607:
From a computational perspective, the formal definition of a straight-line program has some advantages. Firstly, a sequence of abstract expressions requires less memory than terms over the generating set. Secondly, it allows straight-line programs to be constructed in one representation of
153:
Straight-line programs were introduced by Babai and Szemerédi in 1984 as a tool for studying the computational complexity of certain matrix group properties. Babai and Szemerédi prove that every element of a finite group
684:, the group of permutations on six letters, we can take α=(1 2 3 4 5 6) and β=(1 2) as generators. The leftmost column here is an example of a straight-line program to compute (1 2 3)(4 5 6):
2157:
Babai, László, and Endre Szemerédi. "On the complexity of matrix group problems I." Foundations of
Computer Science, 1984. 25th Annual Symposium on Foundations of Computer Science. IEEE, 1984
628:
is the group of symmetries of a hexagon. It can be generated by a 60 degree rotation ρ and one reflection λ. The leftmost column of the following is a straight-line program for λρ:
819:. The online ATLAS of Finite Group Representations provides abstract straight-line programs for computing generating sets of maximal subgroups for many finite simple groups.
334:
2169:Ákos Seress. (2003). Permutation Group Algorithms. . Cambridge Tracts in Mathematics. (No. 152). Cambridge: Cambridge University Press.
2249:
205:
are provided for the group-theoretic functions of multiplication, inversion, and checking for equality with the identity. A
209:
is one which uses only these oracles. Hence, straight-line programs for black box groups are black box algorithms.
793:|). In more detail, SLPs are used to prove that every finite simple group has a first-order description of length
173:
is crucial to many group-theoretic algorithms. It can be stated in terms of SLPs as follows. Given a finite group
860:
has order 25. The following is a straight-line program that computes a generating set for a maximal subgroup E
2185:
1081:. This can be understood as a bound on how hard it is to generate a group element from the generators.
147:
28:
that does not contain any loop or any test, and is formed by a sequence of steps that apply each an
1932:). There is no point in generating the element of maximum length, since it is the identity. Hence
1735:
The next claim is used to show that the cost of every group element is within the required bound.
29:
872:. This straight-line program can be found in the online ATLAS of Finite Group Representations.
213:
817:
Straight-line programs computing generating sets for maximal subgroups of finite simple groups
2244:
212:
Explicit straight-line programs are given for a wealth of finite simple groups in the online
319:
1395:
We now need to verify the following claim to ensure that the process terminates within lg(|
150:, by using SLPs to efficiently encode group elements as words over a given generating set.
71:, is the inverse of a preceding element, or the product of two preceding elements. An SLP
8:
36:
35:
This article is devoted to the case where the allowed operations are the operations of a
2194:
434:
A straight-line program is similar to a derivation in predicate logic. The elements of
782:
438:
correspond to axioms and the group operations correspond to the rules of inference.
2204:
25:
17:
198:
781:. Straight-line programs can be used to study compression of finite groups via
621:
2209:
2180:
2238:
826:
604:⟩. The definition presented above is a common generalisation of this.
2224:
1795:
612:
and evaluated in another. This is an important feature of some algorithms.
197:. The constructive membership problem is often studied in the setting of
1233:) denote the length of a shortest straight-line program that contains
1157:
is constructed by inductively defining an increasing sequence of sets
201:. The elements are encoded by bit strings of a fixed length. Three
2199:
785:. They provide a tool to construct "short" sentences describing
1131:
will be defined during the process). It is usually larger than
1088:) is an integer-valued version of the logarithm function: for
142:, but the length of the corresponding SLP is linear in
39:, that is multiplication and inversion. More specifically a
1055:
The reachability theorem states that, given a finite group
1340:) (which is non-empty) that minimises the "cost increase"
2181:"Describing finite groups by short first-order sentences"
825:: The group Sz(32), belonging to the infinite family of
415:
is the length of a shortest straight-line program over
282:
can be obtained by one of the following three rules:
322:
596:
The original definition appearing in requires that
328:
1388:), effectively making it easier to generate from
2236:
1139:can be expressed as a word of length at most
1029:Second rule iterated: (5) multiplied 14 times
1017:Second rule iterated: (4) multiplied 18 times
2225:"ATLAS of Finite Group Representations - V3"
1107:The idea of the proof is to construct a set
1127:} that will work as a new generating set (
2208:
2198:
189:, find a straight-line program computing
2178:
2165:
2163:
1503:. By the pigeonhole principle there are
805:has a first-order description of length
1830:, then there is an element of the form
1477:. Now suppose for a contradiction that
1050:
2237:
224:
2160:
146:. This has important applications in
875:
687:
631:
441:
427:is not in the subgroup generated by
2055:). By Corollary 2, we need at most
779:Short descriptions of finite groups
106:Intuitively, an SLP computing some
13:
2179:Nies, André; Tent, Katrin (2017).
