946:
432:
implies such a procedure exists, and hence gave a conditional solution to Tarski's problem. Schanuel's conjecture deals with all complex numbers so would be expected to be a stronger result than the decidability of
737:
508:
1038:
180:
261:
142:
725:
45:
690:
606:
460:
339:
422:
207:
984:
393:
310:
76:
632:
227:
534:
284:
554:
371:
1146:
462:, and indeed, Macintyre and Wilkie proved that only a real version of Schanuel's conjecture is required to imply the decidability of this theory.
941:{\displaystyle f_{1}(x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})=\ldots =f_{n}(x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})=0}
466:
1103:
469:
for the decidability of the theory. In their paper, Macintyre and Wilkie showed that an equivalent result to the decidability of
1058:
472:
510:
is what they dubbed the weak
Schanuel's conjecture. This conjecture states that there is an effective procedure that, given
989:
151:
25:
235:
84:
728:
695:
429:
637:
559:
436:
350:
315:
1141:
1091:
398:
144:, with the usual interpretation given to each symbol. It was proved by Tarski that the theory of the
287:
37:
33:
185:
1096:
On the
Decidability of the Real Exponential Field, in: Kreiseliana: about and around Georg Kreisel
954:
376:
293:
59:
611:
349:
The problem can be reduced to finding an effective procedure for determining whether any given
1099:
212:
145:
513:
1117:
1113:
269:
1121:
1109:
1083:
539:
356:
1135:
41:
79:
17:
1087:
29:
266:
He then asked whether this was still the case if one added a unary function
46:
theory of the real numbers (without the exponential function) is decidable
503:{\displaystyle \operatorname {Th} (\mathbb {R} _{\exp })}
465:
Even the real version of
Schanuel's conjecture is not a
229:
there is an effective procedure for determining whether
1019:
992:
957:
740:
698:
640:
614:
562:
542:
516:
475:
439:
401:
379:
359:
318:
296:
272:
238:
215:
188:
154:
87:
62:
1033:{\displaystyle |g(\alpha )|>{\tfrac {1}{\eta }}}
1032:
978:
940:
719:
684:
626:
600:
548:
528:
502:
454:
416:
387:
365:
333:
304:
278:
255:
221:
201:
174:
136:
70:
175:{\displaystyle \operatorname {Th} (\mathbb {R} )}
1098:. Wellesley, MA: A K Peters. pp. 441–467.
1059:"Model theory of the real exponential function"
425:
256:{\displaystyle \mathbb {R} \models \varphi .}
8:
286:to the language that was interpreted as the
137:{\displaystyle L_{\text{or}}=(+,-,<,0,1)}
720:{\displaystyle \alpha \in \mathbb {R} ^{n}}
1018:
1010:
993:
991:
956:
921:
916:
895:
890:
877:
858:
845:
821:
816:
795:
790:
777:
758:
745:
739:
711:
707:
706:
697:
670:
651:
639:
613:
586:
567:
561:
541:
515:
491:
487:
486:
474:
446:
442:
441:
438:
408:
404:
403:
400:
381:
380:
378:
358:
325:
321:
320:
317:
298:
297:
295:
271:
240:
239:
237:
214:
193:
187:
165:
164:
153:
92:
86:
64:
63:
61:
1049:
685:{\displaystyle n,f_{1},\dots ,f_{n},g}
22:Tarski's exponential function problem
7:
601:{\displaystyle f_{1},\dots ,f_{n},g}
556:variables with integer coefficients
455:{\displaystyle \mathbb {R} _{\exp }}
334:{\displaystyle \mathbb {R} _{\exp }}
182:, is decidable. That is, given any
78:is a structure over the language of
373:variables and with coefficients in
345:Conditional and equivalent results
14:
1147:Unsolved problems in mathematics
417:{\displaystyle \mathbb {R} ^{n}}
536:and exponential polynomials in
1011:
1007:
1001:
994:
967:
961:
929:
851:
829:
751:
497:
482:
169:
161:
131:
101:
44:had previously shown that the
1:
1065:. Heidelberg: Springer-Verlag
426:Macintyre & Wilkie (1996)
202:{\displaystyle L_{\text{or}}}
