[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83500-en":3,"doc-seo-83500-105":30,"detail-sidebar-cat-0-en-105":83},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83500,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","Semantic Labelling in Practice","Semantic labelling is a transformation technique for rewrite systems designed to make termination proofs easier while preserving both termination and nontermination. This work addresses the combinatorial difficulty of automating semantic labelling, since the number of candidate algebras grows rapidly. Experiments compare model-finding strategies in their tools Matchbox and MnM: exhaustive enumeration within bounded domains and restricted search spaces, versus semantic context-closure for fixed algebras.","arXiv :2607 .0052 1v 1 [ cs .LO] 1 Jul 2026  \nSemantic Labelling in Practice  \nDieter Hofbauer \\# 􀀚  \nASW Saarland  \nJohannes Waldmann \\# HTWK Leipzig  \n~~ Abstract ~~  \nAutomating semantic labelling for termination proofs is a combinatorially hard problem since the number of algebras grows prohibitively large even for small domains. We report on experiments with our tools Matchbox and MnM, comparing various model-finding strategies: exhaustive enumeration for bounded domain sizes within restricted search spaces, and semantic context-closure for fixed algebras.  \n2012 ACM Subject Classification Theory of computation → Equational logic and rewriting; Theory of computation → Rewrite systems  \nKeywords and phrases termination, string rewriting, term rewriting  \n 1  Introduction  \nSemantic labelling is a transformation technique for rewrite systems, introduced by Zantema [24, 25] . Given a rewrite system, the goal is to find a labelled version for which termination is easier to prove than for the original system. For the transformation to be correct, it must preserve both termination and nontermination. Since the labelled system typically has a larger alphabet, interpretations and other termination methods gain more freedom as they can assign different meanings to the same original symbol, now distinguished by labels. Each semantic labelling is parameterized by an algebra—hence the name—and is correct if that algebra is a model of the rewrite system. Algebras with larger domains yield larger alphabets for the transformed system—as desired—but automating semantic labelling becomes increasingly difficult, due to the combinatorially large number of algebras.  \nIn this paper, we compare different model-finding strategies: exhaustive enumeration for bounded domain sizes within restricted search spaces, and semantic context-closure for fixed algebras. We report on experiments with our tools Matchbox [22]1 and MultumNonMulta (MnM) [10]2 .  \nDefinitions and examples in this paper refer to string rewriting, but all approaches discussed can be directly applied to term rewriting as well. All examples in the directories SRS_Standard and SRS_Relative are taken from the Termination Problems Database3 , and we occasionally refer to results of the annual Termination Competition (TC)4 , where tools are evaluated on benchmark problems.  \nFollowing the seminal work of Zantema [24], numerous extensions and applications of semantic labelling have been proposed. Among these, we mention self-labelling [15]; labelling for rewriting modulo equations [16]; predictive labelling [9, 14], restricting the model property to usable rules, and predictive labelling for innermost termination [21]; root-labelling [18]; modularity results and certification for labelling and unlabelling [19]; SAT-encoding of  \n1 Available at [https://git.imn.htwk-leipzig.de/waldmann/pure-matchbox](https://git.imn.htwk-leipzig.de/waldmann/pure-matchbox).  \n2 Available at [https://hub.docker.com/repositories/dieterhofbauer/multumnonmulta](https://hub.docker.com/repositories/dieterhofbauer/multumnonmulta).  \n3 TPDB, The Termination Problems Database, Version 11 .5 [https://github.com/TermCOMP/TPDB-ARI](https://github.com/TermCOMP/TPDB-ARI).  \n4 For a survey see [https://termination-portal.org/wiki/Termination_Competition_History](https://termination-portal.org/wiki/Termination_Competition_History), and [https://termcomp.github.io/](https://termcomp.github.io/ for)[ for](https://termcomp.github.io/ for) the result data.  \n2 Semantic Labelling in Practice  \nconstraints for labelling, among other proof methods [3, 4] . Various termination tools include implementations of semantic labelling, including Torpa [25], TPA [13], Teparla, Jambox, AProVE [8], and TcT [2] .  \nWe mostly use standard notations from rewriting, formal languages and algebra, as in [6, 20 , 23] . For a a set R of strict rules (denoted by →) and a set S of nonstrict rules (denoted by → = ), we say that R ∪ S is terminating ","cbCaiaifACdUXzNC","https://ap.wps.com/l/cbCaiaifACdUXzNC","pdf",594695,2,1,12,"English","en",105,"# Introduction\n# Semantic Labelling\n## Definitions and Models","[{\"question\":\"What model-finding strategies are compared in the experiments?\",\"answer\":\"The work compares exhaustive enumeration for bounded domain sizes within restricted search spaces, and semantic context-closure for fixed algebras, evaluated using Matchbox and MnM.\"}]",1784188457,30,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"semantic-labelling-in-practice","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/semantic-labelling-in-practice/83500/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"What model-finding strategies are compared in the experiments?","Question",{"text":75,"@type":76},"The work compares exhaustive enumeration for bounded domain sizes within restricted search spaces, and semantic context-closure for fixed algebras, evaluated using Matchbox and MnM.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,114,119,122,126],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":29,"slug":113},"research-report",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]