3748:
2759:
2748:
17:
1638:
697:
3974:
as well as permutation and identification of variables. The main difference is that iterative systems do not necessarily contain all projections. Every clone is an iterative system, and there are 20 non-empty iterative systems which are not clones. (Post also excluded the empty iterative system from
2216:
2532:
1346:
3769:
Composition alone does not allow to generate a nullary function from the corresponding unary constant function, this is the technical reason why nullary functions are excluded from clones in Post's classification. If we lift the restriction, we get more clones. Namely, each clone
3760:
If one only considers clones that are required to contain the constant functions, the classification is much simpler: there are only 7 such clones: UM, Λ, V, U, A, M, and ⊤. While this can be derived from the full classification, there is a simpler proof, taking less than a page.
1038:
3979:, which is an iterative system closed under introduction of dummy variables. There are four closed classes which are not clones: the empty set, the set of constant 0 functions, the set of constant 1 functions, and the set of all constant functions.
1196:
401:
537:
843:
3969:
2611:
2295:
2097:
2410:
1633:{\displaystyle {\begin{aligned}&f(a_{1},\dots ,a_{i-1},c,a_{i+1},\dots ,a_{n})=f(a_{1},\dots ,d,a_{i+1},\dots )\\\Rightarrow &f(b_{1},\dots ,c,b_{i+1},\dots )=f(b_{1},\dots ,d,b_{i+1},\dots )\end{aligned}}}
1925:
2055:
55:, who published a complete description of the lattice in 1941. The relative simplicity of Post's lattice is in stark contrast to the lattice of clones on a three-element (or larger) set, which has the
194:
1351:
896:
1044:
692:{\displaystyle \mathrm {th} _{k}^{n}(x_{1},\dots ,x_{n})={\begin{cases}1&{\text{if }}{\bigl |}\{i\mid x_{i}=1\}{\bigr |}\geq k,\\0&{\text{otherwise.}}\end{cases}}}
237:
740:
3804:
2541:
2225:
2211:{\displaystyle \mathbf {a} ^{1}\land \cdots \land \mathbf {a} ^{k}=\mathbf {0} \ \Rightarrow \ f(\mathbf {a} ^{1})\land \cdots \land f(\mathbf {a} ^{k})=0.}
2527:{\displaystyle \mathbf {a} ^{1}\lor \cdots \lor \mathbf {a} ^{k}=\mathbf {1} \ \Rightarrow \ f(\mathbf {a} ^{1})\lor \cdots \lor f(\mathbf {a} ^{k})=1,}
3975:
the classification, hence his diagram has no least element and fails to be a lattice.) As another alternative, some authors work with the notion of a
3660:
The complete classification of
Boolean clones given by Post helps to resolve various questions about classes of Boolean functions. For example:
1334:
1304:
1265:
1846:
1976:
3774:
in Post's lattice which contains at least one constant function corresponds to two clones under the less restrictive definition:
2700:
The largest clone of all functions is denoted ⊤ = , the smallest clone (which contains only projections) is denoted ⊥ = , and
4016:
116:
1209:
of an arbitrary number of clones is again a clone. It is convenient to denote intersection of clones by simple
1033:{\displaystyle (a_{1},\dots ,a_{n})\leq (b_{1},\dots ,b_{n})\iff a_{i}\leq b_{i}{\text{ for }}i=1,\dots ,n,}
4068:
Aaronson, Scott; Grier, Daniel; Schaeffer, Luke (2015). "The
Classification of Reversible Bit Operations".
422:
56:
1191:{\displaystyle (a_{1},\dots ,a_{n})\land (b_{1},\dots ,b_{n})=(a_{1}\land b_{1},\dots ,a_{n}\land b_{n}).}
883:
4094:
59:, and a complicated inner structure. A modern exposition of Post's result can be found in Lau (2006).
1206:
604:
396:{\displaystyle h(x_{1},\dots ,x_{n})=f(g_{1}(x_{1},\dots ,x_{n}),\dots ,g_{m}(x_{1},\dots ,x_{n})),}
3681:
524:
108:
80:
3692:
426:
48:
838:{\displaystyle \mathrm {maj} =\mathrm {th} _{2}^{3}=(x\land y)\lor (x\land z)\lor (y\land z).}
4027:
509:
501:
3964:{\displaystyle h(x_{1},\dots ,x_{n+m-1})=f(x_{1},\dots ,x_{n-1},g(x_{n},\dots ,x_{n+m-1})),}
2744:, and all its members are finitely generated. All the clones are listed in the table below.
1210:
3998:, Annals of Mathematics studies, no. 5, Princeton University Press, Princeton 1941, 122 pp.
493:
410:. A set of functions closed under composition, and containing all projections, is called a
407:
3794:
Post originally did not work with the modern definition of clones, but with the so-called
3664:
An inspection of the lattice shows that the maximal clones different from ⊤ (often called
8:
469:
4069:
4033:
2741:
2606:{\displaystyle \mathrm {T} _{1}^{\infty }=\bigcap _{k=1}^{\infty }\mathrm {T} _{1}^{k}}
2290:{\displaystyle \mathrm {T} _{0}^{\infty }=\bigcap _{k=1}^{\infty }\mathrm {T} _{0}^{k}}
1758:
functions, i.e., functions which depend on at most one input variable: there exists an
485:
473:
72:
531:) and the constant unary functions 0 and 1. Moreover, we need the threshold functions
4012:
3700:
880:
732:
32:
4009:
Function algebras on finite sets: Basic course on many-valued logic and clone theory
3751:
Subset of Post's lattice showing only the 7 clones containing all constant functions
3715:-formula is satisfiable. Lewis used the description of Post's lattice to show that
2737:
68:
3688:
is functionally complete iff it is not included in one of the five Post's classes.
411:
44:
40:
4099:
4050:
3735:
3626:
2733:
453:
of . For example, are all
Boolean functions, and are the monotone functions.
52:
3747:
1935:} (including the empty conjunction, i.e., the constant 1), and the constant 0.
