[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117836-en":3,"doc-seo-117836-105":30,"detail-sidebar-cat-0-en-105":91},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},117836,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","A Computability Perspective on (Verified) Machine Learning","Machine Learning (ML) is widely viewed as most trustworthy when combined with formal verification, yet the meaning and requirements of “verified ML” remain unclear. This paper provides the first formalisation of ML classifiers and learners using computable analysis, identifying which properties are computable. It defines model-agnostic computational tasks for the proposed verified ML, showing in principle that the tasks can be computed. The approach bridges continuous mathematics behind ML and the discrete foundations of computer science.","A Computability Perspective on (Verified) Machine Learning  \nTonicha Crook [0000􀀀0002􀀀4882􀀀9999], Jay Morgan [0000􀀀0003􀀀3719􀀀362X], Arno Pauly [0000􀀀0002􀀀0173􀀀3295], and Markus Roggenbach [0000􀀀0002􀀀3819􀀀2787]  \nDepartment of Computer Science, Swansea University Swansea, Wales, UK  \n[t.m.crook15@outlook.com](t.m.crook15@outlook.com) [arno.m.pauly@gmail.com](arno.m.pauly@gmail.com)[ ](arno.m.pauly@gmail.com)[m.roggenbach@swansea.ac.uk](m.roggenbach@swansea.ac.uk)[ ](m.roggenbach@swansea.ac.uk)Université de Toulon, Aix Marseille Univ, CNRS, LIS  \nMarseille, France  \n[jay.morgan@univ-tln.fr](jay.morgan@univ-tln.fr)  \nAbstract. In Computer Science there is a strong consensus that it is highly desirable to combine the versatility of Machine Learning (ML) with the assurances formal verification can provide. However, it is unclear what such `verified ML' should look like.  \nThis paper is the first to formalise the concepts of classifiers and learners in ML in terms of computable analysis. It provides results about which properties of classifiers and learners are computable. By doing this we establish a bridge between the continuous mathematics underpinning MLand the discrete setting of most of computer science.  \nWe define the computational tasks underlying the newly suggested verified ML in a model-agnostic way, i.e., they work for all machine learning approaches including, e.g., random forests, support vector machines, and Neural Networks. We show that they are in principle computable.  \nKeywords: Machine Learning 􀀁 adversarial examples 􀀁 formal verification 􀀁 computable analysis  \n1 Introduction  \nMachine Learning (ML) concerns the process of building both predictive and generative models through the use of optimisation procedures. The remarkable success of ML methods in various domains raises the question of how much trust one can put into the responses that an ML model provides. As ML models are also applied in critical domains, some form of verification seems essential (e.g. eloquently argued by Kwiatkowska [9]) .  \nHowever, due to the widespread use of non-discrete mathematics in ML, traditional verification techniques are hard to apply to its artefacts. Furthermore, many ML applications lack specifications in the form of, say, an input/output relationship, on which `classical' verification approaches are often based. A typical example of this would be an ML application that shall decide if a given picture  \n2 T. Crook, J. Morgan, A. Pauly & M. Roggenbach  \ndepicts a cat. Lacking a specification, what kind of properties can be verified? We will take the view that, like in classical verification, it is useful to expand the range of properties beyond simple input/output relations.  \nBy employing the toolset of computable analysis (the field concerned with computation on continuous data types), we are using the same continuous mathematics underpinning the theory of machine learning, and avoids any ad-hoc discretization.  \nWe present an investigation into what kind of verification questions are answerable in principle about ML models – irrespective of the particular ML framework applied. We see these questions as basic building blocks for a future ML property specification language. Discretization, as far as it may be necessary for the sake of efficiency, can then be left to the implementation; without impacting correctness.  \nWe use the language of computable analysis to formally define the computational questions we want to ask. We can prove that they are solvable in general (by exhibiting algorithms for them), while remaining independent of any concrete ML methodology. The semi-decision procedures in this paper are not meant for implementation. We are also not making any claims about computational complexity.  \nOur paper is organised as follows: in Section 2 we provide a gentle summary of our results. In Section 3, we provide definitions and key properties from computable analysis. Section 4 develops our theory with mathematical precision. Finally, S","cbCaidtyTcQR9tG4","https://ap.wps.com/l/cbCaidtyTcQR9tG4","pdf",489436,1,18,"English","en",105,"# Introduction\n# A Gentle Summary of our Results\n## Classifiers\n## Adversarial Examples\n## Learning and Robustness","[{\"question\":\"What problem does the paper address about “verified ML”?\",\"answer\":\"It addresses the lack of clarity about what verified ML should look like and how to formally specify and verify ML components. The paper proposes a computability-based formalisation to make these ideas precise.\"},{\"question\":\"How does the paper formalise classifiers and learners?\",\"answer\":\"It formalises classifiers and learners in terms of computable analysis, using represented spaces to define domains and codomains. This framework supports proving which properties are computable.\"},{\"question\":\"Are the proposed verification tasks tied to a specific ML method?\",\"answer\":\"No. The tasks are defined in a model-agnostic way so they apply to different ML approaches, including random forests, support vector machines, and neural networks. The paper shows these tasks are computable in principle.\"}]","A Computability Perspective on (Verified) Machine Learning | PDF",1785679916,45,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-computability-perspective-on-veried-machine-learning","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/a-computability-perspective-on-veried-machine-learning/117836/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address about “verified ML”?","Question",{"text":75,"@type":76},"It addresses the lack of clarity about what verified ML should look like and how to formally specify and verify ML components. The paper proposes a computability-based formalisation to make these ideas precise.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper formalise classifiers and learners?",{"text":80,"@type":76},"It formalises classifiers and learners in terms of computable analysis, using represented spaces to define domains and codomains. This framework supports proving which properties are computable.",{"name":82,"@type":73,"acceptedAnswer":83},"Are the proposed verification tasks tied to a specific ML method?",{"text":84,"@type":76},"No. The tasks are defined in a model-agnostic way so they apply to different ML approaches, including random forests, support vector machines, and neural networks. 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