{"paper":{"title":"A Circuit Complexity Trichotomy For Regular Languages","arxiv_id":"2611.03615","primary_cat":"cs.CC","submitted_at":"2026-11-02T18:50:25+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":0.86,"id":"s0","own":true,"role":"result","text":"We establish a trichotomy for the non-uniform, unbounded-fan-in circuit complexity of regular languages: the complexity of every regular language lies in exactly one of $O(1)$, $\\Omega(\\log n)\\cap O(\\log n\\log\\log n)$, or $\\Omega(n)\\cap O(n)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.93,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.86,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":true,"role":"contribution","text":"Our main technical contribution is a characterization of the sublinear class in terms of algebra, word congruences, regular expressions, and logic.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":2,"p":0.92,"id":"s2","own":true,"role":"result","text":"This class consists precisely of the regular languages computable by constant-depth circuits of size $O(\\log^2 n)$, or, equivalently, by circuits of size $O(\\log n\\log\\log n)$ and depth $O((\\log\\log n)^2)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.84,"main_claim_p":0.1,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.92,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.01}},{"i":3,"p":0.52,"id":"s3","own":true,"role":"result","text":"Consequently, every regular language of sublinear circuit complexity is definable in $\\mathrm{FO}^2[\\Sigma,<,\\mathrm{mod}]$, which is contained in wire-linear $\\mathbf{AC}^0$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.81,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.63,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.52,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.48}},{"i":4,"p":0.98,"id":"s4","own":true,"role":"result","text":"For each of the three classes, deciding membership is $\\mathbf{PSPACE}$-complete when the language is specified by a nondeterministic finite automaton.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.82,"main_claim_p":0.07,"quantitative":true,"quantitative_p":0.52,"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":5,"p":0.74,"id":"s5","own":true,"role":"result","text":"We also obtain a corresponding classification for regular languages with a neutral 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