1158:
Theories could be decidable yet not admit quantifier elimination. Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory is decidable, and the
1456:
354:
1151:. A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier-free formulas. This technique can be used to show that
986:
896:
1327:
809:
534:. Quantifier-free sentences have no variables, so their validity in a given theory can often be computed, which enables the use of quantifier elimination algorithms to decide validity of sentences.
218:
678:
1302:
390:
1134:
500:
184:
1096:
1021:
55:
1067:
520:
154:
115:
75:
708:
1044:
728:
602:
578:
95:
710:
is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of literals, then if
604:. The theory of linear orders does not have quantifier elimination. However the theory of its universal consequences has the amalgamation property.
1871:(1929). "Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt".
1755:
907:
820:
1451:{\displaystyle \langle \mathbb {N} ,+,<,0,1,(\equiv _{k})_{k>0}\rangle {\text{ where }}x\equiv _{k}y{\text{ iff }}x=y(\mod {k})}
1171:
to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the
739:
349:{\displaystyle \exists x\in \mathbb {R} .(a\neq 0\wedge ax^{2}+bx+c=0)\ \ \Longleftrightarrow \ \ a\neq 0\wedge b^{2}-4ac\geq 0}
1698:
1649:
1624:
1571:
1514:
1199:
626:
1261:
1807:
1778:
1742:
1143:
In early model theory, quantifier elimination was used to demonstrate that various theories possess properties like
527:
472:
If a theory has quantifier elimination, then a specific question can be addressed: Is there a method of determining
430:, as well as many of their combinations such as Boolean algebra with Presburger arithmetic, and term algebras with
1308:: "Presburger arithmetic needs a divisibility (or congruence) predicate '|' to allow quantifier elimination".
542:
Various model-theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.
443:
1964:"A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications"
1831:
364:
1101:
612:
To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an
531:
449:
617:
31:
1187:
461:
1894:(Technical Report). Vol. TR84-639. Ithaca, New York: Dept. of Computer Science, Cornell University.
1726:
1183:
581:
400:
1640:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.).
475:
159:
1910:
1209:
1072:
997:
731:
1168:
613:
129:
37:
1579:
2009:
1321:
1255:
1152:
1049:
554:
505:
396:
139:
100:
60:
1868:
1933:
1144:
686:
457:
203:
1990:
1708:
1659:
8:
1670:
431:
408:
187:
121:
117:?", and the statement without quantifiers can be viewed as the answer to that question.
1848:
1784:
1204:
1029:
713:
587:
563:
419:
125:
80:
23:
553:. Conversely, a model-complete theory, whose theory of universal consequences has the
132:
are thought of as being simpler, with the quantifier-free formulas as the simplest. A
1803:
1774:
1738:
1694:
1645:
1620:
546:
404:
395:
Examples of theories that have been shown decidable using quantifier elimination are
1852:
1788:
1985:
1975:
1948:
1919:
1840:
1823:
1766:
1730:
1704:
1655:
1608:
411:
1952:
1929:
1690:
1641:
1179:
1160:
1148:
191:
1888:
1901:
1718:
1682:
1616:
550:
133:
1980:
1963:
1844:
1770:
1553:
2003:
1819:
1734:
1674:
1575:
1172:
438:
423:
1924:
1905:
1945:
Deciding the First-Order Theory of an
Algebra of Feature Trees with Updates
1678:
427:
415:
207:
27:
1763:
18th Annual IEEE Symposium of Logic in
Computer Science, 2003. Proceedings
1873:
Comptes Rendus du I congrès de Mathématiciens des Pays Slaves, Warszawa
981:{\displaystyle \bigvee _{j=1}^{m}\exists x.\bigwedge _{i=1}^{n}L_{ij}.}
891:{\displaystyle \exists x.\bigvee _{j=1}^{m}\bigwedge _{i=1}^{n}L_{ij}}
1802:(Softcover reprint of the original 1st ed. 1976 ed.). Springer.
