[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81910-en":3,"doc-seo-81910-105":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},81910,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Subcube Stifling","Introduces the subcube stifling number, a combinatorial measure for total Boolean functions that captures the ability to isolate any prescribed point on a small Boolean cube by fixing all remaining coordinates. The definition generalizes the stifling number of prior work by requiring point indicators rather than constant functions. Results include an approximate-degree composition theorem, a probabilistic bound showing random functions have Θ(log n) subcube stifling number with high probability, and explicit linear-code-based examples.","arXiv :2607 .04850v 1 [ cs .CC] 6 Jul 2026  \nSubcube Stifling  \nArjan Cornelissen∗ Nikhil S. Mande† Nithish Raja‡  \nAbstract  \nWe introduce the subcube stifling number, a new combinatorial measure of total Boolean functions. This measure is the largest integer k such that, for every set S of at most k input variables and every assignment b ∈ {0, 1}S , there is a fixing of the variables outside S under which the resulting function on the free variables S is the point indicator I [xS = b] . Equivalently, for every small set of coordinates, the function can isolate any prescribed point of the corresponding Boolean cube by suitably fixing all remaining coordinates. This measure is inspired by the stifling number of Chattopadhyay et al. (ITCS’23); whereas their measure asks for restrictions realizing every constant function, ours asks for restrictions realizing every point indicator. Our results are as follows.  \n• We show that the subcube stifling number gives rise to an approximate-degree composition theorem. Ign, parapptricouxilamra,tif ae deBoogreelean funcompocsteion fs tighstlaty:isfies gdeg(f) = O( p µ(f)), then for every Boolean function  \ngdeg(f ◦ g) = Θ(gdeg(f)gdeg(g)) .  \nThis motivates the study of the subcube stifling number, and in particular the search for functions whose approximate degree is O( p µ(f)) .  \n• We show using a standard probabilistic argument that a random Boolean function on n input bits has subcube stifling number Θ(log n) with high probability.  \n• Chattopadhyay et al. showed that the Majority function has linear stifling number, and no Boolean function has larger stifling number. In contrast, the subcube stifling number of Majority is easily seen to be 0 . This raises the question whether there even exist Boolean functions with linear subcube stifling number.  \nWe show that this is indeed the case. Our examples are obtained from indicators of linear codes over F2 whose minimum distance and dual distance are both linear.  \n• We prove that the functions arising from this linear-code construction do not have approximate degree O( p µ(f)); in fact, they have approximate degree Ω(µ(f)) .  \nA pTheositimainve ansquewerstionwoulledftyiopeneld neisw iwhetnstahnecresthofere extightistapsparoBxiooleamate-ndfunegrcetieocnof wmpositithiogdegn wi(fth) aΘst( pheµo(fut)e)r. function.  \n∗ Simons Institute for the Theory of Computing, University of California, Berkeley, United States of America [ajcornelissen@outlook.com](ajcornelissen@outlook.com)  \n†University of Liverpool, [UK](UK mande@liverpool.ac.uk)[ mande@liverpool.ac.uk](UK mande@liverpool.ac.uk)  \n‡Eindhoven University of Technology, [Netherlands](Netherlands n.r.raja@tue.nl)[ n.r.raja@tue.nl](Netherlands n.r.raja@tue.nl)  \n1 Introduction  \nWe define a combinatorial measure called the subcube stifling number of a total Boolean function f : {0, 1}n →{0, 1}, which for simplicity we denote by µ (f) throughout this paper. Informally µ (f) is the largest value of k such that for all sets S of size at most k and all choices of b ∈ {0, 1}k , there is a way to set the variables outside S such that the restricted function equals the point function I [y = b] . Chattopadhyay et al. [CMSS23] defined a related notion called the stifling number of a Boolean function: it is the largest k such that for every subset S of at most k variables and every b ∈ {0, 1}, there exists a setting of the variables outside S such that the restricted function equals the constant b. Our measure, the subcube stifling number, differs from the stifling number in that it requires the restricted function to behave like an indicator function instead of a constant. We refer the reader to Section 2 for formal definitions of subcube stifling number and stifling number.  \nChattopadhyay et al. [CMSS23] showed a lifting theorem involving stifling number: if the stifling number of a gadget g is large, then the decision tree complexity of f lifts to parity decision tree complexity of f ◦g. 1  \nTheorem ","cbCaibYEMwkkFTZ9","https://ap.wps.com/l/cbCaibYEMwkkFTZ9","pdf",606340,5,1,18,"English","en",105,"# Abstract\n# Introduction\n## Subcube stifling number\n## Stifling number and lifting theorems\n## Approximate degree and composition","[{\"question\":\"What is the subcube stifling number of a total Boolean function?\",\"answer\":\"It is the largest k such that for any set of at most k input variables and any target b on those variables, one can fix all other variables so the restricted function on the remaining variables becomes the point indicator I[xS=b].\"},{\"question\":\"How does the subcube stifling number relate to approximate degree and composition?\",\"answer\":\"The results show that the subcube stifling number leads to an approximate-degree composition theorem, yielding tight scaling for gdeg(f◦g) in terms of gdeg(f) and gdeg(g).\"},{\"question\":\"What do the paper’s results say about random Boolean functions and the Majority function?\",\"answer\":\"A standard probabilistic argument shows random n-bit Boolean functions have subcube stifling number Θ(log n) with high probability. In contrast, Majority has subcube stifling number 0, prompting the construction of functions with linear subcube stifling number via linear codes.\"}]","Subcube Stifling | PDF",1784177008,45,{"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},"subcube-stifling","",{"@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/subcube-stifling/81910/",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 is the subcube stifling number of a total Boolean function?","Question",{"text":77,"@type":78},"It is the largest k such that for any set of at most k input variables and any target b on those variables, one can fix all other variables so the restricted function on the remaining variables becomes the point indicator I[xS=b].","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the subcube stifling number relate to approximate degree and composition?",{"text":82,"@type":78},"The results show that the subcube stifling number leads to an approximate-degree composition theorem, yielding tight scaling for gdeg(f◦g) in terms of gdeg(f) and gdeg(g).",{"name":84,"@type":75,"acceptedAnswer":85},"What do the paper’s results say about random Boolean functions and the Majority function?",{"text":86,"@type":78},"A standard probabilistic argument shows random n-bit Boolean functions have subcube stifling number Θ(log n) with high probability. In contrast, Majority has subcube stifling number 0, prompting the construction of functions with linear subcube stifling number via linear codes.","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"]