{"paper":{"title":"Rational Identity Testing for Noncommutative Circuits is in Polynomial Space","arxiv_id":"2609.37319","primary_cat":"cs.CC","submitted_at":"2026-09-29T11:57:48+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":1.0,"id":"s0","own":false,"role":"background","text":"The rational identity testing problem, RIT, asks whether an input \\emph{rational circuit}, computes the zero element of the free skew field.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.06,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"For rational \\emph{formulas} the problem is known to be in deterministic polynomial time.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.01,"quantitative":true,"quantitative_p":0.64,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.81,"id":"s2","own":false,"role":"background","text":"For rational circuits of size $s$ the associated linear pencil has dimension $2^{O(s)}$, and the known algorithms use exponential time and exponential space.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.07,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.18,"method":0.0,"result":0.0,"background":0.81,"limitation":0.01,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.88,"id":"s3","own":true,"role":"result","text":"We show that RIT for rational circuits over $Q$ and finite fields is in PSPACE.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.92,"quantitative":true,"quantitative_p":0.8,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.88,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.12,"interpretation":0.0}},{"i":4,"p":0.51,"id":"s4","own":true,"role":"interpretation","text":"As a consequence, noncommutative PIT for degree unrestricted noncommutative circuits is also in PSPACE.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.82,"main_claim_p":0.2,"quantitative":true,"quantitative_p":0.66,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.49,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.51}},{"i":5,"p":0.8,"id":"s5","own":true,"role":"method","text":"The proof is based on the following three observations Analyzing the Hrubes-Wigderson construction of a linear pencil for an input rational formula, we obtain a succinctly represented linear pencil $A$ of size $2^{O(s)}$ for the input rational \\emph{circuit} of size $s$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.78,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.87,"role_probabilities":{"gap":0.0,"method":0.8,"result":0.02,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.18,"interpretation":0.0}},{"i":6,"p":0.78,"id":"s6","own":true,"role":"method","text":"More precisely, given indices $i$ and $j$ for the linear pencil we can compute $A_{i,j}$ in space polynomial in $s$ and length of $i,j$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.84,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.87,"role_probabilities":{"gap":0.0,"method":0.78,"result":0.08,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":7,"p":0.48,"id":"s7","own":true,"role":"interpretation","text":"This algorithm essentially gives a succinct presentation of the linear pencil $A$ obtained by the reduction.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.82,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.22,"result":0.2,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.1,"interpretation":0.48}},{"i":8,"p":0.67,"id":"s8","own":false,"role":"background","text":"By the recent theorem of Chatterjee, Ghosh, Gurjar, Raj and Thierauf that deciding whether a symbolic matrix $\\sum_i A_ix_i$ has full noncommutative rank is in NC.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.38,"role_probabilities":{"gap":0.0,"method":0.32,"result":0.0,"background":0.67,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":9,"p":0.95,"id":"s9","own":true,"role":"method","text":"We note that their algorithm is logspace-uniform NC and we use it as a black-box for computing rank of succinctly represented pencil $A$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.54,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.22,"role_probabilities":{"gap":0.0,"method":0.95,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}},{"i":10,"p":0.53,"id":"s10","own":false,"role":"method","text":"Finally, we note a folklore simulation: If a problem is solved by a logspace-uniform family of \\emph{deterministic} Boolean circuits of polylogarithmic depth, and its input is not written down but is presented by a polylogspace subroutine that returns any requested input bit, then the circuit can be evaluated in polylogarithmic space.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.16,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.89,"role_probabilities":{"gap":0.0,"method":0.53,"result":0.01,"background":0.44,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.01}},{"i":11,"p":0.4,"id":"s11","own":false,"role":"interpretation","text":"Hence, simulating a depth $O(\\log^{i}M)$ circuit for succinctly presented inputs of length $M = 2^{\\Theta(s)}$ yields a polynomial space algorithm.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.45,"main_claim_p":0.12,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.33,"result":0.25,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.4}}]},"compiled":{"main_claim":{"p":0.99,"text":"We show that RIT for rational circuits over $Q$ and finite fields is in PSPACE.","location":"abstract 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