15 Search Results for "Pouly, Amaury"

Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2026.2
Moments in Time: Algebraic Analysis for Solvable Loops (Invited Talk)

Authors: Laura Kovács

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
With substantial progress in automated reasoning, algebraic approaches emerged to automatically analyse program loops in an exact manner. In this invited talk, we discuss recent results in characterizing the functional behaviour of loops with polynomial arithmetic and probabilistic updates. This problem remains unsolved even when we restrict consideration to loops that are non-nested, without conditionals, and/or without exit conditions [Ehud Hrushovski et al., 2023; Julian Müllner and others, 2024]. We are motivated by applications of computer-aided verification, in particular to assess the safety, security, and sensitivity of computer systems [M. Z. Kwiatkowska et al., 2011; Gilles Barthe et al., 2012; Gilles Barthe and others, 2018; Marcel Moosbrugger et al., 2023; Alessandro Abate et al., 2023; Andrey Kofnov and others, 2024]. We are interested in modeling, deciding, and solving loop analysis. The key to our work are moment-computable loops [L. Kovács, 2008; Marcel Moosbrugger et al., 2022] which allow us to set limits on what is decidable and solvable in loop analysis. Our approach combines algebra, statistics, and automated reasoning to mechanize loop analysis. Various techniques, such as martingale theory and quantifier elimination, can be seen as examples of moment-computable loop analysis. This talk is structured within three inter-connected parts. We first bring moment-based loop analysis into the landscape of {loop invariant synthesis} and extend moment-computable loops with {termination guarantees}. We next automate the reasoning about (probabilistic) loops by summarizing loop semantics as (probabilistic) algebraic recurrences, whose closed-form solutions capture (higher-order) moments, and hence invariants, among loop variables. These recurrences together with loop tests yield moment-based (super)martingales necessary to prove loop termination and compute probability bounds on termination. We finally describe moment-computable loops whose invariant synthesis {decidable} or as {hard} as open problems, such as the Skolem problem [Graham Everest et al., 2003; Terrence Tao, 2008].
Cite as

Laura Kovács. Moments in Time: Algebraic Analysis for Solvable Loops (Invited Talk). In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kovacs:LIPIcs.STACS.2026.2,
  author =	{Kov\'{a}cs, Laura},
  title =	{{Moments in Time: Algebraic Analysis for Solvable Loops}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{2:1--2:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.2},
  URN =		{urn:nbn:de:0030-drops-254910},
  doi =		{10.4230/LIPIcs.STACS.2026.2},
  annote =	{Keywords: program analysis, algebraic reasoning, symbolic computation, loop invariants}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.29
A Universal Uniform Approximation Theorem for Neural Networks

Authors: Olivier Bournez, Johanne Cohen, and Adrian Wurm

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

Abstract
We show the existence of a fixed recurrent network capable of approximating any computable function with arbitrary precision, provided that an encoding of the function is given in the initial input. While uniform approximation over a compact domain is a well-known property of neural networks, we go further by proving that our network ensures effective uniform approximation - simultaneously ensuring: - Uniform approximation in the sup-norm sense, guaranteeing precision across the compact domain {[0,1]^d}; - Uniformity in the sense of computability theory (also referred to as effectivity or universality), meaning the same network works for all computable functions. Our result is obtained constructively, using original arguments. Moreover, our construction bridges computation theory with neural network approximation, providing new insights into the fundamental connections between circuit complexity and function representation. Furthermore, this connection extends beyond computability to complexity theory. The obtained network is efficient: if a function is computable or approximable in polynomial time in the Turing machine model, then the network requires only a polynomial number of recurrences or iterations to achieve the same level of approximation, and conversely. Moreover, the recurrent network can be assumed to be very narrow, strengthening the link our results and existing models of very deep learning, where uniform approximation properties have already been established.
Cite as

Olivier Bournez, Johanne Cohen, and Adrian Wurm. A Universal Uniform Approximation Theorem for Neural Networks. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bournez_et_al:LIPIcs.MFCS.2025.29,
  author =	{Bournez, Olivier and Cohen, Johanne and Wurm, Adrian},
  title =	{{A Universal Uniform Approximation Theorem for Neural Networks}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{29:1--29:20},
  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.2025.29},
  URN =		{urn:nbn:de:0030-drops-241365},
  doi =		{10.4230/LIPIcs.MFCS.2025.29},
  annote =	{Keywords: Models of computation, Complexity theory, Formal neural networks}
}
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
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.159
The Ultimate Signs of Second-Order Holonomic Sequences

