22:
122:
63:
1983:
3137:
3895:
4068:
742:
2701:
2966:
1278:
3714:
1150:
3900:
1751:
1808:
1225:
1909:
1408:
3255:
1001:
3324:
792:
1360:
1453:
882:
2266:
2900:
1093:
3705:
3051:
1629:
2861:
917:
610:
2140:
948:
3083:
2047:
1947:
1484:
1186:
1308:
822:
678:
2755:
1977:
1830:
381:
3014:
1522:
1035:
2827:
2803:
2779:
666:
315:
2733:
1587:
642:
405:
347:
283:
243:
3160:
3073:
3124:
528:
503:
478:
441:
4317:
4275:
4233:
4191:
3654:
3612:
3570:
3528:
4158:
4130:
4102:
3495:
3467:
3439:
3405:
3378:
3351:
2621:
2594:
2567:
2540:
2513:
2486:
2459:
2432:
2405:
2378:
2351:
2324:
2293:
2194:
2167:
2074:
1687:
1660:
1549:
566:
2637:
2596:
sentry point since, like any diplomatic seat, it should be located outside all other concepts just to ensure that it is not occupied in case of invasion by
2902:. The former uses points to separate sets of concepts, the latter concepts for partitioning sets of points. In particular the following inequality holds (
2912:
140:
3890:{\displaystyle \mathbf {S} (c_{1})=\{x_{1},x_{2}\},\mathbf {S} (c_{2})=\{x_{1}\},\mathbf {S} (c_{3})=\{x_{2}\},\mathbf {S} (c_{4})=\emptyset }
1230:
4063:{\displaystyle \mathbf {S} (c_{1})=\{x_{3}\},\mathbf {S} (c_{2})=\{x_{1}\},\mathbf {S} (c_{3})=\{x_{2}\},\mathbf {S} (c_{4})=\emptyset }
1098:
1692:
4388:
1756:
2631:
The frontier size of the most expensive concept to be sentineled with the least efficient sentineling function, i.e. the quantity
1835:
3178:
1195:
1365:
3260:
754:
325:
a set of points that, since are defined to be external to the concept, prevent it from expanding into another function of
953:
837:
2199:
176:
158:
103:
49:
35:
1319:
2869:
1044:
3673:
3019:
1413:
2835:
353:
from being fully enclosed (invaded) by another concept within the class. Therefore, we call these points either
4393:
201:
of a function denotes the level of informational content, which in turn affects the difficulty of learning the
2079:
1592:
2781:
sentineling in this case the intersections of the concepts with these subsets. Actually, proper subsets of
1993:
737:{\displaystyle {\boldsymbol {S}}:{\mathsf {C}}\cup \{\emptyset ,{\mathfrak {X}}\}\mapsto 2^{\mathfrak {X}}}
1921:
1458:
1160:
887:
575:
1283:
922:
797:
2738:
2295:, provided that these points are not used by the latter for sentineling itself against other concepts.
1960:
1813:
364:
2990:
1493:
1006:
213:
characterize the entire class of functions to which the one we are interested in belongs. Focusing on
2808:
2784:
2760:
647:
296:
2714:
623:
386:
328:
264:
224:
81:
3143:
3056:
3089:
4344:. International Series on Advanced Intelligence. Vol. 5 (2nd ed.). Adelaide: Magill.
206:
3130:(thick circle) are sufficient to prevent a larger circle not containing them from including it
1554:
2972:
85:
41:
4292:
4250:
4208:
4166:
3629:
3587:
3545:
3503:
2696:{\displaystyle \mathrm {D} _{\mathsf {C}}=\sup _{{\boldsymbol {S}},c}\#{\boldsymbol {S}}(c)}
4136:
4108:
4080:
3473:
3445:
3417:
3383:
3356:
3329:
2599:
2572:
2545:
2518:
2491:
2464:
2437:
2410:
2383:
2356:
2329:
2302:
2271:
2172:
2145:
2052:
1665:
1638:
1527:
544:
136:
8:
202:
73:
508:
483:
458:
421:
4370:
4353:
3326:
whose concepts are illustrated in the following scheme, where "+" denotes an element
4365:
214:
190:
3053:, as shown in the picture on left below. Similarly, for the class of segments on
318:
673:
2961:{\displaystyle \mathrm {D} _{\mathsf {C}}\leq \mathrm {D} _{{\mathsf {V}}C}+1}
349:. We may dually define these points in terms of sentinelling a given concept
4382:
2864:
4354:"PAC learning of concept classes through the boundaries of their items"
194:
2805:
may host sentineling tasks that prove harder than those emerging with
3707:. As usual we may have different sentineling functions. A worst case
1455:. This however must be a consequence of the fact that all points of
1273:{\displaystyle {\boldsymbol {S}}'(c)\subseteq {\boldsymbol {S}}(c)}
3136:
3082:
1145:{\displaystyle c_{2}\cap {\boldsymbol {S}}(c_{1})\neq \emptyset }
415:
to be sentineled and internal to at least one other including it,
1746:{\displaystyle c_{1}\subset c_{2}\cup {\boldsymbol {S}}(c_{2})}
1982:
1803:{\displaystyle c_{2}\cap {\boldsymbol {S}}(c_{1})=\emptyset }
1904:{\displaystyle \{c_{1}\}\cup \mathrm {up} (c_{1})-\{c_{2}\}}
1227:
exists satisfying (1) and (2) and having the property that
3250:{\displaystyle {\mathsf {C}}=\{c_{1},c_{2},c_{3},c_{4}\}}
249:
essentially denotes how deeply the class is articulated.
