[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81951-en":3,"doc-seo-81951-105":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},81951,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Generalisation of Baker’s Forcing Method to Arbitrary Prime and NP-hardness of Several p-adic Optimisations","G. D. Baker’s forcing method translates integer optimisation into 2-adic linear regression and establishes NP-hardness for that regression. The paper generalises the forcing technique to a broader class of p-adic optimisation when p is any prime, proving NP-hardness of p-adic linear regression. It further derives NP-hardness results for 2-adic dynamic neural networks and for a partial generalisation tied to van der Put neural networks, termed p-adic smooth neural networks.","arXiv :2607 .06092v 1 [ cs .CC] 7 Jul 2026  \nGeneralisation of Baker’s Forcing Method to Arbitrary Prime and NP-hardness of Several p-adic Optimisations  \nTomoki Mihara  \nAbstract  \nG. D. Baker formulated a forcing method to interpret integer optimisation problem into 2-adic linear regression, and proved the NP-hardness of 2-adic linear regression. We generalise the forcing method to a wider class of p-adic optimisation for the case where p is not necessarily 2, and prove the NP-hardness of p-adic linear regression, the NP-hardness of 2-adic dynamic neural network by S. Albeverio, A. Khrennikov, and B. Tirrozi, and the NP-hardness of a partial generalisation of thep-adic optimisation problem associated to van der Put neural network by G. L. R. N’guessan.  \nContents  \n0 Introduction 1  \n1 Convention 3  \n2 p-adic Optimisation 4  \n3 Forcing Method 5  \n4 NP-hardness 7  \nAcknowledgements 12  \nReferences 12  \n0 Introduction  \nLet p be a prime number. The notion of p-adic numbers is introduced by K. Henselin 1897 in [Hen97], and plays a central role in modern number theory. Recently, the  \np-adic numbers are used also in other branches of science, because of many significant similarities to and differences from the real numbers. See Introductions of [Bra25] and [Mih26-1] for history and basic knowledge respectively on applications of p-adic numbers in computer science.  \nS. Albeverio, A. Khrennikov, and B. Tirrozi formulated and studied 2-adic neural network called 2-adic dynamic neural network in [AKT99] and [KT00] . Starting from these innovative studies, people proposed and studied various optimisation problems related to p-adic numbers: p-adic cellular neural network by W. A. Zúñiga-Galindo, B. A. Zambrano-Luna, and B. Dibba in [ZZ23] and [ZZB24], an alternative of p-adic neural network by an explicit choice of a homeomorphism in Brouwer’s characterisation theorem (cf. [Bro10]) by A. P. Zubarev in [Zub25-1] and [Zub25-2], van der Put neural network by G. L. R. N’guessan in [Ngu25], and p-adic character neural network by us in [Mih26-3] .  \nSince studies of p-adic optimisation just have a short history compared to those of real optimisation, not so many were known for p-adic optimisation before 2020 . For example, E. Amaldi and V. Kann showed that the maximal feasible subsystem problem of linear equations over the finite field Fp is APX-complete, i.e. complete for the class of optimisation problems which allow constant-factor approximations, in [AK95] Proposition A.1. We note that the optimisation problem of modulo p linear regression for the ℓ1-norm of the trivial valuation on Fp is identical to the maximal feasible subsystem problem of linear equations over Fp and is included in the optimisation problem of p-adic linear regression for ℓq-norm of the p-adic valuation on Qp for q ∈ (0, ∞) . See also Introduction of [Mih26-4] for other preceding studies on p-adic optimisation.  \nHowever, as if they synchronised with the evolution of various studies on p-adic neural networks, many results on p-adic optimisation appeared after 2020 . For example, A. F.  \nT. Martins showed polynomial-time algorithms for p-adic linear regression for ℓ∞ -normin [Mar25] . G. D. Baker, S. Mccallum, and D. Pattinson gave a lower bound of the number of sample points in a hyperplane optimal of p-adic linear regression for ℓ1-norm in [BMP25] . Further, G. D. Baker proved that the class of p-adic linear regressions for ℓ 1-norm with varying p is NP-hard in the sense that the maximum cut problem can be reduced to a specific instance of 2-adic linear regressions in [Bak26] . We formulated anew p-adic optimisation problem as an infinitesimal limit of the least squares method, invented and compared various heuristic algorithms for p-adic polynomial regression with thorough analysis of their time and space complexity in [Mih26-1] . Further, we formulated p-adic principal component analyses as dimensionality reduction methods in p-adic vector spaces in [Mih26-2] . M","cbCaitriMMEpJdEE","https://ap.wps.com/l/cbCaitriMMEpJdEE","pdf",124982,5,1,13,"English","en",105,"# Introduction\n## Convention\n## p-adic Optimisation\n## Forcing Method\n## NP-hardness","[{\"question\":\"What does the paper generalise from Baker’s forcing method?\",\"answer\":\"It generalises Baker’s forcing method beyond the 2-adic setting, extending it to a wider class of p-adic optimisation problems for arbitrary primes p.\"},{\"question\":\"Which NP-hardness results are proved in the paper?\",\"answer\":\"The paper proves NP-hardness for p-adic linear regression, for 2-adic dynamic neural networks, and for a partial generalisation associated with van der Put neural networks (p-adic smooth neural networks).\"},{\"question\":\"How is the forcing method used to connect optimisation problems to regression models?\",\"answer\":\"The method re-expresses the optimisation problem in a regression framework (initially 2-adic linear regression in Baker’s work, then extended to p-adic regression), enabling NP-hardness to be established via the regression formulation.\"}]","Generalisation of Baker’s Forcing Method to Arbitrary Prime and NP-hardness of Several p-adic Optimisations | PDF",1784177259,33,{"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},"generalisation-of-bakers-forcing-method-to-arbitrary-prime-and-np-hardness-of-several-p-adic-optimisations","",{"@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/generalisation-of-bakers-forcing-method-to-arbitrary-prime-and-np-hardness-of-several-p-adic-optimisations/81951/",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 does the paper generalise from Baker’s forcing method?","Question",{"text":77,"@type":78},"It generalises Baker’s forcing method beyond the 2-adic setting, extending it to a wider class of p-adic optimisation problems for arbitrary primes p.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"Which NP-hardness results are proved in the paper?",{"text":82,"@type":78},"The paper proves NP-hardness for p-adic linear regression, for 2-adic dynamic neural networks, and for a partial generalisation associated with van der Put neural networks (p-adic smooth neural networks).",{"name":84,"@type":75,"acceptedAnswer":85},"How is the forcing method used to connect optimisation problems to regression models?",{"text":86,"@type":78},"The method re-expresses the optimisation problem in a regression framework (initially 2-adic linear regression in Baker’s work, then extended to p-adic regression), enabling NP-hardness to be established via the regression formulation.","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"]