4088:
2751:
20:
4037:, New York: Gordon and Breach, Part II, "Logic of Sentences", Sec. 3.23,"'N
513:
3696:
3684:
if and only if it generates ⊤, we obtain the following characterization:
481:
107:
denotes the two-element set {0, 1}. Particular
Boolean functions are the
3703:. Consider a restricted version of the problem: for a fixed finite set
3676:, and every proper subclone of ⊤ is contained in one of them. As a set
2758:
2613:= is the set of functions bounded below by a variable: there exists
2297:= is the set of functions bounded above by a variable: there exists
886:
structure. That is, ordering, meets, joins, and other operations on
4074:
516:
461:
421:
be a set of connectives. The functions which can be defined by a
3782:
together with all nullary functions whose unary versions are in
2747:
16:
1920:{\displaystyle f(x_{1},\dots ,x_{n})=\bigwedge _{i\in I}x_{i}}
433:
form a clone , indeed it is the smallest clone which includes
28:
3719:-SAT is NP-complete if the function ↛ can be generated from
3711:-SAT be the algorithmic problem of checking whether a given
3485:
The eight infinite families have actually also members with
2050:{\displaystyle f(x_{1},\dots ,x_{n})=\bigvee _{i\in I}x_{i}}
685:
3755:
2065:} (including the empty disjunction 0), and the constant 1.
731:
is the large conjunction. Of particular importance is the
3798:, which are sets of operations closed under substitution
528:
4030:(1959), rev., Albert Menne, ed. and trans., Otto Bird,
189:{\displaystyle \pi _{k}^{n}(x_{1},\dots ,x_{n})=x_{k},}
4067:
3996:
The two-valued iterative systems of mathematical logic
3560:
The lattice has a natural symmetry mapping each clone
4041:,'" Sec. 3.32, "16 dyadic truth functors", pp. 10-11.
3807:
2544:
2413:
2228:
2100:
1979:
1849:
1349:
1047:
899:
743:
540:
240:
119:
3764:
4057:, Mathematical Systems Theory 13 (1979), pp. 45–53.
3963:
2605:
2526:
2289:
2210:
2049:
1919:
1632:
1190:
1032:
837:
691:
395:
188:
4055:Satisfiability problems for propositional calculi
4086:
3489:= 1, but these appear separately in the table:
1973:. Equivalently, V consists of the disjunctions
890:-ary truth assignments are defined pointwise:
711:is the large disjunction of all the variables
2727:
1676:. Equivalently, the functions expressible as
654:
619:
1260:. Some special clones are introduced below:
649:
624:
2695:when one takes the empty join into account.
2379:when one takes the empty meet into account.
1843:. The clone Λ consists of the conjunctions
977:
973:
4073:
3746:
2757:
2746:
47:on a two-element set {0, 1}, ordered by
15:
3756:Clones requiring the constant functions
4087:
852:(i.e., truth-assignments) as vectors:
2683:. Furthermore, ⊤ can be considered
4011:, Springer, New York, 2006, 668 pp.
3789:
2367:. Furthermore, ⊤ can be considered
1337:functions: the functions satisfying
1201:
13:
2588:
2581:
2557:
2547:
2272:
2265:
2241:
2231:
763:
760:
751:
748:
745:
546:
543:
14:
4111:
3765:Clones allowing nullary functions
62:
2502:
2472:
2451:
2437:
2416:
2189:
2159:
2138:
2124:
2103:
3655:
4060:
4044:
4021:
4001:
3988:
3955:
3952:
3908:
3864:
3855:
3811:
3730:), and in all the other cases
2512:
2497:
2482:
2467:
2458:
2199:
2184:
2169:
2154:
2145:
2015:
1983:
1885:
1853:
1623:
1573:
1564:
1514:
1506:
1499:
1449:
1440:
1358:
1182:
1124:
1118:
1086:
1080:
1048:
974:
970:
938:
932:
900:
829:
817:
811:
799:
793:
781:
593:
561:
387:
384:
352:
330:
298:
285:
276:
244:
167:
135:
1:
3982:
57:cardinality of the continuum
7:
3742:
2762:Central part of the lattice
2732:The set of all clones is a
2397:= is the set of functions
2084:= is the set of functions
227:, we can construct another
10:
4116:
2728:Description of the lattice
456:We use the operations ¬, N
3695:for Boolean formulas is
527:), ?: (the ternary
427:propositional variables
3965:
3752:
3693:satisfiability problem
3629:of a Boolean function
2763:
2755:
2607:
2585:
2528:
2291:
2269:
2212:
2051:
1921:
1634:
1192:
1034:
848:We denote elements of
839:
693:
397:
190:
24:
4028:Jozef Maria Bochenski
3966:
3750:
3682:functionally complete
2761:
2750:
2608:
2565:
2529:
2292:
2249:
2213:
2052:
1922:
1635:
1193:
1035:
840:
694:
510:exclusive disjunction
437:. We call the clone
429:and connectives from
398:
191:
19:
3805:
3707:of connectives, let
2542:
2411:
2226:
2098:
1977:
1847:
1347:
1045:
897:
741:
538:
529:conditional operator
238:
117:
2736:, hence it forms a
2721:constant-preserving
2719:= is the clone of
2668:= consists of the
2602:
2561:
2331:As a special case,
2286:
2245:
1938:V = is the set of
1808:Λ = is the set of
1264:M = is the set of
777:
560:
134:
4034:Mathematical Logic
3961:
3753:
3680:of connectives is
3564:to its dual clone
2764:
2756:
2742:countably infinite
2603:
2586:
2545:
2524:
2287:
2270:
2229:
2208:
2047:
2036:
1917:
1906:
1630:
1628:
1213:, i.e., the clone
1188:
1030:
879:carries a natural
835:
758:
689:
684:
541:
393:
186:
120:
73:logical connective
51:. It is named for
25:
23:of Post's lattice.
