1,038 research outputs found
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Renormalization and Computation II: Time Cut-off and the Halting Problem
This is the second installment to the project initiated in [Ma3]. In the
first Part, I argued that both philosophy and technique of the perturbative
renormalization in quantum field theory could be meaningfully transplanted to
the theory of computation, and sketched several contexts supporting this view.
In this second part, I address some of the issues raised in [Ma3] and provide
their development in three contexts: a categorification of the algorithmic
computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra
renormalization of the Halting Problem.Comment: 28 page
Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
We provide an implicit characterization of polynomial time computation in
terms of ordinary differential equations: we characterize the class
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 . This 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. This extends to deterministic
complexity classes above polynomial time.
This may provide a new perspective on classical complexity, by giving a way
to define complexity classes, like , 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, i.e.~by using the framework of analysis
Classical computing, quantum computing, and Shor's factoring algorithm
This is an expository talk written for the Bourbaki Seminar. After a brief
introduction, Section 1 discusses in the categorical language the structure of
the classical deterministic computations. Basic notions of complexity icluding
the P/NP problem are reviewed. Section 2 introduces the notion of quantum
parallelism and explains the main issues of quantum computing. Section 3 is
devoted to four quantum subroutines: initialization, quantum computing of
classical Boolean functions, quantum Fourier transform, and Grover's search
algorithm. The central Section 4 explains Shor's factoring algorithm. Section 5
relates Kolmogorov's complexity to the spectral properties of computable
function. Appendix contributes to the prehistory of quantum computing.Comment: 27 pp., no figures, amste
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