[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84703-en":3,"doc-seo-84703-105":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},84703,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Finite Observations, Infinite Behaviour: Bicategorical Semantics for Stateful Monoidal Processes","Time-dependent stateful processes evolve internally while an external observer only sees finite input/output measurements that each impose constraints on possible trajectories. The work introduces a semantic construction identifying two stateful processes whenever they satisfy the same observable constraints, independent of hidden internal states. The framework is defined for preorder-enriched monoidal categories with compatible discarding, yielding a discard bicategory that models partial, non-deterministic, probabilistic, and quantum processes. A functorial semantics for free feedback categories is obtained, plus a categorified compactness theorem extending finite observations to infinite closed relations.","arXiv :2607 .03996v 1 [ cs .LO] 4 Jul 2026  \nFinite Observations, Infinite Behaviour: bicategorical semantics for stateful monoidal processes.  \nCole Comfort 1 and Giovanni de Felice2  \n1 Université Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, CentraleSupélec, Laboratoire Méthodes Formelles, 91190 Gif-sur-Yvette, France  \n2 Relational Intelligence Ltd.  \nAbstract  \nTime-dependent processes are often described by machines with an internal state which is updated as time evolves. An external observer cannot see this state and learns about a process only through finite observations of its inputs and outputs, each of which imposes a constraint on the trajectories the process can exhibit. We introduce a semantic construction in which two stateful processes have the same behaviour when they have the same constraints, as determined by finite observations, independent of their internal state. The construction is defined over any preorder-enriched monoidal category with a compatible notion of discarding, which we call a discard bicategory, capturing partial, non-deterministic, probabilistic, and quantum processes. The resulting category of behaviours provides a functorial semantics for free feedback categories in the sense of Katis, Sabadini, and Walters. For non-deterministic systems, we prove a categorified compactness theorem: every compatible family of finite observations between compact Hausdorff spaces extends uniquely and functorially to an infinite closed relation. Restricted to affine relations over finite fields, the compactness theorem recovers Willems’ notion of behaviour for linear time-invariant systems.  \n1 Introduction  \nMany systems in science and engineering run indefinitely: signal flow graphs [60], stochastic processes [26], and quantum channels with memory [40] continuously transform inputs into outputs. A natural model for such systems is a stateful process: one that maintains internal memory coupling past inputs to future outputs. A common specification is a Mealy machine: a transition function f : S × X −→ S × Y which at each time step consumes an input in X , updates an internal state in S, and produces an output in Y [44] . The time evolution of a Mealy machine is obtained by initialising the state, repeatedly feeding the internal state back into the transition function, and discarding the final state:  \n\n| \u003Cbr> |\n| --- |\n|  |\n\n(t)  \n⇝  \n(1)  \nA Mealy machine f together with an initial state s0 ∈ S determines a stream transducer  \nrun (f, s0 ) : XN → YN , defined as the final fixed point of the coinductive equation run (f, s0 )(x0 , x 1 , . . . ) = 􀀀y0 , run (f, s1 )(x1 , x2 , . . . )􀀁 where (s1 , y0 ) = f(s0 , x0 ) .  \nTwo Mealy machines f and g, with initial states s0 and t0 , have the same behaviour when run (f, s0 ) = run (g, t0 ) . This coinductive characterisation is the standard definition of behavioural equivalence in the total deterministic setting. However, as soon as the transition function is  \nallowed to be partial, nondeterministic, probabilistic, or quantum, the coinductive fixed point need not exist, and this notion of behaviour breaks down. This raises the fundamental question that motivates the present paper:  \nWhen do two stateful processes produce the same infinite behaviour?  \nWe seek a denotational answer to this question, in the tradition initiated by Scott [58] . Our approach is process-theoretic [18]: we model processes as morphisms in symmetric monoidal categories, where different choices of base category accommodate partial [17], nondeterministic [13], probabilistic [27], and quantum [2] processes. Following Katis, Sabadini and Walters [35] , we construct feedback categories of stateful processes over these theories and seek a functorial semantics that identifies two such processes whenever they have the same observable behaviour.  \nWhat is behaviour? In Willems’ behavioural approach to systems theory, the behaviour of a time-dependent system is defined to be the set of infinite traje","cbCaipBahuQgaSe1","https://ap.wps.com/l/cbCaipBahuQgaSe1","pdf",1147069,1,54,"English","en",105,"# Introduction\n## Finite observations and observable behaviour\n## Observations as morphisms and discard bicategories\n## Infinite behaviour from finite observations\n## Properties of the construction","[{\"question\":\"How does the paper define when two stateful processes have the same behaviour?\",\"answer\":\"Two stateful processes are identified when they impose the same observable constraints, determined solely by finite input/output observations, regardless of internal state details. The central construction uses equivalence classes of compatible finite-observation families to formalize this.\"},{\"question\":\"Why does the usual coinductive notion of behavioural equivalence fail for general process types?\",\"answer\":\"For partial, non-deterministic, probabilistic, or quantum transition structures, the required coinductive fixed point may not exist. This makes the standard coinductive characterisation of behaviour break down.\"},{\"question\":\"What role does the “discard bicategory” play in the framework?\",\"answer\":\"Observations must be comparable via preorder enrichment and must be restrictable to smaller contexts via discarding morphisms. A monoidal category with these compatible structures is called a discard bicategory, forming the basis for defining behaviour from finite observations.\"}]",1784197748,136,{"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},"finite-observations-infinite-behaviour-bicategorical-semantics-for-stateful-monoidal-processes","",{"@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/finite-observations-infinite-behaviour-bicategorical-semantics-for-stateful-monoidal-processes/84703/",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","2026-07-17","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},"How does the paper define when two stateful processes have the same behaviour?","Question",{"text":75,"@type":76},"Two stateful processes are identified when they impose the same observable constraints, determined solely by finite input/output observations, regardless of internal state details. The central construction uses equivalence classes of compatible finite-observation families to formalize this.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does the usual coinductive notion of behavioural equivalence fail for general process types?",{"text":80,"@type":76},"For partial, non-deterministic, probabilistic, or quantum transition structures, the required coinductive fixed point may not exist. This makes the standard coinductive characterisation of behaviour break down.",{"name":82,"@type":73,"acceptedAnswer":83},"What role does the “discard bicategory” play in the framework?",{"text":84,"@type":76},"Observations must be comparable via preorder enrichment and must be restrictable to smaller contexts via discarding morphisms. A monoidal category with these compatible structures is called a discard bicategory, forming the basis for defining behaviour from finite observations.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]