4095:Universal algebra
4017:978-3-540-36022-3
3796:iterative systems
3790:Iterative systems
3483:
3482:
2772:one of its bases
2754:of Post's lattice
2740:. The lattice is
2647:The special case
2463:
2457:
2352:= is the set of
2150:
2144:
2021:
1891:
1756:essentially unary
1754:= is the set of
1333:= is the set of
1303:= is the set of
1004:
733:majority function
680:
615:
33:universal algebra
4107:
4080:
4079:
4077:
4064:
4058:
4048:
4042:
4025:
4019:
4005:
3999:
3992:
3970:
3968:
3967:
3962:
3951:
3950:
3920:
3919:
3901:
3900:
3876:
3875:
3854:
3853:
3823:
3822:
3729:
3668:) are M, D, A, P
3651:
3647:
3636:
3624:
3581:
3556:
3544:
3533:
3522:
3510:
3499:
2766:
2765:
2738:complete lattice
2718:
2694:
2693:
2682:
2667:
2666:
2665:
2639:
2612:
2610:
2609:
2604:
2601:
2596:
2591:
2584:
2579:
2560:
2555:
2550:
2533:
2531:
2530:
2525:
2511:
2510:
2505:
2481:
2480:
2475:
2461:
2455:
2454:
2446:
2445:
2440:
2425:
2424:
2419:
2396:
2395:
2378:
2377:
2366:
2351:
2350:
2349:
2323:
2296:
2294:
2293:
2288:
2285:
2280:
2275:
2268:
2263:
2244:
2239:
2234:
2217:
2215:
2214:
2209:
2198:
2197:
2192:
2168:
2167:
2162:
2148:
2142:
2141:
2133:
2132:
2127:
2112:
2111:
2106:
2083:
2082:
2057:for all subsets
2056:
2054:
2053:
2048:
2046:
2045:
2035:
2014:
2013:
1995:
1994:
1972:
1927:for all subsets
1926:
1924:
1923:
1918:
1916:
1915:
1905:
1884:
1883:
1865:
1864:
1842:
1804:
1784:
1734:
1639:
1637:
1636:
1631:
1629:
1616:
1615:
1585:
1584:
1557:
1556:
1526:
1525:
1492:
1491:
1461:
1460:
1439:
1438:
1420:
1419:
1395:
1394:
1370:
1369:
1353:
1326:
1296:
1286:
1237:
1202:Naming of clones
1197:
1195:
1194:
1189:
1181:
1180:
1168:
1167:
1149:
1148:
1136:
1135:
1117:
1116:
1098:
1097:
1079:
1078:
1060:
1059:
1039:
1037:
1036:
1031:
1005:
1002:
1000:
999:
987:
986:
969:
968:
950:
949:
931:
930:
912:
911:
874:
844:
842:
841:
836:
776:
771:
766:
754:
730:
729:
710:
709:
698:
696:
695:
690:
688:
687:
681:
678:
658:
657:
642:
641:
623:
622:
616:
613:
592:
591:
573:
572:
559:
554:
549:
402:
400:
399:
394:
383:
382:
364:
363:
351:
350:
329:
328:
310:
309:
297:
296:
275:
274:
256:
255:
195:
193:
192:
187:
182:
181:
166:
165:
147:
146:
133:
128:
102:
95:
69:Boolean function
4115:
4114:
4110:
4109:
4108:
4106:
4105:
4104:
4085:
4084:
4083:
4066:Appendix 12 of
4065:
4061:
4049:
4045:
4026:
4022:
4006:
4002:
3993:
3989:
3985:
3934:
3930:
3915:
3911:
3890:
3886:
3871:
3867:
3837:
3833:
3818:
3814:
3806:
3803:
3802:
3792:
3767:
3758:
3745:
3736:polynomial-time
3728:
3724:
3675:
3671:
3658:
3649:
3646:
3642:
3638:
3634:
3633:. For example,
3622:
3613:
3602:
3593:
3583:
3565:
3554:
3550:
3546:
3543:
3539:
3535:
3532:
3528:
3524:
3520:
3516:
3512:
3509:
3505:
3501:
3498:
3494:
3490:
3469:
3458:
3405:
3394:
3313:
3302:
3275:
3264:
3234:
3209:
3203:
3192:
3170:
3161:
3150:
3128:
3103:
3097:
3084:
3062:
3053:
3040:
3021:
3010:
2980:
2959:
2948:
2937:
2928:
2917:
2895:
2874:
2861:
2850:
2841:
2828:
2799:
2788:
2730:
2717:
2711:
2701:
2692:
2689:
2688:
2687:
2673:
2664:
2661:
2660:
2659:
2654:
2648:
2638:
2622:
2597:
2592:
2587:
2580:
2569:
2556:
2551:
2546:
2543:
2540:
2539:
2506:
2501:
2500:
2476:
2471:
2470:
2450:
2441:
2436:
2435:
2420:
2415:
2414:
2412:
2409:
2408:
2394:
2391:
2390:
2389:
2376:
2373:
2372:
2371:
2357:
2348:
2345:
2344:
2343:
2338:
2332:
2322:
2306:
2281:
2276:
2271:
2264:
2253:
2240:
2235:
2230:
2227:
2224:
2223:
2193:
2188:
2187:
2163:
2158:
2157:
2137:
2128:
2123:
2122:
2107:
2102:
2101:
2099:
2096:
2095:
2081:
2078:
2077:
2076:
2041:
2037:
2025:
2009:
2005:
1990:
1986:
1978:
1975:
1974:
1943:
1911:
1907:
1895:
1879:
1875:
1860:
1856:
1848:
1845:
1844:
1813:
1803:
1794:
1786:
1767:
1741:
1733:
1725:
1716:
1710:
1703:
1696:
1687:
1677:
1627:
1626:
1605:
1601:
1580:
1576:
1546:
1542:
1521:
1517:
1509:
1503:
1502:
1481:
1477:
1456:
1452:
1434:
1430:
1409:
1405:
1384:
1380:
1365:
1361:
1350:
1348:
1345:
1344:
1308:
1288:
1269:
1259:
1250:
1244:
1236:
1227:
1220:
1214:
1204:
1176:
1172:
1163:
1159:
1144:
1140:
1131:
1127:
1112:
1108:
1093:
1089:
1074:
1070:
1055:
1051:
1046:
1043:
1042:
1003: for
1001:
995:
991:
982:
978:
964:
960:
945:
941:
926:
922:
907:
903:
898:
895:
894:
884:Boolean algebra
872:
863:
853:
772:
767:
759:
744:
742:
739:
738:
728:
723:
722:
721:
