{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:X7J2TSP3LTAV3DJBMNHBPRMUNY","short_pith_number":"pith:X7J2TSP3","canonical_record":{"source":{"id":"1112.4295","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2011-12-19T10:51:14Z","cross_cats_sorted":[],"title_canon_sha256":"ceaa58b93fe162479dfe935093cd82586002491bc56c3bf0fe2ffcc87bea2171","abstract_canon_sha256":"60fcc015241a3f1bf0ffa600b4c4c87965bfdae7a44e4fad16140e0f6d2c848d"},"schema_version":"1.0"},"canonical_sha256":"bfd3a9c9fb5cc15d8d21634e17c5946e2163c9792d0239c24c16844b951e10cf","source":{"kind":"arxiv","id":"1112.4295","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.4295","created_at":"2026-05-18T04:06:06Z"},{"alias_kind":"arxiv_version","alias_value":"1112.4295v1","created_at":"2026-05-18T04:06:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4295","created_at":"2026-05-18T04:06:06Z"},{"alias_kind":"pith_short_12","alias_value":"X7J2TSP3LTAV","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"X7J2TSP3LTAV3DJB","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"X7J2TSP3","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:X7J2TSP3LTAV3DJBMNHBPRMUNY","target":"record","payload":{"canonical_record":{"source":{"id":"1112.4295","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2011-12-19T10:51:14Z","cross_cats_sorted":[],"title_canon_sha256":"ceaa58b93fe162479dfe935093cd82586002491bc56c3bf0fe2ffcc87bea2171","abstract_canon_sha256":"60fcc015241a3f1bf0ffa600b4c4c87965bfdae7a44e4fad16140e0f6d2c848d"},"schema_version":"1.0"},"canonical_sha256":"bfd3a9c9fb5cc15d8d21634e17c5946e2163c9792d0239c24c16844b951e10cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:06.675375Z","signature_b64":"Kmn80MjF2eoDcHGLma9ohGCoqLwfLN2QEiWHNOvg+m+HU3WCOcnvhnN4GpdVA/UbmEqhA9JW4Wd6grOVkEj1Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfd3a9c9fb5cc15d8d21634e17c5946e2163c9792d0239c24c16844b951e10cf","last_reissued_at":"2026-05-18T04:06:06.674792Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:06.674792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.4295","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:06:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PR769gTSeNGyjVgkXstZcDHymAyn6iulPtIrRlrYpwUN2Z/Q5suO3MhLSpUMzDd9WeXL4Lv/zJoZ/M86AXMtAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:51:31.990614Z"},"content_sha256":"9086d6265a3be0d511f1e1750236587a8191782c7d1605d305c393d18a17218a","schema_version":"1.0","event_id":"sha256:9086d6265a3be0d511f1e1750236587a8191782c7d1605d305c393d18a17218a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:X7J2TSP3LTAV3DJBMNHBPRMUNY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computing Bits of Algebraic Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Rameshwar Pratap, Samir Datta","submitted_at":"2011-12-19T10:51:14Z","abstract_excerpt":"We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like \\pi, in Logspace.\n  Our main result is that computing a bit of a fixed real algebraic number is in C=NC1\\subseteq Logspace when the bit position has a verbose (unary) representation and in the counting hierarchy when it has a succinct (binary) representation.\n  Our tools are drawn from elementary analysis and numerical analysis, and include the Newton-Raphson method. The proof of our main result is entirel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4295","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:06:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y0LdoPGT1GBb9svD/DnLOkOg2owdfNAmVClEfrekLxWq8WI3tJAmyvu3RYQV59iGYDPMdWYvYQ+7RQv+6KzWCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:51:31.990961Z"},"content_sha256":"906697ba2bfb55fa77efe6344ce29c6c14adc850a3542fcb43ba848803e40655","schema_version":"1.0","event_id":"sha256:906697ba2bfb55fa77efe6344ce29c6c14adc850a3542fcb43ba848803e40655"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X7J2TSP3LTAV3DJBMNHBPRMUNY/bundle.json","state