{"id":"W1522592507","doi":"10.1109/sct.1991.160281","title":"Approximation and small depth Frege proofs","year":2002,"lang":"en","type":"article","venue":"","topic":"Logic, programming, and type systems","field":"Computer Science","cited_by":6,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Toronto","funders":"","keywords":"Pigeonhole principle; Mathematical proof; Proof complexity; Bounding overwatch; Mathematics; Discrete mathematics; Upper and lower bounds; Compact space; Inference; Combinatorics; Proof theory; Combinatorial proof; Function (biology); Rule of inference; Computer science; Pure mathematics; Artificial intelligence","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.005708681,0.001517267,0.001763627,0.002630129,0.001810898,0.003714784,0.003783645,0.002150761,0.009370685],"category_scores_gemma":[0.04229935,0.001935757,0.003011414,0.001888626,0.004608787,0.01784183,0.006806497,0.005930591,0.001599313],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.004593641,"about_ca_system_score_gemma":0.001794736,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.002730027,"about_ca_topic_score_gemma":0.003964885,"domain_scores_codex":[0.9916708,0.002938255,0.0003042989,0.001461032,0.002450927,0.001174565],"domain_scores_gemma":[0.9594424,0.02840606,0.001860608,0.008101783,0.001587367,0.000601782],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.000729871,0.0001451318,0.001422272,0.0004376685,0.0001736961,0.0002280814,0.0005935385,0.03914288,0.007456223,0.8669447,0.00688925,0.07583673],"study_design_scores_gemma":[0.00008439165,0.00009829117,0.0004495553,0.00006971377,0.00008806959,0.0002062848,0.00007144421,0.1088146,0.005733386,0.8713999,0.01293695,0.00004738048],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.04998806,0.00271198,0.9209487,0.002888201,0.0001619722,0.0001826574,0.0002548285,0.002267859,0.02059573],"genre_scores_gemma":[0.5071815,0.001409139,0.4751513,0.001211926,0.0003156622,0.0003314955,0.0004642988,0.0006269802,0.01330781],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.009370685,"threshold_uncertainty_score":0.03332937,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.04169662515343021,"score_gpt":0.2211742414229393,"score_spread":0.1794776162695091,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}