1220:{\displaystyle {\boldsymbol {S}}'\neq {\boldsymbol {S}}}
2863:
is a complexity measure of concept classes dual to the
1986:
A schematic outlook of outer sentineling functionality
1403:{\displaystyle {\boldsymbol {S}}(c)\cap c'=\emptyset }
321:. In this framework, the above function associates to
4295:
4253:
4211:
4169:
4139:
4111:
4083:
3903:
3717:
3676:
3632:
3590:
3548:
3506:
3476:
3448:
3420:
3386:
3359:
3332:
3319:{\displaystyle {\mathfrak {X}}=\{x_{1},x_{2},x_{3}\}}
3263:
3181:
3146:
3092:
3059:
3022:
2993:
2915:
2872:
2838:
2811:
2787:
2763:
2741:
2717:
2640:
2602:
2575:
2548:
2521:
2494:
2467:
2440:
2413:
2386:
2359:
2332:
2305:
2274:
2202:
2175:
2148:
2082:
2055:
1996:
1963:
1924:
1838:
1816:
1759:
1695:
1668:
1641:
1595:
1557:
1530:
1496:
1461:
1416:
1368:
1322:
1286:
1233:
1198:
1163:
1101:
1047:
1009:
956:
925:
890:
840:
800:
787:{\displaystyle c\cap {\boldsymbol {S}}(c)=\emptyset }
757:
681:
650:
626:
578:
547:
541:) is rooted in the inclusion of an augmented concept
511:
486:
461:
424:
389:
367:
331:
299:
267:
227:
533:
they constitute a minimal set with these properties.
131:
may be too technical for most readers to understand
4311:
4269:
4227:
4185:
4152:
4124:
4096:
4062:
3889:
3699:
3648:
3606:
3564:
3522:
3489:
3461:
3433:
3399:
3372:
3345:
3318:
3249:
3154:
3118:
3067:
3045:
3008:
2975:for a recently introduced class complexity index.
2960:
2894:
2855:
2821:
2797:
2773:
2749:
2727:
2695:
2615:
2588:
2561:
2534:
2507:
2480:
2453:
2426:
2399:
2372:
2345:
2318:
2287:
2260:
2188:
2161:
2134:
2068:
2041:
1971:
1941:
1903:
1824:
1802:
1745:
1681:
1654:
1623:
1581:
1543:
1516:
1478:
1447:
1402:
1354:
1302:
1272:
1219:
1180:
1144:
1087:
1029:
996:{\displaystyle c^{+}\subseteq \left(c'\right)^{+}}
995:
942:
911:
876:
816:
786:
736:
660:
636:
604:
560:
522:
497:
472:
435:
399:
375:
341:
309:
277:
237:
285:. Let us focus for a moment on a single function
209:, which is the difficulty to compute a function.
4380:
2659:
877:{\displaystyle c^{+}=c\cup {\boldsymbol {S}}(c)}
615:
4340:Apolloni, B.; Malchiodi, D.; Gaito, S. (2006).
2757:spans also over sentry functions on subsets of
2261:{\displaystyle c_{0}\cup \{x_{1},x_{2},x_{3}\}}
1316:Sentinels are honest guardians. It may be that
3162:and two points needed to sentinel its concepts
1355:{\displaystyle c\subseteq \left(c'\right)^{+}}
411:the sentry points are external to the concept
317:of elements that we may figure as points in a
257:To identify this index we must first define a
2978:
2895:{\displaystyle \mathrm {D} _{{\mathsf {V}}C}}
1088:{\displaystyle c_{2}\in \mathrm {up} (c_{1})}
748:Sentinels are outside the sentineled concept
4027:
4014:
3984:
3971:
3941:
3928:
3854:
3841:
3811:
3798:
3768:
3742:
3700:{\displaystyle \mathrm {D} _{\mathsf {C}}=2}
3313:
3274:
3244:
3192:
3046:{\displaystyle \mathrm {D} _{\mathsf {C}}=2}
2255:
2216:
2036:
1997:
1990:With reference to the picture on the right,
1898:
1885:
1852:
1839:
716:
700:
84:. There might be a discussion about this on
3170:
1448:{\displaystyle c'\not \in \mathrm {up} (c)}
1188:is a minimal set with the above properties
361:; they are assigned by the sentry function
50:Learn how and when to remove these messages
2856:{\displaystyle \mathrm {D} _{\mathsf {C}}}
830:Sentinels are inside the invading concept
4369:
4342:Algorithmic Inference in Machine Learning
4289:
4247:
4205:
4163:
4133:
4105:
4077:
3626:
3584:
3542:
3500:
3470:
3442:
3414:
3148:
3061:
2996:
447:has at least one of the sentry points of
177:Learn how and when to remove this message
159:Learn how and when to remove this message
143:, without removing the technical details.
