{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:T2K6Y5DUO2SKR7J6OJNFAQ3NUP","short_pith_number":"pith:T2K6Y5DU","canonical_record":{"source":{"id":"1312.0355","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-12-02T07:10:02Z","cross_cats_sorted":[],"title_canon_sha256":"c4aa51dc37c73975651ebf6c2c666ca298a55f3c6724640f64d76207aea739de","abstract_canon_sha256":"59321eb2fc6675df86252e6d7cbe4eb423f2ebf4c3b0a8eceb9395ac1072dfac"},"schema_version":"1.0"},"canonical_sha256":"9e95ec747476a4a8fd3e725a50436da3dfab4a98256edd71d9a48e5925c5fbe9","source":{"kind":"arxiv","id":"1312.0355","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.0355","created_at":"2026-05-18T03:05:45Z"},{"alias_kind":"arxiv_version","alias_value":"1312.0355v1","created_at":"2026-05-18T03:05:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.0355","created_at":"2026-05-18T03:05:45Z"},{"alias_kind":"pith_short_12","alias_value":"T2K6Y5DUO2SK","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"T2K6Y5DUO2SKR7J6","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"T2K6Y5DU","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:T2K6Y5DUO2SKR7J6OJNFAQ3NUP","target":"record","payload":{"canonical_record":{"source":{"id":"1312.0355","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-12-02T07:10:02Z","cross_cats_sorted":[],"title_canon_sha256":"c4aa51dc37c73975651ebf6c2c666ca298a55f3c6724640f64d76207aea739de","abstract_canon_sha256":"59321eb2fc6675df86252e6d7cbe4eb423f2ebf4c3b0a8eceb9395ac1072dfac"},"schema_version":"1.0"},"canonical_sha256":"9e95ec747476a4a8fd3e725a50436da3dfab4a98256edd71d9a48e5925c5fbe9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:45.897248Z","signature_b64":"CZogKVxl0c2YPkvpSmuwum9P+8M+yysSZ5HAQKkXOfddEO3VG/9HfQsVi35m8Kg6OBgOdh886Si4brUaGU14Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e95ec747476a4a8fd3e725a50436da3dfab4a98256edd71d9a48e5925c5fbe9","last_reissued_at":"2026-05-18T03:05:45.896762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:45.896762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.0355","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":"2026-05-18T03:05:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ocT5E54jZrHytwmRumX6SpUMyKyyLJxcKt4GuSiHc2Y9jccLiZBWmIXdKbowoZXbJjIBrqvFumdt6huDApwKDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:58:40.172851Z"},"content_sha256":"4f91387426cf895cb4b48a7b06fad3febf135ebd52fa35f83672a4c469c50e42","schema_version":"1.0","event_id":"sha256:4f91387426cf895cb4b48a7b06fad3febf135ebd52fa35f83672a4c469c50e42"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:T2K6Y5DUO2SKR7J6OJNFAQ3NUP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Formulas vs. Circuits for Small Distance Connectivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Benjamin Rossman","submitted_at":"2013-12-02T07:10:02Z","abstract_excerpt":"We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are connected by a path of length at most $k(n)$. This problem is solvable (by the recursive doubling technique) on {\\bf circuits} of depth $O(\\log k)$ and size $O(kn^3)$. In contrast, we show that solving this problem on {\\bf formulas} of depth $\\log n/(\\log\\log n)^{O(1)}$ requires size $n^{\\Omega(\\log k)}$ for all $k(n) \\leq \\log\\log n$. As corollaries:\n  (i)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0355","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":"2026-05-18T03:05:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OUhWt+11wpJ0KU1gtrozPJLKBExr1mRAnZlCaAh61sydOLsQ/U6qO5mAUZjLf4Es3esigsq1hrBKxjVlZS75AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:58:40.173613Z"},"content_sha256":"4de4cd73fc4eca2766662518fdf36361c7aa89ae9df4953e7119febaa40d4967","schema_version":"1.0","event_id":"sha256:4de4cd73fc4eca2766662518fdf36361c7aa89ae9df4953e7119febaa40d4967"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T2K6Y5DUO2SKR7J6OJNFAQ3NUP/bundle.json","state_url":"https://pith.science/pith/T2K6Y5DUO2SKR7J6OJNFAQ3NUP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T2K6Y5DUO2SKR7J6OJNFAQ3NUP/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":"2026-05-25T15:58:40Z","links":{"resolver":"https://pith.science/pith/T2K6Y5DUO2SKR7J6OJNFAQ3NUP","bundle":"https://pith.science/pith/T2K6Y5DUO2SKR7J6OJNFAQ3NUP/bundle.json","state":"https://pith.science/pith/T2K6Y5DUO2SKR7J6OJNFAQ3NUP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T2K6Y5DUO2SKR7J6OJNFAQ3NUP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:T2K6Y5DUO2SKR7J6OJNFAQ3NUP","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":"59321eb2fc6675df86252e6d7cbe4eb423f2ebf4c3b0a8eceb9395ac1072dfac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-12-02T07:10:02Z","title_canon_sha256":"c4aa51dc37c73975651ebf6c2c666ca298a55f3c6724640f64d76207aea739de"},"schema_version":"1.0","source":{"id":"1312.0355","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.0355","created_at":"2026-05-18T03:05:45Z"},{"alias_kind":"arxiv_version","alias_value":"1312.0355v1","created_at":"2026-05-18T03:05:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.0355","created_at":"2026-05-18T03:05:45Z"},{"alias_kind":"pith_short_12","alias_value":"T2K6Y5DUO2SK","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"T2K6Y5DUO2SKR7J6","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"T2K6Y5DU","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:4de4cd73fc4eca2766662518fdf36361c7aa89ae9df4953e7119febaa40d4967","target":"graph","created_at":"2026-05-18T03:05:45Z","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 give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are connected by a path of length at most $k(n)$. This problem is solvable (by the recursive doubling technique) on {\\bf circuits} of depth $O(\\log k)$ and size $O(kn^3)$. In contrast, we show that solving this problem on {\\bf formulas} of depth $\\log n/(\\log\\log n)^{O(1)}$ requires size $n^{\\Omega(\\log k)}$ for all $k(n) \\leq \\log\\log n$. As corollaries:\n  (i)","authors_text":"Benjamin Rossman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-12-02T07:10:02Z","title":"Formulas vs. Circuits for Small Distance Connectivity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0355","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":"sha256:4f91387426cf895cb4b48a7b06fad3febf135ebd52fa35f83672a4c469c50e42","target":"record","created_at":"2026-05-18T03:05:45Z","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":"59321eb2fc6675df86252e6d7cbe4eb423f2ebf4c3b0a8eceb9395ac1072dfac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-12-02T07:10:02Z","title_canon_sha256":"c4aa51dc37c73975651ebf6c2c666ca298a55f3c6724640f64d76207aea739de"},"schema_version":"1.0","source":{"id":"1312.0355","kind":"arxiv","version":1}},"canonical_sha256":"9e95ec747476a4a8fd3e725a50436da3dfab4a98256edd71d9a48e5925c5fbe9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9e95ec747476a4a8fd3e725a50436da3dfab4a98256edd71d9a48e5925c5fbe9","first_computed_at":"2026-05-18T03:05:45.896762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:45.896762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CZogKVxl0c2YPkvpSmuwum9P+8M+yysSZ5HAQKkXOfddEO3VG/9HfQsVi35m8Kg6OBgOdh886Si4brUaGU14Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:45.897248Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.0355","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f91387426cf895cb4b48a7b06fad3febf135ebd52fa35f83672a4c469c50e42","sha256:4de4cd73fc4eca2766662518fdf36361c7aa89ae9df4953e7119febaa40d4967"],"state_sha256":"83d902c0b756bdc92ffba509a9873482e7fec8b64de7f84005d84328e6b3397c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H2h2q7PA25+U3EzmDCfuYs1jnulJWF5rxBsQI3V7b6Lb2s31q6scKQYqJy/5o5B7Wm9ckWACwrBkRiDIxcSuAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T15:58:40.177371Z","bundle_sha256":"dfac1d24b3be7af6c5f8ee9d32d335c8f3a5092aa5e0168144af3f4ef9ce14c7"}}