1929:
1726:. Therefore the question of computability of proofs is not separated from the question of their existence. To overcome this problem it is convenient to consider the restriction of function problems to total relations yielding the class
479:
178:
1439:
1781:
1603:
1564:
430:
521:
1724:
279:
1511:
1476:
1133:
1106:
1079:
1052:
871:
565:
499:
384:
1015:(easy), while the integer factorization problem is believed to be hard for a classical computer. There are several (slightly different) notions of self-reducibility.
126:
1646:
979:
952:
925:
898:
689:
1326:
1686:
1666:
1346:
1297:
1277:
1253:
1233:
1213:
1193:
1173:
1153:
793:
769:
749:
729:
709:
657:
637:
607:
545:
342:
319:
299:
241:
221:
201:
99:
76:
567:, only needs to return "unsatisfiable" or "satisfiable", an FSAT algorithm needs to return some satisfying assignment in the latter case.
435:
1894:
In S. Michaelson and R. Milner, editors, Proceedings of the 3rd
International Colloquium on Automata, Languages, and Programming
134:
547:
is given by tuples of suitably encoded boolean formulas and satisfying assignments. While a SAT algorithm, fed with a formula
1353:
2014:
1996:
1972:
1950:
1751:
17:
1943:
1569:
2036:
2031:
31:
1536:
357:
1001:
if its function variant can be solved in polynomial time using an oracle deciding the original problem. Every
1802:
1008:
575:
571:
389:
106:
102:
1828:
849:
introduced above can be solved using only polynomially many calls to a subroutine which decides the
1937:
504:
1691:
246:
1954:
1024:
79:
1489:
1454:
1111:
1084:
1057:
1030:
856:
550:
484:
369:
1836:
111:
39:
1616:
1807:
957:
930:
903:
876:
662:
1302:
8:
900:
to TRUE and ask again. If the resulting formula is still satisfiable the algorithm keeps
352:
A well-known function problem is given by the
Functional Boolean Satisfiability Problem,
324:
A promise function problem is allowed to do anything (thus may not terminate) if no such
1855:
1671:
1651:
1648:
used to define function problems has the drawback of being incomplete: Not every input
1331:
1282:
1262:
1238:
1218:
1198:
1178:
1158:
1138:
778:
754:
734:
714:
694:
642:
622:
592:
530:
327:
304:
284:
226:
206:
186:
84:
61:
2010:
1992:
1740:
in certain strategic games where a solution is guaranteed to exist. In addition, if
1845:
1792:
797:
587:
47:
1737:
837:, consists of function problems whose solutions can be found in polynomial time.
829:
816:
610:
1850:
1797:
986:
43:
2025:
46:) is expected for every input, but the output is more complex than that of a
659:
which serves as a proof for the 'yes' answer. Thus, the set of these tuples
1011:
is not self-reducible, because deciding whether an integer is prime is in
2002:
1003:
1859:
1892:
Schnorr, C. (1976). "Optimal algorithms for self-reducible problems".
2017:, section 28.10 "The problem classes FP and FNP", pp. 689–694
1989:
Fundamentals of the theory of computation: principles and practice
474:{\displaystyle x_{i}\rightarrow \{{\text{TRUE}},{\text{FALSE}}\}}
50:. For function problems, the output is not simply 'yes' or 'no'.
2007:
Automata, computability and complexity: theory and applications
1736:. This class contains problems such as the computation of pure
1873:
Ko, K. (1983). "On self-reducibility and weak P-selectivity".
833:, which can be thought of as the function class analogue of
1728:
691:
forms a relation, representing the function problem "given
581:
873:
is satisfiable. After that the algorithm can fix variable
574:, which asks for the route taken by the salesman, and the
639:
that is answered 'yes' has a polynomial-size certificate
173:{\displaystyle R\subseteq \Sigma ^{*}\times \Sigma ^{*}.}
1827:
Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin (2004).