104:Learn how and when to remove this message
4351:
4339:
3135:
3081:
2903:
1981:
538:
2743:
2680:
2664:
2626:
2135:{\displaystyle c_{1},c_{2},c_{3},c_{4}}
1965:
1926:
1818:
1774:
1723:
1624:{\displaystyle c',{\boldsymbol {S}}(c)}
1608:
1463:
1370:
1257:
1236:
1213:
1201:
1165:
1116:
861:
765:
683:
505:and distinct from the sentry points of
369:
252:
4381:
4352:Apolloni, B.; Chiaravalli, S. (1997).
3685:
3184:
3031:
2942:
2924:
2882:
2847:
2720:
2649:
2142:. All points are in the gap between a
1524:and not just in avoiding inclusion of
1295:
904:
809:
692:
629:
537:The technical definition coming from (
392:
334:
270:
230:
2042:{\displaystyle \{x_{1},x_{2},x_{3}\}}
744:satisfying the following conditions:
141:make it understandable to non-experts
3075:, as shown in the picture on right.
1942:{\displaystyle {\boldsymbol {S}}(c)}
1479:{\displaystyle {\boldsymbol {S}}(c)}
1181:{\displaystyle {\boldsymbol {S}}(c)}
115:
56:
15:
3266:
2814:
2790:
2766:
1486:are involved in really sentineling
912:{\displaystyle c'\in {\mathsf {C}}}
728:
711:
653:
605:{\displaystyle \left(c'\right)^{+}}
13:
4057:
3884:
3679:
3407:, and ⃝ a sentry point:
3025:
2935:
2918:
2875:
2841:
2676:
2643:
1862:
1859:
1797:
1501:
1498:
1432:
1429:
1397:
1303:{\displaystyle c\in {\mathsf {C}}}
1139:
1065:
1062:
1014:
1011:
943:{\displaystyle c'\not \subseteq c}
817:{\displaystyle c\in {\mathsf {C}}}
781:
703:
572:plus its sentry points by another
302:
14:
4405:
2750:{\displaystyle {\boldsymbol {S}}}
1972:{\displaystyle {\boldsymbol {S}}}
1825:{\displaystyle {\boldsymbol {S}}}
376:{\displaystyle {\boldsymbol {S}}}
31:This article has multiple issues.
4346:Advanced Knowledge International
4034:
3991:
3948:
3905:
3861:
3818:
3775:
3719:
3009:{\displaystyle \mathbb {R} ^{2}}
1911:is a sentry function on this set
1517:{\displaystyle \mathrm {up} (c)}
1030:{\displaystyle \mathrm {up} (c)}
120:
61:
20:
4389:Computational complexity theory
2822:{\displaystyle {\mathfrak {X}}}
2798:{\displaystyle {\mathfrak {X}}}
2774:{\displaystyle {\mathfrak {X}}}
661:{\displaystyle {\mathfrak {X}}}
39:or discuss these issues on the
4051:
4038:
4008:
3995:
3965:
3952:
3922:
3909:
3878:
3865:
3835:
3822:
3792:
3779:
3736:
3723:
2690:
2684:
1936:
1930:
1879:
1866:
1791:
1778:
1740:
1727:
1618:
1612:
1570:
1558:
1511:
1505:
1473:
1467:
1442:
1436:
1380:
1374:
1267:
1261:
1250:
1244:
1175:
1169:
1133:
1120:
1082:
1069:
1024:
1018:
871:
865:
775:
769:
719:
310:{\displaystyle {\mathcal {X}}}
1:
4371:10.1016/S0304-3975(95)00240-5
4333:
2728:{\displaystyle {\mathsf {C}}}
1037:the set of concepts invading
637:{\displaystyle {\mathsf {C}}}
616:Definition of sentry function
400:{\displaystyle {\mathsf {C}}}
342:{\displaystyle {\mathsf {C}}}
278:{\displaystyle {\mathsf {C}}}
238:{\displaystyle {\mathsf {C}}}
4358:Theoretical Computer Science
4326:
4323:
4320:
4284:
4281:
4278:
4242:
4239:
4236:
4200:
4197:
4194:
3663:
3660:
3657:
3621:
3618:
3615:
3579:
3576:
3573:
3537:
3534:
3531:
3155:{\displaystyle \mathbb {R} }
3068:{\displaystyle \mathbb {R} }
7:
4072:
3897:. However a cheaper one is
3409:
3119:{\displaystyle x_{1},x_{2}}
2196:. They avoid inclusion of
2049:is a candidate frontier of
834:Having introduced the sets
10:
4410:
2979:Example: continuous spaces
1810:, then the restriction of
1490:against other concepts in
451:either in the gap between
3380:, "-" an element outside
