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It proves a logical clock is necessary and sufficient for irreflexivity of the “happen before” relation and addresses Lamport clock non-injectivity by proposing a modified injective logical clock that combines the standard clock with node identifiers, improving uniqueness, consistency, integrity, and reliability in decentralized DLT.",{"@graph":14,"@context":72},[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/lamport-spacetime-diagrams-and-logical-clocks/665485/",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/lamport-spacetime-diagrams-and-logical-clocks/665485.png","ImageObject",300,407,{"name":42,"@type":43},"Adam","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"Why can’t physical time reliably order events in asynchronous distributed systems?","Question",{"text":62,"@type":63},"Physical timestamps suffer from clock skew and cannot represent cause-and-effect relationships between events. Therefore they are insufficient for rigorous event ordering and causality determination.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does Lamport spacetime diagrams model execution scenarios?",{"text":67,"@type":63},"The model maps execution events into a logical spacetime using defined event types (local actions, message sending, message receiving) and a “trigger” function to represent message transmission between nodes.",{"name":69,"@type":60,"acceptedAnswer":70},"What problem of the classic Lamport clock does the paper address?",{"text":71,"@type":63},"The paper identifies the classic Lamport clock’s non-injectivity: different events may receive the same timestamp, preventing the clock value from uniquely identifying an event. It then introduces an injective modification to resolve this.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},665485,1793304476,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":33,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":79,"read_time":144},1374404737137,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Lamport Spacetime Diagrams and Logical Clocks ⋆  \nGrygoriy Zholtkevych1,∗,† and Oleksandr Berezovskyi1,*,†  \n1Ivan Franko National University of Lviv, Universytetska Str 1, 79001 Lviv, Ukraine  \nAbstract  \nThis paper addresses the critical challenge of event ordering and causality determination in asynchronous distributed systems, a problem of growing importance with the widespread adoption of Distributed  \nLedger Technology (DLT) . The authors highlight the insufficiency of physical time for ordering transactions due to clock skew and the inability of physical timestamps to reflect cause-and-effect relationships. To provide a rigorous analysis, the paper presents a formal mathematical model for Lamport spacetime diagrams, which visualize execution scenarios by mapping events into logical spacetime. The model defines a set of events (local actions, message sending, and message receiving) and employs a \"trigger\" function to model message transmission between nodes. Central to this framework is the \"happen before\" relation, which serves as a formal model for causality in distributed computations. The authors demonstrate that the existence of a logical clock is a necessary and sufficient condition for the irreflexivity of the \"happen before\" relation, which is essential to guarantee the absence of system anomalies such as deadlocks. While the classic Lamport clock successfully orders events and identifies causal relationships, the paper identifies its primary limitation: non-injectivity. In concurrent systems, different events can receive the same timestamp, preventing the logical clock value from serving as a unique event identifier. To resolve this, the research examines a modified injective logical clock algorithm. This modification operates on a specific time scale by combining the standard logical clock with node identifiers. The authors prove that this modified function is an injective logical clock, ensuring that every distinct event receives a unique timestamp. This enhancement enables logical time to serve as a reliable and unique identifier, significantly improving data consistency, integrity, and functionality within decentralized DLT systems.  \nKeywords  \ndistributed ledger technology, Lamport spacetime diagram, logical clock,“happen before” relation, causality, injectivity  \n1. Introduction  \nNowadays, distributed storage and processing of information have become mainstream due to the growth of data and the spread of cloud technologies.  \nCentralized cloud services managed by large corporations offer convenient and scalable solutions.  \nHowever, this model cannot be considered as entirely trustworthy, particularly when it comes to data registries that must be accessible and reliable for the general public.  \nThe key feature of that approach is that it requires a trusted third party responsible for the integrity, authenticity, and availability of the data.  \nReliance on a single trusted center creates several risks that have been thoroughly studied in the scientific literature.  \nFirst, such a center becomes a single point of failure, making the system vulnerable to technical failures, cyberattacks, or unauthorized access [1] .  \nSecond, it creates an asymmetry of power where a trusted intermediary can censor, manipulate, or block access to data, which is contrary to the principles of openness and transparency [2] .  \n⋆ CMIS’26: The Ninth International Workshop on Computer Modeling and Intelligent Systems, May 5, 2026, Zaporizhzhia, Ukraine  \n1∗ Corresponding author.  \n† These authors contributed equally.  \n [grygoriy](grygoriy.zholtkevych@lnu.edu.ua)[.zholtkevych@lnu.edu.ua](grygoriy.zholtkevych@lnu.edu.ua) (G. Zholtkevych); [oleksandr.berezovskyi@lnu.edu.ua](oleksandr.berezovskyi@lnu.edu.ua) (O. Berezovskyi)  \n 0000-0002-7515-2143 (G. Zholtkevych); 0009-0003-2241-3324 (O. Berezovskyi)  \n Copyright © 2026 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY ","cbCaioJ3uXVeGs1w","https://ap.wps.com/l/cbCaioJ3uXVeGs1w","pdf",409080,11,"English","# Introduction\n## Motivation for distributed ledgers over centralized trust\n## Limitations of physical time for causal ordering\n## Lamport spacetime diagrams and formal event model\n## “Happen before” relation and logical clocks\n## Injective logical clock modification for unique event identifiers","[{\"question\":\"Why can’t physical time reliably order events in asynchronous distributed systems?\",\"answer\":\"Physical timestamps suffer from clock skew and cannot represent cause-and-effect relationships between events. Therefore they are insufficient for rigorous event ordering and causality determination.\"},{\"question\":\"How does Lamport spacetime diagrams model execution scenarios?\",\"answer\":\"The model maps execution events into a logical spacetime using defined event types (local actions, message sending, message receiving) and a “trigger” function to represent message transmission between nodes.\"},{\"question\":\"What problem of the classic Lamport clock does the paper address?\",\"answer\":\"The paper identifies the classic Lamport clock’s non-injectivity: different events may receive the same timestamp, preventing the clock value from uniquely identifying an event. It then introduces an injective modification to resolve this.\"}]","Lamport Spacetime Diagrams and Logical Clocks | PDF",28]