[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82036-en":3,"doc-seo-82036-105":31,"detail-sidebar-cat-0-en-105":92},{"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},82036,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Interval Certifications for Multilayered Perceptrons via Lattice Traversal","提出一种面向AI安全的严格理论框架，用于对抗鲁棒性问题进行形式化刻画。研究表明，对抗鲁棒性可归约为格点遍历问题：格点中的每个元素对应一个区间，即包含输入x的轴对齐超矩形。给定多层感知机分类器，区间I在x∈I且x可在I内任意扰动而不改变预测时为“sound”认证；当x离开I时预测必变则为“complete”认证。文中设计格遍历算子并用于 refine-and-verify 迭代方案，保证sound最大性与complete最小性，并讨论相关优化及实证评估。","arXiv :2607 .08773v 1 [ cs .AI] 9 Apr 2026  \nInterval Certifications for Multilayered Perceptrons via Lattice Traversal  \nMerkouris Papamichail 1 ,2 , Konstantinos Varsos 1 ,2[0000−0002−0833−6890], Giorgos Flouris 1[0000−0002−8937−4118], and João Marques-Silva3 ,4[0000−0002−6632−3086]  \n1 Foundation for Reasearch and Technology-Hellas, Heraklion, Greece  \n{mercoyris,varsosk,[fgeo}@ics.forth.gr](fgeo}@ics.forth.gr)  \n2 University of Crete, Heraklion, Greece  \n3 Catalan Institution for Research and Advanced Studies, Barcelona, Spain  \n[jpms@icrea.cat](jpms@icrea.cat)  \n4 University of Lleida, Lleida, Spain  \nAbstract. In this work we present a rigorous theoretical framework to a foundational problem of AI safety, namely adversarial robustness. In particular, we show that the adversarial robustness problem can be reduced to a lattice traversal problem. Each element of this lattice corresponds to an interval, i.e., an axis-aligned hyper-rectangle, containing an input point x. Consider a multilayered perceptron classifier (MLP) . An interval I constitutes a sound certification if x ∈ I and x can be freely perturbed in I without changing the MLP’s prediction. Complementarily, an interval I constitutes a complete certification if x ∈ I and when x moves outside of I the MLP’s prediction is guaranteed to change. While the sound certification problem corresponds to the well-studied adversarial robustness, complete certifications have not been examined in the literature. We develop lattice traversal operators, which we apply in a refine & verify iterative scheme. Using formal MLP verifiers, sound maximality and complete minimality are guaranteed. Moreover, we examine objective optimization problems. There we discover some interesting asymmetries. For complete certifications, the minimum solution is obtained in polynomial oracle calls. This does not hold for sound certifications, where we prove strong intractability results. Additionally, we examine optimization problems in symmetric intervals (i.e. , ℓ ∞-spheres), where we provide logarithmic algorithms. Finally, we present an empirical evaluation, using the novel ParallelepipedoNN5 system.  \nKeywords: AI-Safety · Adversarial Robustness · NN Verification · Interval Algebra · Lattice Traversal.  \n1 Introduction  \nArtificial Intelligence, mostly driven by deep neural networks (NN), is rapidly becoming part of our everyday life, from recommendation systems in media platforms [39] to large language models chat-bots [4] . Despite these achievements,  \n5 [https://github.com/merkouris148/parallelepipedonn](https://github.com/merkouris148/parallelepipedonn)  \n2 M. Papamichail et al.  \nNNs promise even greater accomplishments by replacing humans in critical, decision-making areas, from driving [18] to healthcare [21] or government administration [40] . However, NNs are brittle, meaning that small, often imperceptible input perturbations can flip their predictions. These inputs are commonly referred to as adversarial examples [9,24,31] .  \nEnsuring a NN’s robustness to adversarial attacks remains a persisting problem for AI safety. The first works on the field focused on exploiting the gradient information to produce adversarial examples, subsequently incorporating them into the learning process, e.g., [9,22] . Nevertheless, these initial attempts fail to solve the problem in its generality [37] . More sophisticated methods utilized the convex relaxation of a NN [6], reducing adversarial robustness to a convex optimization problem. This problem was either solved directly [12,17,19], or in its dual form [33] . Even so, this family of works suffers from low precision, since they rely on a relaxation of the original problem [25,29] .  \nThe hardness of adversarial robustness stems from the NN representation and seems to be deeply rooted in its computational properties. NN’s activation functions introduce nonlinearities that can only be studied using integer constraints. Therefore, NN can only be accur","cbCaipOB0It0hbmk","https://ap.wps.com/l/cbCaipOB0It0hbmk","pdf",1465292,3,1,34,"English","en",105,"# Introduction\n## Adversarial examples and robustness certification\n## Verifiers and computational hardness\n## Interval certifications and lattice structure\n## Lattice traversal operators and iterative schemes","[{\"question\":\"区间认证（interval certification）如何定义 sound 与 complete？\",\"answer\":\"当区间I包含输入x（x∈I），且x在I内可任意扰动仍不改变MLP预测时，I为sound认证；当x∈I且一旦x离开I预测必定发生变化，则I为complete认证。\"},{\"question\":\"为什么将对抗鲁棒性问题归约为“格点遍历”问题？\",\"answer\":\"文中把所有区间认证组织成一个完整格（complete lattice），每个格元素对应一个轴对齐超矩形区间。通过在该格上系统遍历，可以探索并计算对应的最大sound与最小complete区间认证。\"},{\"question\":\"文中在可计算性与优化方面发现了哪些差异？\",\"answer\":\"complete认证的最小解可在多项式级别的oracle调用次数下获得；而sound认证则证明了更强的不可解性（strong intractability）结果。同时，在对称区间（如ℓ∞-球）上给出对数复杂度算法。\"}]","Interval Certifications for Multilayered Perceptrons via Lattice Traversal | PDF",1784177726,86,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":29},"interval-certifications-for-multilayered-perceptrons-via-lattice-traversal","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,51],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":52,"name":13,"@type":44,"position":53},"https://docshare.wps.com/document/interval-certifications-for-multilayered-perceptrons-via-lattice-traversal/82036/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"区间认证（interval certification）如何定义 sound 与 complete？","Question",{"text":76,"@type":77},"当区间I包含输入x（x∈I），且x在I内可任意扰动仍不改变MLP预测时，I为sound认证；当x∈I且一旦x离开I预测必定发生变化，则I为complete认证。","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"为什么将对抗鲁棒性问题归约为“格点遍历”问题？",{"text":81,"@type":77},"文中把所有区间认证组织成一个完整格（complete lattice），每个格元素对应一个轴对齐超矩形区间。通过在该格上系统遍历，可以探索并计算对应的最大sound与最小complete区间认证。",{"name":83,"@type":74,"acceptedAnswer":84},"文中在可计算性与优化方面发现了哪些差异？",{"text":85,"@type":77},"complete认证的最小解可在多项式级别的oracle调用次数下获得；而sound认证则证明了更强的不可解性（strong intractability）结果。同时，在对称区间（如ℓ∞-球）上给出对数复杂度算法。","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"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":107,"slug":139},19,"General","general"]