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The last two examples can be combined: given any o-minimal expansion of the real field (such as the real field with restricted analytic functions), one can define its
Pfaffian closure, which is again an o-minimal structure. (The Pfaffian closure of a structure is, in particular, closed under Pfaffian
642:. A major line of current research is based on discovering expansions of the real ordered field that are o-minimal. Despite the generality of application, one can show a great deal about the geometry of set definable in o-minimal structures. There is a cell decomposition theorem,
575:
611:
added (i.e. analytic functions on a neighborhood of , restricted to ; note that the unrestricted sine function has infinitely many roots, and so cannot be definable in an o-minimal structure.)
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The condition that the interpretation of < be dense is not strictly necessary, but it is known that discrete orders lead to essentially trivial o-minimal structures, see, for example,
767:
in: Lecture notes on o-minimal structures and real analytic geometry, C. Miller, J.-P. Rolin, and P. Speissegger (eds.), Fields
Institute Communications vol. 62, 2012, pp. 179â218.
660:, a property that has been used to guarantee the convergence of some non-smooth optimization methods, such as the stochastic subgradient method (under some mild assumptions).
1399:
455:
O-minimal structures originated in model theory and so have a simpler â but equivalent â definition using the language of model theory. Specifically if
1147:
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1054:
2074:
1191:"Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function"
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O-minimal structures can be defined without recourse to model theory. Here we define a structure on a nonempty set
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Moreover, continuously differentiable definable functions in a o-minimal structure satisfy a generalization of
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The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set.
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of an o-minimal structure is an o-minimal theory. This result is remarkable because, in contrast, the
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897:(PhD thesis). UniversitĂ© Montpellier; Universidad tĂ©cnica Federico Santa MarĂa (Valparaiso, Chili).
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is equivalent to a quantifier-free formula involving only the ordering, also with parameters in
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Descent dynamical systems and algorithms for tame optimization, and multi-objective problems
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467:-structure where < is interpreted to satisfy the axioms of a dense linear order, then (
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The complete theory of dense linear orders in the language with just the ordering.
129:, that is, there may be an elementarily equivalent structure that is not minimal.
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955:. London Mathematical Society Lecture Note Series. Vol. 248. Cambridge:
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chains where arbitrary definable functions are used in place of polynomials.)
398:
has a dense linear order without endpoints on it, say <, then a structure
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1972:
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1186:
834:
Davis, Damek; Drusvyatskiy, Dmitriy; Kakade, Sham; Lee, Jason D. (2020).
604:
33:
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1708:
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1485:
1248:
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is called o-minimal (respect to <) if it satisfies the extra axioms
95:
structures, which are exactly the analogous property down to equality.
925:
3005:
2908:
1961:
1878:
1838:
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471:,<,...) is called an o-minimal structure if for any definable set
1240:
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1142:
1116:
1097:
1068:
2990:
2788:
2236:
1941:
1535:
852:
653:
theorems and a good notion of dimension and Euler characteristic.
638:. Thus the study of o-minimal structures and theories generalises
2586:
1378:
713:
Knight, Pillay and
Steinhorn (1986), Pillay and Steinhorn (1988).
622:. More generally, the complete theory of the real numbers with
593:
1018:. Graduate Texts in Mathematics. Vol. 217. New York, NY:
2130:
1476:
1321:
795:"On gradients of functions definable in o-minimal structures"
614:
The complete theory of the real field with a symbol for the
836:"Stochastic Subgradient Method Converges on Tame Functions"
833:
443:
are precisely the finite unions of intervals and points.
1140:
1044:
570:{\displaystyle X=X_{0}\cup I_{1}\cup \ldots \cup I_{r}.}
515:
459:
is a language including a binary relation <, and (
1271:
Real
Algebraic and Analytic Geometry Preprint Server
984:"Review of "Tame Topology and o-minimal Structures""
117:is o-minimal. It is known that the complete theory
1220:
569:
1148:Transactions of the American Mathematical Society
1103:Transactions of the American Mathematical Society
1055:Transactions of the American Mathematical Society
3036:
634:In the case of RCF, the definable sets are the
67:O-minimality can be regarded as a weak form of
1292:
992:Bulletin of the American Mathematical Society
950:
450:
1199:Journal of the American Mathematical Society
1096:; Pillay, Anand; Steinhorn, Charles (1986).
132:
1484:
1299:
1285:
1143:"Definable Sets in Ordered Structures III"
1141:Pillay, Anand; Steinhorn, Charles (1988).
1045:Pillay, Anand; Steinhorn, Charles (1986).
