{"paper":{"title":"On the rational subset problem for groups","arxiv_id":"math/0602454","primary_cat":"math.GR","submitted_at":"2006-02-21T12:51:24+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.59,"id":"s0","own":true,"role":"contribution","text":"We use language theory to study the rational subset problem for groups and monoids.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.88,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.02,"role_probabilities":{"gap":0.0,"method":0.31,"result":0.0,"background":0.1,"limitation":0.0,"future_work":0.0,"contribution":0.59,"interpretation":0.0}},{"i":1,"p":0.98,"id":"s1","own":true,"role":"result","text":"We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.38,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":2,"p":0.7,"id":"s2","own":true,"role":"result","text":"In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.71,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.7,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.29}},{"i":3,"p":0.51,"id":"s3","own":true,"role":"result","text":"We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.83,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.51,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.49,"interpretation":0.0}},{"i":4,"p":0.99,"id":"s4","own":true,"role":"result","text":"We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for G-automata subsets of M.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.2,"quantitative":false,"quantitative_p":0.18,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":5,"p":0.65,"id":"s5","own":true,"role":"result","text":"It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.16,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.65,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.35}}]},"compiled":{"main_claim":{"p":0.73,"text":"We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups.","location":"abstract s1","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":4864,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.03,"states_limitation":0.14,"human_participants":0.01,"controlled_comparison":0.01,"randomized_assignment":0.01,"states_quantitative_result":0.04,"compares_with_prior_results":0.65,"mentions_code_or_data_availability":0.01},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.69,"text":"We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups.","location":"abstract s1"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":6,"sentences_in_abstract":6},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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