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The question is whether the relation R_0 is primitive positive definable from R_1,...,R_n, i.e., definable by a first-order formula that uses only relation symbols for R_1,..., R_n, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden).\n  We show decidability of this problem for a"},"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":"1012.2381","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-12-10T20:57:44Z","cross_cats_sorted":["cs.LO"],"title_canon_sha256":"cd139e742e1855a2d291fdfcc84fbf630b67190a8d59aaa7995c1514e778b668","abstract_canon_sha256":"b2f46af81a71e887b19f87d135834f31affb51a296b08604e65884dcc25eb8a3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:59.967780Z","signature_b64":"bEiF1Z+mtK0W2ptJFJpMDySNQIhxEL6gF6A7nubruiWAbw9Gu6KZ1HaHz60oflXgFYjpL1Nb+y4vFv6hIMFWCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41cace77903f110d5125c96a038f1446b874ea33bc1af1d980431a313f367e03","last_reissued_at":"2026-05-18T04:00:59.967196Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:59.967196Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decidability of definability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LO"],"primary_cat":"math.LO","authors_text":"Manuel Bodirsky, Michael Pinsker, Todor Tsankov","submitted_at":"2010-12-10T20:57:44Z","abstract_excerpt":"For a fixed countably infinite structure \\Gamma\\ with finite relational signature \\tau, we study the following computational problem: input are quantifier-free \\tau-formulas \\phi_0,\\phi_1,...,\\phi_n that define relations R_0,R_1,...,R_n over \\Gamma. 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