[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-665541-105":3,"detail-sidebar-cat-0-en-105":72,"doc-detail-665541-en":122},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":65,"head_meta":67,"extra_data":69,"updated_unix":71},105,"en","a-first-order-logic-for-string-diagrams","A first-order logic for string diagrams","","Equational reasoning with string diagrams enables intuitive proofs between morphisms in symmetric monoidal categories and can be extended to infinite families via !-box notation. Prior attempts to move beyond pure equational reasoning were limited by the absence of an appropriate logical framework for !-box diagrammatic proofs. This paper introduces !L, a fully fledged first-order logic with conjunction, implication, and universal quantification over !-boxes, supporting formal induction. A standard model for !L is built, and an example proof of a theorem for non-commutative bialgebras is given.",{"@graph":14,"@context":64},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/a-first-order-logic-for-string-diagrams/665541/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/a-first-order-logic-for-string-diagrams/665541.png","ImageObject",300,407,{"name":42,"@type":43},"pop out","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":52,"interactionType":53,"userInteractionCount":30},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58],{"name":59,"@type":60,"acceptedAnswer":61},"What example application is provided for !L?","Question",{"text":62,"@type":63},"The paper builds a standard model for !L and gives an example proof of a theorem for non-commutative bialgebras, which the author notes cannot be obtained by equational reasoning alone.","Answer","https://schema.org",{"og:url":32,"og:type":66,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":68,"canonical":32},"index,follow",{"doc_id":70,"site_id":7},665541,1793304829,{"code":4,"msg":73,"data":74},"success",[75,79,83,87,92,97,102,106,111,114,118],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":76,"show_sort_weight":77,"slug":78},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":80,"show_sort_weight":81,"slug":82},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Exam",70,"exam",{"id":88,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},5,"Comic",60,"comic",{"id":93,"doc_module":4,"doc_module_name":25,"category_name":94,"show_sort_weight":95,"slug":96},6,"Technology",50,"technology",{"id":98,"doc_module":4,"doc_module_name":25,"category_name":99,"show_sort_weight":100,"slug":101},7,"Healthcare",40,"healthcare",{"id":103,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":104,"slug":105},8,30,"research-report",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},9,"Religion & Spirituality",20,"religion-spirituality",{"id":109,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":109,"slug":113},"World Cup","world-cup",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":115,"slug":117},10,"Lifestyle","lifestyle",{"id":119,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":88,"slug":121},19,"General","general",{"code":4,"msg":73,"data":123},{"doc_id":70,"user_id":124,"nickname":42,"user_avatar":125,"doc_module":4,"category_id":103,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":126,"file_id":127,"file_url":128,"file_type":129,"file_size":130,"view_count":30,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":119,"language":131,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":132,"faqs":133,"seo_title":134,"seo_description":12,"update_tm":71,"read_time":135},962090760408,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","A first-order logic for string diagrams Aleks Kissinger 1 and David Quick2  \n1,2 Department of Computer Science  \nUniversity of Oxford, UK  \n{aleks.kissinger,[david.quick}@cs.ox.ac.uk](david.quick}@cs.ox.ac.uk)  \n~~ Abstract ~~  \nEquational reasoning with string diagrams provides an intuitive means of proving equations between morphisms in a symmetric monoidal category. This can be extended to proofs of infinite families of equations using a simple graphical syntax called !-box notation. While this does greatly increase the proving power of string diagrams, previous attempts to go beyond equational reasoning have been largely ad hoc, owing to the lack of a suitable logical framework for diagrammatic proofs involving !-boxes. In this paper, we extend equational reasoning with !-boxes toa fully-fledged first order logic with conjunction, implication, and universal quantification over!-boxes. This logic, called !L, is then rich enough to properly formalise an induction principle for !-boxes. We then build a standard model for !L and give an example proof of a theorem for non-commutative bialgebras using !L, which is unobtainable by equational reasoning alone.  \n1998 ACM Subject Classification D.3.1 Formal Definitions and Theory, F.4.1 Mathematical Logic  \nKeywords and phrases string diagrams, compact closed monoidal categories, abstract tensor systems, first-order logic  \nDigital Object Identifier 10.4230/LIPIcs.CALCO.2015.169  \n 1  Introduction  \nMany processes come with natural notions of parallel and sequential composition. In such cases, it is advantageous to switch from traditional term-based ([i.e. one-dimensional](i.e. one-dimensional)) syntax to the two-dimensional syntax of string diagrams. This diagrams, which consist of boxes (or various other shapes) connected by wires, form a sound and complete language for compositions of morphisms in a monoidal category [8] . Recently, the use of string diagrams has gained much interest in a wide variety of areas, including categorical quantum mechanics [4, 3, 5], computational linguistics [9] and control theory [2, 1] .  \nWhat many of these applications have in common is they make extensive use of equational reasoning for string diagrams. That is, proofs are constructed by starting with a fixed set of diagram equations, e.g.  \n(i)  \n=  \n(ii)   \n =  \n(iii)  \n=  \n(iv)  \n =  \nand using those to construct new equations by substitution of sub-diagrams. For example, the following is a derivation making use of the four rules above:  \n© Aleks Kissinger and David Quick;  \nlicensed under Creative Commons License CC-BY  \n6th International Conference on Algebra and Coalgebra in Computer Science (CALCO’15) . Editors: Larry Moss and Paweł Sobociński; pp. 169–187  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n170 First-order logic for string diagrams  \nHowever, to prove more powerful theorems, one often needs to pass from statements about single diagrams to entire families of diagrams and diagram equations. One way to do this, while staying within the realm of string diagrams is to use !-box notation (pronounced‘bang-box notation’), introduced in [6] and formalised in [11] . In this notation, certain sub-diagrams are wrapped in boxes, which mean ‘repeat this sub-diagram any number of times’. For example, suppose we considered a family of ‘copy’ operations with 1 input and n outputs. Then, if we had some other map with just a single output, we might ask that connecting it to the n-fold ‘copy’ results in n copies. We can represent this family of rules using !-box notation as follows:  \nA A  \n...  \n=   = (1)  \nWhereas the equation on the left is informal, the expression on the right defines a family of equations without ambiguity. Formally, a !-box rule represents a set of string diagram rules obtained by instantiating the !-box, which essentially amounts fixing the number of times to copy each !-box. For example, the inst","cbCairkKT4fjuz6r","https://ap.wps.com/l/cbCairkKT4fjuz6r","pdf",585305,"English","# Introduction\n## Equational reasoning and !-box notation\n## Induction and the need for a logical framework","[{\"question\":\"What example application is provided for !L?\",\"answer\":\"The paper builds a standard model for !L and gives an example proof of a theorem for non-commutative bialgebras, which the author notes cannot be obtained by equational reasoning alone.\"}]","A first-order logic for string diagrams | PDF",48]