[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83401-en":3,"doc-seo-83401-105":30,"detail-sidebar-cat-0-en-105":83},{"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},83401,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","On Constructing Most General Solutions for Parametric Constraints (Extended Preprint)","This work studies theories T with elimination of existential quantifiers, focusing on constraints of the form ∃x1…∃xn ϕ(x1,…,xn,y1,…,ym) where ϕ is a quantifier-free conjunction of literals and y1…ym are treated as parameters. The paper derives necessary and sufficient parameter conditions via quantifier elimination, and shows that suitable “if-then-else” function symbols in models of T enable effective description of most general solutions. Examples illustrate the construction.","arXiv :2607 .08582v 1 [ cs .LO] 9 Jul 2026  \nOn Constructing Most General Solutions for Parametric Constraints (Extended Preprint)  \nViorica Sofronie-Stokkermans [0000−0002−8486−9955]  \nUniversity of Koblenz, Koblenz, Germany  \n[sofronie@uni-koblenz.de](sofronie@uni-koblenz.de)  \nAbstract. Let T be a theory allowing a form of elimination of existential quantifiers (possibly for formulae in a certain class) . We analyze possibilities of constructing (most general) solutions w.r.t. T for formulae of the form ∃x1 ... ∃xn ϕ (x1 ,..., xn , y1 ,..., ym), where ϕ is a quantifierfree conjunction of literals in the signature of T, and the free variables y1 , . . . , ym are regarded as parameters. We show that in the presence of function symbols which describe “if-then-else” constructions in certain models of T, we can describe the most general solution of such formulae, thus generalizing results about the existence of most general unifiers in discriminator varieties. We illustrate the ideas on examples.  \n1 Introduction  \nFinding solutions for parametric constraints is important in many applications. While state of the art SMT-systems can generate models for satisfiable formulae, they usually do not distinguish between parameters (constants/function symbols) and existentially quantified symbols. In this paper we analyze possibilities of constructing parametric solutions and even most general solutions of parametric constraints w.r.t. logical theories T allowing some form of quantifier elimination. The constraints we consider are formulae of the form  \n∃x1 ,..., xn ϕ (x1 ,..., xn , y 1 ,..., ym), (1)  \nwhere ϕ is a conjunction of literals in the signature of T. Our goals are two-fold:(i) Find a necessary and sufficient condition on y 1 , . . . , ym , seen as parameters, under which ∃x1 ,..., xn ϕ (x1 ,..., xn , y 1 ,..., ym) evaluates to true w.r.t. T.  \n(ii) Analyze possibilities of obtaining most general solutions for problems of form (1), i.e. substitutions from which all solutions can be retrieved as instances.  \nWe address (i) by showing that if T allows a form of quantifier elimination for a fragment to which the formula ϕ belongs, the quantifier free formula ψ (y1 ,..., ym) equivalent to ∃x1 ,..., xn ϕ (x1 ,..., xn , y 1 ,..., ym) w.r.t. T is such a necessary and sufficient condition. For (ii), we show that in the presence of suitable “if-then-else” constructions in certain models of T we can effectively describe the most general solution.  \n2 Viorica Sofronie-Stokkermans  \nOur results are inspired by the study of unification in Boolean algebras [4,9,11,3], in varieties generated by primal algebras [10] and, more generally, in discriminator varieties [5] . Consider the case of unification in Boolean algebras i.e. problems of the form ∃x((ta ∧ x) ∨ (tb ∧ ¬x) ≈ 0) , where ta and tb are terms containing other variables, which can be considered to be parametric.  \nIt is known [4,9,11] that for every Boolean algebra B , if a, b ∈ B then the equation (a∧x)∨ (b∧¬x) ≈ 0 has a solution x in B iff a∧b = 0, and in this case the set of solutions in B is the interval S = [b, ¬a] of B. If a∧b = 0 then the set of solutions can also be described using the reproductive solution, F : B → B defined by F (x) = (b ∨ x) ∧ ¬a, which has the properties:  \n(a) F (x) ∈ S for every x ∈ B (i.e. F is a parametric solution);  \n(b) For every x ∈ S we have F (x) = x.  \nFor problems of the form ∃x1..., ∃xn (f(x1...., xn) ≈ 0), where f is an n-ary Boolean function, the variables can be eliminated successively and the reproductive solutions can be reconstructed. If S is the set of all solutions in an algebra B , a reproductive solution is a map F : Bn → Bn with  \n(a) F (x) ∈ S for every x ∈ Bn ; and (b) For every x ∈ S we have F (x) = x.  \nThese ideas have been generalized in [10] to varieties generated by primal algebras and in [5] to discriminator varieties, i.e. classes of algebras generated by a subset K of algebras allowing a switching term, i.e. a term s (x,","cbCaikHJcDSLCxWW","https://ap.wps.com/l/cbCaikHJcDSLCxWW","pdf",570732,2,1,28,"English","en",105,"# Introduction\n## Goals and problem form\n# Related ideas and background","[{\"question\":\"How are most general solutions constructed and why are “if-then-else” operations important?\",\"answer\":\"When suitable “if-then-else” function symbols exist in certain models of T, the paper shows that the most general solution of such parametric constraints can be effectively described, generalizing related results from discriminator varieties.\"}]",1784187258,71,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"on-constructing-most-general-solutions-for-parametric-constraints-extended-preprint","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-constructing-most-general-solutions-for-parametric-constraints-extended-preprint/83401/",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","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How are most general solutions constructed and why are “if-then-else” operations important?","Question",{"text":75,"@type":76},"When suitable “if-then-else” function symbols exist in certain models of T, the paper shows that the most general solution of such parametric constraints can be effectively described, generalizing related results from discriminator varieties.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]