{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:PY2QCHKI4LZGFWLRTPT3BD3ARR","short_pith_number":"pith:PY2QCHKI","canonical_record":{"source":{"id":"1806.00528","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2018-05-30T13:09:33Z","cross_cats_sorted":[],"title_canon_sha256":"aab719d0bc36d64c091a12cf31d630bf2150d6459f48f8056a76f4d023394fc3","abstract_canon_sha256":"0ee7a489daedec9b4647f789e670475b6101dfe2370ccb8a9a088d1caf41e584"},"schema_version":"1.0"},"canonical_sha256":"7e35011d48e2f262d9719be7b08f608c4bf25c4df0366d782571f16689807ba4","source":{"kind":"arxiv","id":"1806.00528","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00528","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00528v1","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00528","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"pith_short_12","alias_value":"PY2QCHKI4LZG","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PY2QCHKI4LZGFWLR","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PY2QCHKI","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:PY2QCHKI4LZGFWLRTPT3BD3ARR","target":"record","payload":{"canonical_record":{"source":{"id":"1806.00528","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2018-05-30T13:09:33Z","cross_cats_sorted":[],"title_canon_sha256":"aab719d0bc36d64c091a12cf31d630bf2150d6459f48f8056a76f4d023394fc3","abstract_canon_sha256":"0ee7a489daedec9b4647f789e670475b6101dfe2370ccb8a9a088d1caf41e584"},"schema_version":"1.0"},"canonical_sha256":"7e35011d48e2f262d9719be7b08f608c4bf25c4df0366d782571f16689807ba4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:03.718708Z","signature_b64":"X2tNi9b3+hja0N3edYBvIhyZ31+4BvWfPSNQEQPNcGPhubbkxTAZAIE8lxVbHVXqlCMmZ+Nb/ohR8NbfWSBcDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e35011d48e2f262d9719be7b08f608c4bf25c4df0366d782571f16689807ba4","last_reissued_at":"2026-05-18T00:14:03.717998Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:03.717998Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.00528","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":"2026-05-18T00:14:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DJ522X3O2wsSr08gmiXO6EEhfzPuWQFL6NI+VeYyyTwWIMyZqM7QawzcLaAFl9om5p3sbPprwwYU0Xh/bYl6CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T21:47:47.487736Z"},"content_sha256":"fb95044529b9e8ca6fb5fc6f4a27d11df46544961b2f3230a627324f9b31e4a8","schema_version":"1.0","event_id":"sha256:fb95044529b9e8ca6fb5fc6f4a27d11df46544961b2f3230a627324f9b31e4a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:PY2QCHKI4LZGFWLRTPT3BD3ARR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Metric-based Approximate Minimization of Markov Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Giorgio Bacci, Giovanni Bacci, Kim G. Larsen, Radu Mardare","submitted_at":"2018-05-30T13:09:33Z","abstract_excerpt":"In this paper, we address the approximate minimization problem of Markov Chains (MCs) from a behavioral metric-based perspective. Specifically, given a finite MC and a positive integer k, we are looking for an MC with at most k states having minimal distance to the original. The metric considered in this work is the bisimilarity distance of Desharnais et al.. For this metric we show that (1) optimal approximations always exist; (2) the problem has a bilinear program characterization; and (3) prove that its threshold problem is in PSPACE and NP-hard. In addition to the bilinear program solution"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00528","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":"2026-05-18T00:14:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P58kSUyQdgMEvoIO9qSb5V7093TDmBeMMa8u9fSxmrfzs0ehYp1xrg7UZNA9HIA8483Ks9Xcc7MkQF1PlKjUAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T21:47:47.488469Z"},"content_sha256":"5e3e4227b4de981a70433597ba5da9a23a47684163bf1a1def5b1597779487e2","schema_version":"1.0","event_id":"sha256:5e3e4227b4de981a70433597ba5da9a23a47684163bf1a1def5b1597779487e2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PY2QCHKI4LZGFWLRTPT3BD3ARR/bundle.json","state_url":"https://pith.science/pith