1731: − 1), a contradiction.
14:
2261:
1981:We now finish the theorem. Since
1609:be the largest integer such that
32:to previously computed elements.
1372:can be written as an element of
1364:is defined in a way so that any
507:is a symbol for some element of
2250:Computational complexity theory
1774:(0)=0 it suffices to show that
773:
171:constructive membership problem
2217:
2172:
2151:
1217:is the group element added to
478:is a sequence of expressions (
55:⟩ is a finite sequence
1:
2186:Israel Journal of Mathematics
2144:
219:
169:An efficient solution to the
829:, has rank 2 via generators
166:|) in every generating set.
7:
2009:can be written in the form
1940:steps suffice. To generate
801:|), and every finite group
615:
63:such that every element of
10:
2266:
2133:| − 1 ≤ (1 + lg|
450:be a finite group and let
423:. The cost is infinite if
233:be a finite group and let
148:computational group theory
134:steps, the word length of
2210:10.1007/s11856-017-1563-2
1271:(0)=0. We define the set
789:(i.e. much shorter than |
1032:Third rule: (11) inverse
99:is encoded by a word in
1978:steps are sufficient.
1303:to take upon the value
1020:Third rule: (7) inverse
593:in the obvious manner.
1071:has a maximum cost of
1038:Second rule: (13).(11)
330:
329:{\displaystyle \cdot }
214:ATLAS of Finite Groups
138:may be exponential in
1636:= 1. It follows that
1451:It is immediate that
1135:, but any element of
1084:Here the function lg(
1035:Second rule: (12).(6)
585:takes upon the value
494:) such that for each
460:straight-line program
331:
267:straight-line program
177: = ⟨
158:has an SLP of length
122:as a group word over
47:) for a finite group
41:straight-line program
22:straight-line program
1802:is connected and if
1051:Reachability theorem
1026:Second rule: (9).(7)
1023:Second rule: (8).(2)
1014:Second rule: (5).(2)
1011:Second rule: (3).(4)
1008:Second rule: (3).(2)
1005:Second rule: (1).(2)
761:Second rule: (6).(6)
758:Second rule: (5).(2)
755:Second rule: (4).(1)
752:Second rule: (3).(2)
749:Second rule: (1).(1)
668:Second rule: (1).(4)
665:Second rule: (3).(2)
662:Second rule: (2).(2)
320:
2077:, and no more than
1743: —
1627:. Assume WLOG that
1490:| < 2|
1407: —
225:Informal definition
207:black box algorithm
2085:steps to generate
2065:steps to generate
1909:steps to generate
1905:It takes at most 2
1768:
1741:
1449:
1405:
1310:Else, choose some
1073:(1 + lg|
589:when evaluated in
392:The straight-line
326:
130:is constructed in
126:; observe that if
103:and its inverses.
67:either belongs to
24:is, informally, a
2127:| + lg|
1766:
1739:
1447:
1403:
1360:By this process,
1047:
1046:
853:has order 25 and
783:first-order logic
770:
769:
677:
676:
659:ρ is a generator.
656:λ is a generator.