719:
708:
705:
704:
703:
702:For example, th
683:
682:
677:
675:
669:
668:
653:
652:
637:
633:
618:
617:
612:
610:
600:
599:
587:
583:
568:
564:
555:
550:
542:
539:
536:
535:
445:, and say that
378:
374:
359:
355:
346:
342:
324:
320:
305:
301:
292:
288:
270:
266:
251:
247:
239:
236:
235:
226:
217:
211:-ary functions
177:
173:
161:
157:
142:
138:
129:
124:
118:
115:
114:
97:
83:
65:
12:
11:
5:
4113:
4103:
4102:
4097:
4082:
4081:
4059:
4043:
4020:
4000:
3986:
3984:
3981:
3972:
3971:
3960:
3957:
3954:
3949:
3946:
3943:
3940:
3937:
3933:
3929:
3926:
3923:
3918:
3914:
3910:
3907:
3904:
3899:
3896:
3893:
3889:
3885:
3882:
3879:
3874:
3870:
3866:
3863:
3860:
3857:
3852:
3849:
3846:
3843:
3840:
3836:
3832:
3829:
3826:
3821:
3817:
3813:
3810:
3791:
3788:
3766:
3763:
3757:
3754:
3744:
3741:
3740:
3739:
3726:
3701:Cook's theorem
3689:
3673:
3669:
3666:Post's classes
3657:
3654:
3644:
3640:
3627:de Morgan dual
3618:
3611:
3598:
3591:
3552:
3548:
3541:
3537:
3530:
3526:
3518:
3514:
3507:
3503:
3496:
3492:
3481:
3480:
3478:
3474:
3473:
3470:
3467:
3463:
3462:
3459:
3456:
3452:
3451:
3448:
3444:
3443:
3440:
3436:
3435:
3432:
3428:
3427:
3414:
3410:
3409:
3406:
3403:
3399:
3398:
3395:
3392:
3388:
3387:
3373:
3369:
3368:
3365:
3361:
3360:
3357:
3353:
3352:
3338:
3334:
3333:
3330:
3326:
3325:
3322:
3318:
3317:
3314:
3311:
3307:
3306:
3303:
3300:
3296:
3295:
3292:
3288:
3287:
3284:
3280:
3279:
3276:
3273:
3269:
3268:
3265:
3262:
3258:
3257:
3254:
3250:
3249:
3235:
3232:
3228:
3227:
3201:
3198:
3190:
3186:
3185:
3171:
3168:
3164:
3163:
3159:
3156:
3148:
3144:
3143:
3129:
3126:
3122:
3121:
3093:
3090:
3082:
3078:
3077:
3063:
3060:
3056:
3055:
3049:
3046:
3038:
3034:
3033:
3030:
3026:
3025:
3022:
3019:
3015:
3014:
3011:
3008:
3004:
3003:
3000:
2996:
2995:
2981:
2978:
2974:
2973:
2957:
2954:
2946:
2942:
2941:
2938:
2935:
2931:
2930:
2926:
2923:
2915:
2911:
2910:
2896:
2893:
2889:
2888:
2870:
2867:
2859:
2855:
2854:
2851:
2848:
2844:
2843:
2837:
2834:
2826:
2822:
2821:
2808:
2804:
2803:
2800:
2797:
2793:
2792:
2789:
2786:
2782:
2781:
2778:
2774:
2773:
2770:
2734:closure system
2729:
2726:
2725:
2724:
2715:
2709:
2697:
2696:
2690:
2662:
2652:
2645:
2634:
2600:
2595:
2590:
2583:
2578:
2575:
2572:
2568:
2564:
2559:
2554:
2549:
2536:
2535:
2534:
2523:
2520:
2517:
2514:
2509:
2504:
2499:
2496:
2493:
2490:
2487:
2484:
2479:
2474:
2469:
2466:
2460:
2453:
2449:
2444:
2439:
2434:
2431:
2428:
2423:
2418:
2403:
2402:
2392:
2381:
2380:
2374:
2346:
2336:
2329:
2318:
2284:
2279:
2274:
2267:
2262:
2259:
2256:
2252:
2248:
2243:
2238:
2233:
2220:
2219:
2218:
2207:
2204:
2201:
2196:
2191:
2186:
2183:
2180:
2177:
2174:
2171:
2166:
2161:
2156:
2153:
2147:
2140:
2136:
2131:
2126:
2121:
2118:
2115:
2110:
2105:
2090:
2089:
2079:
2066:
2044:
2040:
2034:
2031:
2028:
2024:
2020:
2017:
2012:
2008:
2004:
2001:
1998:
1993:
1989:
1985:
1982:
1936:
1914:
1910:
1904:
1901:
1898:
1894:
1890:
1887:
1882:
1878:
1874:
1871:
1868:
1863:
1859:
1855:
1852:
1806:
1799:
1790:
1748:
1747:
1739:
1729:
1721:
1714:
1708:
1701:
1692:
1685:
1642:
1641:
1640:
1625:
1622:
1619:
1614:
1611:
1608:
1604:
1600:
1597:
1594:
1591:
1588:
1583:
1579:
1575:
1572:
1569:
1566:
1563:
1560:
1555:
1552:
1549:
1545:
1541:
1538:
1535:
1532:
1529:
1524:
1520:
1516:
1513:
1510:
1508:
1505:
1504:
1501:
1498:
1495:
1490:
1487:
1484:
1480:
1476:
1473:
1470:
1467:
1464:
1459:
1455:
1451:
1448:
1445:
1442:
1437:
1433:
1429:
1426:
1423:
1418:
1415:
1412:
1408:
1404:
1401:
1398:
1393:
1390:
1387:
1383:
1379:
1376:
1373:
1368:
1364:
1360:
1357:
1354:
1352:
1339:
1338:
1328:
1298:
1255:
1248:
1242:
1238:is denoted by
1232:
1225:
1218:
1203:
1200:
1199:
1198:
1187:
1184:
1179:
1175:
1171:
1166:
1162:
1158:
1155:
1152:
1147:
1143:
1139:
1134:
1130:
1126:
1123:
1120:
1115:
1111:
1107:
1104:
1101:
1096:
1092:
1088:
1085:
1082:
1077:
1073:
1069:
1066:
1063:
1058:
1054:
1050:
1040:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
998:
994:
990:
985:
981:
976:
972:
967:
963:
959:
956:
953:
948:
944:
940:
937:
934:
929:
925:
921:
918:
915:
910:
906:
902:
868:
861:
846:
845:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
775:
770:
765:
762:
757:
753:
750:
747:
724:
715:
706:
700:
699:
686:
676:
674:
671:
670:
667:
664:
661:
656:
651:
