Computing with polynomial ordinary differential equations

Bournez, Olivier; Graça, Daniel S.; Pouly, Amaury

doi:10.1016/j.jco.2016.05.002

Cited by 11 publications

(37 citation statements)

References 12 publications

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“…This notion of computation turns out to be equivalent to various other notions: The following equivalence result is proved in [13]. 3.8 (M , [13]). Let f :⊆ R n → R m .…”

Section: Computable Functionsmentioning

confidence: 96%

“…(1) [11], under revision for publication in Information and Computation, devoted to properties of generable functions. (2) [13], published in Journal of Complexity, devoted to the proof of Proposition 3.8.…”

Section: Related Workmentioning

confidence: 99%

“…They will be a crucial tool used in the paper to simplify the construction of polynomial differential equations. Computable functions were introduced in [13]. This section does not contain any new original result, but only recalls already known results about these classes of functions.…”

Section: 6mentioning

confidence: 99%

“…Indeed, given a solution of a PIVP, the function z = •exp is also solution of a PIVP, but converges exponentially faster. A longer discussion on this topic can be found in [13]. In this section, we recall the main complexity classes and restate the main equivalence theorem.…”

Section: Computable Functionsmentioning

confidence: 99%

“…Without loss of generability we assume that ϒ, Λ, Θ, ℧ and Π are increasing functions. Apply Lemma 38 (AXP time rescaling) of [13] to get that ∐ can be assumed constant. Thus there exists…”

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confidence: 99%

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Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

Bournez

Graça

Pouly

2017

J. ACM

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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. ACM Subject Classification IntroductionThe outcomes of this paper are twofold, and are concerning a priori not closely related topics. Implicit Complexity:Since the introduction of the P and NP complexity classes, much work has been done to build a well-developed complexity theory based on Turing Machines. In particular, classical computational complexity theory is based on limiting resources used by Turing machines, like time and space. Another approach is implicit computational complexity.The term "implicit" in "implicit computational complexity" can sometimes be understood in various ways, but a common point of these characterizations is that they provide (Turing or equivalent) machine-independent alternative definitions of classical complexity. Implicit characterization theory has gained enormous interest in the last decade. This has led to many alternative characterizations of complexity classes using recursive functions, function algebras, rewriting systems, neural networks, lambda calculus and so on.However, most of -if not all -these models or characterizations are essentially discrete: in particular they are based on underlying discrete time models working on objects which are essentially discrete such as words, terms, etc. that can be considered as being defined in a discrete space.Models of computation working on a continuous space have also been considered: they include Blum Shub Smale machines [4], and in some sense Computable A...

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“…This notion of computation turns out to be equivalent to various other notions: The following equivalence result is proved in [13]. 3.8 (M , [13]). Let f :⊆ R n → R m .…”

Section: Computable Functionsmentioning

confidence: 96%

Section: Related Workmentioning

confidence: 99%

Section: 6mentioning

confidence: 99%

Section: Computable Functionsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

Bournez

Graça

Pouly

2017

J. ACM

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Programming with Ordinary Differential Equations: Some First Steps Towards a Programming Language

Bournez

2022

Revolutions and Revelations in Computability

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Various open problems have been recently solved using Ordinary Differential Equation (ODE) programming: basically, ODEs are used to implement various algorithms, including simulation over the continuum of discrete models such as Turing machines, or simulation of discrete time algorithms working over continuous variables. Applications include: Characterization of computability and complexity classes using ODEs [1][2][3][4]; Proof of the existence of a universal (in the sense of Rubel) ODE [5]; Proof of the strong Turing completeness of biochemical reactions [6], or more generally various statements about the completeness of reachability problems (e.g. PTIME-completeness of bounded reachability) for ODEs [7]. It is rather pleasant to explain how this ODE programming technology can be used in many contexts, as ODEs are in practice a kind of universal language used by many experimental sciences, and how consequently we got to these various applications. However, when going to say more about proofs, their authors including ourselves, often feel frustrated: Currently, the proofs are mostly based on technical lemmas and constructions done with ODEs, often mixing both the ideas behind these constructions, with numerical analysis considerations about errors and error propagation in the equations. We believe this is one factor hampering a more widespread use of this technology in other contexts. The current article is born from an attempt to popularize this ODE programming technology to a more general public, and in particular master and even undergraduate students. We show how some constructions can be reformulated using some notations, that can be seen as a pseudo programming language. This provides a way to explain in an easier and modular way the main intuitions behind some of the constructions, focusing on the algorithm design part. We focus here, as an example, on how the proof of the universality of polynomial ODEs (a result due to [8], and fully developed in [2]) can be reformulated and presented.

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On the functions generated by the general purpose analog computer

Bournez¹,

Graça²,

Pouly³

2017

Information and Computation

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We consider the General Purpose Analog Computer (GPAC), introduced by Claude Shannon in 1941 as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later one electronic) machines of that time.The GPAC generates as output univariate functions (i.e. functions f : R → R). In this paper we extend this model by: (i) allowing multivariate functions (i.e. functions f : R n → R m ); (ii) introducing a notion of amount of resources (space) needed to generate a function, which allows the stratification of GPAC generable functions into proper subclasses. We also prove that a wide class of (continuous and discontinuous) functions can be uniformly approximated over their full domain.We prove a few stability properties of this model, mostly stability by arithmetic operations, composition and ODE solving, taking into account the amount of resources needed to perform each operation.We establish that generable functions are always analytic but that they can nonetheless (uniformly) approximate a wide range of nonanalytic functions. Our model and results extend some of the results from (Shannon, 1941) to the multidimensional case, allow one to define classes of functions generated by GPACs which take into account bounded resources, and also strengthen the approximation result from (Shannon, 1941) over a compact domain to a uniform approximation result over unbounded domains.

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Computing with polynomial ordinary differential equations

Cited by 11 publications

References 12 publications

Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

Programming with Ordinary Differential Equations: Some First Steps Towards a Programming Language

On the functions generated by the general purpose analog computer

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