Document

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

We study a precise class of dynamical systems that we call solvable ordinary differential equations. We prove that analog systems mathematically ruled by solvable ordinary differential equations can be used for transfinite computation, solving tasks such as the halting problem for Turing machines and any Turing jump of the halting problem in the hyperarithmetical hierarchy. We prove that the computational power of such analog systems is exactly the one of transfinite computations of the hyperarithmetical hierarchy.
It has been proved recently that polynomial ordinary differential equations correspond unexpectedly naturally to Turing machines. Our results show that the more general exhibited class of solvable ordinary differential equations corresponds, even unexpectedly, naturally to transfinite computations. From a wide philosophical point of view, our results contribute to state that the question of whether such analog systems can be used to solve untractable problems (both for complexity for polynomial systems and for computability for solvable systems) is provably related to the question of the relations between mathematical models, models of physics and our real world.
More technically, we study a precise class of dynamical systems: bounded initial value problems involving ordinary differential equations with a unique solution. We show that the solution of these systems can still be obtained analytically even in the presence of discontinuous dynamics once we carefully select the conditions that describe how discontinuities are distributed in the domain. We call the class of right-hand terms respecting these natural and simple conditions the class of solvable ordinary differential equations. We prove that there is a method for obtaining the solution of such systems based on transfinite recursion and taking at most a countable number of steps. We explain the relevance of these systems by providing several natural examples and showcasing the fact that these solutions can be used to perform limit computations and solve tasks such as the halting problem for Turing machines and any Turing jump of the halting problem in the hyperarithmetical hierarchy.

Olivier Bournez and Riccardo Gozzi. Solving Discontinuous Initial Value Problems with Unique Solutions Is Equivalent to Computing over the Transfinite. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bournez_et_al:LIPIcs.STACS.2024.20, author = {Bournez, Olivier and Gozzi, Riccardo}, title = {{Solving Discontinuous Initial Value Problems with Unique Solutions Is Equivalent to Computing over the Transfinite}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {20:1--20:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.20}, URN = {urn:nbn:de:0030-drops-197308}, doi = {10.4230/LIPIcs.STACS.2024.20}, annote = {Keywords: Analog models, computability, transfinite computations, dynamical systems} }

Document

**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Reasoning about dynamical systems evolving over the reals is well-known to lead to undecidability. In particular, it is known that there cannot be reachability decision procedures for first-order theories over the reals extended with even very basic functions, or for logical theories that reason about real-valued functions, or decision procedures for state reachability. This mostly comes from the fact that reachability for dynamical systems over the reals is fundamentally undecidable, as Turing machines can be embedded into (even very simple) dynamical systems.
However, various results in the literature have shown that decision procedures exist when restricting to robust systems, with a suitably-chosen notion of robustness. In particular, it has been established in the field of verification that if the state reachability is not sensitive to infinitesimal perturbations, then decision procedures for state reachability exist. In the context of logical theories over the reals, it has been established that decision procedures exist if we focus on properties not sensitive to arbitrarily small perturbations. For example by considering properties that are either true or δ-far from being true, for some δ > 0.
In this article, we first propose a unified theory explaining in a uniform framework these statements, that were established in different contexts.
More fundamentally, while all these statements are only about computability issues, we also consider complexity theory aspects. We prove that robustness to some precision is inherently related to the complexity of the decision procedure. When a system is robust, it makes sense to quantify at which level of perturbation it is. We prove that assuming robustness to a polynomial perturbation on precision leads to a characterisation of PSPACE. We prove that assuming robustness to polynomial perturbation on time or length leads to similar statements for PTIME.
In other words, precision on computations is inherently related to space complexity, while length or time of trajectories, is intrinsically related to time complexity. These statements can also be interpreted in relation to several recent results about the computational power of analogue models of computation.