648:
645:
640:
636:
632:
629:
626:
621:
611:
609:
606:
605:
603:
598:
595:
590:
586:
582:
579:
576:
571:
567:
563:
558:
553:
548:
545:
525:nonimplication
404:
403:
392:
389:
386:
381:
377:
373:
370:
367:
362:
358:
354:
349:
345:
341:
338:
335:
332:
327:
323:
319:
316:
313:
308:
304:
300:
295:
291:
287:
284:
281:
278:
273:
269:
265:
262:
259:
254:
250:
246:
243:
231:-ary function
222:
215:
203:-ary function
197:
196:
185:
180:
176:
172:
169:
164:
160:
156:
153:
150:
145:
141:
137:
132:
127:
123:
64:
63:Basic concepts
61:
37:Post's lattice
9:
6:
4:
3:
2:
4112:
4101:
4098:
4096:
4093:
4092:
4090:
4076:
4071:
4063:
4056:
4052:
4047:
4040:
4036:
4035:
4029:
4024:
4018:
4014:
4010:
4004:
3997:
3991:
3987:
3980:
3978:
3958:
3947:
3944:
3941:
3938:
3935:
3931:
3927:
3924:
3921:
3916:
3912:
3905:
3902:
3897:
3894:
3891:
3887:
3883:
3880:
3877:
3872:
3868:
3861:
3858:
3850:
3847:
3844:
3841:
3838:
3834:
3830:
3827:
3824:
3819:
3815:
3808:
3801:
3800:
3799:
3797:
3787:
3785:
3781:
3777:
3773:
3762:
3749:
3737:
3733:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3694:
3690:
3687:
3683:
3679:
3667:
3663:
3662:
3661:
3653:
3632:
3628:
3621:
3617:
3610:
3606:
3601:
3597:
3590:
3586:
3580:
3576:
3572:
3568:
3563:
3558:
3488:
3479:
3476:
3475:
3471:
3465:
3464:
3460:
3454:
3453:
3449:
3446:
3445:
3441:
3438:
3437:
3433:
3430:
3429:
3426:
3422:
3418:
3415:
3412:
3411:
3407:
3401:
3400:
3396:
3390:
3389:
3386:
3382:
3378:
3374:
3371:
3370:
3366:
3363:
3362:
3358:
3355:
3354:
3351:
3347:
3343:
3339:
3336:
3335:
3331:
3328:
3327:
3323:
3320:
3319:
3315:
3309:
3308:
3304:
3298:
3297:
3293:
3290:
3289:
3285:
3282:
3281:
3277:
3271:
3270:
3266:
3260:
3259:
3255:
3252:
3251:
3247:
3243:
3239:
3236:
3230:
3229:
3225:
3221:
3217:
3213:
3207:
3199:
3196:
3188:
3187:
3183:
3179:
3175:
3172:
3166:
3165:
3157:
3154:
3146:
3145:
3141:
3137:
3133:
3130:
3124:
3123:
3119:
3115:
3111:
3107:
3101:
3096:
3091:
3088:
3080:
3079:
3075:
3071:
3067:
3064:
3058:
3057:
3052:
3047:
3044:
3036:
3035:
3031:
3028:
3027:
3023:
3017:
3016:
3012:
3006:
3005:
3001:
2998:
2997:
2993:
2989:
2985:
2982:
2976:
2975:
2971:
2967:
2963:
2955:
2952:
2944:
2943:
2939:
2933:
2932:
2924:
2921:
2913:
2912:
2908:
2904:
2900:
2897:
2891:
2890:
2886:
2882:
2878:
2873:
2868:
2865:
2857:
2856:
2852:
2846:
2845:
2840:
2835:
2832:
2824:
2823:
2820:
2816:
2812:
2809:
2806:
2805:
2801:
2795:
2794:
2790:
2784:
2783:
2779:
2776:
2775:
2771:
2768:
2767:
2760:
2753:
2752:Hasse diagram
2749:
2745:
2743:
2739:
2735:
2722:
2714:
2708:
2704:
2699:
2698:
2686:
2680:
2676:
2671:
2658:
2651:
2646:
2643:
2637:
2633:
2629:
2625:
2620:
2616:
2598:
2593:
2576:
2573:
2570:
2566:
2562:
2552:
2537:
2521:
2518:
2515:
2507:
2494:
2491:
2488:
2485:
2477:
2464:
2447:
2442:
2432:
2429:
2426:
2421:
2407:
2406:
2405:
2404:
2400:
2387:
2383:
2382:
2370:
2364:
2360:
2355:
2342:
2335:
2330:
2327:
2321:
2317:
2313:
2309:
2304:
2300:
2282:
2277:
2260:
2257:
2254:
2250:
2246:
2236:
2221:
2205:
2202:
2194:
2181:
2178:
2175:
2172:
2164:
2151:
2134:
2129:
2119:
2116:
2113:
2108:
2094:
2093:
2092:
2091:
2087:
2075:
2071:
2067:
2064:
2060:
2042:
2038:
2032:
2029:
2026:
2022:
2018:
2010:
2006:
2002:
1999:
1996:
1991:
1987:
1980:
1970:
1966:
1962:
1958:
1954:
1950:
1946:
1941:
1937:
1934:
1930:
1912:
1908:
1902:
1899:
1896:
1892:
1888:
1880:
1876:
1872:
1869:
1866:
1861:
1857:
1850:
1840:
1836:
1832:
1828:
1824:
1820:
1816:
1811:
1807:
1802:
1798:
1793:
1789:
1782:
1778:
1774:
1770:
1765:
1761:
1757:
1753:
1750:
1749:
1745:
1738:
1732:
1728:
1724:
1720:
1713:
1707:
1700:
1695:
1691:
1684:
1680:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1620:
1617:
1612:
1609:
1606:
1602:
1598:
1595:
1592:
1589:
1586:
1581:
1577:
1570:
1567:
1561:
1558:
1553:
1550:
1547:
1543:
1539:
1536:
1533:
1530:
1527:
1522:
1518:
1511:
1496:
1493:
1488:
1485:
1482:
1478:
1474:
1471:
1468:
1465:
1462:
1457:
1453:
1446:
1443:
1435:
1431:
1427:
1424:
1421:
1416:
1413:
1410:
1406:
1402:
1399:
1396:
1391:
1388:
1385:
1381:
1377:
1374:
1371:
1366:
1362:
1355:
1343:
1342:
1341:
1340:
1336:
1332:
1329:
1324:
1320:
1316:
1312:
1306:
1302:
1299:
1295:
1291:
1284:
1280:
1276:
1272:
1267:
1263:
1262:
1261:
1258:
1254:
1247:
1241:
1235:
1231:
1224:
1217:
1212:
1211:juxtaposition
1208:
1185:
1177:
1173:
1169:
1164:
1160:
1156:
1153:
1150:
1145:
1141:
1137:
1132:
1128:
1121:
1113:
1109:
1105:
1102:
1099:
1094:
1090:
1083:
1075:
1071:
1067:
1064:
1061:
1056:
1052:
1041:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
996:
992:
988:
983:
979:
965:
961:
957:
954:
951:
946:
942:
935:
927:
923:
919:
916:
913:
908:
904:
893:
892:
891:
889:
885:
882:
878:
871:
867:
860:
856:
851:
832:
826:
823:
820:
814:
808:
805:
802:
796:
790:
787:
784:
778:
773:
768:
755:
737:
736:
735:
734:
727:
718:
714:
672:
665:
662:
659:
646:
643:
638:
634:
630:
627:
607:
601:
596:
588:
584:
580:
577:
574:
569:
565:
556:
551:
534:
533:
532:
530:
526:
522:
518:
515:
511:
507:
503:
502:biconditional
499:
495:
491:
487:
483:
479:
475:
471:
467:
463:
459:
454:
452:
448:
444:
440:
436:
432:
428:
424:
420:
415:
413:
409:
406:called their
390:
379:
375:
371:
368:
365:
360:
356:
347:
343:
339:
336:
333:
325:
321:
317:
314:
311:
306:
302:
293:
289:
282:
279:
271:
267:
263:
260:
257:
252:
248:
241:
234:
233:
232:
230:
225:
221:
214:
210:
206:
202:
199:and given an
183:
178:
174:
170:
162:
158:
154:
151:
148:
143:
139:
130:
125:
121:
113:
112:
111:
110:
106:
100:
94:
90:
86:
82:
78:
74:
70:
60:
58:
54:
50:
46:
42:
38:
34:
30:
22:
21:Hasse diagram
18:
4062:
4054:
4046:
4038:
4031:
4023:
4008:
4003:
3995:
3994:E. L. Post,
3990:
3977:closed class
3976:
3973:
3795:
3793:
3783:
3779:
3775:
3771:
3768:
3759:
3731:
3720:
3716:
3712:
3708:
3704:
3685:
3677:
3665:
3659:
3656:Applications
3630:
3619:
3615:
3608:
3604:
3599:
3595:
3588:
3584:
3578:
3574:
3570:
3566:
3561:
3559:
3486:
3484:
3424:
3420:
3416:
3384:
3380:
3376:
3349:
3345:
3341:
3245:
3241:
3237:
3223:
3219:
3215:
3211:
3205:
3194:
3181:
3177:
3173:
3152:
3139:
3135:
3131:
3117:
3113:
3109:
3105:
3099:
3094:
3086:
3073:
3069:
3065:
3050:
3042:
2991:
2987:
2983:
2969:
2965:
2961:
2950:
2919:
2906:
2902:
2898:
2884:
2880:
2876:
2871:
2863:
2838:
2830:
2818:
2814:
2810:
2731:
2720:
2712:
2706:
2702:
2684:
2678:
2674:
2670:1-preserving
2669:
2656:
2649:
2641:
2635:
2631:
2627:
2623:
2618:
2614:
2398:
2385:
2368:
2362:
2358:
2354:0-preserving
2353:
2340:
2333:
2325:
2319:
2315:
2311:
2307:
2302:
2298:
2085:
2073:
2069:
2062:
2061:of {1, ...,
2058:
1968:
1964:
1960:
1956:
1952:
1948:
1944:
1939:
1932:
1931:of {1, ...,
1928:
1838:
1834:
1830:
1826:
1822:
1818:
1814:
1809:
1800:
1796:
1791:
1787:
1780:
1776:
1772:
1768:
1763:
1759:
1755:
1751:
1743:
1736:
1730:
1726:
1722:
1718:
1711:
1705:
1698:
1693:
1689:
1682:
1678:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1645:
1330:
1322:
1318:
1314:
1310:
1300:
1293:
1289:
1282:
1278:
1274:
1270:
1256:
1252:
1245:
1239:
1233:
1229:
1222:
1215:
1207:Intersection
1205:
887:
876:
869:
865:
858:
854:
849:
847:
725:
716:
712:
701:
520:
514:Boolean ring
505:
497:
489:
477:
465:
457:
455:
450:
446:
442:
438:
434:
430:
418:
416:
405:
228:
223:
219:
212:
208:
204:
200:
198:
104:
98:
92:
88:
84:
76:
66:
39:denotes the
36:
26:
4051:H. R. Lewis
3697:NP-complete
3002:∧, ∨, 0, 1
2672:functions:
2356:functions:
1942:functions:
1940:disjunctive
1812:functions:
1810:conjunctive
1307:functions:
1268:functions:
494:implication
482:disjunction
470:conjunction
408:composition
109:projections
4089:Categories
4075:1504.05155
4032:Precis of
3983:References
3738:decidable.
2723:functions.
2621:such that
2617:= 1, ...,
2305:such that
2301:= 1, ...,
2222:Moreover,
1766:such that
1762:= 1, ...,
1644:for every
1287:for every
875:. The set
679:otherwise.
3945:−
3925:…
3895:−
3881:…
3848:−
3828:…
3582:}, where
2582:∞
2567:⋂
2558:∞
2492:∨
2489:⋯
2486:∨
2459:⇒
2433:∨
2430:⋯
2427:∨
2401:such that
2266:∞
2251:⋂
2242:∞
2179:∧
2176:⋯
2173:∧
2146:⇒
2120:∧
2117:⋯
2114:∧
2088:such that
2030:∈
2023:⋁
2000:…
1900:∈
1893:⋀
1870:…
1785:whenever
1735:for some
1621:…
1590:…
1562:…
1531:…
1507:⇒
1497:…
1466:…
1425:…
1389:−
1375:…
1305:self-dual
1170:∧
1154:…
1138:∧
1103:…
1084:∧
1065:…
1019:…
989:≤
975:⟺
955:…
936:≤
917:…
824:∧
815:∨
806:∧
797:∨
788:∧
660:≥
631:∣
578:…
439:generated
369:…
337:…
315:…
261:…
152:…
122:π
96:for some
81:operation
53:Emil Post
49:inclusion
4007:D. Lau,
3743:Variants
3734:-SAT is
3614:, ..., ¬
3294:∨, 0, 1
3256:∧, 0, 1
3024:∧, ∨, 1
3013:∧, ∨, 0
2817: :
2813: ?