979:{\displaystyle g(\alpha )=0}
388:{\displaystyle \mathbb {Z} }
305:{\displaystyle \mathbb {R} }
71:{\displaystyle \mathbb {R} }
1063:Encyclopedia of Mathematics
627:{\displaystyle \eta \geq 1}
1163:
222:{\displaystyle \varphi }
1092:Oddifreddi, Piergiorgio
731:solution of the system
529:{\displaystyle n\geq 1}
312:, to get the structure
56:The ordered real field
1034:
980:
942:
721:
686:
628:
608:, produces an integer
602:
550:
530:
504:
456:
418:
389:
367:
351:exponential polynomial
335:
306:
280:
257:
223:
203:
176:
138:
72:
1035:
981:
943:
722:
687:
629:
603:
551:
531:
505:
457:
430:Schanuel's conjecture
419:
390:
368:
336:
307:
281:
279:{\displaystyle \exp }
258:
224:
204:
177:
139:
73:
990:
955:
738:
696:
638:
612:
560:
540:
514:
473:
437:
399:
377:
357:
316:
294:
288:exponential function
270:
236:
213:
186:
152:
85:
60:
34:exponential function
692:, and such that if
467:necessary condition
1030:
1028:
976:
938:
717:
682:
624:
598:
546:
526:
500:
452:
414:
395:has a solution in
385:
363:
331:
302:
276:
253:
219:
199:
172:
134:
68:
32:together with the
1027:
549:{\displaystyle n}
366:{\displaystyle n}
196:
95:
24:asks whether the
1154:
1126:
1125:
1084:Macintyre, Angus
1080:
1074:
1073:
1071:
1070:
1054:
1039:
1037:
1036:
1031:
1029:
1020:
1014:
997:
985:
983:
982:
977:
947:
945:
944:
939:
928:
927:
926:
925:
902:
901:
900:
899:
882:
881:
863:
862:
850:
849:
828:
827:
826:
825:
802:
801:
800:
799:
782:
781:
763:
762:
750:
749:
726:
724:
723:
718:
716:
715:
710:
691:
689:
688:
683:
675:
674:
656:
655:
634:that depends on
633:
631:
630:
625:
607:
605:
604:
599:
591:
590:
572:
571:
555:
553:
552:
547:
535:
533:
532:
527:
509:
507:
506:
501:
496:
495:
490:
461:
459:
458:
453:
451:
450:
445:
423:
421:
420:
415:
413:
412:
407:
394:
392:
391:
386:
384:
372:
370:
369:
364:
340:
338:
337:
332:
330:
329:
324:
311:
309:
308:
303:
301:
285:
283:
282:
277:
262:
260:
259:
254:
243:
228:
226:
225:
220:
208:
206:
205:
200:
198:
197:
194:
181:
179:
178:
173:
168:
143:
141:
140:
135:
97:
96:
93:
77:
75:
74:
69:
67:
1162:
1161:
1157:
1156:
1155:
1153:
1152:
1151:
1132:
1131:
1130:
1129:
1106:
1082:
1081:
1077:
1068:
1066:
1056:
1055:
1051:
1046:
988:
987:
953:
952:
917:
912:
891:
886:
873:
854:
841:
817:
812:
791:
786:
773:
754:
741:
736:
735:
705:
694:
693:
666:
647:
636:
635:
610:
609:
582:
563:
558:
557:
538:
537:
512:
511:
485:
471:
470:
440:
435:
434:
402:
397:
396:
375:
374:
355:
354:
347:
319:
314:
313:
292:
291:
268:
267:
234:
233:
211:
210:
189:
184:
183:
150:
149:
88:
83:
82:
58:
57:
54:
12:
11:
5:
1160:
1158:
1150:
1149:
1144:
1134:
1133:
1128:
1127:
1104:
1075:
1048:
1047:
1045:
1042:
1026:
1023:
1017:
1013:
1009:
1006:
1003:
1000:
996:
975:
972:
969:
966:
963:
960:
949:
948:
937:
934:
931:
924:
920:
915:
911:
908:
905:
898:
894:
889:
885:
880:
876:
872:
869:
866:
861:
857:
853:
848:
844:
840:
837:
834:
831:
824:
820:
815:
811:
808:
805:
798:
794:
789:
785:
780:
776:
772:
769:
766:
761:
757:
753:
748:
744:
714:
709:
704:
701:
681:
678:
673:
669:
665:
662:
659:
654:
650:
646:
643:
623:
620:
617:
597:
594:
589:
585:
581:
578:
575:
570:
566:
545:
525:
522:
519:
499:
494:
489:
484:
481:
478:
449:
444:
411:
406:
383:
362:
346:
343:
328:
323:
300:
275:
264:
263:
252:
249:
246:
242:
218:
192:
171:
167:
163:
160:
157:
133:
130:
127:
124:
121:
118:
115:
112:
109:
106:
103:
100:
91:
66:
53:
50:
13:
10:
9:
6:
4:
3:
2:
1159:
1148:
1145:
1143:
1140:
1139:
1137:
1123:
1119:
1115:
1111:
1107:
1105:9781568810614
1101:
1097:
1093:
1089:
1085:
1079:
1076:
1064:
1060:
1057:Kuhlmann, S.
