{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:B2FIIN7X4Q6DI22THVR2XVJZQT","short_pith_number":"pith:B2FIIN7X","canonical_record":{"source":{"id":"1611.08103","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.AI","submitted_at":"2016-11-24T09:06:57Z","cross_cats_sorted":[],"title_canon_sha256":"fe1314cef08a54557c088dca05b3db981ec950bb93df811fc5f889e73456a637","abstract_canon_sha256":"51abc80822fc8873d8029518df9846314bc902dcbe24297c419a8379e90466ad"},"schema_version":"1.0"},"canonical_sha256":"0e8a8437f7e43c346b533d63abd53984dac5ddfe81529aad47d7f7a4920e8908","source":{"kind":"arxiv","id":"1611.08103","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08103","created_at":"2026-05-18T00:56:44Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08103v1","created_at":"2026-05-18T00:56:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08103","created_at":"2026-05-18T00:56:44Z"},{"alias_kind":"pith_short_12","alias_value":"B2FIIN7X4Q6D","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"B2FIIN7X4Q6DI22T","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"B2FIIN7X","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:B2FIIN7X4Q6DI22THVR2XVJZQT","target":"record","payload":{"canonical_record":{"source":{"id":"1611.08103","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.AI","submitted_at":"2016-11-24T09:06:57Z","cross_cats_sorted":[],"title_canon_sha256":"fe1314cef08a54557c088dca05b3db981ec950bb93df811fc5f889e73456a637","abstract_canon_sha256":"51abc80822fc8873d8029518df9846314bc902dcbe24297c419a8379e90466ad"},"schema_version":"1.0"},"canonical_sha256":"0e8a8437f7e43c346b533d63abd53984dac5ddfe81529aad47d7f7a4920e8908","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:44.380892Z","signature_b64":"dyMcO7xtI3KN8LDFZYwB5tENwxArij2tONIBgWdHFBGcJbxmENl/yJ8cQgbowYVyD7h7QoIo4Ed3xYsAm5fQAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e8a8437f7e43c346b533d63abd53984dac5ddfe81529aad47d7f7a4920e8908","last_reissued_at":"2026-05-18T00:56:44.380162Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:44.380162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.08103","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":"2026-05-18T00:56:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SXGuL7JBctpQzhBOzuJoSy8OUx9qlZ9FrUlpm1kGiFIV5hoEpfzIgIT7gHmBC8GVhthyH9LTc/HWjvyucXcpBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T08:03:57.189713Z"},"content_sha256":"4d25988dc1513d0fb6bbdcc2fd794eb870c9153c8055c07aa08fb672fc5072c0","schema_version":"1.0","event_id":"sha256:4d25988dc1513d0fb6bbdcc2fd794eb870c9153c8055c07aa08fb672fc5072c0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:B2FIIN7X4Q6DI22THVR2XVJZQT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Double-quantitative $\\gamma^{\\ast}-$fuzzy coverings approximation operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.AI","authors_text":"Guangming Lang","submitted_at":"2016-11-24T09:06:57Z","abstract_excerpt":"In digital-based information boom, the fuzzy covering rough set model is an important mathematical tool for artificial intelligence, and how to build the bridge between the fuzzy covering rough set theory and Pawlak's model is becoming a hot research topic. In this paper, we first present the $\\gamma-$fuzzy covering based probabilistic and grade approximation operators and double-quantitative approximation operators. We also study the relationships among the three types of $\\gamma-$fuzzy covering based approximation operators. Second, we propose the $\\gamma^{\\ast}-$fuzzy coverings based multi-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08103","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":"2026-05-18T00:56:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FEKdN0UUS6BP3VQUoeOgAwaavhtprrA9BXeACwvLydTxZZVs7DZiiIOWFsejdpdIDQ9lnLJuJ5u9h9yuQwafCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T08:03:57.190053Z"},"content_sha256":"13ecec42ca564e7def3382acfaa0ba7ca9a6dae3c7ac8dacadbe3c768aa4e95b","schema_version":"1.0","event_id":"sha256:13ecec42ca564e7def3382acfaa0ba7ca9a6dae3c7ac8dacadbe3c768aa4e95b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B2FIIN7X4Q6DI22THVR2XVJZQT/bundle.json","state_url":"https://pith.science/