3140:The class of segments in
205:. This is different from
3171:Example: discrete spaces
1582:{\displaystyle (c')^{+}}
207:computational complexity
1041:, we must have that if
4313:
4312:{\displaystyle c_{4}=}
4271:
4270:{\displaystyle c_{3}=}
4229:
4228:{\displaystyle c_{2}=}
4187:
4186:{\displaystyle c_{1}=}
4154:
4126:
4098:
4064:
3891:
3711:, as illustrated, is:
3701:
3650:
3649:{\displaystyle c_{4}=}
3608:
3607:{\displaystyle c_{3}=}
3566:
3565:{\displaystyle c_{2}=}
3524:
3523:{\displaystyle c_{1}=}
3491:
3463:
3435:
3401:
3374:
3347:
3320:
3251:
3163:
3156:
3131:
3120:
3069:
3047:
3010:
2962:
2896:
2857:
2823:
2799:
2775:
2751:
2729:
2697:
2617:
2590:
2563:
2536:
2509:
2482:
2455:
2428:
2401:
2380:as its own sentinels,
2374:
2347:
2320:
2289:
2262:
2190:
2163:
2136:
2070:
2043:
1987:
1973:
1943:
1905:
1826:
1804:
1747:
1683:
1656:
1625:
1583:
1545:
1518:
1480:
1449:
1404:
1356:
1304:
1274:
1221:
1182:
1146:
1089:
1031:
997:
944:
913:
884:, an invading concept
878:
818:
788:
738:
662:
638:
606:
562:
524:
499:
474:
437:
401:
377:
343:
311:
279:
239:
203:function from examples
4394:Algorithmic inference
4314:
4272:
4230:
4188:
4155:
4153:{\displaystyle x_{3}}
4127:
4125:{\displaystyle x_{2}}
4099:
4097:{\displaystyle x_{1}}
4065:
3892:
3702:
3651:
3609:
3567:
3525:
3492:
3490:{\displaystyle x_{3}}
3464:
3462:{\displaystyle x_{2}}
3436:
3434:{\displaystyle x_{1}}
3402:
3400:{\displaystyle c_{i}}
3375:
3373:{\displaystyle c_{i}}
3348:
3346:{\displaystyle x_{j}}
3321:
3252:
3157:
3139:
3121:
3085:
3070:
3048:
3011:
2973:Rademacher complexity
2963:
2897:
2858:
2824:
2800:
2776:
2752:
2730:
2698:
2618:
2616:{\displaystyle c_{0}}
2591:
2589:{\displaystyle c_{0}}
2564:
2562:{\displaystyle x_{4}}
2537:
2535:{\displaystyle x_{2}}
2510:
2508:{\displaystyle x_{1}}
2483:
2481:{\displaystyle c_{4}}
2456:
2454:{\displaystyle x_{3}}
2429:
2427:{\displaystyle x_{2}}
2402:
2400:{\displaystyle c_{2}}
2375:
2373:{\displaystyle x_{3}}
2348:
2346:{\displaystyle x_{1}}
2321:
2319:{\displaystyle c_{1}}
2290:
2288:{\displaystyle c_{3}}
2263:
2191:
2189:{\displaystyle c_{0}}
2164:
2162:{\displaystyle c_{i}}
2137:
2071:
2069:{\displaystyle c_{0}}
2044:
1985:
1974:
1944:
1906:
1827:
1805:
1748:
1684:
1682:{\displaystyle c_{2}}
1657:
1655:{\displaystyle c_{1}}
1626:
1584:
1546:
1544:{\displaystyle c^{+}}
1519:
1481:
1450:
1405:
1357:
1305:
1275:
1222:
1183:
1147:
1090:
1032:
998:
945:
914:
879:
819:
789:
739:
663:
639:
607:
563:
561:{\displaystyle c^{+}}
525:
500:
475:
438:
402:
378:
344:
312:
280:
245:of Boolean functions
240:
4293:
4251:
4209:
4167:
4137:
4109:
4081:
3901:
3715:
3674:
3630:
3588:
3546:
3504:
3474:
3446:
3418:
3384:
3357:
3330:
3261:
3179:
3144:
3090:
3057:
3020:
2991:
2913:
2870:
2836:
2809:
2785:
2761:
2739:
2715:
2638:
2627:Definition of detail
2600:
2573:
2569:is not allowed as a
2546:
2519:
2492:
2465:
2438:
2411:
2384:
2357:
2330:
2303:
2272:
2200:
2173:
2146:
2080:
2053:
1994:
1961:
1922:
1836:
1814:
1757:
1693:
1666:
1639:
1593:
1589:. Thus if we remove
1555:
1528:
1494:
1459:
1414:
1366:
1320:
1284:
1231:
1196:
1161:
1099:
1045:
1007:
954:
923:
888:
838:
798:
755:
679:
648:
624:
620:For a concept class
576:
545:
509:
484:
459:
422:
407:in such a way that:
387:
365:
329:
297:
265:
253:Technical definition
225:
74:confusing or unclear
2542:analogously. Point
612:in the same class.
