{"paper":{"title":"Rational Reductions and Regular Languages of Constant Circuit Complexity","arxiv_id":"2609.18484","primary_cat":"cs.CC","submitted_at":"2026-09-16T11:14:35+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":0.56,"id":"s0","own":true,"role":"contribution","text":"We study the circuit complexity of regular languages in terms of unbounded fan-in Boolean circuit families.","hedge":{"p":0.84,"label":"asserted"},"own_p":0.77,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.44,"limitation":0.0,"future_work":0.0,"contribution":0.56,"interpretation":0.0}},{"i":1,"p":0.74,"id":"s1","own":true,"role":"result","text":"We characterize the regular languages of constant circuit complexity in terms of the one-variable fragment of first-order logic with regular predicates, in terms of the pseudovariety of stamps $\\mathbf{QEJ}_\\mathbf{1}$, suitable word congruences and regular expressions.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.53,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.74,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.26,"interpretation":0.0}},{"i":2,"p":0.6,"id":"s2","own":true,"role":"contribution","text":"We analogously characterize the neutral letter regular languages of constant circuit complexity.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.94,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.4,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.6,"interpretation":0.0}},{"i":3,"p":0.58,"id":"s3","own":true,"role":"result","text":"Our lower bound result implies that the class of regular languages of sublogarithmic circuit complexity coincides with the one of constant circuit complexity.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.91,"main_claim_p":0.25,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.58,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.42}},{"i":4,"p":0.97,"id":"s4","own":true,"role":"result","text":"In addition we show that deciding whether a regular language, given as a nondeterministic finite automaton, has constant circuit complexity is $\\mathbf{PSPACE}$-complete.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.1,"quantitative":true,"quantitative_p":0.75,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.97,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.03,"interpretation":0.0}},{"i":5,"p":0.86,"id":"s5","own":true,"role":"contribution","text":"We introduce a strong notion of reduction, called rational truth-table reduction, that is tailored towards algebraically defined classes of languages.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.95,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.14,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.86,"interpretation":0.0}},{"i":6,"p":0.93,"id":"s6","own":true,"role":"result","text":"We show that, for a class of functions we call mild, rational truth-table reductions preserve both upper and lower bounds on circuit complexity.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.39,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":7,"p":0.98,"id":"s7","own":true,"role":"result","text":"We show that the class of regular languages, whose circuit complexity is bounded by a mild function, is in fact a length-multiplying variety of languages.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.15,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.01}},{"i":8,"p":0.64,"id":"s8","own":true,"role":"contribution","text":"Slightly extending the class of regular languages of constant circuit complexity, we analogously characterize the class of regular languages that are in the pseudovariety $\\mathbf{QEACom}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.36,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.64,"interpretation":0.0}},{"i":9,"p":1.0,"id":"s9","own":true,"role":"result","text":"For these we derive logarithmic circuit complexity upper 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