pith/B2FIIN7X4Q6DI22THVR2XVJZQT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B2FIIN7X4Q6DI22THVR2XVJZQT/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":"2026-06-05T08:03:57Z","links":{"resolver":"https://pith.science/pith/B2FIIN7X4Q6DI22THVR2XVJZQT","bundle":"https://pith.science/pith/B2FIIN7X4Q6DI22THVR2XVJZQT/bundle.json","state":"https://pith.science/pith/B2FIIN7X4Q6DI22THVR2XVJZQT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B2FIIN7X4Q6DI22THVR2XVJZQT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:B2FIIN7X4Q6DI22THVR2XVJZQT","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":"51abc80822fc8873d8029518df9846314bc902dcbe24297c419a8379e90466ad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.AI","submitted_at":"2016-11-24T09:06:57Z","title_canon_sha256":"fe1314cef08a54557c088dca05b3db981ec950bb93df811fc5f889e73456a637"},"schema_version":"1.0","source":{"id":"1611.08103","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08103","created_at":"2026-05-18T00:56:44Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08103v1","created_at":"2026-05-18T00:56:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08103","created_at":"2026-05-18T00:56:44Z"},{"alias_kind":"pith_short_12","alias_value":"B2FIIN7X4Q6D","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"B2FIIN7X4Q6DI22T","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"B2FIIN7X","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:13ecec42ca564e7def3382acfaa0ba7ca9a6dae3c7ac8dacadbe3c768aa4e95b","target":"graph","created_at":"2026-05-18T00:56:44Z","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":"In digital-based information boom, the fuzzy covering rough set model is an important mathematical tool for artificial intelligence, and how to build the bridge between the fuzzy covering rough set theory and Pawlak's model is becoming a hot research topic. In this paper, we first present the $\\gamma-$fuzzy covering based probabilistic and grade approximation operators and double-quantitative approximation operators. We also study the relationships among the three types of $\\gamma-$fuzzy covering based approximation operators. Second, we propose the $\\gamma^{\\ast}-$fuzzy coverings based multi-","authors_text":"Guangming Lang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.AI","submitted_at":"2016-11-24T09:06:57Z","title":"Double-quantitative $\\gamma^{\\ast}-$fuzzy coverings approximation operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08103","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":"sha256:4d25988dc1513d0fb6bbdcc2fd794eb870c9153c8055c07aa08fb672fc5072c0","target":"record","created_at":"2026-05-18T00:56:44Z","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":"51abc80822fc8873d8029518df9846314bc902dcbe24297c419a8379e90466ad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.AI","submitted_at":"2016-11-24T09:06:57Z","title_canon_sha256":"fe1314cef08a54557c088dca05b3db981ec950bb93df811fc5f889e73456a637"},"schema_version":"1.0","source":{"id":"1611.08103","kind":"arxiv","version":1}},"canonical_sha256":"0e8a8437f7e43c346b533d63abd53984dac5ddfe81529aad47d7f7a4920e8908","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e8a8437f7e43c346b533d63abd53984dac5ddfe81529aad47d7f7a4920e8908","first_computed_at":"2026-05-18T00:56:44.380162Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:44.380162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dyMcO7xtI3KN8LDFZYwB5tENwxArij2tONIBgWdHFBGcJbxmENl/yJ8cQgbowYVyD7h7QoIo4Ed3xYsAm5fQAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:44.380892Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.08103","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4d25988dc1513d0fb6bbdcc2fd794eb870c9153c8055c07aa08fb672fc5072c0","sha256:13ecec42ca564e7def3382acfaa0ba7ca9a6dae3c7ac8dacadbe3c768aa4e95b"],"state_sha256":"16d5f83fa4ada9baef3bd8209c355765c9dca168f77ac058f579f35f12f83267"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ROw3F3Icj1FsdpYCdp9BiGuE/Ujt/eZWSjBklwLgxog6jRQbKRuNTHDmnyG8klPhucyXgwwOHmgxSI1X7iAeAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T08:03:57.192080Z","bundle_sha256":"dafdfaa12557183ea245153cd182ea4b4655ad6a2b44c1a627794723eeef3f9d"}}