1164:
523:
530:
for the theory reduces to deciding the truth of the quantifier-free
202:
An example from high school mathematics says that a single-variable
1889:
Presburger's
Article on Integer Arithmetic: Remarks and Translation
560:
The models of the theory of the universal consequences of a theory
1725:. Encyclopedia of Mathematics and its Applications. Vol. 42.
1689:. Texts in Theoretical Computer Science. An EATCS Series. Berlin:
1947:. International Joint Conference on Automated Reasoning (IJCAR).
1523:, Page 229 describes "the method of eliminating quantification"..
456:
Quantifier elimination can also be used to show that "combining"
522:? If there is such a method we call it a quantifier elimination
437:
Quantifier eliminator for the theory of the real numbers as an
361:
Here the sentence on the left-hand side involves a quantifier
804:{\displaystyle \bigvee _{j=1}^{m}\bigwedge _{i=1}^{n}L_{ij},}
1492:
1490:
1488:
1756:"Structural subtyping of non-recursive types is decidable"
1167:, it is possible to extend the theory with countably many
1502:
392:, whereas the equivalent sentence on the right does not.
1485:
447:; for the theory of the field of real numbers it is the
1800:
Mathematical Logic (Graduate Texts in
Mathematics (37))
1580:"Theorem Proving in Arithmetic without Multiplication"
1473:
1531:
1529:
1330:
1264:
1236:
1104:
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823:
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566:
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478:
367:
221:
162:
142:
103:
83:
63:
40:
1069:
into disjunctive normal form, and use the fact that
1458:. This extension does admit quantifier elimination.
1226:
1224:
673:{\displaystyle \exists x.\bigwedge _{i=1}^{n}L_{i}}
1526:
1461:
1450:
1297:{\displaystyle \langle \mathbb {N} ,+,0,1\rangle }
1296:
1128:
1090:
1061:
1038:
1015:
980:
890:
803:
722:
702:
672:
596:
572:
514:
494:
384:
348:
178:
148:
109:
89:
69:
49:
1138:
730:is a quantifier-free formula, we can write it in
2001:
1221:
136:has quantifier elimination if for every formula
77:" can be viewed as a question "When is there an
620:, that is, show that each formula of the form:
1593:. Edinburgh: Edinburgh University Press: 91–99
467:
460:theories leads to new decidable theories (see
16:Simplification technique in mathematical logic
1668:
1496:
1317:
991:Finally, to eliminate a universal quantifier
1942:
1753:
1395:
1331:
1291:
1265:
1635:
1508:
1867:
1242:
1989:
1979:
1923:
1906:"Elementary properties of Abelian groups"
1886:
1878:
1636:Fried, Michael D.; Jarden, Moshe (2008).
1441:
1440:
1335:
1304:— does not admit quantifier elimination.
1269:
385:{\displaystyle \exists x\in \mathbb {R} }
378:
232:
1900:
1687:Finite model theory and its applications
1685:; Venema, Yde; Weinstein, Scott (2007).
1607:
1520:
1479:
1129:{\displaystyle \lnot \exists x.\lnot F.}
1754:Kuncak, Viktor; Rinard, Martin (2003).
1552:Brown, Christopher W. (July 31, 2002).
22:is a concept of simplification used in
2002:
1818:
1717:
1569:
1535:
1305:
549:theory with quantifier elimination is
526:. If there is such an algorithm, then
34:. Informally, a quantified statement "
1961:
1551:
1230:
1797:
1613:A mathematical introduction to logic
1467:
1943:Jeannerod, Nicolas; Treinen, Ralf.
