[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84613-en":3,"doc-seo-84613-105":29,"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":20,"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":13,"seo_description":14,"update_tm":27,"read_time":28},84613,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Definability of Functional Properties in the Basic Modal-Temporal Language over Ordered Frames","Studies the expressive power of the basic modal-temporal language formed by extending the modal language with Prior’s temporal operators G and H. The work analyzes definability across five order types using two semantic readings: standard (G,H) including the current instant and strict (G*,H*) excluding it. Nine functional properties—such as totality, injectivity, surjectivity, monotonicity, antitonicity, and constancy—are tested over diverse preorder/partial/linear structures, revealing two expressivity levels and showing that non-linearity hinders definability.","arXiv :2607 .0 1 1 10v 1 [ cs .LO] 1 Jul 2026  \nDefinability of Functional Properties in the Basic Modal-Temporal Language over Ordered Frames  \nAlfredo Burrieza  \nUniversity of Malaga  \n[burrieza@uma.es](burrieza@uma.es)  \nAbstract  \nWe study the expressive power of the simplest modal-temporal language, obtained by adding Prior’s temporal operators G and H to the basic modal language with ✷ . This language is the standard bimodal combination of modal and tense logic; under its functional interpretation it is denoted LT × W in the literature. To analyse its definability across five order types, we consider two semantic readings of the temporal operators: the standard reading (G, H), which includes the current instant, and the strict reading (G∗ , H ∗ ), which always excludes it. We examine nine functional properties—totality, non-totality, injectivity, surjectivity, monotonicity, strict monotonicity, antitonicity, strict antitonicity, and constancy—over preorders, strict preorders, partial orders, linear orders, and strict linear orders. Our analysis reveals two different levels of expressive power. In the original multiflow setting (where multiple functions coexist and the modal operators quantify indiscriminately over their images), the language is quite weak; the two readings of G, H coincide. When we restrict the semantics to minimal functional frames (the O2 family), many properties become definable, and the choice of reading becomes crucial: the strict reading can define properties such as injectivity even in reflexive orders. The same definability patterns appear with indexed languages and with the Uniform Domain (U-Dom) condition on the semantics of LT × W . That three such different ways of controlling functional multiplicity lead to identical definability patterns indicates that the expressive limitations of the original framework come from the uncontrolled multiplicity of functions, not from any weakness of the operators. Even after controlling functional multiplicity, a set of properties remains undefinable in all non-linear orders, showing that the lack of connectivity is a fundamental obstacle.  \n1 Introduction  \nWe study the definability of functional properties in the simplest modal-temporal language, obtained by adding the temporal operators G and H to the basic  \n2  \nmodal language with ✷ . This language is the standard bimodal combination of modal and tense logic (see e.g. [14]); in [1] it was given a functional interpretation and denoted LT×W .  \nThroughout the paper we consider two semantic interpretations of the temporal operators G, H: the standard interpretation, which follows the original semantics (including the current instant when a reflexive loop is present), and the strict interpretation, which always excludes the current instant. For clarity, we keep the notation G, H for the standard interpretation, and denote the strict interpretation by G∗ , H ∗ . This notational convention allows us to compare both interpretations without mixing them within the same formula.  \nIn [1, 3], the study of functional properties using LT×W was restricted to strict linear orders and global constraints—such as totality or the Uniform Domain (U-Dom) property. Two questions remained open: whether these properties are definable in isolation, and how their definability varies across a wider range of order structures  \nThe present study addresses these questions by providing a systematic account of the expressive limits of LT×W across a wide range of order structures, including (strict) linear orders, partial orders, and (strict) preorders. We examine these functional properties independently of the global conditions previously required, such as totality or the Uniform Domain (U-Dom) property, to determine if they are definable in isolation. The functional properties under study are: totality, non-totality, injectivity, surjectivity, monotonicity, strict monotonicity, antitonicity, strict antitonicity, and constancy.  \nRat","cbCaifZXEAGrk75n","https://ap.wps.com/l/cbCaifZXEAGrk75n","pdf",711907,1,42,"English","en",105,"# Abstract\n# Introduction\n## Language and semantic interpretations\n## Functional properties studied\n## Two-stage analysis approach","[{\"question\":\"What language and temporal operators are investigated in the paper?\",\"answer\":\"The paper studies the simplest modal-temporal language obtained by adding Prior’s temporal operators G and H to a basic modal language with the modality operator ✷.\"},{\"question\":\"How do the standard and strict readings of G and H differ?\",\"answer\":\"The standard reading (G,H) includes the current instant when a reflexive loop is present, while the strict reading (G*,H*) always excludes the current instant.\"},{\"question\":\"Which kinds of order structures are used to test definability, and what is the key limitation found?\",\"answer\":\"Definability is examined over preorders, strict preorders, partial orders, linear orders, and strict linear orders. The analysis finds that a set of properties remains undefinable in all non-linear orders, indicating that lack of connectivity is a fundamental obstacle.\"}]",1784197128,106,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"definability-of-functional-properties-in-the-basic-modal-temporal-language-over-ordered-frames","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/definability-of-functional-properties-in-the-basic-modal-temporal-language-over-ordered-frames/84613/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-20","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What language and temporal operators are investigated in the paper?","Question",{"text":75,"@type":76},"The paper studies the simplest modal-temporal language obtained by adding Prior’s temporal operators G and H to a basic modal language with the modality operator ✷.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the standard and strict readings of G and H differ?",{"text":80,"@type":76},"The standard reading (G,H) includes the current instant when a reflexive loop is present, while the strict reading (G*,H*) always excludes the current instant.",{"name":82,"@type":73,"acceptedAnswer":83},"Which kinds of order structures are used to test definability, and what is the key limitation found?",{"text":84,"@type":76},"Definability is examined over preorders, strict preorders, partial orders, linear orders, and strict linear orders. 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