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

We prove that functions over the reals computable in polynomial time can becharacterised using discrete ordinary differential equations (ODE), also knownas finite differences. We also provide a characterisation of functionscomputable in polynomial space over the reals.  In particular, this covers space complexity, while existing characterisationswere only able to cover time complexity, and were restricted to functions overthe integers. We prove furthermore that no artificial sign or test function isneeded even for time complexity.  At a technical level, this is obtained by proving that Turing machines can besimulated with analytic discrete ordinary differential equations. We believethis result opens the way to many applications, as it opens the possibility ofprogramming with ODEs, with an underlying well-understood time and spacecomplexity.

Simulation of Turing machines with analytic discrete ODEs: FPTIME and  FPSPACE over the reals characterised with discrete ordinary differential  equations