1200:Cylindrical algebraic decomposition
537:
206:has a real root if and only if its
13:
1117:
1108:
1105:
1076:
1053:
1046:is quantifier-free, we transform
1001:
932:
824:
630:
368:
222:
41:
14:
2021:
1554:"What is Quantifier Elimination"
1968:Mathematics in Computer Science
1824:"Linear Quantifier Elimination"
1436:
156:, there exists another formula
130:depth of quantifier alternation
1991:11858/00-001M-0000-002C-A3B5-B
1832:Journal of Automated Reasoning
1445:
1433:
1380:
1366:
1311:
1248:
1139:Relationship with decidability
607:
557:, has quantifier elimination.
297:
288:
239:
1:
1544:
1953:10.1007/978-3-319-94205-6_29
1887:Stansifer, Ryan (Sep 1984).
1615:(2nd ed.). Boston, MA:
1188:differentially closed fields
616:applied to a conjunction of
495:{\displaystyle \alpha _{QF}}
186:without quantifiers that is
179:{\displaystyle \alpha _{QF}}
32:theoretical computer science
7:
1193:
1184:algebraically closed fields
1091:{\displaystyle \forall x.F}
1016:{\displaystyle \forall x.F}
468:Algorithms and decidability
444:Fourier–Motzkin elimination
401:algebraically closed fields
197:
10:
2026:
1881:for an English translation
1727:Cambridge University Press
1981:10.1007/s11786-017-0319-z
1845:10.1007/s10817-010-9183-0
1771:10.1109/LICS.2003.1210049
1163:of its valid formulas is
450:Tarski–Seidenberg theorem
50:{\displaystyle \exists x}
1798:Monk, J. Donald (2012).
1735:10.1017/CBO9780511551574
1215:
1925:10.4064/fm-41-2-203-271
1911:Fundamenta Mathematicae
1509:Fried & Jarden 2008
1210:Conjunction elimination
1062:{\displaystyle \lnot F}
732:disjunctive normal form
515:{\displaystyle \alpha }
462:Feferman–Vaught theorem
149:{\displaystyle \alpha }
120:One way of classifying
110:{\displaystyle \ldots }
70:{\displaystyle \ldots }
1962:Sturm, Thomas (2017).
1452:
1298:
1130:
1092:
1063:
1040:
1017:
982:
961:
931:
892:
874:
853:
814:and use the fact that
805:
784:
763:
724:
704:
674:
659:
614:existential quantifier
598:
574:
516:
496:
439:ordered additive group
386:
350:
180:
150:
111:
91:
71:
51:
20:Quantifier elimination
1570:Cooper, D.C. (1972).
1453:
1322:Presburger arithmetic
1320:, p. 20) define
1299:
1256:Presburger arithmetic
1153:Presburger arithmetic
1131:
1093:
1064:
1041:
1018:
983:
941:
911:
893:
854:
833:
806:
764:
743:
725:
705:
703:{\displaystyle L_{i}}
675:
639:
599:
575:
555:amalgamation property
517:
497:
397:Presburger arithmetic
387:
351:
181:
151:
128:. Formulas with less
112:
92:
72:
52:
1671:Kolaitis, Phokion G.
1587:Machine Intelligence
1328:
1262:
1102:
1073:
1050:
1030:
998:
908:
821:
740:
714:
687:
627:
588:
564:
506:
476:
365:
219:
204:quadratic polynomial
160:
140:
124:is by the amount of
101:
81:
61:
38:
1869:Presburger, MojĹĽesz
1765:. pp. 96–107.
1318:Grädel et al. (2007
420:dense linear orders
1497:Grädel et al. 2007
1448:
1294:
1205:Elimination theory
1126:
1088:
1059:
1036:
1013:
978:
888:
801:
720:
700:
670:
594:
580:are precisely the
570:
512:
492:
405:real closed fields
382:
346:
176:
146:
107:
87:
67:
47:
24:mathematical logic
1700:978-3-540-00428-8
1677:; Maarten, Marx;
1651:978-3-540-77269-9
1626:978-0-12-238452-3
1609:Enderton, Herbert
1422:
1401:
1400: where
1175:of the formula).