1211:
1160:
1115:
1098:"Definable Sets in Ordered Structures II"
1067:
1004:
851:
810:
141:in a set-theoretic manner, as a sequence
1047:"Definable Sets in Ordered Structures I"
890:
840:Foundations of Computational Mathematics
383:. A subset is called definable if it is
792:
479:there are finitely many open intervals
3037:
1306:
1221:Denef, J.; van den Dries, L. (1989). "
1185:
1013:
981:
953:Tame Topology and o-minimal Structures
1280:
907:
327:, we consider the smallest structure
125:of a minimal structure need not be a
910:"An Invitation to Tame Optimization"
585:Examples of o-minimal theories are:
499: âȘ {±â} and a finite set
294:is the projection map on the first
13:
1225:-adic and real subanalytic sets".
891:Garrigos, Guillaume (2015-11-02).
379:is called a set of parameters for
75:is o-minimal if and only if every
14:
3071:
1256:
362:-definable if it is contained in
339:such that every finite subset of
158: = 0,1,2,... such that
3018:
901:
765:Pfaffian sets and o-minimality,
884:
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1:
2979:History of mathematical logic
1213:10.1090/S0894-0347-96-00216-0
1016:Model theory: An introduction
1006:10.1090/S0273-0979-00-00866-1
944:
799:Annales de l'Institut Fourier
2904:Primitive recursive function
1264:Model Theory preprint server
914:SIAM Journal on Optimization
410:the set < (={(
52:(with parameters taken from
7:
951:van den Dries, Lou (1998).
793:Kurdyka, Krzysztof (1998).
773:10.1007/978-1-4614-4042-0_5
663:
603:The complete theory of the
580:
91:. This is analogous to the
20:, and more specifically in
10:
3076:
1968:SchröderâBernstein theorem
1695:Monadic predicate calculus
1354:Foundations of mathematics
957:Cambridge University Press
862:10.1007/s10208-018-09409-5
685:Weakly o-minimal structure
451:Model theoretic definition
3014:
3001:Philosophy of mathematics
2950:Automated theorem proving
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2827:
2659:
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2121:
2097:
2075:Von NeumannâBernaysâGödel
2020:
1914:
1818:
1716:
1707:
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1569:
1475:
1397:
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439:the definable subsets of
700:
133:Set-theoretic definition
3055:Real algebraic geometry
2651:Self-verifying theories
2472:Tarski's axiomatization
1423:Tarski's undefinability
1418:incompleteness theorems
680:Strongly minimal theory
675:Real algebraic geometry
640:real algebraic geometry
408:
127:strongly minimal theory
3025:Mathematics portal
2636:Proof of impossibility
2284:propositional variable
1594:Propositional calculus
1014:Marker, David (2002).
982:Marker, David (2000).
658:Ćojasiewicz inequality
571:
69:quantifier elimination
2894:Kolmogorov complexity
2847:Computably enumerable
2747:Model complete theory
2539:Principia Mathematica
1599:Propositional formula
1428:BanachâTarski paradox
1228:Annals of Mathematics
908:Ioffe, A. D. (2009).
572:
40:if and only if every
36:by < is called an
2842:ChurchâTuring thesis
2829:Computability theory
2038:continuum hypothesis
1556:Square of opposition
1414:Gödel's completeness
763:Patrick Speisseger,
616:exponential function
513:
387:-definable for some
2996:Mathematical object
2887:P versus NP problem
2852:Computable function
2646:Reverse mathematics
2572:Logical consequence
2449:primitive recursive
2444:elementary function
2217:Free/bound variable
2070:TarskiâGrothendieck
1589:Logical connectives
1519:Logical equivalence
1369:Logical consequence
783:Marker (2002) p.103
38:o-minimal structure
32:,<,...) that is
3045:Mathematical logic
2794:Transfer principle
2757:Semantics of logic
2742:Categorical theory
2718:Non-standard model
2232:Logical connective
1359:Information theory
1308:Mathematical logic
754:Marker (2002) p.99
722:Marker (2002) p.81
636:semialgebraic sets
624:Pfaffian functions
609:analytic functions
598:real closed fields
567:
298:coordinates, then
83:and parameters in
18:mathematical logic
3032:
3031:
2964:Abstract category
2767:Theories of truth
2577:Rule of inference
2567:Natural deduction
2548:
2547:
2093:
2092:
1798:Cartesian product
1703:
1702:
1609:Many-valued logic
1584:Boolean functions
1467:Russell's paradox
1442:diagonal argument
1339:First-order logic
1029:978-0-387-98760-6
966:978-0-521-59838-5
926:10.1137/080722059
670:Semialgebraic set
463:,<,...) is an
3067:
3023:
3022:
2974:History of logic
2969:Category of sets
2862:Decision problem
2641:Ordinal analysis
2582:Sequent calculus
2480:Boolean algebras
2420:
2419:
2394:
2365:logical/constant
2119:
2118:
2105:
2028:ZermeloâFraenkel
1779:Set operations:
1714:
1713:
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1482:
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1462:LöwenheimâSkolem
1349:Formal semantics
1301:
1294:
1287:
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1215:
1206:(4): 1051â1095.
