Polynomial differential equations compute all real computable functions on computable compact intervals
TLDR
It is shown that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals, and that all real computable functions over compact intervals can be defined by GPACs.About:
This article is published in Journal of Complexity.The article was published on 2007-06-01 and is currently open access. It has received 74 citations till now. The article focuses on the topics: Computable analysis & Computable function.read more
Citations
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Book ChapterDOI
A Survey on Continuous Time Computations
TL;DR: An overview of theories of continuous time computation is provided, which allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuously time analog models.
Book ChapterDOI
Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs
TL;DR: The strong (uniform computability) Turing completeness of chemical reaction networks over a finite set of molecular species under the differential semantics is derived, solving a long standing open problem.
Journal ArticleDOI
The Structure of Differential Invariants and Differential Cut Elimination
TL;DR: This work refutes the differential cut elimination hypothesis and shows that, unlike standard cuts, differential cuts are fundamental proof principles that strictly increase the deductive power, and proves that the deduction power increases further when adding auxiliary differential variables to the dynamics.
Journal ArticleDOI
The Structure of Differential Invariants and Differential Cut Elimination
TL;DR: In this paper, the deductive power of differential cuts and differential invariants with auxiliary differential variables was analyzed and it was shown that differential cuts are fundamental proof principles for hybrid systems verification.
References
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Book
Ordinary differential equations
TL;DR: The fourth volume in a series of volumes devoted to self-contained and up-to-date surveys in the theory of ODEs was published by as discussed by the authors, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience.
Journal ArticleDOI
On Computable Numbers, with an Application to the Entscheidungsproblem
TL;DR: This chapter discusses the application of the diagonal process of the universal computing machine, which automates the calculation of circle and circle-free numbers.
Book
Ordinary Differential Equations
TL;DR: In this paper, differential equations on manifolds have been used to prove linear systems and linear systems are proven to be linear systems with differential equations on manifolds.
Book
Computable Analysis : An Introduction
TL;DR: This book provides a solid fundament for studying various aspects of computability and complexity in analysis and is written in a style suitable for graduate-level and senior students in computer science and mathematics.