arXiv subjects

Olivier Bournez

Publications and source records attributed to Olivier Bournez.

At least 19 recordsLinked to original sources

arXiv2026

Fractal Gadgets for Neural Networks: The Complexity of the Narrow Regime

We study the verification problem for deep narrow ReLU neural networks: given a network of bounded width computing a piecewise-affine map on [0,1], does some input satisfy a prescribed output constraint? Classical NP-hardness proofs for ReLU verification use one neuron per Boolean variable and say nothing about networks of small constant width, while width-1 networks are easy to verify. We show that verification of ReLU networks is NP-complete at width 4 for arbitrary inputs in [0,1]. When inputs are restricted to a natural discrete encoding set, NP-completeness already holds at width 3. Together with polynomial-time decidability at width 1, this leaves open only width 2 on the encoding set, and widths 2 and 3 on [0,1]. The technical core is a fractal preprocessing gadget: a width-2 ReLU subnetwork whose iterate vanishes precisely near a finite Cantor-like subset of [0,1] with 2^n points. It reduces verification of a continuous function on [0,1] to verification on 2^n discrete points without increasing the width, and is the missing ingredient for width-bounded hardness reductions. The same construction yields further results at width 3 on the encoding set: the universal problem is coNP-complete, counting zeros is #P-complete, a majority variant is PP-complete, and approximating the minimum output within a constant gap inherited from Max-3Sat is NP-hard. The NP, coNP and inapproximability results lift to all of [0,1] at width 4; lifting counting and majority, and lifting at width 3, remain open.

cs.CC↗

arXiv2026

Relating the Computational and Logical Difficulty of Solving ODEs: From Polynomial to Discontinuous Right-Hand Sides

When a computer algebra system fails to solve an Ordinary Differential Equation, is this a limitation of its implementation, or a genuine computational barrier? Three traditions bear on the question. Modern computer algebra algorithms can be extremely efficient: Newton-type methods solve polynomial ODEs over $\mathbb{Q}[[X]]$ in quasi-linear time. Analog models of computation has shown that polynomial ODEs and Turing machines are two presentations of the same phenomenon, with solution length acting as time and precision as space. Computable analysis shows that ODEs can be intrinsically hard -- undecidable, even $\mathsf{PSPACE}$-complete, over compact domains. Comparing these traditions is natural and necessary, yet such comparisons routinely reduce to comparisons of encodings rather than of underlying algorithmic content. We argue that reverse mathematics provides a representation-invariant lens in which algorithmic content is compared directly. We prove that every level of the Big Five hierarchy is inhabited by a natural statement from classical ODE theory, as an exact equivalence: the regularity of $f$ is an intrinsic algorithmic invariant placing the initial value problem $y'(t)=f(t,y(t))$, $y(t_0)=y_0$, into one of several computational strata, ranging from polynomial-time solvability to transfinite computation. The resulting stratification acts as a practical diagnostic common to the three traditions. By abstracting from representation, it separates fundamental barriers from the technical shortcomings of symbolic solvers, the artefacts of analog encodings, and the effectivity constraints of computable analysis, identifying the intrinsic parameters (length bounds, radii of convergence, moduli of continuity) under which feasibility is restored.

cs.SC↗

arXiv2026

Primitive Recursion without Composition: Dynamical Characterizations, from Neural Networks to Polynomial ODEs

What do recurrent neural networks, polynomial ODEs, and discrete polynomial maps each bring to computation, and what do they lack? All three operate over the continuum--real-valued states evolved by real-valued dynamics--even when the target functions are discrete. We study them through primitive recursion. We prove that primitive recursion admits equivalent characterizations in all three frameworks: bounded iteration of a fixed recurrent ReLU network, robust computation by a fixed polynomial ODE, and iteration of a fixed polynomial map with an externally supplied step-size parameter. In each, the time bound is itself primitive recursive, composition emerges from the dynamics rather than as a closure rule, and inputs are raw integer vectors. Every primitive recursive function is first compiled into bounded iteration of a single threshold-affine normal form, then interpreted as a ReLU computation and as a polynomial ODE. The equivalences expose a structural asymmetry: no fixed polynomial map can round uniformly to the nearest integer or realize exact phase selection--operations polynomial ODEs perform robustly via continuous-time flow. Each formalism compensates for a limitation the others lack: the ReLU gate provides exact branching, continuous time provides autonomous rounding and control, and the step-size parameter recovers both at the cost of discretization precision. This opens dynamical characterizations of subrecursive hierarchies and complexity classes by restricting time bounds, polynomial degrees, or discretization resources within one framework. More broadly, these models do not compute by composing subroutines: they shape the trajectory of a dynamical system through clocks, phase selectors, and error correction built into the dynamics. This differs structurally from symbolic programming, and our theorem gives a precise framework to study the difference.