Authors: Fugen Hagihara and Akitoshi Kawamura

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

Abstract
A real-valued sequence f = {f(n)}_{n ∈ ℕ} is said to be second-order holonomic if it satisfies a linear recurrence f (n + 2) = P (n) f (n + 1) + Q (n) f (n) for all sufficiently large n, where P, Q ∈ ℝ(x) are rational functions. We study the ultimate sign of such a sequence, i.e., the repeated pattern that the signs of f (n) follow for sufficiently large n. For each P, Q we determine all ultimate signs that f can have, and show how they partition the space of initial values of f. This completes the prior work by Neumann, Ouaknine and Worrell, who have settled some restricted cases. As a corollary, it follows that when P, Q have rational coefficients, f either has an ultimate sign of length 1, 2, 3, 4, 6, 8 or 12, or never falls into a repeated sign pattern. We also give a partial algorithm that finds the ultimate sign of f (or tells that there is none) in almost all cases.
Cite as

Fugen Hagihara and Akitoshi Kawamura. The Ultimate Signs of Second-Order Holonomic Sequences. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 159:1-159:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hagihara_et_al:LIPIcs.ICALP.2025.159,
  author =	{Hagihara, Fugen and Kawamura, Akitoshi},
  title =	{{The Ultimate Signs of Second-Order Holonomic Sequences}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{159:1--159:20},
  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.2025.159},
  URN =		{urn:nbn:de:0030-drops-235363},
  doi =		{10.4230/LIPIcs.ICALP.2025.159},
  annote =	{Keywords: Holonomic sequences, ultimate signs, Skolem Problem, Positivity Problem}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.154
Submonoid Membership in n-Dimensional Lamplighter Groups and S-Unit Equations

Authors: Ruiwen Dong

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

Abstract
We show that Submonoid Membership is decidable in n-dimensional lamplighter groups (ℤ/pℤ) ≀ ℤⁿ for any prime p and integer n. More generally, we show decidability of Submonoid Membership in semidirect products of the form 𝒴 ⋊ ℤⁿ, where 𝒴 is any finitely presented module over the Laurent polynomial ring 𝔽_p[X₁^{±}, …, X_n^{±}]. Combined with a result of Shafrir (2024), this gives the first example of a group G and a finite index subgroup G̃ ≤ G, such that Submonoid Membership is decidable in G̃ but undecidable in G. To obtain our decidability result, we reduce Submonoid Membership in 𝒴 ⋊ ℤⁿ to solving S-unit equations over 𝔽_p[X₁^{±}, …, X_n^{±}]-modules. We show that the solution set of such equations is effectively p-automatic, extending a result of Adamczewski and Bell (2012). As an intermediate result, we also obtain that the solution set of the Knapsack Problem in 𝒴 ⋊ ℤⁿ is effectively p-automatic.
Cite as

Ruiwen Dong. Submonoid Membership in n-Dimensional Lamplighter Groups and S-Unit Equations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 154:1-154:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dong:LIPIcs.ICALP.2025.154,
  author =	{Dong, Ruiwen},
  title =	{{Submonoid Membership in n-Dimensional Lamplighter Groups and S-Unit Equations}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{154:1--154:20},
  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.2025.154},
  URN =		{urn:nbn:de:0030-drops-235316},
  doi =		{10.4230/LIPIcs.ICALP.2025.154},
  annote =	{Keywords: Submonoid Membership, lamplighter groups, S-unit equations, p-automatic sets, Knapsack in groups}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.37
On the Existential Theory of the Reals Enriched with Integer Powers of a Computable Number

Authors: Jorge Gallego-Hernández and Alessio Mansutti

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
This paper investigates ∃ℝ(ξ^ℤ), that is the extension of the existential theory of the reals by an additional unary predicate ξ^ℤ for the integer powers of a fixed computable real number ξ > 0. If all we have access to is a Turing machine computing ξ, it is not possible to decide whether an input formula from this theory is satisfiable. However, we show an algorithm to decide this problem when - ξ is known to be transcendental, or - ξ is a root of some given integer polynomial (that is, ξ is algebraic). In other words, knowing the algebraicity of ξ suffices to circumvent undecidability. Furthermore, we establish complexity results under the proviso that ξ enjoys what we call a polynomial root barrier. Using this notion, we show that the satisfiability problem of ∃ℝ(ξ^ℤ) is - in ExpSpace if ξ is an algebraic number, and - in 3Exp if ξ is a logarithm of an algebraic number, Euler’s e, or the number π, among others. To establish our results, we first observe that the satisfiability problem of ∃ℝ(ξ^ℤ) reduces in exponential time to the problem of solving quantifier-free instances of the theory of the reals where variables range over ξ^ℤ. We then prove that these instances have a small witness property: only finitely many integer powers of ξ must be considered to find whether a formula is satisfiable. Our complexity results are shown by relying on well-established machinery from Diophantine approximation and transcendental number theory, such as bounds for the transcendence measure of numbers. As a by-product of our results, we are able to remove the appeal to Schanuel’s conjecture from the proof of decidability of the entropic risk threshold problem for stochastic games with rational probabilities, rewards and threshold [Baier et al., MFCS, 2023]: when the base of the entropic risk is e and the aversion factor is a fixed algebraic number, the problem is (unconditionally) in Exp.
Cite as

