[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665538-105":59,"doc-detail-665538-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","categorical-e-graphs-for-lambda-calculi","Categorical E-Graphs for Lambda Calculi","","Categorical E-Graphs for Lambda Calculi proposes a categorical semantics for equality graphs (e-graphs) that natively support variable binding. It interprets e-graphs as string diagrams in symmetric monoidal semilattice-enriched categories, then equips them with a monoidal closed structure to handle lambda-calculus constructs. The paper gives a concrete representation via hierarchical hypergraphs and introduces a DPO rewriting system, proving equivalence between term rewriting and DPO rewriting while absorbing the symmetric monoidal equations. The approach is compared with slotted e-graphs.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/categorical-e-graphs-for-lambda-calculi/665538/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/categorical-e-graphs-for-lambda-calculi/665538.png","ImageObject",300,407,{"name":92,"@type":93},"Alexandar","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem do the authors target in traditional e-graphs for lambda calculi?","Question",{"text":112,"@type":113},"Traditional e-graphs lack native support for variable binding, which limits applicability to programming languages. The work addresses binding by extending e-graphs from a categorical perspective.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are e-graphs interpreted in this paper’s approach?",{"text":117,"@type":113},"The paper interprets e-graphs as string diagrams, using morphisms in symmetric monoidal semilattice-enriched categories and extending this with a monoidal closed structure.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the role of hierarchical hypergraphs and DPO rewriting?",{"text":121,"@type":113},"Hierarchical hypergraphs provide a concrete encoding of string diagrams and binding. Equational reasoning is formalized as double-pushout (DPO) rewriting, and the paper establishes equivalence between term rewriting and DPO rewriting.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665538,1793304820,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},962090768658,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","arXiv :2505 .00807v2 [ cs .LO] 25 Jun 2026  \nCategorical E-Graphs for Lambda Calculi  \nAleksei Tiurin Dan R. Ghica Nick Hu  \nUniversity of Birmingham University of Birmingham Kyoto University  \nand  \nHuawei Central Software Institute  \n∗  \nEquality saturation is a powerful technique for program optimisation and reasoning, driven by the use of equivalence classes of terms under rewrite rules. These equivalence classes lie at the root of data structures called equality graphs (e-graphs) . Despite their numerous advantages, until recently e-graphs lacked native support for variable binding, limiting their applicability to programming languages. We propose to address this problem from a categorical perspective by extending the interpretation of e-graphs as string diagrams, namely morphisms in symmetric monoidal semilatticeenriched categories which we additionally equip with a monoidal closed structure. We further define a concrete representation using hierarchical hypergraphs, and introduce a corresponding double-pushout (DPO) rewriting system. Finally, we establish the equivalence between term rewriting and DPOrewriting, with the combinatorial model inherently absorbing the equations of the symmetric monoidal structure. Our approach, specifically designed for lambda calculi, is compared and contrasted with slotted e-graphs—an alternative method for incorporating variable binding within e-graphs.  \n1 Introduction  \n1.1 Motivation and contribution  \nE-graphs (short for equality graphs) [34] are an efficient data structure for non-destructive rewriting, incorporating equivalence representation, used in automated theorem proving, compiler optimisation, and program analysis. They mitigate the phase-ordering problem of term rewriting: the way rewrites are scheduled may enable or block further rewrites due to introduction or elimination of redexes, resulting ina suboptimal result. This is achieved by making rewrites on e-graphs non-destructive: an application of a rewrite rule, instead of replacing the left-hand side by the right-hand side, adds the latter to thee-graph so that all pre-existing redexes are protected and new ones are introduced. Exponential growth of the e-graph is avoided by the clever use of sharing of nodes.  \nE-graphs were first introduced in the automated theorem proving literature in the 1980s [26], but recent years have seen renewed interest, particularly for optimisation and synthesis using the technique of equality saturation [32, 15, 34], wherein rewrites to an e-graph are repeated until a fixpoint (or timeout) is reached. We are interested in using e-graphs in the context of functional programming languages, which builds upon an internal representation of the 􀁟-calculus. Kœhler et al. [23] pioneered work in this area by identifying two key, interconnected challenges—handling binding and ensuring capture-avoiding substitution—when applying equality saturation to an encoding of the 􀁟-calculus in e-graphs.  \nIn the textual representation of 􀁟-terms, 􀁕-equivalent terms may be distinct and thus cannot be shared. De Bruijn indices [13] syntactically identify 􀁕-equivalent terms, but the same variable can receive different indices depending on its scope, again interfering with sharing. As a simple example, 􀁟􀀵 .􀀵 ((􀁟􀁇 .􀀵 􀁇)2) is encoded as 􀁟.%0((􀁟.%1%0)2), where the two occurrences of 􀀵 are represented differently and cannot be shared.  \n∗ This work was supported by EPSRC grants EP/V001612/1 and EP/Y010035/1 .  \nSubmitted to:  \nA simple but theoretically [1] and practically [28] important extension of the 􀁟-calculus consists of adding let-bindings or explicit substitutions to the syntax. Terms are identified up to permutations of non-interacting substitutions, called “graph equivalence” [4] . Modelling this in conventional e-graphs requires additional nodes for explicit substitutions and additional rewrite rules to capture graphical equivalence, polluting the equality saturation state space with bureaucracy rather than","cbCait5JitdWnSur","https://ap.wps.com/l/cbCait5JitdWnSur","pdf",1096378,38,"English","# Introduction\n## Motivation and contribution\n## String diagrams\n## E-graphs","[{\"question\":\"What problem do the authors target in traditional e-graphs for lambda calculi?\",\"answer\":\"Traditional e-graphs lack native support for variable binding, which limits applicability to programming languages. The work addresses binding by extending e-graphs from a categorical perspective.\"},{\"question\":\"How are e-graphs interpreted in this paper’s approach?\",\"answer\":\"The paper interprets e-graphs as string diagrams, using morphisms in symmetric monoidal semilattice-enriched categories and extending this with a monoidal closed structure.\"},{\"question\":\"What is the role of hierarchical hypergraphs and DPO rewriting?\",\"answer\":\"Hierarchical hypergraphs provide a concrete encoding of string diagrams and binding. Equational reasoning is formalized as double-pushout (DPO) rewriting, and the paper establishes equivalence between term rewriting and DPO rewriting.\"}]","Categorical E-Graphs for Lambda Calculi | PDF",96]