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Given binary relations $R, S$, a componentwise reducibility is defined by $ R\\le S \\iff \\ex f \\, \\forall x, y \\, [xRy \\lra f(x) Sf(y)]. $ Here $f$ is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and $f$ must be computable. We show that there is a $\\Pi_1$-complete equivalence relation, but no $\\Pi k$-complete for $k \\ge 2$.\n  We show that $\\Sigma k$ preorders arising naturally in the above-mentioned areas are $\\Sigma k$-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1302.0580","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2013-02-04T04:02:20Z","cross_cats_sorted":[],"title_canon_sha256":"13296688e1e91b1949c4ee610d73aa3b224644434af87042ef53e3ff10211c9a","abstract_canon_sha256":"29ad43fc53f40cd5525547c27bd3c454b41356903b7a9879ff300e1e0e58415e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:02.567445Z","signature_b64":"jyWuYeAQowVmK7Uaor9ooh9A22DPVUeJbHx3fw13qBOuJIjhlsZhXjlzKhycMOdeaToa8p1TGH2ocFqHUjg5CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36087714eb8811df4f4b8740031e1fb5cf1075bb1c1eb04542a6fbb401fc8a95","last_reissued_at":"2026-05-18T00:24:02.567025Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:02.567025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Complexity of equivalence relations and preorders from computability theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Andre Nies, Egor Ianovski, Keng Meng Ng, Russell Miller","submitted_at":"2013-02-04T04:02:20Z","abstract_excerpt":"We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. 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