English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Verification of discrete time or continuous time dynamical systems over thereals is known to be undecidable. It is however known that undecidability doesnot hold for various classes of systems: if robustness is defined as the factthat reachability relation is stable under infinitesimal perturbation, thentheir reachability relation is decidable. In other words, undecidabilityimplies sensitivity under infinitesimal perturbation, a property usually notexpected in systems considered in practice, and hence can be seen (somehowinformally) as an artefact of the theory, that always assumes exactness. In asimilar vein, it is known that, while undecidability holds for logical formulasover the reals, it does not hold when considering delta-undecidability: onemust determine whether a property is true, or $\delta$-far from being true. Wefirst extend the previous statements to a theory for general (discrete time,continuous-time, and even hybrid) dynamical systems, and we relate the twoapproaches. We also relate robustness to some geometric properties ofreachability relation. But mainly, when a system is robust, it then makes senseto quantify at which level of perturbation. We prove that assuming robustnessto polynomial perturbations on precision leads to reachability verifiable incomplexity class PSPACE, and even to a characterization of this complexityclass. We prove that assuming robustness to polynomial perturbations on time orlength of trajectories leads to similar statements, but with PTIME. It has beenrecently unexpectedly shown that the length of a solution of a polynomialordinary differential equation corresponds to a time of computation: PTIMEcorresponds to solutions of polynomial differential equations of polynomiallength. Our results argue that the answer is given by precision: spacecorresponds to the involved precision.

Measuring robustness of dynamical systems. Relating time and space to length and precision