1098:is equivalent to
1039:{\displaystyle F}
901:is equivalent to
723:{\displaystyle F}
597:{\displaystyle T}
584:of the models of
573:{\displaystyle T}
305:
302:
296:
293:
210:is non-negative:
90:{\displaystyle x}
2017:
1995:
1993:
1983:
1974:(3–4): 483–502.
1956:
1937:
1927:
1895:
1893:
1879:Stansifer (1984)
1876:
1862:
1860:
1859:
1828:
1813:
1792:
1760:
1748:
1712:
1663:
1638:Field arithmetic
1630:
1602:
1600:
1598:
1584:
1572:Meltzer, Bernard
1564:
1562:
1560:
1539:
1533:
1524:
1518:
1512:
1506:
1500:
1494:
1483:
1477:
1471:
1465:
1459:
1457:
1455:
1454:
1449:
1423:
1420:
1415:
1414:
1402:
1399:
1394:
1393:
1378:
1377:
1338:
1315:
1309:
1303:
1301:
1300:
1295:
1272:
1252:
1246:
1240:
1234:
1228:
1135:
1133:
1132:
1127:
1097:
1095:
1094:
1089:
1068:
1066:
1065:
1060:
1045:
1043:
1042:
1037:
1022:
1020:
1019:
1014:
987:
985:
984:
979:
974:
973:
960:
955:
930:
925:
897:
895:
894:
889:
887:
886:
873:
868:
852:
847:
810:
808:
807:
802:
797:
796:
783:
778:
762:
757:
729:
727:
726:
721:
709:
707:
706:
701:
699:
698:
679:
677:
676:
671:
669:
668:
658:
653:
603:
601:
600:
595:
579:
577:
576:
571:
538:Related concepts
521:
519:
518:
513:
501:
499:
498:
493:
491:
490:
412:Boolean algebras
391:
389:
388:
383:
381:
355:
353:
352:
347:
327:
326:
303:
300:
294:
291:
266:
265:
235:
185:
183:
182:
177:
175:
174:
155:
153:
152:
147:
116:
114:
113:
108:
96:
94:
93:
88:
76:
74:
73:
68:
56:
54:
53:
48:
2025:
2024:
2020:
2019:
2018:
2016:
2015:
2014:
2000:
1999:
1998:
1902:Szmielew, Wanda
1891:
1857:
1855:
1826:
1810:
1781:
1758:
1745:
1719:Hodges, Wilfrid
1701:
1691:Springer-Verlag
1683:Vardi, Moshe Y.
1669:Grädel, Erich;
1652:
1642:Springer-Verlag
1627:
1596:
1594:
1582:
1558:
1556:
1547:
1542:
1534:
1527:
1519:
1515:
1507:
1503:
1495:
1486:
1478:
1474:
1466:
1462:
1421: iff
1419:
1410:
1406:
1398:
1383:
1379:
1373:
1369:
1334:
1329:
1326:
1325:
1316:
1312:
1268:
1263:
1260:
1259:
1253:
1249:
1243:Presburger 1929
1241:
1237:
1229:
1222:
1218:
1196:
1180:Nullstellensatz
1141:
1103:
1100:
1099:
1074:
1071:
1070:
1051:
1048:
1047:
1031:
1028:
1027:
999:
996:
995:
966:
962:
956:
945:
926:
915:
909:
906:
905:
879:
875:
869:
858:
848:
837:
822:
819:
818:
789:
785:
779:
768:
758:
747:
741:
738:
737:
715:
712:
711:
694:
690:
688:
685:
684:
664:
660:
654:
643:
628:
625:
624:
610:
589:
586:
585:
565:
562:
561:
540:
507:
504:
503:
483:
479:
477:
474:
473:
470:
377:
366:
363:
362:
322:
318:
261:
257:
231:
220:
217:
216:
200:
167:
163:
161:
158:
157:
141:
138:
137:
102:
99:
98:
82:
79:
78:
62:
59:
58:
39:
36:
35:
17:
12:
11:
5:
2023:
2013:
2012:
1997:
1996:
1958:
1957:
1939:
1938:
1918:(2): 203–271.
