[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118872-en":3,"doc-seo-118872-105":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},118872,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Algebraic Dynamical Systems in Machine Learning - Term Rewriting and Category-Theoretic Compositionality","Algebraic Dynamical Systems in Machine Learning develops a term-rewriting based algebraic analogue of dynamical systems. A recursive function applied to the output of an iterated rewriting system defines a formal model class, embedding major dynamic-model architectures such as recurrent neural networks, graph neural networks, and diffusion models. Framed in category theory, the construction yields a language for compositionality, and a template for generalising dynamic models to structured or non-numerical learning problems, including hybrid symbolic-numeric models.","Algebraic Dynamical Systems in Machine Learning  \nIolo Jones1,2 · Jerry Swan2 · Jeﬀrey Giansiracusa1  \nReceived: 16 June 2023 / Accepted: 13 December 2023 © The Author(s) 2024  \nAbstract  \nWe introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system deﬁnes a formal class of models into which all the main architectures for dynamic machine learning models (including recurrent neural networks, graph neural networks, and diffusion models) can be embedded. Considered in category theory, we also show that these algebraic models are a natural language for describing the compositionality of dynamic models. Furthermore, we propose that these models provide a template for the generalisation of the above dynamic models to learningproblems on structured or non-numerical data, including‘hybrid symbolicnumeric’models.  \nKeywords Machine learning · Dynamical systems · Term rewriting · Functional programming · Compositionality  \n1 Introduction  \nThe relationship between the structure of a model and its observable behaviour is central to many areas of applied mathematics. In linguistics and computer science, there are corresponding notions of the syntax and semantics of an expression or program. In machine learning, behaviour is determined via learned parameters while the structure of a model is typically considered to be prescribed by hyperparameters. This is often categoriﬁed in the context of functional programming, where programs are viewed as maps in a category of data types. Here the syntax is speciﬁed by an algebraic data type, on which, in a general setting, recursively-deﬁned functions determine the semantics [26] . These perspectives can  \nCommunicated by Stefan Milius.  \nB  \n1  \n2  \nIolo Jones  \n[iolo.j.jones@durham.ac.uk](iolo.j.jones@durham.ac.uk)  \nJerry Swan  \n[jerry@hylomorph-solutions.com](jerry@hylomorph-solutions.com)  \nJeffrey Giansiracusa  \n[jeffrey.giansiracusa@durham.ac.uk](jeffrey.giansiracusa@durham.ac.uk)  \nDurham University, Durham, UK Hylomorph Solutions, Glasgow, UK  \n1 3  \nbe combined in formal machine learning theory, where the categorical perspective forms a basis for describing ‘compositionality’: the properties of a model’s components which are preserved under composition. Compositional modelling provides vital support for safe and causal inference, where unconstrained neural approaches are known to be lacking [11] .  \nIn this paper, we develop this theory to include the increasingly popular class of models based on dynamical systems. We describe these models via universal algebra [5] and category theory and show that term rewriting systems[2]are the exact algebraic analogue ofdynamical systems, but also explicitly encode the structure of the model in their expression. The rewrite rule corresponds to this syntax or structure, while the semantics are speciﬁed by a recursive function on that algebraic structure. The categorical setting also allows us to talk, in full generality, about which properties of models are preserved under composition. We use this to prove that rewriting models are naturally compositional, in the sense that the corresponding dynamical system will lift to any category with the appropriate structure.  \n1.1 Structural Constraints in Machine Learning  \nA proper appreciation of the role played by structural constraints requires a brief summary of the history of Artiﬁcial Intelligence (AI) . Ever since its inception [29], the ﬁeld of artiﬁcial intelligence has been split between ostensibly-competing symbolic and connectionist perspectives. The symbolic approach was initially favoured, exempliﬁed by so-called ‘Good Old Fashioned AI’ (GOFAI) [33], that typically attempts to model the world in terms of rules which manipulate opaque symbols via formalisms such as predicate calculus. Such approaches were ill-suited for modelling the noisy real world and suffered from the ‘knowledge elicitati","cbCailayx3COOvrs","https://ap.wps.com/l/cbCailayx3COOvrs","pdf",422295,1,32,"English","en",105,"# Abstract\n# Introduction\n## Structural Constraints in Machine Learning","[{\"question\":\"What algebraic structure is used to model dynamical systems in this work?\",\"answer\":\"The paper introduces an algebraic analogue of dynamical systems based on term rewriting systems, with semantics given by a recursive function on the algebraic structure.\"},{\"question\":\"How are existing dynamic machine learning architectures related to the proposed framework?\",\"answer\":\"A recursive function applied to outputs of iterated rewriting systems defines a formal class into which main dynamic ML architectures, including recurrent neural networks, graph neural networks, and diffusion models, can be embedded.\"},{\"question\":\"Why is the category-theoretic perspective important here?\",\"answer\":\"Considering the models in category theory lets the paper describe, in general terms, which properties are preserved under composition, showing rewriting models are naturally compositional.\"}]","Algebraic Dynamical Systems in Machine Learning - Term Rewriting and Category-Theoretic Compositionality | PDF",1785720723,81,{"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},"algebraic-dynamical-systems-in-machine-learning-term-rewriting-and-category-theoretic-compositionality","",{"@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/algebraic-dynamical-systems-in-machine-learning-term-rewriting-and-category-theoretic-compositionality/118872/",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","2026-08-03",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 algebraic structure is used to model dynamical systems in this work?","Question",{"text":75,"@type":76},"The paper introduces an algebraic analogue of dynamical systems based on term rewriting systems, with semantics given by a recursive function on the algebraic structure.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are existing dynamic machine learning architectures related to the proposed framework?",{"text":80,"@type":76},"A recursive function applied to outputs of iterated rewriting systems defines a formal class into which main dynamic ML architectures, including recurrent neural networks, graph neural networks, and diffusion models, can be embedded.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is the category-theoretic perspective important here?",{"text":84,"@type":76},"Considering the models in category theory lets the paper describe, in general terms, which properties are preserved under composition, showing rewriting models are naturally compositional.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]