2640:for all
2384:For any
2324:for all
2068:For any
1717:+ ... +
1266:monotone
1228:∩ ... ∩
720:, and th
614:if
517:addition
462:negation
103:, where
75:, is an
3723:(i.e.,
3625:is the
3594:, ...,
3573:|
3332:maj, ¬
1688:, ...,
881:product
864:, ...,
519:), ↛, L
504:), +, J
496:), ↔, E
488:), →, C
476:), ∨, A
464:), ∧, K
449:is the
423:formula
218:, ...,
43:of all
41:lattice
4015:
3778:, and
3648:, and
3222:) for
3116:) for
2462:
2456:
2388:≥ 1, T
2149:
2143:
1664:, and
1335:affine
425:using
207:, and
45:clones
4100:Logic
4070:arXiv
3650:M = M
3643:) = T
3635:Λ = V
3603:) = ¬
3551:= MPT
3450:0, 1
3434:¬, 0
3367:↔, 0
3340:maj,
3316:∨, 1
3305:∨, 0
3278:∧, 1
3267:∧, 0
3210:maj,
3184:), 1
3104:maj,
3076:), 0
3032:∧, ∨
2802:∧, →
2791:∨, +
2780:∨, ¬
2769:clone
2681:) = 1
2365:) = 0
2072:≥ 1,
451:basis
412:clone
79:-ary
71:, or
29:logic
4013:ISBN
3691:The
3555:= MP
3540:= MP
3529:= MP
3517:= PT
3359:maj
3226:= 2
3208:≥ 3,
3204:for
3162:, 1
3120:= 2
3102:≥ 3,
3098:for
3054:, 0
2929:, →
2842:, ↛
2630:) ≥
2538:and
2314:) ≤
1963:) ∨
1955:) =
1833:) ∧
1825:) =
1775:) =
1697:) =
1317:) =
1277:) ≤
486:join
474:meet
417:Let
31:and
3725:⊇ T
3699:by
3672:, P
3569:= {
3547:MPT
3521:= P
3506:= P
3495:= P
3375:¬,
3240:∨ (
3231:MPT
3214:∨ (
3197:≥ 2
3189:MPT
3176:∨ (
3155:≥ 2
3134:∧ (
3125:MPT
3108:∧ (
3089:≥ 2
3081:MPT
3068:∧ (
3045:≥ 2
2986:∨ (
2964:∨ (
2953:≥ 2
2922:≥ 2
2901:∧ (
2879:∧ (
2866:≥ 2
2833:≥ 2
1742:,
1251:...
857:= (
523:, (
512:or
500:, (
492:, (
484:or
480:, (
472:or
468:, (
460:, (
441:by
101:≥ 1
27:In
4091::
4053:,
3786:.
3652:.
3639:(T
3637:,
3607:(¬
3577:∈
3557:.
3545:,
3536:MT
3534:,
3525:MT
3523:,
3513:PT
3511:,
3500:,
3472:1
3466:UP
3461:0
3455:UP
3447:UM
3442:¬
3439:UD
3423:+
3419:+
3413:AP
3408:↔
3402:AP
3397:+
3391:AP
3383:+
3379:+
3372:AD
3356:DM
3348:+
3344:+
3337:DP
3324:∨
3321:VP
3310:VP
3299:VP
3286:∧
3283:ΛP
3272:ΛP
3261:ΛP
3248:)
3244:∧
3218:∧
3200:th
3193:,
3180:∧
3167:MT
3158:th
3151:,
3147:MT
3142:)
3138:∨
3112:∨
3092:th
3085:,
3072:∨
3059:MT
3048:th
3041:,
3037:MT
3029:MP
3018:MP
3007:MP
2994:)
2990:+
2977:PT
2972:)
2968:+
2960:,
2956:th
2949:,
2945:PT
2940:→
2925:th
2918:,
2909:)
2905:→
2892:PT
2887:)
2883:→
2875:,
2869:th
2862:,
2858:PT
2853:↛
2836:th
2829:,
2705:=
2655:=
2339:=
2206:0.
1951:∨
1821:∧
1795:=
1704:+
1672:∈
1668:,
1660:∈
1656:,
1652:,
1648:≤
1321:(¬
1292:≤
1221:∩
521:pq
506:pq
498:pq
490:pq
478:pq
466:pq
414:.
91:→
87::
67:A
35:,
4078:.
4072::
4039:p
3959:,
3956:)
3953:)
3948:1
3942:m
3939:+
3936:n
3932:x
3928:,
3922:,
3917:n
3913:x
3909:(
3906:g
3903:,
3898:1
3892:n
3888:x
3884:,
3878:,
3873:1
3869:x
3865:(
3862:f
3859:=
3856:)
3851:1
3845:m
3842:+
3839:n
3835:x
3831:,
3825:,
3820:1
3816:x
3812:(
3809:h
3784:C
3780:C
3776:C
3772:C
3732:B
3727:0
3721:B
3717:B
3713:B
3709:B
3705:B
3686:B
3678:B
3674:1
3670:0
3645:1
3641:0
3631:f
3623:)
3620:n
3616:x
3612:1
3609:x
3605:f
3600:n
3596:x
3592:1
3589:x
3587:(
3585:f
3579:C
3575:f
3571:f
3567:C
3562:C
3553:1
3549:0
3542:1
3538:1
3531:0
3527:0
3519:1
3515:0
3508:1
3504:1
3502:T
3497:0
3493:0
3491:T
3487:k
3477:⊥
3468:1
3457:0
3431:U
3425:z
3421:y
3417:x
3404:1
3393:0
3385:z
3381:y
3377:x
3364:A
3350:z
3346:y
3342:x
3329:D
3312:1
3301:0
3291:V
3274:1
3263:0
3253:Λ
3246:z
3242:y
3238:x
3233:1
3224:k
3220:z
3216:y
3212:x
3206:k
3202:2
3195:k
3191:1
3182:z
3178:y
3174:x
3169:1
3160:2
3153:k
3149:1
3140:z
3136:y
3132:x
3127:0
3118:k
3114:z
3110:y
3106:x
3100:k
3095:k
3087:k
3083:0
3074:z
3070:y
3066:x
3061:0
3051:k
3043:k
3039:0
3020:1
3009:0
2999:M
2992:z
2988:y
2984:x
2979:1
2970:z
2966:y
2962:x
2958:2
2951:k
2947:1
2936:1
2934:T
2927:2
2920:k
2916:1
2914:T
2907:z
2903:y
2899:x
2894:0
2885:z
2881:y
2877:x
2872:k
2864:k
2860:0
2849:0
2847:T
2839:k
2831:k
2827:0
2825:T
2819:z
2815:y
2811:x
2807:P
2798:1
2796:P
2787:0
2785:P
2777:⊤
2716:1
2713:P
2710:0
2707:P
2703:P
2691:1
2685:T
2679:1
2677:(
2675:f
2663:1
2657:T
2653:1
2650:P
2644:.
