{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T05:10:19Z","timestamp":1648876219288},"reference-count":18,"publisher":"Association for Computing Machinery (ACM)","issue":"1","funder":[{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"crossref","award":["EP\/S03238X\/1"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ACM Trans. Comput. Logic"],"published-print":{"date-parts":[[2022,1,31]]},"abstract":"Fixed-point logic with rank (FPR) is an extension of fixed-point logic with counting (FPC) with operators for computing the rank of a matrix over a finit field. The expressive power of FPR properly extends that of FPC and is contained in P, but it is not known if that containment is proper. We give a circuit characterization for FPR in terms of families of symmetric circuits with rank gates, along the lines of that for FPC given by Anderson and Dawar in 2017. This requires the development of a broad framework of circuits in which the individual gates compute functions that are not symmetric (i.e., invariant under all permutations of their inputs). This framework also necessitates the development of novel techniques to prove the equivalence of circuits and logic. Both the framework and the techniques are of greater generality than the main result.<\/jats:p>","DOI":"10.1145\/3476227","type":"journal-article","created":{"date-parts":[[2021,11,22]],"date-time":"2021-11-22T17:43:55Z","timestamp":1637603035000},"page":"1-35","source":"Crossref","is-referenced-by-count":0,"title":["Symmetric Circuits for Rank Logic"],"prefix":"10.1145","volume":"23","author":[{"given":"Anuj","family":"Dawar","sequence":"first","affiliation":[{"name":"University of Cambridge, Cambridge, United Kingdom"}]},{"given":"Gregory","family":"Wilsenach","sequence":"additional","affiliation":[{"name":"University of Cambridge, Cambridge, United Kingdom"}]}],"member":"320","reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00224-016-9692-2"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(99)00005-6"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/2728816.2728820"},{"key":"e_1_3_2_5_2","doi-asserted-by":"crossref","first-page":"23","volume-title":"Topics in Theoretical Computer Science","author":"Dawar A.","year":"2016","DOI":"10.1007\/978-3-319-28678-5_2"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.1109\/LICS.2009.24"},{"key":"e_1_3_2_7_2","doi-asserted-by":"crossref","first-page":"251","volume-title":"Automata, Languages, and Programming","author":"Dawar A.","year":"2012","DOI":"10.1007\/978-3-642-31585-5_25"},{"key":"e_1_3_2_8_2","first-page":"Article 20, 16","volume-title":"Proceedings of the 27th EACSL Annual Conference on Computer Science Logic (CSL\u201918)","author":"Dawar A.","year":"2018"},{"key":"e_1_3_2_9_2","first-page":"Article 36, 18","volume-title":"Proceedings of the 27th International Colloquium on Automata, Languages, and Programming (ICALP\u201920)","author":"Dawar A.","year":"2020"},{"issue":"2","key":"e_1_3_2_10_2","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/S0019-9958(86)80006-7","article-title":"Definability by constant-depth polynomial-size circuits","volume":"70","author":"Denenberg L.","year":"1986","journal-title":"Information and Control"},{"key":"e_1_3_2_11_2","doi-asserted-by":"crossref","volume-title":"Permutation Groups","author":"Dixon J. 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