{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:LCFOYYYNFEWTJUAMWFNVG7F3TM","short_pith_number":"pith:LCFOYYYN","canonical_record":{"source":{"id":"1310.7041","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-10-25T21:00:23Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c1a1c692f3781bbeffb39640a0ef03a63b3b82bff8f2d615677f53d93c3f11d5","abstract_canon_sha256":"205a5d8dee7fb2be3de8253122433a30456cf9d1123eaae0c89ba57f329f7816"},"schema_version":"1.0"},"canonical_sha256":"588aec630d292d34d00cb15b537cbb9b13987dc1f6bedcf6e0ae98b8d70aab21","source":{"kind":"arxiv","id":"1310.7041","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.7041","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"arxiv_version","alias_value":"1310.7041v1","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.7041","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"pith_short_12","alias_value":"LCFOYYYNFEWT","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LCFOYYYNFEWTJUAM","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LCFOYYYN","created_at":"2026-05-18T12:27:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:LCFOYYYNFEWTJUAMWFNVG7F3TM","target":"record","payload":{"canonical_record":{"source":{"id":"1310.7041","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-10-25T21:00:23Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c1a1c692f3781bbeffb39640a0ef03a63b3b82bff8f2d615677f53d93c3f11d5","abstract_canon_sha256":"205a5d8dee7fb2be3de8253122433a30456cf9d1123eaae0c89ba57f329f7816"},"schema_version":"1.0"},"canonical_sha256":"588aec630d292d34d00cb15b537cbb9b13987dc1f6bedcf6e0ae98b8d70aab21","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:38.626628Z","signature_b64":"yimPklxGjVA/GaMNbq5qDtA8v4yY6ZBp64Ef/82//9kyQ6V9b/OILMTURtz8v/JB6iSEQf2IdlXWPHgRXmYUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"588aec630d292d34d00cb15b537cbb9b13987dc1f6bedcf6e0ae98b8d70aab21","last_reissued_at":"2026-05-18T00:57:38.626091Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:38.626091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.7041","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":"2026-05-18T00:57:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"50sJFN1JbYhwzI+HHVwVj+E6tLbuergC/m1PGe1yBobFi6S+pAh0DcOjwPFRgBTY4W8cSnPUDRES/KONIEsvDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T01:26:56.828908Z"},"content_sha256":"e5d259f92241eeeacc0ec6678b4a8a93cbb5d7f29ea0091b4e6b6d681a818e63","schema_version":"1.0","event_id":"sha256:e5d259f92241eeeacc0ec6678b4a8a93cbb5d7f29ea0091b4e6b6d681a818e63"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:LCFOYYYNFEWTJUAMWFNVG7F3TM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A complete classification of equational classes of threshold functions included in clones","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Erkko Lehtonen, Karsten Sch\\\"olzel, Miguel Couceiro","submitted_at":"2013-10-25T21:00:23Z","abstract_excerpt":"The class of threshold functions is known to be characterizable by functional equations or, equivalently, by pairs of relations, which are called relational constraints. It was shown by Hellerstein that this class cannot be characterized by a finite number of such objects. In this paper, we investigate classes of threshold functions which arise as intersections of the class of all threshold functions with clones of Boolean functions, and provide a complete classification of such intersections in respect to whether they have finite characterizations. Moreover, we provide a characterizing set of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7041","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":"2026-05-18T00:57:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d6zENLer9XgFPeIExCY+IyT1PAtRBd6srgu5yIgqnVEjMM3Ct1HWeamB7goVsj+GwHv7vNewYq0VI8kvw1LTAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T01:26:56.829517Z"},"content_sha256":"f39fffe67c302551ebcef3052110cf7221b7120222cb87d592bcb2e66a986ab4","schema_version":"1.0","event_id":"sha256:f39fffe67c302551ebcef3052110cf7221b7120222cb87d592bcb2e66a986ab4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LCFOYYYNFEWTJUAMWFNVG7F3TM/bundle.json","state_url