{"paper":{"title":"On the Size Complexity and Decidability of First-Order Progression","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"First-order progression for local-effect, normal, and acyclic actions grows only polynomially under reasonable assumptions.","cross_cats":[],"primary_cat":"cs.AI","authors_text":"Daxin Liu, Jens Classen","submitted_at":"2026-05-12T19:40:45Z","abstract_excerpt":"Progression, the task of updating a knowledge base to reflect action effects, generally requires second-order logic. Identifying first-order special cases, by restricting either the knowledge base or action effects, has long been a central topic in reasoning about actions. It is known that local-effect, normal, and acyclic actions, three increasingly expressive classes, admit first-order progression. However, a systematic analysis of the size of such progressions, crucial for practical applications, has been missing. In this paper, using the framework of Situation Calculus, we show that under "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"under reasonable assumptions, first-order progression for these action classes grows only polynomially. Moreover, we show that when the KB belongs to decidable fragments such as two-variable first-order logic or universal theories with constants, the progression remains within the same fragment, ensuring decidability and practical applicability.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The unspecified 'reasonable assumptions' on the knowledge base and action effects that enable the polynomial size bound; these restrictions to local-effect, normal, or acyclic actions are load-bearing but their precise scope is not detailed in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"First-order progression for local-effect, normal, and acyclic actions is polynomial-sized and preserves decidable fragments such as two-variable logic.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"First-order progression for local-effect, normal, and acyclic actions grows only polynomially under reasonable assumptions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8e1b0d9e323869e6d4428ad78698ea5a747e81a22e7ad506df511b06d08dca6a"},"source":{"id":"2605.12691","kind":"arxiv","version":1},"verdict":{"id":"77a7df88-6df5-4ade-8d40-5851337a5bc9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:13:17.308661Z","strongest_claim":"under reasonable assumptions, first-order progression for these action classes grows only polynomially. Moreover, we show that when the KB belongs to decidable fragments such as two-variable first-order logic or universal theories with constants, the progression remains within the same fragment, ensuring decidability and practical applicability.","one_line_summary":"First-order progression for local-effect, normal, and acyclic actions is polynomial-sized and preserves decidable fragments such as two-variable logic.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The unspecified 'reasonable assumptions' on the knowledge base and action effects that enable the polynomial size bound; these restrictions to local-effect, normal, or acyclic actions are load-bearing but their precise scope is not detailed in the abstract.","pith_extraction_headline":"First-order progression for local-effect, normal, and acyclic actions grows only polynomially under reasonable assumptions."},"references":{"count":24,"sample":[{"doi":"","year":1935,"title":"Untersuchungen ¨uber das Eliminationsproblem der mathematischen Logik","work_id":"df0567b8-d14e-4ce8-bbd1-8ca296bcecef","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"[Arenaset al., 2018 ] Marcelo Arenas, Jorge A. Baier, Juan S. Navarro, and Sebastian Sardi ˜na. On the progres- sion of situation calculus universal theories with constants. In Michael Thielscher, Fra","work_id":"42dba9b1-2e28-4576-807b-77384465f412","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1928,"title":"Zum Entscheidungsproblem der mathematis- chen Logik.Mathematische Annalen, 99:342–372,","work_id":"66f2b18c-5aa0-49ea-81f3-818364e2d884","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"On the undecidability of the situation calculus extended with description logic ontologies","work_id":"2126fec4-825c-4f53-bc2a-1962ce822db0","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Corr ˆea and Giuseppe De Giacomo","work_id":"e788ced6-3af2-483b-9271-23b439ca2b21","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":24,"snapshot_sha256":"b10eb8c4135047ff7333d8ffe64ff66c256297e94a2ef83ff80f25c3f710ee1d","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"}