[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81947-en":3,"doc-seo-81947-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81947,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","A Lower Bound for Read-Once Parity Branching Programs","Proves an Ω~(n^2) lower bound for read-once parity branching programs computing an explicit Boolean function on n variables. Establishes a new best bound over the previously known Ω(n^1.5) result. The argument reduces the Boolean lower-bound task to an appropriate lower bound in algebraic circuit complexity, leveraging how a natural “algebrization” yields multilinear polynomials determined by agreement on the Boolean cube, enabling exact identification of the induced polynomial.","arXiv :2607 .05944v 1 [ cs .CC] 7 Jul 2026  \nA Lower Bound for Read-Once Parity Branching Programs  \nBen Lee Volk∗  \nAbstract  \nWe prove an ˜Ω(n2 ) lower bound for read-once parity branching program˜ s computing an  \nexplicit boolean function on n variables. The previous best lower bound was Ω(n1.5 ) . Our lower bound is proved by reducing the problem to a lower bound in algebraic circuit complexity.  \n1 Introduction  \nAlgebraic complexity is a beautiful and mathematically rich area that studies the complexity of symbolic computation of polynomials. Virtually all the known algorithms for algebraic problems (such as computing the determinant or permanent, multiplying matrices, or computing the discrete Fourier transform) are naturally modeled using algebraic models. One of its raisons d’ˆetre, however, is also the hope that lower bounds in the algebraic model will inspire lower boundsin the arguably more natural, and definitely more common, boolean models of computation. Along line of work on lower bounds for algebraic models has had numerous successes, such as, to give a non-exhaustive list, super-polynomial lower bounds for monotone circuits [JS82], noncommutative formulas [Nis91], multilinear formulas [Raz09, RY09], and bounded-depth circuits [LST25, For24]; and super-linear lower bounds for circuits [Str73, BS83], algebraic branching programs and formulas [CKSV22, Kal85] . More comprehensive surveys of lower bounds in algebraic complexity are [Sap15, SY10] .  \nThese lower bounds use the syntactic nature of the computation. For some, it is not clear what the analogous boolean model is, and for some the corresponding lower bounds for boolean models have been in fact known even earlier.  \nMotivated by considerations from proof complexity, there has been some work on functional lower bounds for algebraic circuits [GR00, FSTW21 , FKS16 , HLT24] . These are lower bounds for algebraic models that do not apply only to a single polynomial, but rather to a set of polynomials all computing the same function over some limited domain.  \nIn this paper, we give an instance in which one can prove a lower bound on a bona fide boolean model of computation by reducing to a lower bound on an algebraic model of computation. One of the main obstacles to obtaining lower bounds on boolean circuits using lower boundson algebraic circuits is that boolean circuits can exploit boolean identities that do not hold in the algebraic setting. One can trivially convert a boolean circuit C computing a function g : {0, 1}n → {0, 1} to an algebraic circuit over F2 gate-by-gate (say by replacing AND gates with multiplication gates and NOT gates with gates that add, modulo 2, the boolean value ‘1’), and the resulting algebraic circuit computes a polynomial that agrees with g on the boolean cube. But its specific form depends on the circuit C: as a trivial example, the boolean function g (x) = x is functionally identical to the function g (x) = x ∧ x, but the straightforward way alluded to above for converting a boolean circuit computing x ∧ x to a polynomial would result  \n∗ Efi Arazi School of Computer Science, Reichman University, Israel. Email: benleevolk@gmail .com. The research leading to these results has received funding from the Israel Science Foundation (grant number 843/23) .  \nin the polynomial x2 , which is distinct from the polynomial x. Therefore, a lower bound on algebraic circuits computing a specific polynomial doesn’t rule out the possibility that there’s a different efficient way to compute the same function over the boolean domain.  \nThe driving force behind our method is that some boolean models of computation yield multilinear polynomials when one applies the natural transformation that “algebrizes” them. Since two multilinear polynomials that agree on Fn2 are identical, we can deduce exactly which polynomial is obtained after this transformation, and prove (syntactic) lower bounds for this polynomial. The easy proofs for these observations","cbCaiamxJe0OUM24","https://ap.wps.com/l/cbCaiamxJe0OUM24","pdf",357748,5,1,14,"English","en",105,"# Introduction\n## Read-Once Parity Branching Programs","[{\"question\":\"What lower bound is proved for read-once parity branching programs?\",\"answer\":\"An Ω~(n^2) lower bound is proved for read-once parity branching programs computing a specific explicit Boolean function on n variables.\"},{\"question\":\"How does the paper improve on the previous best result?\",\"answer\":\"The previous best lower bound was Ω(n^1.5), and the new result strengthens it to Ω~(n^2).\"},{\"question\":\"What is the main proof strategy used to obtain the Boolean lower bound?\",\"answer\":\"The method reduces the problem to a lower bound in algebraic circuit complexity, using the fact that the algebrization of certain Boolean models produces multilinear polynomials whose identity is determined by agreement on Fn_2.\"}]","A Lower Bound for Read-Once Parity Branching Programs | PDF",1784177222,35,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"a-lower-bound-for-read-once-parity-branching-programs","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/a-lower-bound-for-read-once-parity-branching-programs/81947/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What lower bound is proved for read-once parity branching programs?","Question",{"text":77,"@type":78},"An Ω~(n^2) lower bound is proved for read-once parity branching programs computing a specific explicit Boolean function on n variables.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the paper improve on the previous best result?",{"text":82,"@type":78},"The previous best lower bound was Ω(n^1.5), and the new result strengthens it to Ω~(n^2).",{"name":84,"@type":75,"acceptedAnswer":85},"What is the main proof strategy used to obtain the Boolean lower bound?",{"text":86,"@type":78},"The method reduces the problem to a lower bound in algebraic circuit complexity, using the fact that the algebrization of certain Boolean models produces multilinear polynomials whose identity is determined by agreement on Fn_2.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":20,"slug":139},19,"General","general"]