1053:
1050:
1043:
1041:
1024:
1021:
1015:
1004:
998:
973:
970:
964:
958:
935:
932:
922:
918:
913:
909:
906:
903:
896:
892:
887:
883:
878:
874:
870:
867:
864:
859:
855:
846:
842:
838:
835:
832:
822:
818:
813:
809:
806:
803:
796:
792:
787:
783:
778:
774:
770:
767:
764:
759:
755:
746:
742:
734:
733:
732:
730:
712:
702:
699:
679:
676:
671:
667:
663:
660:
657:
652:
648:
644:
641:
621:
618:
615:
595:
592:
587:
583:
579:
576:
573:
568:
564:
543:
523:
520:
517:
492:
479:
476:
468:
463:
447:
431:
427:
409:
360:
352:
344:
342:
326:
289:
273:
250:
247:
244:
232:
231:
230:
216:
190:
158:
155:
147:
128:
125:
122:
119:
116:
113:
110:
107:
104:
98:
89:
81:
80:ordered rings
51:
49:
47:
43:
42:Alfred Tarski
39:
35:
31:
27:
23:
19:
1142:Model theory
1095:
1088:Wilkie, Alex
1078:
1067:. Retrieved
1062:
1052:
951:then either
950:
729:non-singular
464:
428:showed that
348:
265:
55:
30:real numbers
21:
18:model theory
15:
52:The problem
1136:Categories
1122:0896.03012
1069:2024-08-07
1044:References
209:-sentence
146:real field
1025:η
1005:α
965:α
907:…
868:…
836:…
807:…
768:…
703:∈
700:α
661:…
619:≥
616:η
577:…
521:≥
480:
248:φ
245:⊨
217:φ
159:
111:−
38:decidable
1090:(1996).
1114:1435773
1094:(ed.).
28:of the
1120:
1112:
1102:
26:theory
727:is a
1100:ISBN
1016:>
117:<
1118:Zbl
986:or
493:exp
448:exp
424:.
353:in
327:exp
290:on
274:exp
36:is
16:In
1138::
1116:.
1110:MR
1108:.
1086:;
1061:.
1040:.
477:Th
341:.
195:or
156:Th
148:,
94:or
48:.
40:.
20:,
1124:.
1072:.
1022:1
1012:|
1008:)
1002:(
999:g
995:|
974:0
971:=
968:)
962:(
959:g
936:0
933:=
930:)
923:n
919:x
914:e
910:,
904:,
897:1
893:x
888:e
884:,
879:n
875:x
871:,
865:,
860:1
856:x
852:(
847:n
843:f
839:=
833:=
830:)
823:n
819:x
814:e
810:,
804:,
797:1
793:x
788:e
784:,
779:n
775:x
771:,
765:,
760:1
756:x
752:(
747:1
743:f
713:n
708:R
680:g
677:,
672:n
668:f
664:,
658:,
653:1
649:f
645:,
642:n
622:1
596:g
593:,
588:n
584:f
580:,
574:,
569:1
565:f
544:n
524:1
518:n
498:)
488:R
483:(
443:R
410:n
405:R
382:Z
361:n
322:R
299:R
251:.
241:R
191:L
170:)
166:R
162:(
132:)
129:1
126:,
123:0
120:,
114:,
108:,
105:+
102:(
99:=
90:L
65:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.