853:
problem: An algorithm can first ask whether the formula
356:
for short. The problem, which is closely related to the
1135:
if there exists polynomially-time computable functions
1007:
problem is self-reducible. It is conjectured that the
1434:{\displaystyle (f(x),y)\in S\implies (x,g(x,y))\in R.}
1754:
1694:
1674:
1654:
1619:
1572:
1539:
1492:
1457:
1356:
1334:
1305:
1285:
1265:
1241:
1221:
1201:
1181:
1161:
1141:
1114:
1087:
1060:
1033:
1027:
much like decision problems: Given function problems
960:
933:
906:
879:
859:
781:
757:
737:
717:
697:
665:
645:
625:
595:
553:
533:
507:
487:
438:
392:
372:
330:
307:
287:
249:
229:
209:
189:
137:
114:
87:
64:
807:
can be thought of as the function class analogue of
1018:
1907:Selman, A. (1988). "Natural self-reducible sets".
1826:
1775:
1718:
1680:
1660:
1640:
1597:
1558:
1505:
1470:
1433:
1340:
1320:
1291:
1271:
1247:
1227:
1207:
1187:
1167:
1147:
1127:
1100:
1073:
1046:
973:
946:
919:
892:
865:
787:
763:
743:
723:
703:
683:
651:
631:
601:
559:
539:
515:
493:
473:
424:
378:
336:
313:
293:
273:
235:
215:
195:
172:
120:
93:
70:
2023:
362:decision problem, can be formulated as follows:
1776:{\displaystyle \mathbf {NP} ={\textbf {co-NP}}}
1448:problems analogous to the NP-complete problem:
1598:{\displaystyle \mathbf {FP} =\mathbf {FNP} }
468:
452:
1608:
1559:{\displaystyle \mathbf {P} =\mathbf {NP} }
1391:
1387:
1973:Learn how and when to remove this message
1849:
523:or decide that no such assignment exists.
1936:This article includes a list of general
985:is solvable in polynomial time using an
582:Relationship to other complexity classes
1891:
1875:Journal of Computer and System Sciences
771:". This function problem is called the
14:
2024:
1906:
815:problems can be efficiently (i.e., in
578:, which asks for the list of factors.
981:has to be FALSE and continues. Thus,
819:in terms of the length of the input)
1922:
840:
53:
1987:Raymond Greenlaw, H. James Hoover,
1768:
1444:It is therefore possible to define
927:fixed to TRUE and continues to fix
570:Other notable examples include the
425:{\displaystyle x_{1},\ldots ,x_{n}}
24:
1942:it lacks sufficient corresponding
1872:
1820:
1494:
1459:
1116:
1089:
1062:
1035:
823:, but not necessarily efficiently
281:, the algorithm produces one such
158:
145:
115:
25:
2048:
1927:
1759:
1756:
1591:
1588:
1585:
1577:
1574:
1552:
1549:
1541:
1019:Reductions and complete problems
32:computational complexity theory
1900:
1885:
1866:
1707:
1695:
1635:
1623:
1419:
1416:
1404:
1392:
1388:
1378:
1369:
1363:
1357:
1315:
1309:
678:
666:
449:
262:
250:
13:
1:
1813:
1803:Counting problem (complexity)
1009:integer factorization problem
576:integer factorization problem
516:{\displaystyle {\text{TRUE}}}
27:Type of computational problem
1175:such that for all instances
954:, otherwise it decides that
42:where a single output (of a
7:
1851:10.4007/annals.2004.160.781
1786:
1533:problem, and it holds that
993:. In general, a problem in
572:travelling salesman problem
347:
301:, and if there are no such
10:
2053:
1719:{\displaystyle (x,y)\in R}
1513:. The complexity class of
795:; it belongs to the class
527:In this case the relation
274:{\displaystyle (x,y)\in R}
1991:, Morgan Kaufmann, 1998,
1909:SIAM Journal on Computing
1023:Function problems can be
845:Observe that the problem
827:. In contrast, the class
223:such that there exists a
1748:problem it follows that
1506:{\displaystyle \Pi _{R}}
1471:{\displaystyle \Pi _{R}}
1128:{\displaystyle \Pi _{S}}
1101:{\displaystyle \Pi _{R}}
1074:{\displaystyle \Pi _{S}}
1047:{\displaystyle \Pi _{R}}
866:{\displaystyle \varphi }
619:, each problem instance
560:{\displaystyle \varphi }
494:{\displaystyle \varphi }
379:{\displaystyle \varphi }
366:Given a boolean formula
2009:, Prentice Hall, 2008,
1957:more precise citations.