_url":"https://pith.science/pith/X7J2TSP3LTAV3DJBMNHBPRMUNY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X7J2TSP3LTAV3DJBMNHBPRMUNY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T16:51:31Z","links":{"resolver":"https://pith.science/pith/X7J2TSP3LTAV3DJBMNHBPRMUNY","bundle":"https://pith.science/pith/X7J2TSP3LTAV3DJBMNHBPRMUNY/bundle.json","state":"https://pith.science/pith/X7J2TSP3LTAV3DJBMNHBPRMUNY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X7J2TSP3LTAV3DJBMNHBPRMUNY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:X7J2TSP3LTAV3DJBMNHBPRMUNY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"60fcc015241a3f1bf0ffa600b4c4c87965bfdae7a44e4fad16140e0f6d2c848d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2011-12-19T10:51:14Z","title_canon_sha256":"ceaa58b93fe162479dfe935093cd82586002491bc56c3bf0fe2ffcc87bea2171"},"schema_version":"1.0","source":{"id":"1112.4295","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.4295","created_at":"2026-05-18T04:06:06Z"},{"alias_kind":"arxiv_version","alias_value":"1112.4295v1","created_at":"2026-05-18T04:06:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4295","created_at":"2026-05-18T04:06:06Z"},{"alias_kind":"pith_short_12","alias_value":"X7J2TSP3LTAV","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"X7J2TSP3LTAV3DJB","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"X7J2TSP3","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:906697ba2bfb55fa77efe6344ce29c6c14adc850a3542fcb43ba848803e40655","target":"graph","created_at":"2026-05-18T04:06:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like \\pi, in Logspace.\n  Our main result is that computing a bit of a fixed real algebraic number is in C=NC1\\subseteq Logspace when the bit position has a verbose (unary) representation and in the counting hierarchy when it has a succinct (binary) representation.\n  Our tools are drawn from elementary analysis and numerical analysis, and include the Newton-Raphson method. The proof of our main result is entirel","authors_text":"Rameshwar Pratap, Samir Datta","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2011-12-19T10:51:14Z","title":"Computing Bits of Algebraic Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4295","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9086d6265a3be0d511f1e1750236587a8191782c7d1605d305c393d18a17218a","target":"record","created_at":"2026-05-18T04:06:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"60fcc015241a3f1bf0ffa600b4c4c87965bfdae7a44e4fad16140e0f6d2c848d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2011-12-19T10:51:14Z","title_canon_sha256":"ceaa58b93fe162479dfe935093cd82586002491bc56c3bf0fe2ffcc87bea2171"},"schema_version":"1.0","source":{"id":"1112.4295","kind":"arxiv","version":1}},"canonical_sha256":"bfd3a9c9fb5cc15d8d21634e17c5946e2163c9792d0239c24c16844b951e10cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bfd3a9c9fb5cc15d8d21634e17c5946e2163c9792d0239c24c16844b951e10cf","first_computed_at":"2026-05-18T04:06:06.674792Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:06.674792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Kmn80MjF2eoDcHGLma9ohGCoqLwfLN2QEiWHNOvg+m+HU3WCOcnvhnN4GpdVA/UbmEqhA9JW4Wd6grOVkEj1Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:06.675375Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.4295","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9086d6265a3be0d511f1e1750236587a8191782c7d1605d305c393d18a17218a","sha256:906697ba2bfb55fa77efe6344ce29c6c14adc850a3542fcb43ba848803e40655"],"state_sha256":"6521fb189f74879239993fad3ec513a0e4e7cfae8b2d8ba51e0900546c19f7b8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mka8tdI4KfKnh98h1tf3U1mTBU9fpEPnV6QPPbnid6zsEqEArf4ixNFIOC3+fTicuRqOn9OyO2d1QLnDh0HKAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T16:51:31.992913Z","bundle_sha256":"d38f071eb3f03afcd81bf21316b87bd5e753e8537224f1f09646fdf4e6465ef7"}}