[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82405-en":3,"doc-seo-82405-105":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},82405,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Solving the Reachability Problem for Branching Vector Addition Systems via Semilinear Inductive Invariants","Solving the reachability problem for branching vector addition systems (BVAS), the work addresses a long-standing open question by leveraging semilinear inductive invariants. The key result establishes a separating semilinear invariant when a target configuration is unreachable: an inductive invariant excludes that configuration. This separation property enables a straightforward enumerative decision algorithm that searches for such invariants. The paper also situates BVAS within related models from concurrency, logics, and automata theory.","arXiv :2607 .09558v 1 [ cs .LO] 10 Jul 2026  \nSolving the Reachability Problem for Branching Vector Addition Systems via Semilinear Inductive Invariants  \nCLOTILDE BIZIÈRE, LaBRI, University of Bordeaux, CNRS, Bordeaux INP, France and Institute of Informatics, University of Warsaw, Poland  \nJÉRÔME LEROUX, LaBRI, University of Bordeaux, CNRS, Bordeaux INP, France GRÉGOIRE SUTRE, LaBRI, University of Bordeaux, CNRS, Bordeaux INP, France  \nIn this paper, we solve the reachability problem for branching vector addition systems (BVAS), a long standing open problem. Our approach is based on semilinear inductive invariants. More precisely, we prove that ifa configuration of a BVAS is not reachable, then there exists an inductive invariant, given as a semilinear set, that does not contain this configuration. Based on this property, we deduce a very simple (enumerative) algorithm solving the reachability problem for BVAS.  \n1 Introduction  \nContext. Branching vector addition systems (BVAS) are a computational model for the distribution of additive resources through branching structures. The model can be traced back to the work of Rambow [35] in computational linguistics. To overcome the limitations of context-free grammarsas a model of natural-language syntax, Rambow introduced multiset-valued linear index grammars (MV-LIG), in which nonterminals carry multisets of resources that are distributed among the children of a derivation node. When studying the language-emptiness problem for such grammars, the left-to-right order of derivation trees becomes irrelevant. The resulting abstraction is precisely a BVAS, and language emptiness can equivalently be viewed as a reachability problem.  \nA decade later, the same model was independently rediscovered in two seemingly unrelated contexts. Verma and Goubault-Larrecq [40] used BVAS to study a class of equational tree automata arising in the analysis of cryptographic protocols, obtaining decidability results for a lossy variant of the model. Around the same time, de Groote, Guillaume, and Salvati [9] proposed BVAS reachability as a natural automata-theoretic reformulation of provability in multiplicative exponential linear logic (MELL) . In this setting, BVAS executions capture the flow of resources through proof trees.  \nSince then, BVAS and closely related models have appeared in a variety of areas of theoretical computer science, including logics over data trees [1, 4, 17], timed pushdown systems [6], verification of concurrent systems [5], and the semantics of higher-order functional languages [7] .  \nFormally, a BVAS is a finite sequence B := (∆1, . . . , ∆ 􀁁) of finite subsets of Z􀀳 . Each set ∆ 􀀽 in this sequence contains actions of arity 􀀽 . Intuitively, such an action v ∈ ∆ 􀀽 may be viewed as therewriting rule x1, . . . , x 􀀽 → v + x1 + · · · + x 􀀽 with formal parameters x1, . . . , x 􀀽 . Configurations ofBare vectors in N􀀳 , and executions are finite trees labeled by configurations in which every internal node is obtained by summing its 􀀽 ≥ 1 children and adding an action from ∆ 􀀽. The reachability problem asks, given a BVAS, a target configuration and a finite set of initial configurations, whether there exists an execution with root labeled by the target and leaves labeled by initial configurations. The goal of this paper is to provide a solution to this long standing open problem.  \nDecidability of the VAS Reachability Problem. Vector addition systems (VAS), or equivalently Petri nets, are obtained as the special case where 􀁁 = 1. Executions then degenerate into sequences of configurations, and reachability asks whether a given target configuration can be obtained from an initial one by repeatedly applying additive actions from ∆1 .  \nAuthors’ Contact Information: Clotilde Bizière, LaBRI, University of Bordeaux, CNRS, Bordeaux INP, Talence, France and Institute of Informatics, University of Warsaw, Warsaw, Poland; Jérôme Leroux, LaBRI, University of Bordeaux, CNRS, Bordeaux INP, Tale","cbCaibdriX5vFunT","https://ap.wps.com/l/cbCaibdriX5vFunT","pdf",714031,1,32,"English","en",105,"# Introduction\n## Background and Formal Model\n## Decidability of the VAS Reachability Problem\n## Challenges of BVAS Reachability","[{\"question\":\"What problem does the paper solve for branching vector addition systems (BVAS)?\",\"answer\":\"It solves the BVAS reachability problem: determining whether a target configuration can be obtained from given initial configurations via a branching execution.\"},{\"question\":\"How do semilinear inductive invariants contribute to the solution?\",\"answer\":\"If a configuration is unreachable, the paper proves the existence of an inductive invariant given as a semilinear set that contains the initials but excludes the unreachable configuration.\"},{\"question\":\"What algorithmic outcome is derived from the separation property?\",\"answer\":\"The paper derives a simple enumerative algorithm that, by enumerating executions and semilinear inductive invariants in parallel, decides reachability for BVAS.\"}]",1784180157,81,{"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},"solving-the-reachability-problem-for-branching-vector-addition-systems-via-semilinear-inductive-invariants","",{"@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/solving-the-reachability-problem-for-branching-vector-addition-systems-via-semilinear-inductive-invariants/82405/",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},"What problem does the paper solve for branching vector addition systems (BVAS)?","Question",{"text":75,"@type":76},"It solves the BVAS reachability problem: determining whether a target configuration can be obtained from given initial configurations via a branching execution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do semilinear inductive invariants contribute to the solution?",{"text":80,"@type":76},"If a configuration is unreachable, the paper proves the existence of an inductive invariant given as a semilinear set that contains the initials but excludes the unreachable configuration.",{"name":82,"@type":73,"acceptedAnswer":83},"What algorithmic outcome is derived from the separation property?",{"text":84,"@type":76},"The paper derives a simple enumerative algorithm that, by enumerating executions and semilinear inductive invariants in parallel, decides reachability for 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