1195:
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1119:
1089:
1071:
1051:
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988:
978:
938:
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920:(4): 1894â1917.
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812:10.5802/aif.1638
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690:C-minimal theory
620:Wilkie's theorem
607:with restricted
576:
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563:
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544:
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531:
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426: <
375:); in that case
343:is contained in
107:o-minimal theory
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3010:
2955:Category theory
2945:Algebraic logic
2928:
2899:Lambda calculus
2837:Church encoding
2823:
2799:Truth predicate
2655:
2621:Complete theory
2544:
2413:
2409:
2405:
2400:
2392:
2112: and
2108:
2103:
2089:
2065:New Foundations
2033:axiom of choice
2016:
1978:Gödel numbering
1918: and
1910:
1814:
1699:
1649:
1630:
1579:Boolean algebra
1565:
1529:Equiconsistency
1494:Classical logic
1471:
1452:Halting problem
1440: and
1416: and
1404: and
1403:
1398:Theorems (
1393:
1310:
1305:
1259:
1241:10.2307/1971463
1193:
1162:10.2307/2000920
1117:10.2307/2000053
1069:10.2307/2000052
1049:
1030:
1020:Springer-Verlag
986:
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34:totally ordered
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2940:Abstract logic
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2857:Computable set
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2752:Satisfiability
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2669:Interpretation
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2490:minimal axioms
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2180:Formation rule
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2099:Formal systems
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1988:Large cardinal
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1753:Extensionality
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1748:Ordinal number
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1432:Cantor's
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1257:External links
1255:
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1155:(2): 469â476.
1138:
1110:(2): 593â605.
1090:
1062:(2): 565â592.
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999:(3): 351â357.
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174:of subsets of
165:
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71:. A structure
56:) is a finite
24:, an infinite
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2914:Recursive set
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2723:of arithmetic
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2630:from ZFC
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2529:non-Euclidean
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2406:Example
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2050:KripkeâPlatek
2048:
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2023:
2019:
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1900:
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1894:constructible
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1597:
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1572:
1571:Propositional
1568:
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1499:Logical truth
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1235:(1): 79â138.
1234:
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1094:Knight, Julia
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695:Tame topology
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335:) containing
334:
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319:For a subset
313:
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272: â
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197: Ă
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81:free variable
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65:
63:
59:
55:
51:
48: â
47:
43:
39:
35:
31:
27:
23:
19:
3050:Model theory
3016:
2814:Ultraproduct
2661:Model theory
2626:Independence
2562:Formal proof
2554:Proof theory
2537:
2510:
2467:real numbers
2439:second-order
2350:Substitution
2227:Metalanguage
2168:conservative
2141:Axiom schema
2085:Constructive
2055:MorseâKelley
2021:Set theories
2000:Aleph number
1993:inaccessible
1899:Grothendieck
1783:intersection
1670:Higher-order
1658:Second-order
1604:Truth tables
1561:Venn diagram
1344:Formal proof
1270:
1263:
1232:
1226:
1222:
1203:
1197:
1187:Wilkie, A.J.
1152:
1146:
1107:
1101:
1059:
1053:
1015:
996:
990:
952:
917:
913:
903:
893:
886:
843:
839:
829:
802:
798:
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779:
764:
759:
750:
727:
718:
709:
655:
633:
584:
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136:
118:
114:
106:
102:
97:
88:
84:
72:
66:
64:and points.
