{"paper":{"title":"A foundational characterization of Hoare Logic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A partial-correctness assertion for an iterative program is provable in Hoare logic if and only if it is provable in second-order logic with first-order comprehension.","cross_cats":["math.LO"],"primary_cat":"cs.LO","authors_text":"Daniel Leivant","submitted_at":"2026-05-13T17:51:13Z","abstract_excerpt":"We show that a partial-correctness assertion about an iterative program is provable in Hoare Logic iffit is provable in standard second-order logic with comprehension restricted to first-order predicates. This equivalence was claimed twice in the past, both with faulty proofs, and seems to be the first foundational characterization of Hoare Logic."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"a partial-correctness assertion about an iterative program is provable in Hoare Logic iff it is provable in standard second-order logic with comprehension restricted to first-order predicates.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The equivalence depends on the precise syntactic definitions of Hoare logic for iterative programs and on the exact restriction of comprehension to first-order predicates; any deviation in these definitions would invalidate the claimed if-and-only-if.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Partial-correctness assertions for iterative programs are provable in Hoare logic if and only if they are provable in second-order logic with comprehension restricted to first-order predicates.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A partial-correctness assertion for an iterative program is provable in Hoare logic if and only if it is provable in second-order logic with first-order comprehension.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f8b8b95a3f86985cd91af61bb7ec6a69306f1c3314b49883aa9e77d87b080917"},"source":{"id":"2605.13944","kind":"arxiv","version":1},"verdict":{"id":"333e2f73-08d5-4c69-9fb5-09c2859dc221","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:40:04.722172Z","strongest_claim":"a partial-correctness assertion about an iterative program is provable in Hoare Logic iff it is provable in standard second-order logic with comprehension restricted to first-order predicates.","one_line_summary":"Partial-correctness assertions for iterative programs are provable in Hoare logic if and only if they are provable in second-order logic with comprehension restricted to first-order predicates.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The equivalence depends on the precise syntactic definitions of Hoare logic for iterative programs and on the exact restriction of comprehension to first-order predicates; any deviation in these definitions would invalidate the claimed if-and-only-if.","pith_extraction_headline":"A partial-correctness assertion for an iterative program is provable in Hoare logic if and only if it is provable in second-order logic with first-order comprehension."},"references":{"count":216,"sample":[{"doi":"","year":null,"title":"Stephen Simpson , title=","work_id":"32922cba-c5ba-4574-ba32-f086f91d1260","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Dynamic Logic","work_id":"d1b04a75-ee0a-4c7a-9f41-df293f3d4d53","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Arithmetices principia, novo methodo exposita","work_id":"8ddfcebb-6139-4a23-84e4-cd245e902779","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1931,"title":"van Heijenoort","work_id":"62044c68-d023-4fd4-a37e-c44d97e8fa21","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Ramified recurrence and computational complexity I : Word recurrence and poly-time","work_id":"28fdb9a5-5620-4806-ad21-9d2f07475645","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":216,"snapshot_sha256":"d31674114d4b670adbbbee16a244b35e2afe7b61fac5235f0f18cde26f41fbbc","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"9fc7cfa8567581c8579cfc969c4d12bcee38cb2177ed73c6ffd95779e0313518"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}