cs.CC↗

arXiv2025

Toward higher-order infinite time Turing machines: simulational $Γ$-machines

Infinite time Turing machines (ITTMs) have been introduced by Hamkins and Lewis in their seminal article arXiv:math/9808093. The strength of the model comes from a limit rule which allows the ITTM to compute through ordinal stages. This rule is simple to describe and another rule would lead to a different model of ordinal computation. The aim of this article is to define a collection of limit rules for which the models of infinite Turing machine they induce have nice properties, akin to that of the ITTM. Through the analysis of the universal ITTM and of its preponderant role in the study of the ITTM, we devise a set of four constraints. A limit rule satisfying those constraints yields a model of infinite machine for which we can define a universal machine. Adding a fifth constraint, we show that any limit rule definable in set theory which meets those constraints produces a model of infinite machine with the desired properties. Among those, the fact that the supremum of the writable ordinal matches that of the clockable ordinal. That is, with the usual notations, the equality $λ= γ$ holds for any of those limit rules. Eventually, we provide a counter-example to show that the four first constraints alone are not sufficient.

math.LO↗

arXiv2024

Set Descriptive Complexity of Solvable Functions

In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique evolution. They correspond to a class of systems for which a transfinite method exist to compute the solution. We also presented several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems might describe ordinal Turing computations. In the current article, we study in more depth solvable systems, using tools from descriptive set theory. By establishing a correspondence with the class of well-founded trees, we construct a coanalytic ranking over the set of solvable functions and discuss its relation with other existing rankings for differentiable functions, in particular with the Kechris-Woodin, Denjoy and Zalcwasser ranking. We prove that our ranking is unbounded below the first uncountable ordinal.

cs.CC↗

arXiv2024

Solvable Initial Value Problems Ruled by Discontinuous Ordinary Differential Equations

We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call solvable intitial value problems and we prove that for this class of problems the unique solution can always be obtained analytically via transfinite recursion. We present several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems is related to ordinal Turing computations.

cs.CC↗

arXiv2024

The complexity of computing in continuous time: space complexity is precision

Models of computations over the integers are equivalent from a computability and complexity theory point of view by the 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). For example, the GPAC model of Shannon 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. 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. We propose and prove an algebraic characterisation of FPSPACE, using continuous ODEs. Recent papers proposed algebraic characterisations of polynomial-time and -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. 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.

cs.CC↗

arXiv2024

Simulation of Turing machines with analytic discrete ODEs: FPTIME and FPSPACE over the reals characterised with discrete ordinary differential equations

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. We prove furthermore 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.

cs.CC↗

arXiv2024

The domino problem is decidable for robust tilesets

One of the most fundamental problems in tiling theory is the domino problem: given a set of tiles and tiling rules, decide if there exists a way to tile the plane using copies of tiles and following their rules. The problem is known to be undecidable in general and even for sets of Wang tiles, which are unit square tiles wearing colours on their edges which can be assembled provided they share the same colour on their common edge, as proven by Berger in the 1960s. In this paper, we focus on Wang tilesets. We prove that the domino problem is decidable for robust tilesets, i.e. tilesets that either cannot tile the plane or can but, if so, satisfy some particular invariant provably. We establish that several famous tilesets considered in the literature are robust. We give arguments that this is true for all tilesets unless they are produced from non-robust Turing machines: a Turing machine is said to be non-robust if it does not halt and furthermore does so non-provably. As a side effect of our work, we provide a sound and relatively complete method for proving that a tileset can tile the plane. Our analysis also provides explanations for the observed similarities between proofs in the literature for various tilesets, as well as of phenomena that have been observed experimentally in the systematic study of tilesets using computer methods.

cs.DM↗

arXiv2023

Measuring robustness of dynamical systems. Relating time and space to length and precision

