[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82759-en":3,"doc-seo-82759-105":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},82759,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","Second-Order Quantifier Elimination and Uniform Interpolation for Bath Path Logic and the Ordered Fragment","Second-order quantifier elimination and uniform interpolation are studied for basic path logic and for the ordered fragment of first-order logic, both arising from functional translations of modal logic. Binary resolution is shown to decide uniform interpolation and to compute uniform interpolants for basic path logic. Using constant Skolemisation, sentences in the ordered fragment are transformed into basic path logic while preserving logical consequences. The SCAN algorithm’s search space is characterized and termination yields decision of second-order quantifier elimination for this class; extracting uniform interpolants from SCAN’s output in the ordered fragment remains open.","arXiv :2607 .03645v 1 [ cs .LO] 3 Jul 2026  \nSecond-Order Quantifier Elimination and Uniform Interpolation for Bath Path Logic and the Ordered  \nFragment  \nRenate A. Schmidt 1* and Hongkai Yin2  \n1* Department of Computer Science, The University of Manchester,  \nOxford Road, Manchester, M13 9PL, UK.  \n2 Department of Philosophy, Central European University, Quellenstraße  \n51, Wien, 1100, Austria.  \nAbstract  \nWe consider and extend results on basic path logic and the ordered fragment of first-order logic, both of which originate from the functional translation of modal logic. Basic path logic is a subclass of the clausal class for the ∃ ∗∀∗-fragment and has the remarkable property that binary resolution decides it. This decidability result and the consequence finding completeness of binary resolution allows us to observe that binary resolution also decides uniform interpolation and computes uniform interpolants for basic path logic. By introducing constant Skolemisation, we show that sentences of the ordered fragment can be transformed into basic path logic, and this transformation preserves logical consequences in the ordered fragment. We characterise the search space of the SCAN algorithm on the clausal form of the ordered fragment by a variation of basic path logic and prove that SCAN terminates on this class, and therefore it decides second-order quantifier elimination for this class. It remains unclear whether uniform interpolants in the ordered fragment can be extracted from the output of SCAN.  \nKeywords: First-order logic, Second-order quantifier elimination, Uniform  \ninterpolation, SCAN, Resolution  \n1  \n1 Introduction  \nIn this paper we investigate the application of the SCAN algorithm [1] and resolution in solving second-order quantifier elimination and computing uniform interpolants for basic path logic [2, 3] .  \nWe say that a clause is prefix stable for variables if for any variable in the clause, its prefix in any argument sequence (in which it occurs) is the same [4] . For example, the first clause below is prefix stable for variables and the second is not (where y has two different prefixes):  \nP (x, y) ∨ Q(x, y, a)  \nP (x, y) ∨ Q(a, y, z) .  \nBy basic path logic (BPL) we understand the clausal class in which: (i) all argument sequences contain only variables and constants, and (ii) all clauses have prefix stability for variables. This implies BPL is a subclass of the clausal class for the ∃ ∗∀∗-fragment of first-order logic.  \nBasic path logic has been introduced as the clausal class associated with modal logic KD based on the world path semantics and the functional translation to firstorder logic [2, 3] . Basic path logic is computationally well-behaved: it is decidable by resolution without special refinements (no ordering- or selection-based restriction of inferences is needed) [3], which implies that any sound and refutationally complete ordering- and/or selection-based refinement provides a decision procedure for this class.1 Further, the performance of resolution theorem provers is superior for modal logic problems when they are mapped to basic path logic than when they are mapped to the guarded fragment using the standard relational translation [6, 7] .  \nThe ordered fragment of first-order logic consists of those formulas in which, roughly speaking: (i) the argument sequence of each predicate is of the form (x1 , x2 ,..., xn ); and (ii) if i \u003C j, no quantifier binding xj out-scopes any quantifier binding xi. For example, the following sentence belongs to the fragment:  \n∀x1 ∃x2 (R(x1 , x2 ) ∧ ∀x3 ¬S (x1 , x2 , x3 )) .  \nThe ordered fragment first appeared in the “fluted schema” introduced by W. V. Quine [8, 9], but the formal definition we follow was given, independently of Quine’s work, by A. Herzig [10], where the term “ordered formula” was coined. Herzig proposed alinear-time reduction from satisfiability of ordered formulas to satisfiability in KD , which implies that the former is decidable in ","cbCaissKsPYCtkp2","https://ap.wps.com/l/cbCaissKsPYCtkp2","pdf",607065,1,27,"English","en",105,"# Introduction\n## Prefix stability and basic path logic\n## Ordered fragment of first-order logic\n## Second-order quantifier elimination and uniform interpolants\n## Connection to SCAN and resolution","[{\"question\":\"What does the paper show about binary resolution for basic path logic?\",\"answer\":\"Binary resolution decides uniform interpolation for basic path logic and computes uniform interpolants for it, using properties of decidability and completeness linked to binary resolution.\"},{\"question\":\"How are ordered-fragment sentences related to basic path logic?\",\"answer\":\"By introducing constant Skolemisation, sentences in the ordered fragment can be transformed into basic path logic, and the transformation preserves logical consequences within the ordered fragment.\"},{\"question\":\"What role does the SCAN algorithm play in deciding second-order quantifier elimination?\",\"answer\":\"The paper characterizes SCAN’s search space on the clausal form of the ordered fragment via a variation of basic path logic, then proves SCAN terminates on this class, which therefore decides second-order quantifier elimination for it.\"}]",1784182747,68,{"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},"second-order-quantifier-elimination-and-uniform-interpolation-for-bath-path-logic-and-the-ordered-fragment","",{"@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/second-order-quantifier-elimination-and-uniform-interpolation-for-bath-path-logic-and-the-ordered-fragment/82759/",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 does the paper show about binary resolution for basic path logic?","Question",{"text":75,"@type":76},"Binary resolution decides uniform interpolation for basic path logic and computes uniform interpolants for it, using properties of decidability and completeness linked to binary resolution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are ordered-fragment sentences related to basic path logic?",{"text":80,"@type":76},"By introducing constant Skolemisation, sentences in the ordered fragment can be transformed into basic path logic, and the transformation preserves logical consequences within the ordered fragment.",{"name":82,"@type":73,"acceptedAnswer":83},"What role does the SCAN algorithm play in deciding second-order quantifier elimination?",{"text":84,"@type":76},"The paper characterizes SCAN’s search space on the clausal form of the ordered fragment via a variation of basic path logic, then proves SCAN terminates on this class, which therefore decides second-order 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