442:Formal definition
261:) of elements of
2257:
2229:
2228:
2221:
2215:
2214:
2212:
2202:
2176:
2170:
2167:
2158:
2155:
2140:
2138:
2132:
2126:
2084:
2064:
2039:
2038:
2020:
2019:
1939:
1866:
1761:
1744:
1502:
1500:
1489:
1476:
1474:
1464:| ≤ 2|
1463:
1442:
1440:
1430:| = 2|
1429:
1408:
1208:∈ {0,1}}, where
1148:
1146:
1080:
1078:
876:
746:β is a generator
743:α is a generator
688:
632:
407:) of an element
379:
378:
335:
333:
332:
327:
199:black box groups
79:a group element
18:computer science
2265:
2264:
2260:
2259:
2258:
2256:
2255:
2254:
2235:
2234:
2233:
2232:
2223:
2222:
2218:
2177:
2173:
2168:
2161:
2156:
2152:
2147:
2134:
2128:
2122:
2101:
2078:
2056:
2046:
2037:
2034:
2033:
2032:
2027:
2018:
2015:
2014:
2013:
1953:
1946:
1933:
1915:
1903:
1896:
1873:
1844:
1837:
1831:
1764:
1745:
1742:
1733:
1714:
1693:
1684:
1677:
1670:
1663:
1653:
1644:
1635:
1626:
1617:
1604:
1595:
1586:
1579:
1569:
1562:
1555:
1545:
1538:
1531:
1516:
1509:
1491:
1480:
1478:
1465:
1454:
1452:
1445:
1431:
1420:
1418:
1406:
1399:|) many steps:
1319:
1266:
1256:
1247:
1216:
1207:
1198:
1189:
1182:
1142:
1140:
1126:
1117:
1074:
1072:
1053:
1048:
1002:is a generator.
996:is a generator.
871:
867:
863:
776:
771:
683:
678:
627:
618:
584:
563:
554:
545:
528:
519:
506:
493:
484:
470:computing some
454:be a subset of
444:
377:
372:
371:
370:
365:
343:
321:
318:
317:
316:
307:
293:
281:
260:
251:
241:. A sequence
237:be a subset of
227:
222:
118:way of storing
59:of elements of
51:= ⟨
12:
11:
5:
2263:
2253:
2252:
2247:
2231:
2230:
2216:
2171:
2159:
2149:
2148:
2146:
2143:
2121:− 1 ≤ lg|
2044:
2035:
2025:
2016:
1951:
1944:
1913:
1894:
1871:
1842:
1835:
1765:
1737:
1710:
1689:
1682:
1675:
1668:
1658:
1649:
1640:
1631:
1622:
1613:
1600:
1591:
1584:
1574:
1567:
1560:
1550:
1543:
1536:
1529:
1514:
1507:
1446:
1401:
1358:
1357:
1314:
1308:
1262:
1252:
1245:
1225:-th step. Let
1212:
1203:
1194:
1187:
1180:
1122:
1115:
1052:
1049:
1045:
1044:
1040:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
997:
989:
988:
987:
972:
962:
955:
948:
933:
923:
916:
909:
904:
899:
894:
889:
884:
874:
869:
865:
861:
775:
772:
768:
767:
763:
762:
759:
756:
753:
750:
747:
744:
739:
738:
737:
736:(1 2 3)(4 5 6)
734:
731:
730:(1 4)(2 5 3 6)
728:
725:
724:(1 3 5)(2 4 6)
722:
719:
714:
713:
712:
709:
706:
703:
700:
697:
694:
686:
681:
675:
674:
670:
669:
666:
663:
660:
657:
652:
651:
650:
647:
644:
641:
638:
630:
625:
622:dihedral group
617:
614:
580:
559:
550:
541:
529:,-1) for some
524:
515:
502:
489:
482:
443:
440:
390:
389:
373:
361:
356:
339:
325:
312:
303:
298:
289:
277:
256:
249:
226:
223:
221:
218:
9:
6:
4:
3:
2:
2262:
2251:
2248:
2246:
2243:
2242:
2240:
2226:
2220:
2211:
2206:
2201:
2196:
2192:
2188:
2187:
2182:
2175:
2166:
2164:
2154:
2150:
2142:
2137:
2131:
2125:
2120:
2116:
2112:
2108:
2104:
2098:
2096:
2092:
2088:
2082:
2076:
2072:
2068:
2063:
2059:
2054:
2050:
2043:
2031:
2024:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1984:
1979:
1977:
1973:
1969:
1965:
1961:
1957:
1950:
1943:
1937:
1931:
1927:
1923:
1919:
1912:
1908:
1902:
1900:
1893:
1889:
1885:
1881:
1877:
1870:
1864:
1860:
1856:
1852:
1848:
1841:
1834:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1781:
1777:
1773:
1763:
1760:
1756:
1752:
1748:
1736:
1732:
1730:
1726:
1722:
1718:
1713:
1709:
1705:
1701:
1697:
1692:
1688:
1681:
1674:
1667:
1661:
1657:
1652:
1648:
1643:
1639:
1634:
1630:
1625:
1621:
1616:
1612:
1608:
1605:∈ {0,1}. Let
1603:
1599:
1594:
1590:
1583:
1577:
1573:
1566:
1559:
1553:
1549:
1542:
1535:
1528:
1524:
1520:
1513:
1506:
1498:
1494:
1487:
1483:
1472:
1468:
1461:
1457:
1444:
1438:
1434:
1427:
1423:
1416:
1412:
1400:
1398:
1393:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1355:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1317:
1313:
1309:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1277:
1276:
1275:recursively:
1274:
1270:
1265:
1260:
1255:
1251:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1215:
1211:
1206:
1202:
1197:
1193:
1186:
1179:
1175:
1171:
1166:
1164:
1160:
1156:
1152:
1145:
1138:
1134:
1130:
1125:
1121:
1114:
1110:
1105:
1103:
1099:
1095:
1091:
1087:
1082:
1077:
1070:
1066:
1062:
1059:generated by
1058:
1043:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
991:
990:
985:
981:
977:
973:
971:
967:
963:
960:
956:
953:
949:
946:
942:
938:
934:
932:
928:
924:
921:
917:
914:
910:
908:
905:
903:
900:
898:
895:
893:
890:
888:
885:
883:
880:
879:
878:
877:
873:
859:
856:
852:
849:has order 5,
848:
845:has order 4,
844:
841:has order 2,
840:
836:
832:
828:
827:Suzuki groups
824:
820:
818:
814:
812:
808:
804:
800:
796:
792:
788:
784:
780:
766:
760:
757:
754:
751:
748:
745:
742:
741:
740:
735:
733:(1 4 2 5 3 6)
732:
729:
727:(1 3 5 2 4 6)
726:
723:
720:
718:(1 2 3 4 5 6)
717:
716:
715:
710:
707:
704:
701:
698:
695:
692:
691:
690:
689:
685:
673:
667:
664:
661:
658:
655:
654:
653:
648:
645:
642:
639:
636:
635:
634:
633:
629:
623:
613:
611:
605:
603:
599:
594:
592:
588:
583:
579:
575:
571:
567:
562:
558:
553:
549:
544:
540:
536:
532:
527:
523:
518:
514:
510:
505:
501:
497:
492:
488:
481:
477:
473:
469:
465:
461:
457:
453:
449:
439:
437:
432:
430:
426:
422:
418:
414:
410:
406:
402:
398:
395:
387:
383:
376:
369:
364:
360:
357:
355:
351:
347:
342:
338:
323:
315:
311:
306:
302:
299:
297:
292:
288:
285:
284:
283:
280:
276:
272:
268:
264:
259:
255:
248:
244:
240:
236:
232:
217:
215:
210:
208:
204:
200:
196:
192:
188:
185: ∈
184:
181:⟩ and
180:
176:
172:
167:
165:
161:
157:
151:
149:
145:
141:
137:
133:
129:
125:
121:
117:
113:
110: ∈
109:
104:
102:
98:
94:
91: ∈
90:
86:
83: ∈
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
33:
31:
27:
23:
19:
2245:Group theory
2219:
2190:
2184:
2174:
2153:
2135:
2129:
2123:
2118:
2114:
2110:
2106:
2102:
2099:
2094:
2090:
2086:
2080:
2074:
2070:
2066:
2061:
2057:
2052:
2048:
2041:
2029:
2022:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1980:
1975:
1971:
1967:
1963:
1959:
1955:
1948:
1941:
1935:
1929:
1925:
1921:
1917:
1910:
1906:
1904:
1898:
1891:
1887:
1883:
1879:
1875:
1868:
1862:
1858:
1854:
1850:
1846:
1839:
1832:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1796:Cayley graph
1791:
1787:
1783:
1779:
1775:
1771:
1769:
1758:
1754:
1750:
1746:
1738:
1734:
1728:
1724:
1720:
1716:
1711:
1707:
1703:
1699:
1695:
1690:
1686:
1679:
1672:
1665:
1659:
1655:
1650:
1646:
1641:
1637:
1632:
1628:
1623:
1619:
1614:
1610:
1606:
1601:
1597:
1592:
1588:
1581:
1575:
1571:
1564:
1557:
1551:
1547:
1540:
1533:
1526:
1522:
1518:
1511:
1504:
1496:
1492:
1485:
1481:
1470:
1466:
1459:
1455:
1450:
1436:
1432:
1425:
1421:
1414:
1410:
1402:
1396:
1394:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1361:
1359:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1315:
1311:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1272:
1268:
1263:
1258:
1253:
1249:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1213:
1209:
1204:
1200:
1195:
1191:
1184:
1177:
1173:
1169:
1167:
1162:
1158:
1154:
1150:
1143:
1136:
1132:
1128:
1123:
1119:
1112:
1108:
1106:
1101:
1100: : 2 ≤
1097:
1093:
1089:
1085:
1083:
1075:
1068:
1064:
1060:
1056:
1054:
1041:
999:
993:
983:
979:
975:
969:
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951:
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940:
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926:
919:
912:
906:
901:
896:
891:
886:
881:
857:
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850:
