[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81581-en":3,"doc-seo-81581-105":30,"detail-sidebar-cat-0-en-105":95},{"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},81581,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Unification of Deterministic Higher-Order Patterns","Sound and complete unification for deterministic higher-order patterns is presented for simply-typed lambda terms introduced by Yokoyama et al., together with a deterministic matching problem. The procedure is a special case of full higher-order unification, where flex-flex pairs are solved in a most general manner. It also generalizes functions-as-constructors higher-order unification by removing a global restriction on variable arguments. The remaining unifiers may form infinite minimal complete sets, so decidability stays open.","arXiv :2601 . 142 1 1v4 [ cs .LO] 10 Jul 2026  \nUnification of Deterministic Higher-Order  \nPatterns  \nJohannes Niederhauser (B) [0000−0002−8662−6834] and Aart Middeldorp [0000−0001−7366−8464]  \nDepartment of Computer Science, University of Innsbruck, Innsbruck, Austria {johannes.niederhauser,[aart.middeldorp}@uibk.ac.at](aart.middeldorp}@uibk.ac.at)  \nAbstract. We present a sound and complete unification procedure for deterministic higher-order patterns, a class of simply-typed lambda terms introduced by Yokoyama et al. which comes with a deterministic matching problem. Our unification procedure can be seen as a special case of full higher-order unification where flex-flex pairs can be solved in a most general way. Moreover, our method generalizes Libal and Miller’s recent functions-as-constructors higher-order unification (FCU) by dropping their global restriction on variable arguments, thereby losing the property that every solvable problem has a most general unifier. In fact, minimal complete sets of unifiers of deterministic higher-order patterns may be infinite, so decidability of the unification problem remains an open question.  \n1 Introduction  \nGiven two simply-typed λ-terms s and t, higher-order unification tries to answer the question whether they can be made equivalent up to βη-conversion by applying some substitution θ to both of them, i.e. , sθ ↔ η tθ . The first practical higher-order unification procedure has been introduced by Huet in 1975 [8] . Since then, higher-order unification has evolved into an extensively studied topic which plays a central role in research areas such as higher-order automated theorem proving [1, 3, 16], interactive theorem proving [13], higherorder logic programming [11] and higher-order rewriting [10] . In contrast to firstorder unification, which is decidable and unitary (most general unifiers exist), higher-order unification is undecidable and infinitary (infinitely many incomparable unifiers may exist) [5] . Since neither of these properties makes higher-order unification a well-behaved subroutine for tasks such as theorem proving, more well-behaved subclasses of higher-order unification have been identified and investigated in the past. One important subclass is higher-order pattern unification due to Miller [11] where the arguments of free variables are restricted to η-equivalent representations of distinct bound variables. Even though this rules out many simply-typed λ-terms, the unification procedure is sufficient for many practical purposes [10, 11] while being both decidable and unitary. Furthermore, higher-order pattern unification has linear time complexity [14] .  \n2 J. Niederhauser and A. Middeldorp  \nExample 1 (taken from [12]) . Let c be a constant and F , G be variables. The higher-order patterns λx, y.F x and λx, y.c (G y x) are unifiable with θ = {F 7→λz.c (H z), G 7→ λz 1 , z2.H z2 } where H is a fresh variable:  \n(λx, y.F x)θ =β λx, y.c (H x) =β (λx, y.c (G y x))θ  \nIn fact, θ is a most general unifier.  \nRecently, Libal and Miller have generalized higher-order pattern unification to functions-as-constructors higher-order unification (FCU) [9] . In this subclass, arguments of free variables may contain constants. In contrast to Miller’s patterns, it is not enough to demand distinctness of arguments of a free variable. Instead, no argument of a free variable may be a higher-order subterm of another argument of the same free variable. In addition, it is required that an argument of a free variable is not a strict higher-order subterm of any argument of any variable in the unification problem (global restriction) in order to retain unicity and decidability of the unification problem.  \nExample 2 (taken from [9]) . Let fst, snd be constants and X , Y be variables. The simply-typed λ-terms λl1 , l2.X (fstl1 )(fst (sndl1 )) and λl1 , l2.snd (Y (fstl2 )(fstl1 )) are unifiable with θ = {X 7→ λz 1 , z2.snd (H z1 ) , Y 7→ λz 1 , z2.H z2 } where H is afresh variable:  \n(λl1","cbCainPq44UYqoE0","https://ap.wps.com/l/cbCainPq44UYqoE0","pdf",549485,3,1,22,"English","en",105,"# Introduction\n## Background on higher-order unification\n## Pattern unification and its properties\n## Functions-as-constructors higher-order unification (FCU)\n## Deterministic higher-order patterns (DHPs) and relation to DSPs","[{\"question\":\"What problem does the document address?\",\"answer\":\"It addresses unification for deterministic higher-order patterns—simply-typed lambda terms with a deterministic matching problem—seeking a sound and complete unification procedure.\"},{\"question\":\"How does the proposed unification relate to full higher-order unification?\",\"answer\":\"The method can be viewed as a special case of full higher-order unification, where flex-flex pairs can be solved in a most general way.\"},{\"question\":\"What change is made compared to functions-as-constructors higher-order unification (FCU)?\",\"answer\":\"It generalizes FCU by dropping FCU’s global restriction on variable arguments, which removes the guarantee that every solvable problem has a most general unifier.\"},{\"question\":\"Is the decidability of unification settled for deterministic higher-order patterns?\",\"answer\":\"No. Minimal complete sets of unifiers of deterministic higher-order patterns may be infinite, so the decidability of the unification problem remains an open question.\"}]",1784174472,55,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":28},"unification-of-deterministic-higher-order-patterns","",{"@graph":36,"@context":89},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/unification-of-deterministic-higher-order-patterns/81581/",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","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the document address?","Question",{"text":75,"@type":76},"It addresses unification for deterministic higher-order patterns—simply-typed lambda terms with a deterministic matching problem—seeking a sound and complete unification procedure.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed unification relate to full higher-order unification?",{"text":80,"@type":76},"The method can be viewed as a special case of full higher-order unification, where flex-flex pairs can be solved in a most general way.",{"name":82,"@type":73,"acceptedAnswer":83},"What change is made compared to functions-as-constructors higher-order unification (FCU)?",{"text":84,"@type":76},"It generalizes FCU by dropping FCU’s global restriction on variable arguments, which removes the guarantee that every solvable problem has a most general unifier.",{"name":86,"@type":73,"acceptedAnswer":87},"Is the decidability of unification settled for deterministic higher-order patterns?",{"text":88,"@type":76},"No. Minimal complete sets of unifiers of deterministic higher-order patterns may be infinite, so the decidability of the unification problem remains an open question.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & 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