2642:a
2636:i
2632:a
2628:a
2626:(
2624:f
2619:n
2615:i
2599:k
2594:1
2589:T
2577:1
2574:=
2571:k
2563:=
2553:1
2548:T
2522:,
2519:1
2516:=
2513:)
2508:k
2503:a
2498:(
2495:f
2483:)
2478:1
2473:a
2468:(
2465:f
2452:1
2448:=
2443:k
2438:a
2422:1
2417:a
2399:f
2393:1
2386:k
2375:0
2369:T
2363:0
2361:(
2359:f
2347:0
2341:T
2337:0
2334:P
2328:.
2326:a
2320:i
2316:a
2312:a
2310:(
2308:f
2303:n
2299:i
2283:k
2278:0
2273:T
2261:1
2258:=
2255:k
2247:=
2237:0
2232:T
2203:=
2200:)
2195:k
2190:a
2185:(
2182:f
2170:)
2165:1
2160:a
2155:(
2152:f
2139:0
2135:=
2130:k
2125:a
2109:1
2104:a
2086:f
2080:0
2074:T
2070:k
2063:n
2059:I
2043:i
2039:x
2033:I
2027:i
2019:=
2016:)
2011:n
2007:x
2003:,
1997:,
1992:1
1988:x
1984:(
1981:f
1971:)
1969:b
1967:(
1965:f
1961:a
1959:(
1957:f
1953:b
1949:a
1947:(
1945:f
1933:n
1929:I
1913:i
1909:x
1903:I
1897:i
1889:=
1886:)
1881:n
1877:x
1873:,
1867:,
1862:1
1858:x
1854:(
1851:f
1841:)
1839:b
1837:(
1835:f
1831:a
1829:(
1827:f
1823:b
1819:a
1817:(
1815:f
1805:.
1801:i
1797:b
1792:i
1788:a
1783:)
1781:b
1779:(
1777:f
1773:a
1771:(
1769:f
1764:n
1760:i
1752:U
1746:.
1744:a
1740:0
1737:a
1731:n
1727:x
1723:n
1719:a
1715:1
1712:x
1709:1
1706:a
1702:0
1699:a
1694:n
1690:x
1686:1
1683:x
1681:(
1679:f
1674:2
1670:d
1666:c
1662:2
1658:b
1654:a
1650:n
1646:i
1624:)
1618:,
1613:1
1610:+
1607:i
1603:b
1599:,
1596:d
1593:,
1587:,
1582:1
1578:b
1574:(
1571:f
1568:=
1565:)
1559:,
1554:1
1551:+
1548:i
1544:b
1540:,
1537:c
1534:,
1528:,
1523:1
1519:b
1515:(
1512:f
1500:)
1494:,
1489:1
1486:+
1483:i
1479:a
1475:,
1472:d
1469:,
1463:,
1458:1
1454:a
1450:(
1447:f
1444:=
1441:)
1436:n
1432:a
1428:,
1422:,
1417:1
1414:+
1411:i
1407:a
1403:,
1400:c
1397:,
1392:1
1386:i
1382:a
1378:,
1372:,
1367:1
1363:a
1359:(
1356:f
1331:A
1327:.
1325:)
1323:a
1319:f
1315:a
1313:(
1311:f
1309:¬
1301:D
1297:.
1294:b
1290:a
1285:)
1283:b
1281:(
1279:f
1275:a
1273:(
1271:f
1257:k
1253:C
1249:2
1246:C
1243:1
1240:C
1234:k
1230:C
1226:2
1223:C
1219:1
1216:C
1186:.
1183:)
1178:n
1174:b
1165:n
1161:a
1157:,
1151:,
1146:1
1142:b
1133:1
1129:a
1125:(
1122:=
1119:)
1114:n
1110:b
1106:,
1100:,
1095:1
1091:b
1087:(
1081:)
1076:n
1072:a
1068:,
1062:,
1057:1
1053:a
1049:(
1028:,
1025:n
1022:,
1016:,
1013:1
1010:=
1007:i
997:i
993:b
984:i
980:a
971:)
966:n
962:b
958:,
952:,
947:1
943:b
939:(
933:)
928:n
924:a
920:,
914:,
909:1
905:a
901:(
888:n
877:2
873:)
870:n
866:a
862:1
859:a
855:a
850:2
833:.
830:)
827:z
821:y
818:(
812:)
809:z
803:x
800:(
794:)
791:y
785:x
782:(
779:=
774:3
769:2
764:h
761:t
756:=
752:j
749:a
746:m
726:n
717:i
713:x
707:1
673:0
666:,
663:k
655:|
650:}
647:1
644:=
639:i
635:x
628:i
625:{
620:|
608:1
602:{
597:=
594:)
589:n
585:x
581:,
575:,
570:1
566:x
562:(
557:n
552:k
547:h
544:t
508:(
458:p
447:B
443:B
435:B
431:B
419:B
391:,
388:)
385:)
380:n
376:x
372:,
366:,
361:1
357:x
353:(
348:m
344:g
340:,
334:,
331:)
326:n
322:x
318:,
312:,
307:1
303:x
299:(
294:1
290:g
286:(
283:f
280:=
277:)
272:n
268:x
264:,
258:,
253:1
249:x
245:(
242:h
229:n
224:m
220:g
216:1
213:g
209:n
205:f
201:m
184:,
179:k
175:x
171:=
168:)
163:n
159:x
155:,
149:,
144:1
140:x
136:(
131:n
126:k
105:2
99:n
93:2
89:2
85:f
77:n
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