[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83994-en":3,"doc-seo-83994-105":30,"detail-sidebar-cat-0-en-105":92},{"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},83994,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Extending the Ginsburg–Spanier Theorem to Functions and Mixed Arithmetic","Studies sets and functions definable in three additive logics: Presburger arithmetic FO(Z,+,≤), the real additive theory FO(R,+,≤), and the mixed additive theory FO(R,Z,+,≤). The Ginsburg–Spanier theorem characterizes Presburger definable sets as semi-linear; the work extends this by giving an exact algebraic description of definable functions. Presburger and real theories yield precisely piecewise-linear functions. For the mixed theory, semi-polinear sets are introduced and shown to match mixed-definable sets, while mixed-definable functions are exactly piecewise-simple. Proofs are purely algebraic, using linear algebra and the Ginsburg–Spanier theorem, and correct a prior error.","EXTENDING THE GINSBURG–SPANIER THEOREM TO FUNCTIONS  \nAND MIXED ARITHMETIC  \nALAIN FINKEL  AND JÉRÔME LEROUX   \narXiv :2607 .0570 1v2 [ cs .LO] 8 Jul 2026  \na Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190, Gif-sur-Yvette, France [e-mail address](e-mail address: finkel@ens-paris-saclay.fr)[: finkel@ens-paris-saclay.fr](e-mail address: finkel@ens-paris-saclay.fr)  \nb LaBRI, Université de Bordeaux, CNRS, LaBRI, 33405, Talence, France e-mail address: [leroux@labri.fr](leroux@labri.fr)  \nAbstract . We study sets and functions definable in three classical additive theories: Presburger arithmetic (first-order logic over the integers with addition and order), the real additive theory (first-order logic over the reals with addition and order), and the mixed additive theory (first-order logic over both reals and integers with addition and order) .  \nThe Ginsburg–Spanier theorem characterizes the sets definable in Presburger arithmetic as exactly the semi-linear sets. We extend this characterization in two directions.  \nFirst, we show that the functions definable in Presburger arithmetic are exactly the piecewise linear functions, and that the same holds for the real additive theory. The proofs are direct algebraic arguments using only a stability lemma and the Ginsburg– Spanier theorem.  \nSecond, we introduce semi-polinear sets as the analogue of semi-linear sets for the mixed additive theory, and prove that the mixed-definable sets are exactly the semi-polinear sets. We further show that the functions definable in the mixed additive theory are exactly the piecewise-simple functions, a new class of functions that are linear separately in the integer part and in the fractional part of the argument, but with potentially different linear coefficients for each part.  \nThese algebraic characterizations unify the three theories in a single geometric framework. The proofs are purely algebraic, using only linear algebra and the Ginsburg–Spanier theorem, without reference to automata, quantifier elimination, or machines. We also correct an error in a prior characterization of mixed-definable sets.  \n1. Introduction  \nContext. Presburger arithmetic FO(Z, + , ≤) [Pre29] is the decidable first-order theory of the integers with addition and order. The Ginsburg–Spanier theorem [GS66] characterizes its definable sets as exactly the semi-linear sets, i.e., finite unions of linear sets. The real additive theory FO(R, + , ≤) is decidable via Fourier–Motzkin elimination, with definable sets being exactly the finite unions of polyhedral convex sets [Wei88] . The mixed additive theory FO(R, Z, + , ≤) was shown decidable by Weispfenning [Wei99] via quantifier elimination, and later by Boigelot, Jodogne and Wolper [BJW05] via an alternative approach using weak Büchi automata.  \nPreprint submitted to © Finkel and Leroux  \nLogical Methods in Computer Science ⃝CC Creative Commons  \nConcerning Presburger functions, prior work studied computability by flat (without nested loops) counter automata [MR67, Che76, Gur85] or by reversal-bounded counter automata [GI79], closure under composition [IL81], and evaluation in linear space and quadratic time [CI83] .  \nMotivation. The three additive theories FO(Z, + , ≤) , FO(R, + , ≤), and FO(R, Z, + , ≤) are well-studied logics with applications across verification, model checking, and constraint solving. A natural question is: what are the functions definable in these logics, described in purely geometric terms? Prior characterizations of Presburger-definable functions all relied on operational or logical descriptions — counter automata [MR67, Che76, GI79], closure under composition [IL81], or logical normal forms — and the same holds for the mixed theory via quantifier elimination [Wei99] or Büchi automata [BJW05] . Surprisingly, a direct geometric answer was missing from the literature.  \n\n|  | FO(Z, + , ≤) | FO(R, + , ≤) | FO(R, Z, + , ≤) |\n| --- | --- | --- | --- |\n| Decidability | [Pre29], 2-EXPSPACE [FR7","cbCaih1PzUQZW9PW","https://ap.wps.com/l/cbCaih1PzUQZW9PW","pdf",467274,5,1,12,"English","en",105,"# Introduction\n## Context and Motivation\n## Related Work\n## Contributions and Result Depth","[{\"question\":\"What logics does the paper study, and what do they assume about addition and order?\",\"answer\":\"It studies Presburger arithmetic FO(Z,+,≤), the real additive theory FO(R,+,≤), and the mixed additive theory FO(R,Z,+,≤). Each is a first-order logic over integers, reals, or both, with addition and order.\"},{\"question\":\"How does the paper extend the Ginsburg–Spanier theorem?\",\"answer\":\"It extends the Ginsburg–Spanier characterization from semi-linear definable sets to algebraic characterizations of definable functions. For Presburger and real theories, it shows definable functions are exactly piecewise-linear, and for the mixed theory it introduces semi-polinear sets and characterizes definable functions as piecewise-simple.\"},{\"question\":\"What is a piecewise-simple function in the mixed additive theory?\",\"answer\":\"Piecewise-simple functions are linear separately in the integer part and in the fractional part of the argument, allowing potentially different linear coefficients for each part. The paper proves that this class matches exactly the functions definable in FO(R,Z,+,≤).\"}]",1784191914,30,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"extending-the-ginsburgspanier-theorem-to-functions-and-mixed-arithmetic","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/extending-the-ginsburgspanier-theorem-to-functions-and-mixed-arithmetic/83994/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What logics does the paper study, and what do they assume about addition and order?","Question",{"text":76,"@type":77},"It studies Presburger arithmetic FO(Z,+,≤), the real additive theory FO(R,+,≤), and the mixed additive theory FO(R,Z,+,≤). Each is a first-order logic over integers, reals, or both, with addition and order.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the paper extend the Ginsburg–Spanier theorem?",{"text":81,"@type":77},"It extends the Ginsburg–Spanier characterization from semi-linear definable sets to algebraic characterizations of definable functions. For Presburger and real theories, it shows definable functions are exactly piecewise-linear, and for the mixed theory it introduces semi-polinear sets and characterizes definable functions as piecewise-simple.",{"name":83,"@type":74,"acceptedAnswer":84},"What is a piecewise-simple function in the mixed additive theory?",{"text":85,"@type":77},"Piecewise-simple functions are linear separately in the integer part and in the fractional part of the argument, allowing potentially different linear coefficients for each part. 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