383:to each concept of
82:clarify the article
4309:
4267:
4225:
4183:
4150:
4122:
4094:
4060:
3887:
3697:
3646:
3604:
3562:
3520:
3487:
3459:
3431:
3397:
3370:
3343:
3316:
3247:
3164:
3152:
3132:
3116:
3065:
3043:
3006:
2958:
2892:
2853:
2819:
2795:
2771:
2747:
2725:
2693:
2675:
2613:
2586:
2559:
2532:
2505:
2478:
2451:
2424:
2397:
2370:
2343:
2316:
2285:
2258:
2186:
2159:
2132:
2066:
2039:
1988:
1969:
1939:
1901:
1822:
1800:
1743:
1679:
1652:
1631:remains unchanged
1621:
1579:
1541:
1514:
1476:
1445:
1400:
1352:
1300:
1270:
1217:
1178:
1142:
1085:
1027:
993:
940:
909:
874:
814:
784:
734:
658:
634:
602:
558:
523:{\displaystyle c'}
520:
498:{\displaystyle c'}
495:
473:{\displaystyle c'}
470:
436:{\displaystyle c'}
433:
397:
373:
339:
307:
275:
235:
211:Complexity indices
4331:
4330:
3668:
3667:
3168:
3167:
2658:
293:defined on a set
215:Boolean functions
187:
186:
179:
169:
168:
161:
114:
113:
106:
54:
4401:
4375:
4373:
4348:
4318:
4316:
4315:
4310:
4305:
4304:
4276:
4274:
4273:
4268:
4263:
4262:
4234:
4232:
4231:
4226:
4221:
4220:
4192:
4190:
4189:
4184:
4179:
4178:
4159:
4157:
4156:
4151:
4149:
4148:
4131:
4129:
4128:
4123:
4121:
4120:
4103:
4101:
4100:
4095:
4093:
4092:
4073:
4069:
4067:
4066:
4061:
4050:
4049:
4037:
4026:
4025:
4007:
4006:
3994:
3983:
3982:
3964:
3963:
3951:
3940:
3939:
3921:
3920:
3908:
3896:
3894:
3893:
3888:
3877:
3876:
3864:
3853:
3852:
3834:
3833:
3821:
3810:
3809:
3791:
3790:
3778:
3767:
3766:
3754:
3753:
3735:
3734:
3722:
3710:
3706:
3704:
3703:
3698:
3690:
3689:
3688:
3682:
3655:
3653:
3652:
3647:
3642:
3641:
3613:
3611:
3610:
3605:
3600:
3599:
3571:
3569:
3568:
3563:
3558:
3557:
3529:
3527:
3526:
3521:
3516:
3515:
3496:
3494:
3493:
3488:
3486:
3485:
3468:
3466:
3465:
3460:
3458:
3457:
3440:
3438:
3437:
3432:
3430:
3429:
3410:
3406:
3404:
3403:
3398:
3396:
3395:
3379:
3377:
3376:
3371:
3369:
3368:
3352:
3350:
3349:
3344:
3342:
3341:
3325:
3323:
3322:
3317:
3312:
3311:
3299:
3298:
3286:
3285:
3270:
3269:
3256:
3254:
3253:
3248:
3243:
3242:
3230:
3229:
3217:
3216:
3204:
3203:
3188:
3187:
3161:
3159:
3158:
3153:
3151:
3125:
3123:
3122:
3117:
3115:
3114:
3102:
3101:
3078:
3077:
3074:
3072:
3071:
3066:
3064:
3052:
3050:
3049:
3044:
3036:
3035:
3034:
3028:
3015:
3013:
3012:
3007:
3005:
3004:
2999:
2967:
2965:
2964:
2959:
2951:
2950:
2946:
2945:
2938:
2929:
2928:
2927:
2921:
2901:
2899:
2898:
2893:
2891:
2890:
2886:
2885:
2878:
2862:
2860:
2859:
2854:
2852:
2851:
2850:
2844:
2828:
2826:
2825:
2820:
2818:
2817:
2804:
2802:
2801:
2796:
2794:
2793:
2780:
2778:
2777:
2772:
2770:
2769:
2756:
2754:
2753:
2748:
2746:
2734:
2732:
2731:
2726:
2724:
2723:
2702:
2700:
2699:
2694:
2683:
2674:
2667:
2654:
2653:
2652:
2646:
2622:
2620:
2619:
2614:
2612:
2611:
2595:
2593:
2592:
2587:
2585:
2584:
2568:
2566:
2565:
2560:
2558:
2557:
2541:
2539:
2538:
2533:
2531:
2530:
2514:
2512:
2511:
2506:
2504:
2503:
2487:
2485:
2484:
2479:
2477:
2476:
2460:
2458:
2457:
2452:
2450:
2449:
2433:
2431:
2430:
2425:
2423:
2422:
2406:
2404:
2403:
2398:
2396:
2395:
2379:
2377:
2376:
2371:
2369:
2368:
2352:
2350:
2349:
2344:
2342:
2341:
2325:
2323:
2322:
2317:
2315:
2314:
2294:
2292:
2291:
2286:
2284:
2283:
2267:
2265:
2264:
2259:
2254:
2253:
2241:
2240:
2228:
2227:
2212:
2211:
2195:
2193:
2192:
2187:
2185:
2184:
2168:
2166:
2165:
2160:
2158:
2157:
2141:
2139:
2138:
2133:
2131:
2130:
2118:
2117:
2105:
2104:
2092:
2091:
2075:
2073:
2072:
2067:
2065:
2064:
2048:
2046:
2045:
2040:
2035:
2034:
2022:
2021:
2009:
2008:
1978:
1976:
1975:
1970:
1968:
1948:
1946:
1945:
1940:
1929:
1910:
1908:
1907:
1902:
1897:
1896:
1878:
1877:
1865:
1851:
1850:
1831:
1829:
1828:
1823:
1821:
1809:
1807:
1806:
1801:
1790:
1789:
1777:
1769:
1768:
1752:
1750:
1749:
1744:
1739:
1738:
1726:
1718:
1717:
1705:
1704:
1688:
1686:
1685:
1680:
1678:
1677:
1661:
1659:
1658:
1653:
1651:
1650:
1630:
1628:
1627:
1622:
1611:
1603:
1588:
1586:
1585:
1580:
1578:
1577:
1568:
1550:
1548:
1547:
1542:
1540:
1539:
1523:
1521:
1520:
1515:
1504:
1485:
1483:
1482:
1477:
1466:
1454:
1452:
1451:
1446:
1435:
1424:
1409:
1407:
1406:
1401:
1393:
1373:
1361:
1359:
1358:
1353:
1351:
1350:
1345:
1341:
1309:
1307:
1306:
1301:
1299:
1298:
1279:
1277:
1276:
1271:
1260:
1243:
1239:
1226:
1224:
1223:
1218:
1216:
1208:
1204:
1187:
1185:
1184:
1179:
1168:
1151:
1149:
1148:
1143:
1132:
1131:
1119:
1111:
1110:
1094:
1092:
1091:
1086:
1081:
1080:
1068:
1057:
1056:
1036:
1034:
1033:
1028:
1017:
1002:
1000:
999:
994:
992:
991:
986:
982:
966:
965:
949:
947:
946:
941:
933:
918:
916:
915:
910:
908:
907:
898:
883:
881:
880:
875:
864:
850:
849:
823:
821:
820:
815:
813:
812:
793:
791:
790:
785:
768:
743:
741:
740:
735:
733:
732:
731:
715:
714:
696:
695:
686:
667:
665:
664:
659:
657:
656:
643:
641:
640:
635:
633:
632:
611:
609:
608:
603:
601:
600:
595:
591:
567:
565:
564:
559:
557:
556:
529:
527:
526:
521:
519:
504:
502:
501:
496:
494:
479:
477:
476:
471:
469:
442:
440:
439:
434:
432:
406:
404:
403:
398:
396:
395:
382:
380:
379:
374:
372:
348:
346:
345:
340:
338:
337:
316:
314:
313:
308:
306:
305:
284:
282:
281:
276:
274:
273:
244:
242:
241:
236:
234:
233:
199:complexity index
191:computer science
182:
175:
164:
157:
153:
150:
144:
124:
123:
116:
109:
102:
98:
95:
89:
65:
64:
57:
46:
24:
23:
16:
4409:
4408:
4404:
4403:
4402:
4400:
4399:
4398:
4379:
4378:
4364:(1–2): 91–120.
