4 Search Results for "Sablik, Mathieu"

Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.129
The Complexity of Computing in Continuous Time: Space Complexity Is Precision

Authors: Manon Blanc and Olivier Bournez

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Models of computations over the integers are equivalent from a computability and complexity theory point of view by the (effective) Church-Turing thesis. It is not possible to unify discrete-time models over the reals. The situation is unclear but simpler for continuous-time models, as there is a unifying mathematical model, provided by ordinary differential equations (ODEs). Each model corresponds to a particular class of ODEs. For example, the General Purpose Analog Computer model of Claude Shannon, introduced as a mathematical model of analogue machines (Differential Analyzers), is known to correspond to polynomial ODEs. However, the question of a robust complexity theory for such models and its relations to classical (discrete) computation theory is an old problem. There was some recent significant progress: it has been proved that (classical) time complexity corresponds to the length of the involved curves, i.e. to the length of the solutions of the corresponding polynomial ODEs. The question of whether there is a simple and robust way to measure space complexity remains. We argue that space complexity corresponds to precision and conversely. Concretely, we propose and prove an algebraic characterisation of FPSPACE, using continuous ODEs. Recent papers proposed algebraic characterisations of polynomial-time and polynomial-space complexity classes over the reals, but with a discrete-time: those algebras rely on discrete ODE schemes. Here, we use classical (continuous) ODEs, with the classic definition of derivation and hence with the more natural context of continuous-time associated with ODEs. We characterise both the case of polynomial space functions over the integers and the reals. This is done by proving two inclusions. The first is obtained using some original polynomial space method for solving ODEs. For the other, we prove that Turing machines, with a proper representation of real numbers, can be simulated by continuous ODEs and not just discrete ODEs. A major consequence is that the associated space complexity is provably related to the numerical stability of involved schemas and the associated required precision. We obtain that a problem can be solved in polynomial space if and only if it can be simulated by some numerically stable ODE, using a polynomial precision.
Cite as

Manon Blanc and Olivier Bournez. The Complexity of Computing in Continuous Time: Space Complexity Is Precision. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 129:1-129:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blanc_et_al:LIPIcs.ICALP.2024.129,
  author =	{Blanc, Manon and Bournez, Olivier},
  title =	{{The Complexity of Computing in Continuous Time: Space Complexity Is Precision}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{129:1--129:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.129},
  URN =		{urn:nbn:de:0030-drops-202722},
  doi =		{10.4230/LIPIcs.ICALP.2024.129},
  annote =	{Keywords: Models of computation, Ordinary differential equations, Real computations, Analog computations, Complexity theory, Implicit complexity, Recursion scheme}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.26
Domino Problem Under Horizontal Constraints

Authors: Nathalie Aubrun, Julien Esnay, and Mathieu Sablik

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

Abstract
The Domino Problem on ℤ² asks if it is possible to tile the plane with a given set of Wang tiles; it is a classical decision problem which is known to be undecidable. The purpose of this article is to parameterize this problem to explore the frontier between decidability and undecidability. To do so we fix some horizontal constraints H on the tiles and consider a new Domino Problem DP_H: given a vertical constraint, is it possible to tile the plane? We characterize the nearest-neighbor horizontal constraints where DP_H is decidable using graphs combinatorics.
Cite as

Nathalie Aubrun, Julien Esnay, and Mathieu Sablik. Domino Problem Under Horizontal Constraints. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{aubrun_et_al:LIPIcs.STACS.2020.26,
  author =	{Aubrun, Nathalie and Esnay, Julien and Sablik, Mathieu},
  title =	{{Domino Problem Under Horizontal Constraints}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{26:1--26:15},
  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.2020.26},
  URN =		{urn:nbn:de:0030-drops-118875},
  doi =		{10.4230/LIPIcs.STACS.2020.26},
  annote =	{Keywords: Dynamical Systems, Symbolic Dynamics, Subshifts, Wang tiles, Undecidability, Domino Problem, Combinatorics, Tilings, Subshifts of Finite Type}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.110
Algorithmic Complexity for the Realization of an Effective Subshift By a Sofic

Authors: Mathieu Sablik and Michael Schraudner

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

Abstract
Realization of d-dimensional effective subshifts as projective sub-actions of d + d'-dimensional sofic subshifts for d' >= 1 is now well known [Hochman, 2009; Durand/Romashchenko/Shen, 2012; Aubrun/Sablik, 2013]. In this paper we are interested in qualitative aspects of this realization. We introduce a new topological conjugacy invariant for effective subshifts, the speed of convergence, in view to exhibit algorithmic properties of these subshifts in contrast to the usual framework that focuses on undecidable properties.
Cite as

Mathieu Sablik and Michael Schraudner. Algorithmic Complexity for the Realization of an Effective Subshift By a Sofic. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 110:1-110:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{sablik_et_al:LIPIcs.ICALP.2016.110,
  author =	{Sablik, Mathieu and Schraudner, Michael},
  title =	{{Algorithmic Complexity for the Realization of an Effective Subshift By a Sofic}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{110:1--110:14},
  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.2016.110},
  URN =		{urn:nbn:de:0030-drops-62454},
  doi =		{10.4230/LIPIcs.ICALP.2016.110},
  annote =	{Keywords: Subshift, computability, time complexity, space complexity, tilings}
}
Document
DOI: 10.4230/LIPIcs.STACS.2009.1833
An Order on Sets of Tilings Corresponding to an Order on Languages

Authors: Nathalie Aubrun and Mathieu Sablik

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Abstract
Traditionally a tiling is defined with a finite number of finite forbidden patterns. We can generalize this notion considering any set of patterns. Generalized tilings defined in this way can be studied with a dynamical point of view, leading to the notion of subshift. In this article we establish a correspondence between an order on subshifts based on dynamical transformations on them and an order on languages of forbidden patterns based on computability properties.
Cite as

Nathalie Aubrun and Mathieu Sablik. An Order on Sets of Tilings Corresponding to an Order on Languages. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 99-110, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{aubrun_et_al:LIPIcs.STACS.2009.1833,
  author =	{Aubrun, Nathalie and Sablik, Mathieu},
  title =	{{An Order on Sets of Tilings Corresponding to an Order on Languages}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{99--110},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1833},
  URN =		{urn:nbn:de:0030-drops-18336},
  doi =		{10.4230/LIPIcs.STACS.2009.1833},
  annote =	{Keywords: Tiling, Subshift, Turing machine with oracle, Subdynamics}
}
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