[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85396-en":3,"doc-seo-85396-105":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},85396,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Explorable Parity Automata","Explorable automata are introduced over finite and infinite words as a generalisation of History-Deterministic (HD) automata, allowing non-deterministic choices to be resolved by finitely many simultaneous runs rather than a single on-the-fly run. Recognition of HD parity automata of fixed index among explorable ones is shown to be in PTime. Explorable automata recognition is ExpTime-complete for finite words and for parity automata up to index [0,2], and ω-explorable automata extend the model to countably many runs, yielding ExpTime-completeness for safety and coBüchi acceptance.","arXiv :2410 .23 187v 3 [ cs .FL] 13 Jul 2026  \nEXPLORABLE PARITY AUTOMATA  \nEMILE HAZARD a , OLIVIER IDIR  b , AND DENIS KUPERBERG  a  \ne-mail address: [emile.hazard@ens-lyon.fr](emile.hazard@ens-lyon.fr)  \ne-mail address: [olivier.idir@irif.fr](olivier.idir@irif.fr)  \ne-mail address: [denis.kuperberg@ens-lyon.fr](denis.kuperberg@ens-lyon.fr)  \na CNRS, LIP, ENS Lyon, France  \nb IRIF, Universit´e Paris-Cit´e, France  \nAbstract. We define the class of explorable automata on finite or infinite words. This is a generalisation of History-Deterministic (HD) automata, where this time non-deterministic choices can be resolved by building finitely many simultaneous runs instead of just one.  \nWe show that recognizing HD parity automata of fixed index among explorable ones isin PTime, thereby giving a strong link between the two notions. We then show that recognizing explorable automata is ExpTime-complete, in the case of finite words or parity automata up to index [0 , 2] . Additionally, we define the notion of ω-explorable automata on infinite words, where countably many runs can be used to resolve the non-deterministic choices. We show ExpTime-completeness for ω-explorability of automata on infinite words for the safety and coB¨uchi acceptance conditions. We finally characterize the expressivity of (ω-)explorable automata with respect to the parity index hierarchy.  \n1. Introduction  \nIn several fields of theoretical science, the tension between deterministic and non-deterministic models is a source of fundamental open questions, and has led to important lines of research. The most famous of this kind is the P vs NP question in complexity theory. This paper aims at further investigating the frontier between determinism and non-determinism in automata theory. Although Non-deterministic and Deterministic Finite Automata (NFA and DFA) are known to be equivalent in terms of expressive power, many subtle questions remain about the cost of determinism, and a deep understanding of non-determinism will be needed to solve them.  \nOne of the approaches investigating non-determinism in automata is the study of History-Deterministic (HD) automata, introduced in [HP06] under the name Good-ForGames (GFG) automata. An automaton is HD if, when reading input letters one by one, its non-determinism can be resolved on-the-fly without any need to guess the future. This constitutes a model that is intermediary between non-determinism and determinism, and can sometimes bring the best of both worlds. Like deterministic automata, HD automata on infinite words retain good properties such as their soundness with respect to composition with games, making them appropriate for use in Church synthesis algorithms [HP06] . On the other hand, like non-deterministic automata, they can be exponentially more succinct  \nthan deterministic ones [KS15] . There is a very active line of research trying to understand the various properties of HD automata, see e.g. [AK22, BKLS20 , BL22 , Cas23] for some of the recent developments. The terminology history-deterministic, was introduced originally in the theory of regular cost functions [Col09] . The name “history-deterministic” corresponds to the above intuition of solving non-determinism on-the-fly, while the earlier name of“good-for-games” refers to sound composition with games. These two notions may actually differ in some quantitative frameworks, but coincide on Boolean automata [BL21], and have been used interchangeably in most of the literature on the topic. In this paper, since we are mainly interested in resolving the non-determinism on-the-fly, we choose the HD denomination to emphasize this aspect 1.  \nThe goal of this paper is to pursue this line of research by introducing and studying the class of explorable automata on finite and infinite words. The intuition behind explorability is to limit the amount of non-determinism required by the automaton to accept its language, in a more permissive way than HD automata. If","cbCaiugNA7qtiviZ","https://ap.wps.com/l/cbCaiugNA7qtiviZ","pdf",475151,1,33,"English","en",105,"# Introduction\n## Relationship to History-Deterministic (HD) automata\n## Definition of k-explorable and explorable automata\n## Complexity and decidability results for finite words\n## Extension to ω-explorable automata on infinite words","[{\"question\":\"What are explorable automata and how do they generalize HD automata?\",\"answer\":\"Explorable automata over finite or infinite words restrict how much non-determinism is used to accept a language. They generalize History-Deterministic (HD) automata by allowing non-deterministic choices to be resolved using finitely many simultaneous runs instead of just one.\"},{\"question\":\"What is the complexity of recognizing HD parity automata of fixed index within explorable automata?\",\"answer\":\"Recognizing HD parity automata of fixed index among explorable ones is in PTime.\"},{\"question\":\"How does ω-explorability extend the model to infinite words, and what are the complexity results?\",\"answer\":\"ω-explorability allows countably many runs to resolve non-deterministic choices on infinite words. The ω-explorability problem is ExpTime-complete for safety and coBüchi acceptance conditions, and expressivity is characterized with respect to the parity index hierarchy.\"}]",1784203109,83,{"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},"explorable-parity-automata","",{"@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/explorable-parity-automata/85396/",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 are explorable automata and how do they generalize HD automata?","Question",{"text":75,"@type":76},"Explorable automata over finite or infinite words restrict how much non-determinism is used to accept a language. They generalize History-Deterministic (HD) automata by allowing non-deterministic choices to be resolved using finitely many simultaneous runs instead of just one.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the complexity of recognizing HD parity automata of fixed index within explorable automata?",{"text":80,"@type":76},"Recognizing HD parity automata of fixed index among explorable ones is in PTime.",{"name":82,"@type":73,"acceptedAnswer":83},"How does ω-explorability extend the model to infinite words, and what are the complexity results?",{"text":84,"@type":76},"ω-explorability allows countably many runs to resolve non-deterministic choices on infinite words. 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