":"https://pith.science/pith/LCFOYYYNFEWTJUAMWFNVG7F3TM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LCFOYYYNFEWTJUAMWFNVG7F3TM/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":"2026-05-27T01:26:56Z","links":{"resolver":"https://pith.science/pith/LCFOYYYNFEWTJUAMWFNVG7F3TM","bundle":"https://pith.science/pith/LCFOYYYNFEWTJUAMWFNVG7F3TM/bundle.json","state":"https://pith.science/pith/LCFOYYYNFEWTJUAMWFNVG7F3TM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LCFOYYYNFEWTJUAMWFNVG7F3TM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:LCFOYYYNFEWTJUAMWFNVG7F3TM","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":"205a5d8dee7fb2be3de8253122433a30456cf9d1123eaae0c89ba57f329f7816","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-10-25T21:00:23Z","title_canon_sha256":"c1a1c692f3781bbeffb39640a0ef03a63b3b82bff8f2d615677f53d93c3f11d5"},"schema_version":"1.0","source":{"id":"1310.7041","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.7041","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"arxiv_version","alias_value":"1310.7041v1","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.7041","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"pith_short_12","alias_value":"LCFOYYYNFEWT","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LCFOYYYNFEWTJUAM","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LCFOYYYN","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:f39fffe67c302551ebcef3052110cf7221b7120222cb87d592bcb2e66a986ab4","target":"graph","created_at":"2026-05-18T00:57:38Z","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 class of threshold functions is known to be characterizable by functional equations or, equivalently, by pairs of relations, which are called relational constraints. It was shown by Hellerstein that this class cannot be characterized by a finite number of such objects. In this paper, we investigate classes of threshold functions which arise as intersections of the class of all threshold functions with clones of Boolean functions, and provide a complete classification of such intersections in respect to whether they have finite characterizations. Moreover, we provide a characterizing set of","authors_text":"Erkko Lehtonen, Karsten Sch\\\"olzel, Miguel Couceiro","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-10-25T21:00:23Z","title":"A complete classification of equational classes of threshold functions included in clones"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7041","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:e5d259f92241eeeacc0ec6678b4a8a93cbb5d7f29ea0091b4e6b6d681a818e63","target":"record","created_at":"2026-05-18T00:57:38Z","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":"205a5d8dee7fb2be3de8253122433a30456cf9d1123eaae0c89ba57f329f7816","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-10-25T21:00:23Z","title_canon_sha256":"c1a1c692f3781bbeffb39640a0ef03a63b3b82bff8f2d615677f53d93c3f11d5"},"schema_version":"1.0","source":{"id":"1310.7041","kind":"arxiv","version":1}},"canonical_sha256":"588aec630d292d34d00cb15b537cbb9b13987dc1f6bedcf6e0ae98b8d70aab21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"588aec630d292d34d00cb15b537cbb9b13987dc1f6bedcf6e0ae98b8d70aab21","first_computed_at":"2026-05-18T00:57:38.626091Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:57:38.626091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yimPklxGjVA/GaMNbq5qDtA8v4yY6ZBp64Ef/82//9kyQ6V9b/OILMTURtz8v/JB6iSEQf2IdlXWPHgRXmYUAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:57:38.626628Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.7041","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e5d259f92241eeeacc0ec6678b4a8a93cbb5d7f29ea0091b4e6b6d681a818e63","sha256:f39fffe67c302551ebcef3052110cf7221b7120222cb87d592bcb2e66a986ab4"],"state_sha256":"ce9b1812c6586c3fd6fa39fbae0761cb3c76d9890b1af4a257dee103d27f476c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bJboVMYM7hx6wDcfALtwwJRxq01fCdqBgppmeQxcfB6Jrf3urJdSX3Mhpxa4Xgr32XJ8tYmFNQlNMaMqbXNTCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T01:26:56.832668Z","bundle_sha256":"2d96932ac2bc6acb82f2226752439ecc1ab495bc1f48357237cbb7426df643c0"}}