/PY2QCHKI4LZGFWLRTPT3BD3ARR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PY2QCHKI4LZGFWLRTPT3BD3ARR/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":"2026-05-23T21:47:47Z","links":{"resolver":"https://pith.science/pith/PY2QCHKI4LZGFWLRTPT3BD3ARR","bundle":"https://pith.science/pith/PY2QCHKI4LZGFWLRTPT3BD3ARR/bundle.json","state":"https://pith.science/pith/PY2QCHKI4LZGFWLRTPT3BD3ARR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PY2QCHKI4LZGFWLRTPT3BD3ARR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PY2QCHKI4LZGFWLRTPT3BD3ARR","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":"0ee7a489daedec9b4647f789e670475b6101dfe2370ccb8a9a088d1caf41e584","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2018-05-30T13:09:33Z","title_canon_sha256":"aab719d0bc36d64c091a12cf31d630bf2150d6459f48f8056a76f4d023394fc3"},"schema_version":"1.0","source":{"id":"1806.00528","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00528","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00528v1","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00528","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"pith_short_12","alias_value":"PY2QCHKI4LZG","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PY2QCHKI4LZGFWLR","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PY2QCHKI","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:5e3e4227b4de981a70433597ba5da9a23a47684163bf1a1def5b1597779487e2","target":"graph","created_at":"2026-05-18T00:14:03Z","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 this paper, we address the approximate minimization problem of Markov Chains (MCs) from a behavioral metric-based perspective. Specifically, given a finite MC and a positive integer k, we are looking for an MC with at most k states having minimal distance to the original. The metric considered in this work is the bisimilarity distance of Desharnais et al.. For this metric we show that (1) optimal approximations always exist; (2) the problem has a bilinear program characterization; and (3) prove that its threshold problem is in PSPACE and NP-hard. In addition to the bilinear program solution","authors_text":"Giorgio Bacci, Giovanni Bacci, Kim G. Larsen, Radu Mardare","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2018-05-30T13:09:33Z","title":"On the Metric-based Approximate Minimization of Markov Chains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00528","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:fb95044529b9e8ca6fb5fc6f4a27d11df46544961b2f3230a627324f9b31e4a8","target":"record","created_at":"2026-05-18T00:14:03Z","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":"0ee7a489daedec9b4647f789e670475b6101dfe2370ccb8a9a088d1caf41e584","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2018-05-30T13:09:33Z","title_canon_sha256":"aab719d0bc36d64c091a12cf31d630bf2150d6459f48f8056a76f4d023394fc3"},"schema_version":"1.0","source":{"id":"1806.00528","kind":"arxiv","version":1}},"canonical_sha256":"7e35011d48e2f262d9719be7b08f608c4bf25c4df0366d782571f16689807ba4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7e35011d48e2f262d9719be7b08f608c4bf25c4df0366d782571f16689807ba4","first_computed_at":"2026-05-18T00:14:03.717998Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:03.717998Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X2tNi9b3+hja0N3edYBvIhyZ31+4BvWfPSNQEQPNcGPhubbkxTAZAIE8lxVbHVXqlCMmZ+Nb/ohR8NbfWSBcDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:03.718708Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.00528","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb95044529b9e8ca6fb5fc6f4a27d11df46544961b2f3230a627324f9b31e4a8","sha256:5e3e4227b4de981a70433597ba5da9a23a47684163bf1a1def5b1597779487e2"],"state_sha256":"7260d1a032d8ca5881cb6138c9f1c3b7ef69ab0ff62a56b01062052d2bc3c2d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9y51BFAsEj1C6+Dgua4s1w+EwNFAsqzDmDujQSuzsgtF9Rkl4HKld14T0/K8FHdl84ftbqU6u0ps5SG0CtzPAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T21:47:47.492363Z","bundle_sha256":"ad7c99929a89aec419a5a36316952e890b7a17bbf656416bce67f8d37044d557"}}