{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:7HJO7SD7DKWUR254JDPIC23PQA","short_pith_number":"pith:7HJO7SD7","schema_version":"1.0","canonical_sha256":"f9d2efc87f1aad48ebbc48de816b6f803abad7b5b875ace223382f19bc562b79","source":{"kind":"arxiv","id":"2309.15026","version":4},"attestation_state":"computed","paper":{"title":"Instance complexity of Boolean functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Symmetric Boolean functions have instance complexity fully characterized, with only Parity and its complement achieving value 1.","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Alison Hsiang-Hsuan Liu, Nikhil S. Mande","submitted_at":"2023-09-26T15:56:14Z","abstract_excerpt":"In the area of query complexity of Boolean functions, the most widely studied cost measure of an algorithm is the worst-case number of queries made by it on an input. Motivated by the most natural cost measure studied in online algorithms, the competitive ratio, we consider a different cost measure for query algorithms for Boolean functions that captures the ratio of the cost of the algorithm and the cost of an optimal algorithm that knows the input in advance. The cost of an algorithm is its largest cost over all inputs. Grossman, Komargodski and Naor [ITCS'20] introduced this measure for Boo"},"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":"2309.15026","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2023-09-26T15:56:14Z","cross_cats_sorted":[],"title_canon_sha256":"bfc65f203104222fa67f3fab0511b2f8e5f7e57153d058cf696442aa9c850d95","abstract_canon_sha256":"acdb715f5e831d63ad557dfa9b4cd663254b753f6a3c890461d05f8bbbb934e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T14:15:49.441121Z","signature_b64":"QhWYr7jsJcuNXpepN9dsvZRB9OPbrhs1kfoS4TGKfJTpVKzByqL2R2vhNfsDZIxCojmZ9uymaaNn85w1f8JwDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9d2efc87f1aad48ebbc48de816b6f803abad7b5b875ace223382f19bc562b79","last_reissued_at":"2026-06-24T14:15:49.440618Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T14:15:49.440618Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Instance complexity of Boolean functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Symmetric Boolean functions have instance complexity fully characterized, with only Parity and its complement achieving value 1.","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Alison Hsiang-Hsuan Liu, Nikhil S. Mande","submitted_at":"2023-09-26T15:56:14Z","abstract_excerpt":"In the area of query complexity of Boolean functions, the most widely studied cost measure of an algorithm is the worst-case number of queries made by it on an input. Motivated by the most natural cost measure studied in online algorithms, the competitive ratio, we consider a different cost measure for query algorithms for Boolean functions that captures the ratio of the cost of the algorithm and the cost of an optimal algorithm that knows the input in advance. The cost of an algorithm is its largest cost over all inputs. Grossman, Komargodski and Naor [ITCS'20] introduced this measure for Boo"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We complement the above-mentioned result of Grossman et al. by completely characterizing the instance complexity of symmetric Boolean functions. As a corollary we conclude that the only symmetric Boolean functions with instance complexity 1 are the Parity function and its complement. We show that the above-mentioned ratio is linear in the input size for both of these functions, while we exhibit algorithms for which the instance complexity is a constant.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The measure of instance complexity is taken directly from the definition introduced by Grossman et al. (the max over inputs of algorithm cost divided by optimal known-input cost) and the standard decision-tree model for Boolean functions; this premise enters when the paper invokes the prior definition to derive the symmetric characterization and the separation for Greater-Than and Odd-Max-Bit.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Instance complexity of symmetric Boolean functions is completely characterized with only Parity and complement at value 1; Greater-Than and Odd-Max-Bit have constant IC while QC/C_min is linear.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Symmetric Boolean functions have instance complexity fully characterized, with only Parity and its complement achieving value 1.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d75f898ddd64e51168fc12e1383828a1b0c70b376e0d249a985df6219e719435"},"source":{"id":"2309.15026","kind":"arxiv","version":4},"verdict":{"id":"255f4da9-0999-460f-bb20-71275fae0c59","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T07:22:37.974305Z","strongest_claim":"We complement the above-mentioned result of Grossman et al. by completely characterizing the instance complexity of symmetric Boolean functions. As a corollary we conclude that the only symmetric Boolean functions with instance complexity 1 are the Parity function and its complement. We show that the above-mentioned ratio is linear in the input size for both of these functions, while we exhibit algorithms for which the instance complexity is a constant.","one_line_summary":"Instance complexity of symmetric Boolean functions is completely characterized with only Parity and complement at value 1; Greater-Than and Odd-Max-Bit have constant IC while QC/C_min is linear.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The measure of instance complexity is taken directly from the definition introduced by Grossman et al. (the max over inputs of algorithm cost divided by optimal known-input cost) and the standard decision-tree model for Boolean functions; this premise enters when the paper invokes the prior definition to derive the symmetric characterization and the separation for Greater-Than and Odd-Max-Bit.","pith_extraction_headline":"Symmetric Boolean functions have instance complexity fully characterized, with only Parity and its complement achieving value 1."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2309.15026/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2309.15026","created_at":"2026-06-24T14:15:49.440677+00:00"},{"alias_kind":"arxiv_version","alias_value":"2309.15026v4","created_at":"2026-06-24T14:15:49.440677+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.15026","created_at":"2026-06-24T14:15:49.440677+00:00"},{"alias_kind":"pith_short_12","alias_value":"7HJO7SD7DKWU","created_at":"2026-06-24T14:15:49.440677+00:00"},{"alias_kind":"pith_short_16","alias_value":"7HJO7SD7DKWUR254","created_at":"2026-06-24T14:15:49.440677+00:00"},{"alias_kind":"pith_short_8","alias_value":"7HJO7SD7","created_at":"2026-06-24T14:15:49.440677+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA","json":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA.json","graph_json":"https://pith.science/api/pith-number/7HJO7SD7DKWUR254JDPIC23PQA/graph.json","events_json":"https://pith.science/api/pith-number/7HJO7SD7DKWUR254JDPIC23PQA/events.json","paper":"https://pith.science/paper/7HJO7SD7"},"agent_actions":{"view_html":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA","download_json":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA.json","view_paper":"https://pith.science/paper/7HJO7SD7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2309.15026&json=true","fetch_graph":"https://pith.science/api/pith-number/7HJO7SD7DKWUR254JDPIC23PQA/graph.json","fetch_events":"https://pith.science/api/pith-number/7HJO7SD7DKWUR254JDPIC23PQA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA/action/storage_attestation","attest_author":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA/action/author_attestation","sign_citation":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA/action/citation_signature","submit_replication":"https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA/action/replication_record"}},"created_at":"2026-06-24T14:15:49.440677+00:00","updated_at":"2026-06-24T14:15:49.440677+00:00"}