1609:Total function problems
1517:problems is denoted by
1215:and possible solutions
811:, in that solutions of
615:. By the definition of
121:{\displaystyle \Sigma }
2037:Functions and mappings
2032:Computational problems
1777:
1720:
1682:
1662:
1642:
1641:{\displaystyle R(x,y)}
1599:
1560:
1507:
1472:
1435:
1342:
1322:
1293:
1273:
1249:
1229:
1209:
1189:
1169:
1149:
1129:
1102:
1075:
1048:
975:
948:
921:
894:
867:
789:
765:
745:
725:
705:
685:
653:
633:
603:
586:Consider an arbitrary
561:
541:
517:
495:
475:
426:
380:
338:
315:
295:
275:
237:
217:
197:
174:
122:
95:
72:
1837:Annals of Mathematics
1778:
1721:
1683:
1663:
1643:
1600:
1561:
1508:
1473:
1436:
1343:
1323:
1294:
1274:
1250:
1230:
1210:
1190:
1170:
1150:
1130:
1103:
1076:
1049:
976:
974:{\displaystyle x_{1}}
949:
947:{\displaystyle x_{2}}
922:
920:{\displaystyle x_{1}}
895:
893:{\displaystyle x_{1}}
868:
790:
766:
746:
731:, find a certificate
726:
706:
686:
684:{\displaystyle (x,y)}
654:
634:
604:
562:
542:
518:
496:
476:
432:, find an assignment
427:
381:
339:
316:
296:
276:
238:
218:
198:
175:
123:
96:
73:
58:A functional problem
40:computational problem
1808:Optimization problem
1752:
1692:
1672:
1652:
1617:
1570:
1537:
1525:. Hence the problem
1490:
1482:if every problem in
1455:
1354:
1332:
1321:{\displaystyle f(x)}
1303:
1283:
1263:
1239:
1219:
1199:
1179:
1159:
1139:
1112:
1085:
1058:
1031:
958:
931:
904:
877:
857:
779:
755:
735:
715:
695:
663:
643:
623:
593:
551:
531:
505:
485:
436:
390:
370:
328:
305:
285:
247:
227:
207:
187:
183:An algorithm solves
135:
112:
85:
62:
203:if for every input
1773:
1716:
1678:
1668:has a counterpart
1658:
1638:
1595:
1556:
1503:
1486:can be reduced to
1468:
1431:
1338:
1318:
1289:
1269:
1245:
1225:
1205:
1185:
1165:
1145:
1125:
1098:
1071:
1044:
971:
944:
917:
890:
863:
785:
761:
741:
721:
701:
681:
649:
629:
599:
557:
537:
513:
491:
471:
422:
376:
334:
311:
291:
271:
233:
213:
193:
170:
118:
91:
68:
1983:
1982:
1975:
1770:
1732:as a subclass of
1681:{\displaystyle y}
1661:{\displaystyle x}
1341:{\displaystyle S}
1292:{\displaystyle R}
1272:{\displaystyle x}
1248:{\displaystyle S}
1228:{\displaystyle y}
1208:{\displaystyle R}
1188:{\displaystyle x}
1168:{\displaystyle g}
1148:{\displaystyle f}
841:Self-reducibility
788:{\displaystyle L}
764:{\displaystyle x}
744:{\displaystyle y}
724:{\displaystyle L}
704:{\displaystyle x}
652:{\displaystyle y}
632:{\displaystyle x}
602:{\displaystyle L}
540:{\displaystyle R}
511:
466:
458:
337:{\displaystyle y}
314:{\displaystyle y}
294:{\displaystyle y}
236:{\displaystyle y}
216:{\displaystyle x}
196:{\displaystyle P}
94:{\displaystyle R}
71:{\displaystyle P}
54:Formal definition
18:Function problems
16:(Redirected from
2044:
1978:
1971:
1967:
1964:
1958:
1953:this article by
1944:inline citations
1931:
1930:
1923:
1917:
1916:
1904:
1898:
1897:
1889:
1883:
1882:
1870:
1864:
1863:
1853:
1833:
1829:"PRIMES is in P"
1824:
1793:Decision problem
1782:
1780:
1779:
1774:
1772:
1771:
1762:
1725:
1723:
1722:
1717:
1687:
1685:
1684:
1679:
1667:
1665:
1664:
1659:
1647:
1645:
1644:
1639:
1604:
1602:
1601:
1596:
1594:
1580:
1565:
1563:
1562:
1557:
1555:
1544:
1512:
1510:
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1504:
1502:
1501:
1477:
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1474:
1469:
1467:
1466:
1440:
1438:
1437:
1432:
1347:
1345:
1344:
1339:
1327:
1325:
1324:
1319:
1299:-solution, then
1298:
1296:
1295:
1290:
1278:
1276:
1275:
1270:
1255:, it holds that
1254:
1252:
1251:
1246:
1234:
1232:
1231:
1226:
1214:
1212:
1211:
1206:
1194:
1192:
1191:
1186:
1174:
1172:
1171:
1166:
1154:
1152:
1151:
1146:
1134:
1132:
1131:
1126:
1124:
1123:
1107:
1105:
1104:
1099:
1097:
1096:
1080:
1078:
1077:
1072:
1070:
1069:
1053:
1051:
1050:
1045:
1043:
1042:
980:
978:
977:
972:
970:
969:
953:
951:
950:
945:
943:
942:
926:
924:
923:
918:
916:
915:
899:
897:
896:
891:
889:
888:
872:
870:
869:
864:
794:
792:
791:
786:
773:function variant
770:
768:
767:
762:
750:
748:
747:
742:
730:
728:
727:
722:
710:
708:
707:
702:
690:
688:
687:
682:
658:
656:
655:
650:
638:
636:
635:
630:
608:
606:
605:
600:
588:decision problem
566:
564:
563:
558:
546:
544:
543:
538:
522:
520:
519:
514:
512:
509:
500:
498:
497:
492:
480:
478:
477:
472:
467:
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447:
431:
429:
428:
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401:
385:
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377:
343:
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335:
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202:
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179:
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171:
166:
165:
153:
152:
127:
125:
124:
119:
105:of an arbitrary
100:
98:
97:
92:
78:is defined by a
77:
75:
74:
69:
48:decision problem
36:function problem
21:
2052:
2051:
2047:
2046:
2045:
2043:
2042:
2041:
2022:
2021:
2020:
1999:, p. 45-51
1979:
1968:
1962:
1959:
1949:Please help to
1948:
1932:
1928:
1921:
1920:
1905:
1901:
1890:
1886:
1871:
1867:
1831:
1825:
1821:
1816:
1789:
1767:
1766:
1755:
1753:
1750:
1749:
1738:Nash equilibria
1693:
1690:
1689:
1673:
1670:
1669:
1653:
1650:
1649:
1618:
1615:
1614:
1611:
1584:
1573:
1571:
1568:
1567:
1566:if and only if
1548:
1540:
1538:
1535:
1534:
1497:
1493:
1491:
1488:
1487:
1462:
1458:
1456:
1453:
1452:
1355:
1352:
1351:
1333:
1330:
1329:
1304:
1301:
1300:
1284:
1281:
1280:
1264:
1261:
1260:
1240:
1237:
1236:
1220:
1217:
1216:
1200:
1197:
1196:
1180:
1177:
1176:
1160:
1157:
1156:
1140:
1137:
1136:
1119:
1115:
1113:
1110:
1109:
1092:
1088:
1086:
1083:
1082:
1065:
1061:
1059:
1056:
1055:
1038:
1034:
1032:
1029:
1028:
1021:
965:
961:
959:
956:
955:
938:
934:
932:
929:
928:
911:
907:
905:
902:
901:
884:
880:
878:
875:
874:
858:
855:
854:
843:
817:polynomial time
780:
777:
776:
756:
753:
752:
736:
733:
732:
716:
713:
712:
696:
693:
692:
664:
661:
660:
644:
641:
640:
624:
621:
620:
594:
591:
590:
584:
552:
549:
548:
532:
529:
528:
508:
506:
503:
502:
486:
483:
482:
463:
455:
443:
439:
437:
434:
433:
416:
412:
397:
393:
391:
388:
387:
386:with variables
371:
368:
367:
350:
329:
326:
325:
306:
303:
302:
286:
283:
282:
248:
245:
244:
228:
225:
224:
208:
205:
204:
188:
185:
184:
161:
157:
148:
144:
136:
133:
132:
113:
110:
109:
86:
83:
82:
63:
60:
59:
56:
28:
23:
22:
15:
12:
11:
5:
2050:
2040:
2039:
2034:
2019:
2018:
2000:
1984:
1981:
1980:
1935:
1933:
1926:
1919:
1918:
1899:
1884:
1865:
1844:(2): 781–793.