Verification of discrete time or continuous time dynamical systems over the reals is known to be undecidable. It is however known that undecidability does not hold for various classes of systems: if robustness is defined as the fact that reachability relation is stable under infinitesimal perturbation, then their reachability relation is decidable. In other words, undecidability implies sensitivity under infinitesimal perturbation, a property usually not expected in systems considered in practice, and hence can be seen (somehow informally) as an artefact of the theory, that always assumes exactness. In a similar vein, it is known that, while undecidability holds for logical formulas over the reals, it does not hold when considering delta-undecidability: one must determine whether a property is true, or $δ$-far from being true. We first extend the previous statements to a theory for general (discrete time, continuous-time, and even hybrid) dynamical systems, and we relate the two approaches. We also relate robustness to some geometric properties of reachability relation. But mainly, when a system is robust, it then makes sense to quantify at which level of perturbation. We prove that assuming robustness to polynomial perturbations on precision leads to reachability verifiable in complexity class PSPACE, and even to a characterization of this complexity class. We prove that assuming robustness to polynomial perturbations on time or length of trajectories leads to similar statements, but with PTIME. It has been recently unexpectedly shown that the length of a solution of a polynomial ordinary differential equation corresponds to a time of computation: PTIME corresponds to solutions of polynomial differential equations of polynomial length. Our results argue that the answer is given by precision: space corresponds to the involved precision.

cs.CC↗

arXiv2022

Polynomial time computable functions over the reals characterized using discrete ordinary differential equations

The class of functions from the integers to the integers computable in polynomial time has been characterized recently using discrete ordinary differential equations (ODE), also known as finite differences. In the framework of ordinary differential equations, this is very natural to try to extend the approach to classes of functions over the reals, and not only over the integers. Recently, an extension of previous characterization was obtained for functions from the integers to the reals, but the method used in the proof, based on the existence of a continuous function from the integers to a suitable discrete set of reals, cannot extend to functions from the reals to the reals, as such a function cannot exist for clear topological reasons. In this article, we prove that this is indeed possible to provide an elegant and simple algebraic characterization of functions from the reals to the reals: we provide a characterization of such functions as the smallest class of functions that contains some basic functions, and that is closed by composition, linear length ODEs, and a natural effective limit schema. This is obtained using an alternative proof technique based on the construction of specific suitable functions defined recursively, and a barycentric method. Furthermore, we also extend previous characterizations in several directions: First, we prove that there is no need of multiplication. We prove a normal form theorem, with a nice side effect related to formal neural networks. Indeed, given some fixed error and some polynomial time t(n), our settings produce effectively some neural network that computes the function over its domain with the given precision, for any t(n)-polynomial time computable function f .

cs.CC↗

arXiv2022

Surreal fields stable under exponential, logarithmic, derivative and anti-derivative functions

The class of surreal numbers, denoted by $\textbf{No}$, initially proposed by Conway, is a universal ordered field in the sense that any ordered field can be embedded in it. They include in particular the real numbers and the ordinal numbers. They have strong relations with other fields such as field of transseries. Following Gonshor, surreal numbers can be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. $\textbf{No}$ can actually be seen as an elegant (generalized) power series field with real coefficients, namely Hahn series with exponents in $\textbf{No}$ itself. Some years ago, Berarducci and Mantova considered derivation over the surreal numbers, seeing them as germs of functions, in correspondence to transseries. In this article, following our previous work, we exhibit a sufficient condition on the structure of a surreal field to be stable under all operations among exponential, logarithm, derivation and anti-derivation. Motivated, in the long term, by computability considerations, we also provide a non-trivial application of this theorem: the existence of a pretty reasonable field that only requires ordinals up to $ε_ω$, which is far smaller than $ω_1^{CK}$ (resp. $ω_1$), the first non-computable (resp. uncountable) ordinal.

math.LO↗

arXiv2022

A characterization of polynomial time computable functions from the integers to the reals using discrete ordinary differential equations

In a recent article, the class of functions from the integers to the integers computable in polynomial time has been characterized using discrete ordinary differential equations (ODE), also known as finite differences. Doing so, we pointed out the fundamental role of linear (discrete) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming. In this article, we extend the approach to a characterization of functions from the integers to the reals computable in polynomial time in the sense of computable analysis. In particular, we provide a characterization of such functions in terms of the smallest class of functions that contains some basic functions, and that is closed by composition, linear length ODEs, and a natural effective limit schema.

cs.CC↗

arXiv2022

A characterization of functions over the integers computable in polynomial time using discrete differential equations

