Limitations of the invertible-map equivalences

Thisnotedrawsconclusionsthatarisebycombiningtworecentpapers, byAnujDawar, ErichGradelandWiedPakusa, publishedatICALP2019, andbyMoritzLichter, publishedatLICS2021. Inbothpapers, themaintechnicalresultsrelyonthecombinatorialandalgebraicanalysisoftheinvertiblemapequivalences=IMk,QoncertainvariantsofCai-Furer-Immermanstructures( CFI-structuresforshort). These=IMk,Q-equivalences, foranaturalnumberkandasetofprimesQ, refinethewellknownWeisfeiler-Lemanequivalencesusedinalgorithmsforgraphisomorphism. TheintuitionisthattwographsG=IMk,QHcannotbedistinguishedbyiterativerefinementsofequivalencesonktuplesdefinedvialinearoperatorsonvectorspacesoverfieldsofcharacteristicp. Q. Inthefirstpaperithasbeenshown, usingconsiderablealgebraicmachinery, thatforaprimeq/. Q, the=IMk,Qequivalencesarenotstrongenoughtodistinguishbetweennon-isomorphicCFI-structuresoverthefieldFq. Inthesecondpaper, asimilarbutnotidenticalconstructionforCFI-structuresovertheringsZ2ihas, againbyratherinvolvedcombinatorialandalgebraicarguments, beenshowntobeindistinguishablewithrespectto=IMk,{2}. Togetherwithanearlierworkonranklogic, thissecondresultsufficestoseparateranklogicfrompolynomialtime. WeshowherethatthetwoapproachescanbeunifiedtoprovethatCFIstructuresovertheringsZ2iareinfactindistinguishablewithrespectto= IMk,P, forthesetPofallprimes. Inparticular, thisimpliesthefollowingtworesults. First, thereisnofixedksuchthattheinvertiblemapequivalence= IMk,Pcoincideswithisomorphismonallfinitegraphs. Second, noextensionoffixed-pointlogicbylinearalgebraicoperatorsoverfieldscancapturepolynomialtime.