Manon Blanc and Olivier Bournez. Quantifiying the Robustness of Dynamical Systems. Relating Time and Space to Length and Precision. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blanc_et_al:LIPIcs.CSL.2024.17, author = {Blanc, Manon and Bournez, Olivier}, title = {{Quantifiying the Robustness of Dynamical Systems. Relating Time and Space to Length and Precision}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {17:1--17:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello 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.CSL.2024.17}, URN = {urn:nbn:de:0030-drops-196609}, doi = {10.4230/LIPIcs.CSL.2024.17}, annote = {Keywords: Computability, Complexity theory, Computable analysis, Verification, Decision, Robustness, Dynamical Systems, Models of computation, Analogue Computations} }

Document

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

We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals. In particular, this covers space complexity, while existing characterisations were only able to cover time complexity, and were restricted to functions over the integers, and we prove that no artificial sign or test function is needed even for time complexity. At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity.

Manon Blanc and Olivier Bournez. A Characterisation of Functions Computable in Polynomial Time and Space over the Reals with Discrete Ordinary Differential Equations: Simulation of Turing Machines with Analytic Discrete ODEs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blanc_et_al:LIPIcs.MFCS.2023.21, author = {Blanc, Manon and Bournez, Olivier}, title = {{A Characterisation of Functions Computable in Polynomial Time and Space over the Reals with Discrete Ordinary Differential Equations: Simulation of Turing Machines with Analytic Discrete ODEs}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {21:1--21:15}, 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.2023.21}, URN = {urn:nbn:de:0030-drops-185554}, doi = {10.4230/LIPIcs.MFCS.2023.21}, annote = {Keywords: Discrete ordinary differential equations, Finite Differences, Implicit complexity, Recursion scheme, Ordinary differential equations, Models of computation, Analog Computations} }

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Invited Talk

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

Ordinary Differential Equations (ODEs) appear to be a universally adopted and very natural way for expressing properties for continuous time dynamical systems. They are intensively used, in particular in applied sciences. There exists an abundant literature about the hardness of solving ODEs with numerical methods.
We adopt a dual view: we consider ODEs as a way to program or to describe our mathematical/computer science world. We survey several results considering ODEs under this computational perspective, with a computability and complexity theory point of view. In particular, we provide various reasons why polynomial ODEs should be considered as the continuous time analog of Turing machines for continuous-time computations, or should be used as a way to talk about mathematical logic.
This has already applications in various fields: determining whether analog models of computation can compute faster than classical digital models of computation; solving complexity issues for computations with biochemical reactions in bioinformatics; machine independent characterizations of various computability and complexity classes such as PTIME or NPTIME, or proof of the existence of a universal polynomial ordinary differential equation whose solutions can approximate any continuous function if provided with a suitable well-chosen initial condition.

Olivier Bournez. Computability, Complexity and Programming with Ordinary Differential Equations (Invited Talk). In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 3:1-3:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bournez:LIPIcs.STACS.2020.3, author = {Bournez, Olivier}, title = {{Computability, Complexity and Programming with Ordinary Differential Equations}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {3:1--3:13}, 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.2020.3}, URN = {urn:nbn:de:0030-drops-118642}, doi = {10.4230/LIPIcs.STACS.2020.3}, annote = {Keywords: Ordinary differential equations, Models of computation, Analog Computations, discrete ordinary differential equations, Implicit complexity, recursion scheme} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several implicit characterizations of complexity and computability classes.
The proposed framework presents an original point of view on complexity and computation classes. It unifies several constructions that have been proposed for characterizing these classes including classical approaches in implicit complexity using restricted recursion schemes, as well as recent characterizations of computability and complexity by classes of continuous ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory.
At a more technical point of view, this paper points out the fundamental role of linear (discrete) ordinary differential equations and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms.

Olivier Bournez and Arnaud Durand. Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bournez_et_al:LIPIcs.MFCS.2019.23, author = {Bournez, Olivier and Durand, Arnaud}, title = {{Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.23}, URN = {urn:nbn:de:0030-drops-109670}, doi = {10.4230/LIPIcs.MFCS.2019.23}, annote = {Keywords: Implicit complexity, discrete ordinary differential equations, recursion scheme} }

Document

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

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).

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

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

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.

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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