1818:
1817:
1815:
1812:
1811:
1810:
1805:
1800:
1798:Search problem
1795:
1788:
1785:
1765:
1761:
1758:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1677:
1657:
1637:
1634:
1631:
1628:
1625:
1622:
1610:
1607:
1593:
1590:
1587:
1583:
1579:
1576:
1554:
1551:
1547:
1543:
1500:
1496:
1465:
1461:
1442:
1441:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1390:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1349:
1337:
1317:
1314:
1311:
1308:
1288:
1268:
1244:
1224:
1204:
1184:
1164:
1144:
1122:
1118:
1095:
1091:
1068:
1064:
1041:
1037:
1020:
1017:
999:self-reducible
968:
964:
941:
937:
914:
910:
887:
883:
862:
842:
839:
784:
760:
740:
720:
700:
680:
677:
674:
671:
668:
648:
628:
598:
583:
580:
556:
536:
525:
524:
490:
470:
462:
454:
451:
446:
442:
419:
415:
411:
408:
405:
400:
396:
375:
349:
346:
333:
321:, it rejects.
310:
290:
270:
267:
264:
261:
258:
255:
252:
232:
212:
192:
181:
180:
169:
164:
160:
156:
151:
147:
143:
140:
117:
90:
67:
55:
52:
44:total function
26:
9:
6:
4:
3:
2:
2049:
2038:
2035:
2033:
2030:
2029:
2027:
2016:
2015:0-13-228806-0
2012:
2008:
2004:
2001:
1998:
1997:1-55860-474-X
1994:
1990:
1986:
1985:
1977:
1974:
1966:
1956:
1952:
1946:
1945:
1939:
1934:
1925:
1924:
1915:(5): 989–996.
1914:
1910:
1903:
1895:
1888:
1881:(2): 209–221.
1880:
1876:
1869:
1861:
1857:
1852:
1847:
1843:
1839:
1838:
1830:
1823:
1819:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1790:
1784:
1763:
1747:
1744:contains any
1743:
1739:
1735:
1731:
1730:
1713:
1710:
1704:
1701:
1698:
1675:
1655:
1632:
1629:
1626:
1620:
1613:The relation
1606:
1581:
1545:
1532:
1528:
1524:
1520:
1516:
1498:
1485:
1481:
1463:
1449:
1447:
1428:
1425:
1422:
1413:
1410:
1407:
1401:
1398:
1395:
1384:
1381:
1375:
1372:
1366:
1360:
1350:
1335:
1312:
1306:
1286:
1266:
1258:
1257:
1256:
1242:
1222:
1202:
1182:
1162:
1142:
1120:
1093:
1066:
1039:
1026:
1016:
1014:
1010:
1006:
1005:
1000:
996:
992:
988:
984:
966:
962:
939:
935:
912:
908:
885:
881:
860:
852:
848:
838:
836:
832:
831:
826:
822:
818:
814:
810:
806:
802:
800:
799:
782:
774:
758:
738:
718:
698:
675:
672:
669:
646:
626:
618:
614:
613:
609:in the class
596:
589:
579:
577:
573:
568:
554:
534:
501:evaluates to
488:
460:
444:
440:
417:
413:
409:
406:
403:
398:
394:
373:
365:
364:
363:
361:
360:
355:
345:
331:
322:
308:
288:
268:
265:
259:
256:
253:
230:
210:
190:
167:
162:
154:
149:
141:
138:
131:
130:
129:
108:
104:
88:
81:
65:
51:
49:
45:
41:
37:
33:
19:
2006:
1988:
1969:
1963:October 2015
1960:
1941:
1912:
1908:
1902:
1893:
1887:
1878:
1874:
1868:
1841:
1835:
1822:
1746:FNP-complete
1745:
1741:
1733:
1727:
1612:
1531:FNP-complete
1530:
1526:
1522:
1518:
1515:FNP-complete
1514:
1483:
1480:FNP-complete
1479:
1450:
1446:FNP-complete
1445:
1443:
1081:we say that
1022:
1012:
1002:
998:
994:
990:
982:
850:
846:
844:
834:
828:
824:
820:
812:
808:
804:
803:
796:
772:
616:
611:
585:
569:
526:
358:
353:
351:
323:
182:
57:
35:
29:
2003:Elaine Rich
1955:introducing
1529:is also an
1108:reduces to
1004:NP-complete
243:satisfying
2026:Categories
1938:references
1896:: 322–337.
