Recursion theory on the reals and continuous-time computation
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A case of recursive functions on the reals analogous to the classical recursive functionson the natural numbers, corresponding to a conceptual analog computer that operates in continuous time is defined, and this class turns out to be surprisingly large, and includes many functions which are uncomputable in the traditional sense.About:
This article is published in Theoretical Computer Science.The article was published on 1996-08-03 and is currently open access. It has received 170 citations till now. The article focuses on the topics: μ operator & μ-recursive function.read more
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Computational Mechanics: Pattern and Prediction, Structure and Simplicity
TL;DR: This paper showed that the causal-state representation of ∈-machine is the minimal one consistent with accurate prediction and established several results on ∈machine optimality and uniqueness and on how ∆-machines compare to alternative representations.
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Between order and chaos
TL;DR: A completely ordered universe is as unexciting as an entirely disordered one as mentioned in this paper, and interesting "complex" phenomena arise in a middle ground between order and chaos, and tools have been developed to quantify structural complexity and to automatically discover patterns hidden between the two.
Causal architecture, complexity and self-organization in time series and cellular automata
TL;DR: This work develops computational mechanics for four increasingly sophisticated types of process—memoryless transducers, time series, transducers with memory, and cellular automata, and proves the optimality and uniqueness of the e-machine's representation of the causal architecture, and gives reliable algorithms for pattern discovery.
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Theory of hybrid dynamical systems and its applications to biological and medical systems.
Kazuyuki Aihara,Hideyuki Suzuki +1 more
TL;DR: This Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine.
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Book
Computation: Finite and Infinite Machines
TL;DR: In this article, the authors present an abstract theory that categorically and systematically describes what all these machines can do and what they cannot do, giving sound theoretical or practical grounds for each judgment, and the abstract theory tells us in no uncertain terms that the machines' potential range is enormous and that its theoretical limitations are of the subtlest and most elusive sort.
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On a theory of computation and complexity over the real numbers: $NP$- completeness, recursive functions and universal machines
Book
Classical recursion theory
TL;DR: Theories of Recursive functions, Hierarchies of recursive functions, and Arithmetical sets: Recursively enumerable sets.
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Unpredictability and undecidability in dynamical systems
TL;DR: It is shown that motion with as few as three degrees of freedom can be equivalent to a Turing machine, and so be capable of universal computation.