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830:
822:
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816:
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810:
806:
802:
798:
794:
790:
786:
778:
777:
774:Applications
764:
679:
671:
619:
609:
606:
601:
597:
595:
590:
586:
581:
577:
576:, such that
573:
569:
565:
560:
556:
551:
547:
542:
538:
534:
530:
525:
521:
516:
512:
508:
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490:
486:
479:
475:
471:
467:
463:
459:
455:
451:
447:
445:
435:
433:
428:
424:
420:
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412:
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400:
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336:
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274:
270:
266:
262:
257:
253:
246:
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238:
234:
230:
228:
211:
206:
202:
194:
190:
186:
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178:
174:
170:
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163:
159:
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152:
143:
139:
135:
131:
127:
123:
119:
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564:) for some
75:is said to
2239:Categories
2193:: 85–115.
2145:References
2100:Therefore
1299:, declare
1153:. The set
1092:≥1 let lg(
462:of length
419:computing
220:Definition
193:over
2200:1409.8390
1587:for some
1525:+1) with
1307:and stop.
600:=⟨
380:for some
344:for some
324:⋅
116:efficient
30:operation
1706:. Hence
1261:(0) = {1
1199: :
1096:) = max{
837:, where
711:αβαβαβαβ
616:Examples
273:if each
95:, where
2139:|)
1740:Claim 2
1694:, with
1404:Claim 1
1257:}. Let
1221:at the
1141:2|
1079:|)
1063:, each
823:Example
203:oracles
114:is an
77:compute
26:program
2001:, any
1890:) and
1794:. The
1782:+1) -
1770:Since
1501:|
1479:|
1475:|
1453:|
1441:|
1419:|
1348:+1) −
1267:} and
1147:|
1042:
980:ababbb
970:ababbb
907:ababbb
765:
672:
2195:arXiv
2089:from
2028:with
1867:with
1806:<
1790:) ≤ 2
1767:Proof
1702:<
1685:·...·
1664:·...·
1570:·...·
1546:·...·
1448:Proof
1417:then
1413:<
1248:,...,
1241:) = {
1190:·...·
1176:) = {
1149:over
1118:,...,
984:ababb
976:ababb
966:ababb
959:ababb
952:ababb
902:ababb
809:(log|
797:(log|
721:(1 2)
572:<
537:, or
533:<
511:, or
485:,...,
466:over
384:<
352:<
269:over
265:is a
252:,...,
162:(log|
37:group
2113:) ≤
2073:) =
1997:) =
1974:), 2
1826:) ≠
1753:) ≤
1295:) =
1168:Let
855:abab
833:and
813:|).
708:αβαβ
680:In S
620:The
458:. A
446:Let
394:cost
229:Let
20:, a
2205:doi
2191:221
2097:).
2083:− 1
1798:of
1723:−1)
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1462:+1)
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1409:If
1279:If
1165:).
1111:= {
1104:}.
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920:abb
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705:αβα
546:= (
520:= (
245:= (
87:if
45:SLP
16:In
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2162:^
2141:.
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1762:.
1757:−
1715:∈
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1645:=
1618:≠
1580:=
1578:+1
1556:=
1554:+1
1532:=
1517:∈
1443:.
1392:.
1368:∈
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1320:∈
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868:⋊C
866:32
864:·E
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702:αβ
649:λρ
626:12
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474:∈
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411:∈
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308:=
294:∈
216:.
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43:(
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