Jorge Gallego-Hernández and Alessio Mansutti. On the Existential Theory of the Reals Enriched with Integer Powers of a Computable Number. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gallegohernandez_et_al:LIPIcs.STACS.2025.37,
  author =	{Gallego-Hern\'{a}ndez, Jorge and Mansutti, Alessio},
  title =	{{On the Existential Theory of the Reals Enriched with Integer Powers of a Computable Number}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.37},
  URN =		{urn:nbn:de:0030-drops-228635},
  doi =		{10.4230/LIPIcs.STACS.2025.37},
  annote =	{Keywords: Theory of the reals with exponentiation, decision procedures, computability}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.52
On Polynomial-Time Decidability of k-Negations Fragments of FO Theories (Extended Abstract)

Authors: Christoph Haase, Alessio Mansutti, and Amaury Pouly

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
This paper introduces a generic framework that provides sufficient conditions for guaranteeing polynomial-time decidability of fixed-negation fragments of first-order theories that adhere to certain fixed-parameter tractability requirements. It enables deciding sentences of such theories with arbitrary existential quantification, conjunction and a fixed number of negation symbols in polynomial time. It was recently shown by Nguyen and Pak [SIAM J. Comput. 51(2): 1-31 (2022)] that an even more restricted such fragment of Presburger arithmetic (the first-order theory of the integers with addition and order) is NP-hard. In contrast, by application of our framework, we show that the fixed negation fragment of weak Presburger arithmetic, which drops the order relation from Presburger arithmetic in favour of equality, is decidable in polynomial time.
Cite as

Christoph Haase, Alessio Mansutti, and Amaury Pouly. On Polynomial-Time Decidability of k-Negations Fragments of FO Theories (Extended Abstract). In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{haase_et_al:LIPIcs.MFCS.2023.52,
  author =	{Haase, Christoph and Mansutti, Alessio and Pouly, Amaury},
  title =	{{On Polynomial-Time Decidability of k-Negations Fragments of FO Theories (Extended Abstract)}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.52},
  URN =		{urn:nbn:de:0030-drops-185869},
  doi =		{10.4230/LIPIcs.MFCS.2023.52},
  annote =	{Keywords: first-order theories, arithmetic theories, fixed-parameter tractability}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.4
Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding

Authors: Divesh Aggarwal, Yanlin Chen, Rajendra Kumar, and Yixin Shen

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. 1) A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4 ≤ q ≤ √n, our algorithm takes q^{13n+o(n)} time and requires poly(n)⋅ q^{16n/q²} memory. This tradeoff which ranges from enumeration (q = √n) to sieving (q constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. 2) A quantum algorithm that runs in time 2^{0.9533n+o(n)} and requires 2^{0.5n+o(n)} classical memory and poly(n) qubits. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [Divesh Aggarwal et al., 2015] that has a time and space complexity 2^{n+o(n)}. 3) A classical algorithm for SVP that runs in time 2^{1.741n+o(n)} time and 2^{0.5n+o(n)} space. This improves over an algorithm of [Yanlin Chen et al., 2018] that has the same space complexity. The time complexity of our classical and quantum algorithms are expressed using a quantity related to the kissing number of a lattice. A known upper bound of this quantity is 2^{0.402n}, but in practice for most lattices, it can be much smaller and even 2^o(n). In that case, our classical algorithm runs in time 2^{1.292n} and our quantum algorithm runs in time 2^{0.750n}.
Cite as

Divesh Aggarwal, Yanlin Chen, Rajendra Kumar, and Yixin Shen. Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{aggarwal_et_al:LIPIcs.STACS.2021.4,
  author =	{Aggarwal, Divesh and Chen, Yanlin and Kumar, Rajendra and Shen, Yixin},
  title =	{{Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.4},
  URN =		{urn:nbn:de:0030-drops-136494},
  doi =		{10.4230/LIPIcs.STACS.2021.4},
  annote =	{Keywords: Lattices, Shortest Vector Problem, Discrete Gaussian Sampling, Time-Space Tradeoff, Quantum computation, Bounded distance decoding}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.36
Reachability in Dynamical Systems with Rounding