1814:References
1688:such that
1451:A problem
1348:-solution.
997:is called
481:such that
1711:∈
1495:Π
1460:Π
1423:∈
1389:⟹
1382:∈
1117:Π
1090:Π
1063:Π
1036:Π
989:deciding
861:φ
555:φ
489:φ
450:→
407:…
374:φ
266:∈
163:∗
159:Σ
155:×
150:∗
146:Σ
142:⊆
116:Σ
1787:See also
821:verified
348:Examples
344:exists.
107:alphabet
80:relation
1951:improve
1860:3597229
1328:has an
1279:has an
1025:reduced
103:strings
2013:
1995:
1940:, but
1858:
987:oracle
1856:JSTOR
1832:(PDF)
1769:co-NP
1519:FNP-C
825:found
465:FALSE
101:over
38:is a
2011:ISBN
1993:ISBN
1742:TFNP
1729:TFNP
1527:FSAT
1523:FNPC
1155:and
1054:and
983:FSAT
847:FSAT
751:for
510:TRUE
457:TRUE
354:FSAT
34:, a
1846:doi
1842:160
1734:FNP
1521:or
1484:FNP
1478:is
1259:If
1235:of
1195:of
991:SAT
851:SAT
813:FNP
805:FNP
798:FNP
775:of
711:in
359:SAT
30:In
2028::
2005:,
1913:17
1911:.
1879:26
1877:.
1854:.
1840:.
1834:.
1783:.
1605:.
995:NP
830:FP
809:NP
801:.
617:NP
612:NP
128::
1976:)
1970:(
1965:)
1961:(
1947:.
1862:.
1848::
1764:=
1760:P
1757:N
1714:R
1708:)
1705:y
1702:,
1699:x
1696:(
1676:y
1656:x
1636:)
1633:y
1630:,
1627:x
1624:(
1621:R
1592:P
1589:N
1586:F
1582:=
1578:P
1575:F
1553:P
1550:N
1546:=
1542:P
1499:R
1464:R
1429:.
1426:R
1420:)
1417:)
1414:y
1411:,
1408:x
1405:(
1402:g
1399:,
1396:x
1393:(
1385:S
1379:)
1376:y
1373:,
1370:)
1367:x
1364:(
1361:f
1358:(
1336:S
1316:)
1313:x
1310:(
1307:f
1287:R
1267:x
1243:S
1223:y
1203:R
1183:x
1163:g
1143:f
1121:S
1094:R
1067:S
1040:R
1013:P
967:1
963:x
940:2
936:x
913:1
909:x
886:1
882:x
835:P
783:L
759:x
739:y
719:L
699:x
679:)
676:y
673:,
670:x
667:(
647:y
627:x
597:L
535:R
469:}
461:,
453:{
445:i
441:x
418:n
414:x
410:,
404:,
399:1
395:x
332:y
309:y
289:y
269:R
263:)
260:y
257:,
254:x
251:(
231:y
211:x
191:P
168:.
139:R
89:R
66:P
20:)
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