This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations 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) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms.

cs.LO↗

arXiv2022

Surreal fields stable under exponential and logarithmic functions

Surreal numbers, have a very rich and elegant theory. This class of numbers, denoted by No, includes simultaneously the ordinal numbers and the real numbers, and forms a universal huge real closed field: It is universal in the sense that any real closed field can be embedded in it. Following Gonshor, surreal numbers can also be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. No can actually also be seen as an elegant particular (generalized) power series field with real coefficients, namely Hahn series with exponents in No itself. It can also be considered as a particular field of transseries, providing tools to do some analysis and asymptotic analysis for functions over the continuum, providing natural concepts for discussing hyperexponential or sublogarithm functions, and their asymptotics. In this article, we consider stability of subfields of No under exponential and logarithmic functions. Namely, we consider the set surreal numbers whose signs sequences have length less than some ordinal λ. Extending the discussion from van den Dries and Ehrlich, we show that is stable by exponential and logarithm iff λ is some ε-number. Motivated in a longer term by computability issues using ordinal machines, we consider subfields stables by exponential and logarithmic functions defined by Hahn series that does not require to go up to cardinal lengths and exponents. We prove that No can be expressed as a strict hierarchy of subfields stable by exponential and logarithmic functions. This provides many explicit examples of subfields of No stable by exponential and logarithmic functions, and does not require to go up to a cardinal λ to provide such examples.

math.LO↗

arXiv2020

A Universal Ordinary Differential Equation

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 $φ$ on the reals, and for any positive continuous function $ε(t)$, it has a $\mathcal{C}^\infty$ solution with $| y(t) - φ(t) | < ε(t)$ 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, Rubel's DAE \emph{never} has a unique solution, even with a finite number of conditions of the form $y^{(k_i)}(a_i)=b_i$. The question whether one can require the solution that approximates $φ$ 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 \textbf{fixed} polynomial ODE such that for any $φ$ and $ε(t)$ there exists some initial condition that yields a solution that is $ε$-close to $φ$ at all times. In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.

math.CA↗

arXiv2020

Continuous Ordinary Differential Equations and Transfinite Computations

We consider Continuous Ordinary Differential Equations (CODE) y'=f(y), where f is a continuous function. They are known to always have solutions for a given initial condition y(0)=y0, these solutions being possibly non unique. We restrict to our attention to a class of continuous functions, that we call greedy: they always admit unique greedy solutions, i.e. going in greedy way in some fixed direction. We prove that they can be seen as models of computation over the ordinals and conversely in a very strong sense. In particular, for such ODEs, to a greedy trajectory can be associated some ordinal corresponding to some time of computation, and conversely models of computation over the ordinals can be associated to some CODE. In particular, analyzing reachability for one or the other concept with respect to greedy trajectories has the same hardness. This also brings new perspectives on analysis in Mathematics, by providing ways to translate results for ITTMs to CODEs. This also extends some recent results about the relations between ordinary differential equations and Turing machines, and more widely with (generalized) computability theory.

cs.CC↗

arXiv2019

Homonym Population Protocols, or Providing a Small Space of Computation Using a Few Identifiers

Population protocols have been introduced by Angluin et al. as a model in which n passively mobile anonymous finite-state agents stably compute a predicate on the multiset of their inputs via interactions by pairs. The model has been extended by Guerraoui and Ruppert to yield the community protocol models where agents have unique identifiers but may only store a finite number of the identifiers they already heard about. The population protocol models can only compute semi-linear predicates, whereas in the community protocol model the whole community of agents provides collectively the power of a Turing machine with a O(n log n) space. We consider variations on the above models and we obtain a whole landscape that covers and extends already known results: By considering the case of homonyms, that is to say the case when several agents may share the same identifier, we provide a hierarchy that goes from the case of no identifier (i.e. a single one for all, i.e. the population protocol model) to the case of unique identifiers (i.e. community protocol model). We obtain in particular that any Turing Machine on space O(logO(1) n) can be simulated with at least O(logO(1) n) identifiers, a result filling a gap left open in all previous studies. Our results also extend and revisit in particular the hierarchy provided by Chatzigiannakis et al. on population protocols carrying Turing Machines on limited space, solving the problem of the gap left by this work between per-agent space o(log log n) (proved to be equivalent to population protocols) and O(log n) (proved to be equivalent to Turing machines).

cs.CC↗