doi:10.1007/s00224-022-10096-7engLutz, Jack H.Lutz, NeilMayordomo, ElviraDimension and the structure of complexity classesART-2023-130096We prove three results on the dimension structure of complexity classes. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances characterize the resource-bounded dimension of a set X of languages in terms of the relativized resource-bounded dimensions of the individual elements of X, provided that the former resource bound is large enough to parametrize the latter. Thus for example, the dimension of a class X of languages in EXP is characterized in terms of the relativized p-dimensions of the individual elements of X. Every language that is =mP-reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resource-bounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is =mP-hard for NP. If the set of all disjoint pairs of NP languages has dimension 1 in the set of all disjoint pairs of EXP languages, then NP has positive dimension in EXP.2023http://zaguan.unizar.es/record/13131410.1007/s00224-022-10096-7http://zaguan.unizar.es/record/131314oai:zaguan.unizar.es:131314info:eu-repo/grantAgreement/ES/DGA/T64-20Rinfo:eu-repo/grantAgreement/ES/MICIU/PID2019-104358RB-I00THEORY OF COMPUTING SYSTEMS 67 (2023), 473-490All rights reservedhttp://www.europeana.eu/rights/rr-f/info:eu-repo/semantics/openAccess