Computability and Computational Complexity of the Evolution of Nonlinear Dynamical Systems

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

doi:10.1007/978-3-642-39053-1_2

Cited by 4 publications

(3 citation statements)

References 41 publications

(41 reference statements)

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“…The relationship between dynamical systems and computability has been studied before by Bournez [11,10], Blondel [9], Moore [32] and by Fredkin, Margolus and Toffoli [22,30], among others. That emergence is a consequence of incomputability has been proposed by Cooper [19].…”

Section: Introductionmentioning

confidence: 99%

Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence

2018

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Is undecidability a requirement for open-ended evolution (OEE)? Using methods derived from algorithmic complexity theory, we propose robust computational definitions of open-ended evolution and the adaptability of computable dynamical systems. Within this framework, we show that decidability imposes absolute limits on the stable growth of complexity in computable dynamical systems. Conversely, systems that exhibit (strong) open-ended evolution must be undecidable, establishing undecidability as a requirement for such systems. Complexity is assessed in terms of three measures: sophistication, coarse sophistication, and busy beaver logical depth. These three complexity measures assign low complexity values to random (incompressible) objects. As time grows, the stated complexity measures allow for the existence of complex states during the evolution of a computable dynamical system. We show, however, that finding these states involves undecidable computations. We conjecture that for similar complexity measures that assign low complexity values, decidability imposes comparable limits on the stable growth of complexity, and that such behavior is necessary for nontrivial evolutionary systems. We show that the undecidability of adapted states imposes novel and unpredictable behavior on the individuals or populations being modeled. Such behavior is irreducible. Finally, we offer an example of a system, first proposed by Chaitin, that exhibits strong OEE.

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Section: Introductionmentioning

confidence: 99%

Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence

2018

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“…However, rigorous algorithmic solutions, and even establishing cases of provable incomputability, lies within the core expertise of Recursive Analysis (Brattka and Yoshikawa 2006;Sun et al 2015;Weihrauch and Zhong 2002, 2005. Concerning the refined view of computational complexity, investigations have recently moved to ordinary differential equations (Bournez et al 2013;Kawamura 2010).…”

Section: Recap On Computability and Complexity In Analysismentioning

confidence: 99%

On the computational complexity of the Dirichlet Problem for Poisson's Equation

Kawamura

Steinberg

Ziegler

2016

Math. Struct. Comp. Sci.

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The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense ‘complete’ for the complexity class ${\#\mathcal{P}}$ and thus as hard or easy as parametric Riemann integration (Friedman 1984; Ko 1991. Complexity Theory of Real Functions).

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“…The issue of computability in the context of nonlinear dynamics has recently received considerable attention; see for example [5,4] and references therein. An important implication of this work is that the topological structure of invariant sets need not be computable.…”

Section: Introductionmentioning

confidence: 99%

Lattice Structures for Attractors II

Kalies

Mischaikow

Vandervorst³

2015

Found Comput Math

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ABSTRACT. The algebraic structure of the attractors in a dynamical system determine much of its global dynamics. The collection of all attractors has a natural lattice structure, and this structure can be detected through attracting neighborhoods, which can in principle be computed. Indeed, there has been much recent work on developing and implementing general computational algorithms for global dynamics, which are capable of computing attracting neighborhoods efficiently. Here we address the question of whether all of the algebraic structure of attractors can be captured by these methods.

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Computability and Computational Complexity of the Evolution of Nonlinear Dynamical Systems

Cited by 4 publications

References 41 publications

Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence

Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence

On the computational complexity of the Dirichlet Problem for Poisson's Equation

Lattice Structures for Attractors II

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