• Publications
  • Influence
Robust Simulations of Turing Machines with Analytic Maps and Flows
TLDR
In this paper, we show that closed-form analytic maps and flows can simulate Turing machines in an error-robust manner. Expand
  • 41
  • 4
  • PDF
Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
The outcomes of this article are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize theExpand
  • 18
  • 3
  • PDF
Computability with polynomial differential equations
TLDR
In this paper, we show that there are initial value problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. Expand
  • 55
  • 2
  • PDF
Polynomial differential equations compute all real computable functions on computable compact intervals
TLDR
We show that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Expand
  • 59
  • 1
  • PDF
Effective Computability of Solutions of Differential Inclusions The Ten Thousand Monkeys Approach
TLDR
We show that the solution of a locally Lip- schitz differential equation is computable even if the function is not effectively locally Lipschitz, and recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is. Expand
  • 24
  • 1
  • PDF
Effective Computability of Solutions of Ordinary Differential Equations The Thousand Monkeys Approach
  • P. Collins, D. Graça
  • Mathematics, Computer Science
  • Electron. Notes Theor. Comput. Sci.
  • 1 December 2008
TLDR
We show that the solution of a differential equation defined by a locally Lipschitz function is computable even if the right-hand side is not effectively locally Lippingchitz. Expand
  • 16
  • 1
  • PDF
On the Functions Generated by the General Purpose Analog Computer
TLDR
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 electronic) machines of that time. Expand
  • 12
  • 1
  • PDF
Computability and Computational Complexity of the Evolution of Nonlinear Dynamical Systems
Nonlinear dynamical systems abound as models of natural phenomena. They are often characterized by highly unpredictable behaviour which is hard to analyze as it occurs, for example, in chaoticExpand
  • 5
  • 1
  • PDF
The connection between computability of a nonlinear problem and its linearization: The Hartman-Grobman theorem revisited
As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics Pour-El and Richards (1989) [17], Pour-El and Richards asked, ''What is the connectionExpand
  • 4
  • 1
  • PDF
Computing geometric Lorenz attractors with arbitrary precision
TLDR
We show that geometric Lorenz attractors are computable, as well as their physical measures. Expand
  • 6
  • 1
  • PDF