1897:
1896:
1883:
1882:
1864:
1863:
1839:(2): 189–212.
1820:Nipkow, Tobias
1815:
1814:
1808:
1794:
1793:
1779:
1750:
1749:
1743:
1714:
1713:
1699:
1675:Libkin, Leonid
1665:
1664:
1650:
1632:
1631:
1625:
1617:Academic Press
1604:
1603:
1576:Michie, Donald
1566:
1565:
1548:
1546:
1543:
1541:
1540:
1525:
1513:
1511:, p. 171.
1501:
1484:
1482:, p. 188.
1472:
1470:, p. 240.
1460:
1447:
1444:
1439:
1435:
1432:
1429:
1426:
1418:
1413:
1409:
1405:
1397:
1392:
1389:
1386:
1382:
1376:
1372:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1337:
1333:
1310:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1271:
1267:
1247:
1235:
1219:
1217:
1214:
1213:
1212:
1207:
1202:
1195:
1192:
1173:free variables
1155:is decidable.
1140:
1137:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1087:
1084:
1081:
1078:
1058:
1055:
1035:
1024:
1023:
1012:
1009:
1006:
1003:
989:
988:
977:
972:
969:
965:
959:
954:
951:
948:
944:
940:
937:
934:
929:
924:
921:
918:
914:
899:
898:
885:
882:
878:
872:
867:
864:
861:
857:
851:
846:
843:
840:
836:
832:
829:
826:
812:
811:
800:
795:
792:
788:
782:
777:
774:
771:
767:
761:
756:
753:
750:
746:
719:
697:
693:
681:
680:
667:
663:
657:
652:
649:
646:
642:
638:
635:
632:
609:
606:
593:
569:
551:model complete
539:
536:
511:
489:
486:
482:
469:
466:
424:abelian groups
380:
376:
373:
370:
359:
358:
357:
356:
345:
342:
339:
336:
333:
330:
325:
321:
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290:
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281:
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264:
260:
256:
253:
250:
247:
244:
241:
238:
234:
230:
227:
224:
199:
196:
194:this theory).
173:
170:
166:
145:
126:quantification
106:
86:
66:
46:
43:
15:
9:
6:
4:
3:
2:
2022:
2011:
2008:
2007:
2005:
1992:
1987:
1982:
1977:
1973:
1969:
1965:
1960:
1959:
1954:
1950:
1946:
1941:
1940:
1935:
1931:
1926:
1921:
1917:
1913:
1912:
1907:
1903:
1899:
1898:
1890:
1885:
1884:
1880:
1874:
1870:
1866:
1865:
1854:
1850:
1846:
1842:
1838:
1834:
1833:
1825:
1821:
1817:
1816:
1811:
1809:9781468494549
1805:
1801:
1796:
1795:
1790:
1786:
1782:
1780:0-7695-1884-2
1776:
1772:
1768:
1764:
1757:
1752:
1751:
1746:
1744:9780521304429
1740:
1736:
1732:
1728:
1724:
1720:
1716:
1715:
1710:
1706:
1702:
1696:
1692:
1688:
1684:
1680:
1679:Spencer, Joel
1676:
1672:
1667:
1666:
1661:
1657:
1653:
1647:
1643:
1639:
1634:
1633:
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683:where each
608:Basic ideas
547:first-order
1858:2022-11-12
1709:1133.03001
1660:1145.12001
1545:References
1231:Brown 2002
188:equivalent
97:such that
57:such that
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1597:30 August
1559:30 August
1468:Monk 2012
1408:≡
1396:⟩
1371:≡
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1266:⟨
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1165:countable
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532:sentences
524:algorithm
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