{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:UY3Q2DLAFFLNERWL44KAAW6RB4","short_pith_number":"pith:UY3Q2DLA","canonical_record":{"source":{"id":"1201.3543","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-01-17T16:09:13Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"27a993c7f9b60a52cd01be61d7aadc2fb4f3690727703f8ba114d42142236339","abstract_canon_sha256":"ff13edab8cbec3222eb6346c732d7bbc21e3faed50d9565e2b23547b382cf96f"},"schema_version":"1.0"},"canonical_sha256":"a6370d0d602956d246cbe714005bd10f154a5cd5d2999323cd57c910c928acdf","source":{"kind":"arxiv","id":"1201.3543","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3543","created_at":"2026-05-18T02:32:46Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3543v2","created_at":"2026-05-18T02:32:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3543","created_at":"2026-05-18T02:32:46Z"},{"alias_kind":"pith_short_12","alias_value":"UY3Q2DLAFFLN","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"UY3Q2DLAFFLNERWL","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"UY3Q2DLA","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:UY3Q2DLAFFLNERWL44KAAW6RB4","target":"record","payload":{"canonical_record":{"source":{"id":"1201.3543","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-01-17T16:09:13Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"27a993c7f9b60a52cd01be61d7aadc2fb4f3690727703f8ba114d42142236339","abstract_canon_sha256":"ff13edab8cbec3222eb6346c732d7bbc21e3faed50d9565e2b23547b382cf96f"},"schema_version":"1.0"},"canonical_sha256":"a6370d0d602956d246cbe714005bd10f154a5cd5d2999323cd57c910c928acdf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:46.611447Z","signature_b64":"lR01ZUCzjj077uUXMaTdfnqJ3Xzn947XSKfEhB/U8WIyZW/yUBWm8B09Q4+R6qSgVRSyI3D2zdqcGTY/wasdAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6370d0d602956d246cbe714005bd10f154a5cd5d2999323cd57c910c928acdf","last_reissued_at":"2026-05-18T02:32:46.610948Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:46.610948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.3543","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:32:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"++sLG7BQ+qzlpqslimz7ttXiq3KW3gZ1Qs64h6LcC9ySLAkSx3yRYQ6OiRowyQYj3k0xQeGqABwMMEu80coDBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:35:34.102774Z"},"content_sha256":"282f3a45be5293d7a255591ff2575654e6a7a001d6cf8cadb332571a5728701f","schema_version":"1.0","event_id":"sha256:282f3a45be5293d7a255591ff2575654e6a7a001d6cf8cadb332571a5728701f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:UY3Q2DLAFFLNERWL44KAAW6RB4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.OC","authors_text":"Jean-Luc Marichal, Pierre Mathonet","submitted_at":"2012-01-17T16:09:13Z","abstract_excerpt":"The Banzhaf power and interaction indexes for a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficients in the standard least squares approximation of the function by a pseudo-Boolean function of a specified degree. We first observe that this property still holds if we consider approximations by pseudo-Boolean functions depending only on specified variables. We then show that the Banzhaf influence index can also be obtained from the latter approximation problem. Considering certain weighted versions of this approximation problem, we introduce a class of weighte"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3543","kind":"arxiv","version":2},"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":"2026-05-18T02:32:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H46xK2h429RNlmWtBrqZvXEGr9toeLEDd9tUlOIbHPj3N+F5bWNpxP8RJ8sRJc4IR5Sdije/0FFeZQ9ADlODCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:35:34.103164Z"},"content_sha256":"7212fa1df06fc461a479d31d9df1b4c42ae43628b20a3965c35a43069f3690e6","schema_version":"1.0","event_id":"sha256:7212fa1df06fc461a479d31d9df1b4c42ae43628b20a3965c35a43069f3690e6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UY3Q2DLAFFLNERWL44KAAW6RB4/bundle.json","state_url":"https://pith.science/pith/UY3Q2DLAFFLNERWL44KAAW6RB4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UY3Q2DLAFFLNERWL44KAAW6RB4/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":"2026-05-27T05:35:34Z","links":{"resolver":"https://pith.science/pith/UY3Q2DLAFFLNERWL44KAAW6RB4","bundle":"https://pith.science/pith/UY3Q2DLAFFLNERWL44KAAW6RB4/bundle.json","state":"https://pith.science/pith/UY3Q2DLAFFLNERWL44KAAW6RB4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UY3Q2DLAFFLNERWL44KAAW6RB4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:UY3Q2DLAFFLNERWL44KAAW6RB4","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":"ff13edab8cbec3222eb6346c732d7bbc21e3faed50d9565e2b23547b382cf96f","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-01-17T16:09:13Z","title_canon_sha256":"27a993c7f9b60a52cd01be61d7aadc2fb4f3690727703f8ba114d42142236339"},"schema_version":"1.0","source":{"id":"1201.3543","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3543","created_at":"2026-05-18T02:32:46Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3543v2","created_at":"2026-05-18T02:32:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3543","created_at":"2026-05-18T02:32:46Z"},{"alias_kind":"pith_short_12","alias_value":"UY3Q2DLAFFLN","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"UY3Q2DLAFFLNERWL","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"UY3Q2DLA","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:7212fa1df06fc461a479d31d9df1b4c42ae43628b20a3965c35a43069f3690e6","target":"graph","created_at":"2026-05-18T02:32:46Z","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":"The Banzhaf power and interaction indexes for a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficients in the standard least squares approximation of the function by a pseudo-Boolean function of a specified degree. We first observe that this property still holds if we consider approximations by pseudo-Boolean functions depending only on specified variables. We then show that the Banzhaf influence index can also be obtained from the latter approximation problem. Considering certain weighted versions of this approximation problem, we introduce a class of weighte","authors_text":"Jean-Luc Marichal, Pierre Mathonet","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-01-17T16:09:13Z","title":"Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3543","kind":"arxiv","version":2},"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":"sha256:282f3a45be5293d7a255591ff2575654e6a7a001d6cf8cadb332571a5728701f","target":"record","created_at":"2026-05-18T02:32:46Z","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":"ff13edab8cbec3222eb6346c732d7bbc21e3faed50d9565e2b23547b382cf96f","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-01-17T16:09:13Z","title_canon_sha256":"27a993c7f9b60a52cd01be61d7aadc2fb4f3690727703f8ba114d42142236339"},"schema_version":"1.0","source":{"id":"1201.3543","kind":"arxiv","version":2}},"canonical_sha256":"a6370d0d602956d246cbe714005bd10f154a5cd5d2999323cd57c910c928acdf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a6370d0d602956d246cbe714005bd10f154a5cd5d2999323cd57c910c928acdf","first_computed_at":"2026-05-18T02:32:46.610948Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:46.610948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lR01ZUCzjj077uUXMaTdfnqJ3Xzn947XSKfEhB/U8WIyZW/yUBWm8B09Q4+R6qSgVRSyI3D2zdqcGTY/wasdAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:46.611447Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.3543","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:282f3a45be5293d7a255591ff2575654e6a7a001d6cf8cadb332571a5728701f","sha256:7212fa1df06fc461a479d31d9df1b4c42ae43628b20a3965c35a43069f3690e6"],"state_sha256":"76a03b3160aa5d9720e00a15b222343c17743626d488724c3b12c30c1f9cb375"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lpwKacusmTrSLvAUgTGUILFhzIenzgSxBvF6PooSYbkiaIdfIegxDSwdCGYJ+UDhu1DkLIkhHxvALJtoz4vYAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:35:34.106959Z","bundle_sha256":"433455532a9b381e18fab9fae89c9ef30b584b89ccb17af260cd3adc10ff8297"}}