Authors: Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel Lefaucheux, Joël Ouaknine, Amaury Pouly, David Purser, and Markus A. Whiteland

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

Abstract
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.
Cite as

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
DOI: 10.4230/LIPIcs.CONCUR.2020.32
Algebraic Invariants for Linear Hybrid Automata

Authors: Rupak Majumdar, Joël Ouaknine, Amaury Pouly, and James Worrell

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

Abstract
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.
Cite as

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
Invited Talk
DOI: 10.4230/LIPIcs.CONCUR.2019.3
Program Invariants (Invited Talk)

Authors: Joël Ouaknine

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

Abstract
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.
Cite as

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
DOI: 10.4230/LIPIcs.ICALP.2017.116
A Universal Ordinary Differential Equation

Authors: Olivier Bournez and Amaury Pouly

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

Abstract
An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function phi on the reals, and for any positive continuous function epsilon(t), it has a C^infinity solution with | y(t) - phi(t) | < epsilon(t) for all t. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, while these results may seem very surprising, their proofs are quite simple and are frustrating for a computability theorist, or for people interested in modeling systems in experimental sciences. First, the involved notions of universality is far from usual notions of universality in computability theory. In particular, the proofs heavily rely on the fact that constructed DAE does not have unique solutions for a given initial data. This is very different from usual notions of universality where one would expect that there is clear unambiguous notion of evolution for a given initial data, for example as in computability theory. Second, the proofs usually rely on solutions that are piecewise defined. Hence they cannot be analytic, while analycity is often a key expected property in experimental sciences. Third, the proofs of these results can be interpreted more as the fact that (fourth-order) polynomial algebraic differential equations is a too loose a model compared to classical ordinary differential equations. In particular, one may challenge whether the result is really a universality result. The question whether one can require the solution that approximates phi to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a fixed polynomial ODE such that for any phi and epsilon(t) there exists some initial condition that yields a solution that is epsilon-close to phi at all times. The proof uses ordinary differential equation programming. We believe it sheds some light on computability theory for continuous-time models of computations. It also demonstrates that ordinary differential equations are indeed universal in the sense of Rubel and hence suffer from the same problem as DAEs for modelization: a single equation is capable of modelling any phenomenon with arbitrary precision, meaning that trying to fit a model based on polynomial DAEs or ODEs is too general (if ithas a sufficient dimension).
Cite as

Olivier Bournez and Amaury Pouly. A Universal Ordinary Differential Equation. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 116:1-116:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bournez_et_al:LIPIcs.ICALP.2017.116,
  author =	{Bournez, Olivier and Pouly, Amaury},
  title =	{{A Universal Ordinary Differential Equation}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{116:1--116:14},
  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.2017.116},
  URN =		{urn:nbn:de:0030-drops-74335},
  doi =		{10.4230/LIPIcs.ICALP.2017.116},
  annote =	{Keywords: Ordinary Differential Equations, Universal Differential Equations, Analog Models of Computation, Continuous-Time Models of Computation, Computabilit}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.29
Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem

Authors: Nathanaël Fijalkow, Pierre Ohlmann, Joël Ouaknine, Amaury Pouly, and James Worrell

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2016.29
Model Checking Flat Freeze LTL on One-Counter Automata

Authors: Antonia Lechner, Richard Mayr, Joël Ouaknine, Amaury Pouly, and James Worrell

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.109
Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length: The General Purpose Analog Computer and Computable Analysis Are Two Efficiently Equivalent Models of Computations

Authors: Olivier Bournez, Daniel S. Graça, and Amaury Pouly

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

Abstract
The outcomes of this paper are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous (time and space) elegant and simple characterization of P. We believe it is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis. Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations. Continuous-Time Models of Computation. Our results can also be interpreted in terms of analog computers or analog model of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both at the computability and complexity level, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both at a computability and at a computational complexity level.
Cite as

Olivier Bournez, Daniel S. Graça, and Amaury Pouly. Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length: The General Purpose Analog Computer and Computable Analysis Are Two Efficiently Equivalent Models of Computations. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 109:1-109:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bournez_et_al:LIPIcs.ICALP.2016.109,
  author =	{Bournez, Olivier and Gra\c{c}a, Daniel S. and Pouly, Amaury},
  title =	{{Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length: The General Purpose Analog Computer and Computable Analysis Are Two Efficiently Equivalent Models of Computations}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{109:1--109:15},
  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.2016.109},
  URN =		{urn:nbn:de:0030-drops-62445},
  doi =		{10.4230/LIPIcs.ICALP.2016.109},
  annote =	{Keywords: Analog Models of Computation, Continuous-Time Models of Computation, Computable Analysis, Implicit Complexity, Computational Complexity, Ordinary Diff}
}
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