53:
49:
45:
37:
29:
22:model theory
15:
2924:Type theory
2872:undecidable
2804:Truth value
2691:equivalence
2370:non-logical
1983:Enumeration
1973:Isomorphism
1920:cardinality
1904:Von Neumann
1869:Ultrafilter
1834:Uncountable
1768:equivalence
1685:Quantifiers
1675:Fixed-point
1644:First-order
1524:Consistency
1509:Proposition
1486:Traditional
1457:Lindström's
1447:Compactness
1389:Type theory
1334:Cardinality
350:. A subset
3039:Categories
2735:elementary
2428:arithmetic
2296:Quantifier
2274:functional
2146:Expression
1864:Transitive
1808:identities
1793:complement
1726:hereditary
1709:Set theory
1179:0707.03024
1134:0662.03024
1086:0662.03023
1038:1003.03034
975:0953.03045
945:References
853:1804.07795
605:real field
506:such that
358:is called
221:the set {(
3006:Supertask
2909:Recursion
2867:decidable
2701:saturated
2679:of models
2602:deductive
2597:axiomatic
2517:Hilbert's
2504:Euclidean
2485:canonical
2408:axiomatic
2340:Signature
2269:Predicate
2158:Extension
2080:Ackermann
2005:Operation
1884:Universal
1874:Recursive
1849:Singleton
1844:Inhabited
1829:Countable
1819:Types of
1803:power set
1773:partition
1690:Predicate
1636:Predicate
1551:Syllogism
1541:Soundness
1514:Inference
1504:Tautology
1406:paradoxes
934:1052-6234
870:1615-3375
821:0373-0956
592:RCF, the
552:∪
549:…
546:∪
533:∪
430:}) is in
109:if every
79:with one
62:intervals
42:definable
26:structure
3060:Topology
2991:Logicism
2984:timeline
2960:Concrete
2819:Validity
2789:T-schema
2782:Kripke's
2777:Tarski's
2772:semantic
2762:Strength
2711:submodel
2706:spectrum
2674:function
2522:Tarski's
2511:Elements
2498:geometry
2454:Robinson
2375:variable
2360:function
2333:spectrum
2323:Sentence
2279:variable
2222:Language
2175:Relation
2136:Automata
2126:Alphabet
2110:language
1964:-jection
1942:codomain
1928:Function
1889:Universe
1859:Infinite
1763:Relation
1546:Validity
1536:Argument
1434:theorem,
1189:(1996).
664:See also
581:Examples
257:} is in
2933:Related
2730:Diagram
2628: (
2607:Hilbert
2592:Systems
2587:Theorem
2465:of the
2410:systems
2190:Formula
2185:Grammar
2101: (
2045:General
1758:Forcing
1743:Element
1663:Monadic
1438:paradox
1379:Theorem
1315:General
1249:1971463
1171:2000920
1126:2000053
1078:2000052
878:5025719
743:0943306
736:0899083
648:Verdier
644:Whitney
209:are in
205: Ă
93:minimal
77:formula
44:subset
2696:finite
2459:Skolem
2412:
2387:Theory
2355:Symbol
2345:String
2328:atomic
2205:ground
2200:closed
2195:atomic
2151:ground
2114:syntax
2010:binary
1937:domain
1854:Finite
1619:finite
1477:Logics
1436:
1384:Theory
1247:
1177:
1169:
1132:
1124:
1084:
1076:
1036:
1026:
973:
963:
932:
876:
868:
819:
626:added.
594:theory
486:,...,
105:is an
100:theory
2686:Model
2434:Peano
2291:Proof
2131:Arity
2060:Naive
1947:image
1879:Fuzzy
1839:Empty
1788:union
1733:Class
1374:Model
1364:Lemma
1322:Axiom
1245:JSTOR
1194:(PDF)
1167:JSTOR
1122:JSTOR
1074:JSTOR
1050:(PDF)
987:(PDF)
874:S2CID
848:arXiv
701:Notes
228:,...,
193:then
170:is a
111:model
58:union
2809:Type
2612:list
2416:list
2393:list
2382:Term
2316:rank
2210:open
2104:list
1916:Maps
1821:sets
1680:Free
1650:list
1400:list
1327:list
1024:ISBN
961:ISBN
930:ISSN
866:ISSN
817:ISSN
738:and
646:and
282:and
201:and
2496:of
2478:of
2426:of
1958:Sur
1932:Map
1739:Ur-
1721:Set
1237:doi
1233:128
1208:doi
1175:Zbl
1157:doi
1153:309
1130:Zbl
1112:doi
1108:295
1082:Zbl
1064:doi
1060:295
1034:Zbl
1001:doi
971:Zbl
922:doi
858:doi
807:doi
769:doi
618:by
596:of
495:in
402:on
394:If
354:of
323:of
268:if
180:if
154:),
113:of
60:of
16:In
3041::
2882:NP
2506::
2500::
2430::
2107:),
1962:Bi
1954:In
1243:.
1231:.
1202:.
1196:.
1173:.
1165:.
1151:.
1145:.
1128:.
1120:.
1106:.
1100:.
1080:.
1072:.
1058:.
1052:.
1032:.
1022:.
997:37
995:.
989:.
969:.
959:.
928:.
918:19
916:.
912:.
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