4336:
4300:
4296:
4294:
4291:
4290:
4258:
4254:
4252:
4249:
4248:
4216:
4212:
4210:
4207:
4206:
4174:
4170:
4168:
4165:
4164:
4144:
4140:
4138:
4135:
4134:
4116:
4112:
4110:
4107:
4106:
4088:
4084:
4082:
4079:
4078:
4045:
4041:
4033:
4021:
4017:
4002:
3998:
3990:
3978:
3974:
3959:
3955:
3947:
3935:
3931:
3916:
3912:
3904:
3902:
3899:
3898:
3872:
3868:
3860:
3848:
3844:
3829:
3825:
3817:
3805:
3801:
3786:
3782:
3774:
3762:
3758:
3749:
3745:
3730:
3726:
3718:
3716:
3713:
3712:
3708:
3684:
3683:
3678:
3677:
3675:
3672:
3671:
3670:This class has
3637:
3633:
3631:
3628:
3627:
3595:
3591:
3589:
3586:
3585:
3553:
3549:
3547:
3544:
3543:
3511:
3507:
3505:
3502:
3501:
3481:
3477:
3475:
3472:
3471:
3453:
3449:
3447:
3444:
3443:
3425:
3421:
3419:
3416:
3415:
3391:
3387:
3385:
3382:
3381:
3364:
3360:
3358:
3355:
3354:
3337:
3333:
3331:
3328:
3327:
3307:
3303:
3294:
3290:
3281:
3277:
3265:
3264:
3262:
3259:
3258:
3238:
3234:
3225:
3221:
3212:
3208:
3199:
3195:
3183:
3182:
3180:
3177:
3176:
3173:
3147:
3145:
3142:
3141:
3110:
3106:
3097:
3093:
3091:
3088:
3087:
3060:
3058:
3055:
3054:
3030:
3029:
3024:
3023:
3021:
3018:
3017:
3000:
2995:
2994:
2992:
2989:
2988:
2981:
2941:
2940:
2939:
2934:
2933:
2923:
2922:
2917:
2916:
2914:
2911:
2910:
2881:
2880:
2879:
2874:
2873:
2871:
2868:
2867:
2846:
2845:
2840:
2839:
2837:
2834:
2833:
2813:
2812:
2810:
2807:
2806:
2789:
2788:
2786:
2783:
2782:
2765:
2764:
2762:
2759:
2758:
2742:
2740:
2737:
2736:
2719:
2718:
2716:
2713:
2712:
2679:
2663:
2662:
2648:
2647:
2642:
2641:
2639:
2636:
2635:
2629:
2607:
2603:
2601:
2598:
2597:
2580:
2576:
2574:
2571:
2570:
2553:
2549:
2547:
2544:
2543:
2526:
2522:
2520:
2517:
2516:
2499:
2495:
2493:
2490:
2489:
2472:
2468:
2466:
2463:
2462:
2445:
2441:
2439:
2436:
2435:
2418:
2414:
2412:
2409:
2408:
2391:
2387:
2385:
2382:
2381:
2364:
2360:
2358:
2355:
2354:
2337:
2333:
2331:
2328:
2327:
2310:
2306:
2304:
2301:
2300:
2299:we expect that
2279:
2275:
2273:
2270:
2269:
2249:
2245:
2236:
2232:
2223:
2219:
2207:
2203:
2201:
2198:
2197:
2180:
2176:
2174:
2171:
2170:
2153:
2149:
2147:
2144:
2143:
2126:
2122:
2113:
2109:
2100:
2096:
2087:
2083:
2081:
2078:
2077:
2060:
2056:
2054:
2051:
2050:
2030:
2026:
2017:
2013:
2004:
2000:
1995:
1992:
1991:
1964:
1962:
1959:
1958:
1925:
1923:
1920:
1919:
1914:
1892:
1888:
1873:
1869:
1858:
1846:
1842:
1837:
1834:
1833:
1817:
1815:
1812:
1811:
1785:
1781:
1773:
1764:
1760:
1758:
1755:
1754:
1734:
1730:
1722:
1713:
1709:
1700:
1696:
1694:
1691:
1690:
1673:
1669:
1667:
1664:
1663:
1646:
1642:
1640:
1637:
1636:
1634:
1607:
1596:
1594:
1591:
1590:
1573:
1569:
1561:
1556:
1553:
1552:
1535:
1531:
1529:
1526:
1525:
1497:
1495:
1492:
1491:
1462:
1460:
1457:
1456:
1428:
1417:
1415:
1412:
1411:
1386:
1369:
1367:
1364:
1363:
1346:
1334:
1330:
1329:
1321:
1318:
1317:
1312:
1294:
1293:
1285:
1282:
1281:
1256:
1235:
1234:
1232:
1229:
1228:
1212:
1200:
1199:
1197:
1194:
1193:
1191:
1164:
1162:
1159:
1158:
1154:
1127:
1123:
1115:
1106:
1102:
1100:
1097:
1096:
1076:
1072:
1061:
1052:
1048:
1046:
1043:
1042:
1010:
1008:
1005:
1004:
987:
975:
971:
970:
961:
957:
955:
952:
951:
926:
924:
921:
920:
903:
902:
891:
889:
886:
885:
860:
845:
841:
839:
836:
835:
833:
826:
808:
807:
799:
796:
795:
764:
756:
753:
752:
751:
727:
726:
722:
710:
709:
691:
690:
682:
680:
677:
676:
670:sentry function
652:
651:
649:
646:
645:
628:
627:
625:
622:
621:
618:
596:
584:
580:
579:
577:
574:
573:
552:
548:
546:
543:
542:
512:
510:
507:
506:
487:
485:
482:
481:
462:
460:
457:
456:
425:
423:
420:
419:
391:
390:
388:
385:
384:
368:
366:
363:
362:
333:
332:
330:
327:
326:
319:Euclidean space
301:
300:
298:
295:
294:
269:
268:
266:
263:
262:
259:sentry function
255:
229:
228:
226:
223:
222:
183:
172:
171:
170:
165:
154:
148:
145:
137:help improve it
134:
125:
121:
110:
99:
93:
90:
79:
66:
62:
25:
21:
12:
11:
5:
4407:
4397:
4396:
4391:
4377:
4376:
4349:
4335:
4332:
4329:
4328:
4325:
4322:
4319:
4308:
4303:
4299:
4287:
4286:
4283:
4280:
4277:
4266:
4261:
4257:
4245:
4244:
4241:
4238:
4235:
4224:
4219:
4215:
4203:
4202:
4199:
4196:
4193:
4182:
4177:
4173:
4161:
4160:
4147:
4143:
4132:
4119:
4115:
4104:
4091:
4087:
4076:
4059:
4056:
4053:
4048:
4044:
4040:
4036:
4032:
4029:
4024:
4020:
4016:
4013:
4010:
4005:
4001:
3997:
3993:
3989:
3986:
3981:
3977:
3973:
3970:
3967:
3962:
3958:
3954:
3950:
3946:
3943:
3938:
3934:
3930:
3927:
3924:
3919:
3915:
3911:
3907:
3886:
3883:
3880:
3875:
3871:
3867:
3863:
3859:
3856:
3851:
3847:
3843:
3840:
3837:
3832:
3828:
3824:
3820:
3816:
3813:
3808:
3804:
3800:
3797:
3794:
3789:
3785:
3781:
3777:
3773:
3770:
3765:
3761:
3757:
3752:
3748:
3744:
3741:
3738:
3733:
3729:
3725:
3721:
3696:
3693:
3687:
3681:
3666:
3665:
3662:
3659:
3656:
3645:
3640:
3636:
3624:
3623:
3620:
3617:
3614:
3603:
3598:
3594:
3582:
3581:
3578:
3575:
3572:
3561:
3556:
3552:
3540:
3539:
3536:
3533:
3530:
3519:
3514:
3510:
3498:
3497:
3484:
3480:
3469:
3456:
3452:
3441:
3428:
3424:
3413:
3394:
3390:
3367:
3363:
3340:
3336:
3315:
3310:
3306:
3302:
3297:
3293:
3289:
3284:
3280:
3276:
3273:
3268:
3246:
3241:
3237:
3233:
3228:
3224:
3220:
3215:
3211:
3207:
3202:
3198:
3194:
3191:
3186:
3172:
3169:
3166:
3165:
3150:
3133:
3113:
3109:
3105:
3100:
3096:
3063:
3042:
3039:
3033:
3027:
3003:
2998:
2987:of circles in
2980:
2977:
2969:
2968:
2957:
2954:
2949:
2944:
2937:
2932:
2926:
2920:
2889:
2884:
2877:
2849:
2843:
2816:
2792:
2768:
2745:
2722:
2705:
2704:
2692:
2689:
2686:
2682:
2678:
2673:
2670:
2666:
2661:
2657:
2651:
2645:
2628:
2625:
2610:
2606:
2583:
2579:
2556:
2552:
2529:
2525:
2502:
2498:
2475:
2471:
2448:
2444:
2421:
2417:
2394:
2390:
2367:
2363:
2340:
2336:
2313:
2309:
2282:
2278:
2257:
2252:
2248:
2244:
2239:
2235:
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539:Apolloni 2006
532:
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480:, or outside
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418:each concept
417:
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359:sentry points
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129:This article
127:
118:
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87:
86:the talk page
83:
77:
75:
70:This article
68:
59:
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53:
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44:
43:
38:
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32:
27:
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286:
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173:
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149:January 2022
146:
130:
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94:January 2022
91:
80:Please help
71:
47:
40:
34:
33:Please help
30:
3086:Two points
3016:has detail
2832:The detail
1003:. Denoting
644:on a space
568:made up of
221:of a class
4383:Categories
4334:References
3175:The class
2971:See also
2707:is called
2297:Vice versa
1280:for every
443:including
195:statistics
189:In modern
76:to readers
36:improve it
4058:∅
3885:∅
2931:≤
2677:#
2214:∪
1883:−
1856:∪
1798:∅
1771:∩
1720:∪
1707:⊂
1635:Whenever
1398:∅
1384:∩
1327:⊆
1291:∈
1254:⊆
1210:≠
1140:∅
1137:≠
1113:∩
1059:∈
968:⊆
900:∈
858:∪
805:∈
782:∅
762:∩
720:↦
704:∅
698:∪
355:sentinels
42:talk page
3126:outside
2829:itself.
2076:against
1951:frontier
1601:′
1566:′
1426:∉
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935:⊈
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896:′
794:for all
589:′
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1949:is the
1095:, then
291:concept
135:Please
72:may be
2983:Class
2709:detail
2488:uses
2407:uses
219:detail
217:, the
197:, the
2326:uses
1957:upon
672:is a
530:, and
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2169:and
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950:and
668:, a
455:and
193:and
4366:doi
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