafe points, it is decidable whether there exists ε > 0 such that all orbits that begin in the ε-ball around S avoid T.
Cite as

Toghrul Karimov. Verification of Linear Dynamical Systems via O-Minimality of the Real Numbers. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 163:1-163:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{karimov:LIPIcs.ICALP.2025.163,
  author =	{Karimov, Toghrul},
  title =	{{Verification of Linear Dynamical Systems via O-Minimality of the Real Numbers}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{163:1--163:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.163},
  URN =		{urn:nbn:de:0030-drops-235401},
  doi =		{10.4230/LIPIcs.ICALP.2025.163},
  annote =	{Keywords: Linear dynamical systems, reachability problems, o-minimality}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.40
The Pseudo-Reachability Problem for Diagonalisable Linear Dynamical Systems

Authors: Julian D'Costa, Toghrul Karimov, Rupak Majumdar, Joël Ouaknine, Mahmoud Salamati, and James Worrell

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.34
The Pseudo-Skolem Problem is Decidable

Authors: Julian D'Costa, Toghrul Karimov, Rupak Majumdar, Joël Ouaknine, Mahmoud Salamati, Sadegh Soudjani, and James Worrell

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

Abstract
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.
Cite as

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}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2020.133
On Decidability of Time-Bounded Reachability in CTMDPs

Authors: Rupak Majumdar, Mahmoud Salamati, and Sadegh Soudjani

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

Abstract
We consider the time-bounded reachability problem for continuous-time Markov decision processes. We show that the problem is decidable subject to Schanuel’s conjecture. Our decision procedure relies on the structure of optimal policies and the conditional decidability (under Schanuel’s conjecture) of the theory of reals extended with exponential and trigonometric functions over bounded domains. We further show that any unconditional decidability result would imply unconditional decidability of the bounded continuous Skolem problem, or equivalently, the problem of checking if an exponential polynomial has a non-tangential zero in a bounded interval. We note that the latter problems are also decidable subject to Schanuel’s conjecture but finding unconditional decision procedures remain longstanding open problems.
Cite as

Rupak Majumdar, Mahmoud Salamati, and Sadegh Soudjani. On Decidability of Time-Bounded Reachability in CTMDPs. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 133:1-133:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{majumdar_et_al:LIPIcs.ICALP.2020.133,
  author =	{Majumdar, Rupak and Salamati, Mahmoud and Soudjani, Sadegh},
  title =	{{On Decidability of Time-Bounded Reachability in CTMDPs}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{133:1--133:19},
  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.2020.133},
  URN =		{urn:nbn:de:0030-drops-125408},
  doi =		{10.4230/LIPIcs.ICALP.2020.133},
  annote =	{Keywords: CTMDP, Time bounded reachability, Continuous Skolem Problem, Schanuel’s Conjecture}
}
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