{"paper":{"title":"A Homological Theory of Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DM","cs.LG","math.CO"],"primary_cat":"math.AC","authors_text":"Greg Yang","submitted_at":"2017-01-09T18:58:27Z","abstract_excerpt":"In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes $A \\not= B$ especially when $A$ is known to be a subset of $B$. In this paper we introduce a homological theory of functions that can be used to establish complexity separations, while also providing other interesting consequences. We propose to associate a topological space $S_A$ to each class of functions $A$, such that, to separate complexity classes $A \\subseteq B'$, it suffices to observe a change in \"the number of holes\", i.e. homo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02302","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}