{"paper":{"title":"Relativizing Small Complexity Classes and their Theories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Klaus Aehlig, Phuong Nguyen, Stephen Cook","submitted_at":"2012-04-24T22:44:53Z","abstract_excerpt":"Existing definitions of the relativizations of \\NCOne, \\L\\ and \\NL\\ do not preserve the inclusions $\\NCOne \\subseteq \\L$, $\\NL\\subseteq \\ACOne$. We start by giving the first definitions that preserve them. Here for \\L\\ and \\NL\\ we define their relativizations using Wilson's stack oracle model, but limit the height of the stack to a constant (instead of $\\log(n)$). We show that the collapse of any two classes in $\\{\\ACZm, \\TCZ, \\NCOne, \\L, \\NL\\}$ implies the collapse of their relativizations. Next we exhibit an oracle $\\alpha$ that makes $\\ACk(\\alpha)$ a proper hierarchy. This strengthens and c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.5508","kind":"arxiv","version":1},"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"}