2544:) that, when given the answers to the queries, will produce the final answer of the reduction. In a weak truth table reduction, the reduction uses the oracle answers as a basis for further computation depending on the given answers (but not using the oracle). Equivalently, a weak truth table reduction is one for which the use of the reduction is bounded by a computable function. For this reason, weak truth table reductions are sometimes called "bounded Turing" reductions.
2635:
These reductions are stronger in the sense that they provide a finer distinction into equivalence classes, and satisfy more restrictive requirements than Turing reductions. Consequently, such reductions are harder to find. There may be no way to build a many-one reduction from one set to another even
3179:
847:
1606:
1388:
1273:
437:
875:
1210:
2548:
The second way to produce a stronger reducibility notion is to limit the computational resources that the program implementing the Turing reduction may use. These limits on the
996:
966:
799:
324:. However, because the oracle machine may query the oracle a large number of times, the resulting algorithm may require more time asymptotically than either the algorithm for
667:
1120:
1092:
1044:
903:
2858:
736:
1938:
1905:
1833:
1800:
1767:
1734:
1673:
is Turing reducible to every other set. Because any computable set can be computed with no oracle, it can be computed by an oracle machine that ignores the given oracle.
1651:
1338:
936:
700:
517:
2076:
2042:
1864:
1701:
2805:
1541:
1498:
1455:
1999:
1420:
1305:
1155:
2523:
2474:
3017:
1964:
2987:
2882:
2825:
2782:
2756:
2736:
2712:
2692:
2666:
2624:
2604:
2584:
2494:
2445:
2425:
2405:
2382:
2356:
2324:
2304:
2284:
2264:
2244:
2216:
2196:
2176:
2156:
2136:
2116:
1016:
766:
633:
609:
481:
457:
362:
342:
322:
302:
282:
262:
242:
218:
198:
171:
147:
123:
103:
79:
59:
3049:
2334:
There are two common ways of producing reductions stronger than Turing reducibility. The first way is to limit the number and manner of oracle queries.
2648:, a Turing reduction is the most general form of an effectively calculable reduction. Nevertheless, weaker reductions are also considered. Set
3229:
3112:
3200:
807:
3207:
3186:
2952:
2540:
must present all of its oracle queries at the same time. In a truth table reduction, the reduction also gives a boolean function (a
1122:. Insufficient because it may still be the case that, the language accepted by the machine is not itself recursively enumerable.
3252:
2549:
1546:
3037:
2760:
1343:
1098:
condition for a machine to be computationally universal, is that the machine's halting problem be Turing-complete for
577:
3105:
2562:
370:
2645:
1215:
1059:
408:
852:
2945:
The
Undecidable—Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions
1164:
2886:
971:
941:
3224:
2963:
2536:
2530:
771:
390:
377:
The first formal definition of relative computability, then called relative reducibility, was given by
3257:
3098:
638:
31:
1101:
1073:
1025:
884:
3193:
3132:
3080:
2961:
S. C. Kleene and E. L. Post, 1954. "The upper semi-lattice of degrees of recursive unsolvability".
2927:
2830:
2628:
2385:
1055:
705:
2975:
2360:
1910:
1877:
1805:
1772:
1739:
1706:
1623:
1310:
1067:
908:
672:
527:
489:
2054:
2020:
1842:
1679:
3152:
2948:
2785:
739:
2790:
2158:
in only finitely many steps, it can only make finitely many queries of membership in the set
2996:
2715:
2670:
1511:
1468:
1425:
174:
39:
1972:
1393:
1278:
1133:
3030:
2499:
2450:
1063:
365:
1943:
3147:
3142:
3024:
2971:
2903:
2867:
2810:
2767:
2741:
2721:
2697:
2677:
2651:
2609:
2589:
2569:
2553:
2479:
2430:
2410:
2390:
2367:
2341:
2309:
2289:
2269:
2249:
2229:
2201:
2181:
2161:
2141:
2121:
2101:
1670:
1001:
751:
618:
594:
523:
466:
442:
386:
382:
347:
327:
307:
287:
267:
247:
227:
203:
183:
156:
132:
108:
88:
82:
64:
44:
3075:
3246:
3137:
3041:
3021:, ser. 2 v. 45, pp. 161–228. Reprinted in "The Undecidable", M. Davis ed., 1965.
1871:
744:
3001:
2079:
17:
1500:
ignores its input and merely simulates the computation of the machine with index
1058:
in the sense of computational universality. Specifically, a Turing machine is a
3162:
3121:
2086:
2045:
1458:
378:
3027:, 1967. Theory of recursive functions and effective computability. McGraw-Hill.
2976:"Recursively enumerable sets of positive integers and their decision problems"
2958:
S. C. Kleene, 1952. Introduction to
Metamathematics. Amsterdam: North-Holland.
2891:
150:
2089:, but the Turing jump of a set is never Turing reducible to the original set.
3085:
2496:. Such a function can be used to generate a Turing reduction (by computing
394:
221:
126:
2552:
of the reduction are important when studying subrecursive classes such as
1054:
Turing completeness, as just defined above, corresponds only partially to
1867:
1157:
denote the set of input values for which the Turing machine with index
3081:
University of
Cambridge, Andrew Pitts, Tobias Kohn: Computation Theory
1653:. The reductions presented here are not only Turing reductions but
842:{\displaystyle {\mathcal {X}}\subseteq {\mathcal {P}}(\mathbb {N} )}
3076:
NIST Dictionary of
Algorithms and Data Structures: Turing reduction
2286:
was queried by the reduction while determining the membership of
3094:
3090:
2980:
2226:
of a reduction is the function that sends each natural number
1307:
denotes an effective pairing function). A reduction showing
1050:
Relation of Turing completeness to computational universality
397:
used the term "Turing reducibility" to refer to the concept.
1107:
1079:
1031:
983:
953:
890:
823:
813:
1066:(i.e., the set of inputs for which it eventually halts) is
3033:, 1987. Recursively enumerable sets and degrees, Springer.
2525:, querying the oracle, and then interpreting the result).
559:
If there is an oracle machine that, when run with oracle
364:. A Turing reduction in which the oracle machine runs in
1543:
either halts on every input or halts on no input. Thus
3015:
A. Turing, 1939. "Systems of logic based on ordinals."
2051:
There are infinite decreasing sequences of sets under
125:(Rogers 1967, Soare 1987). It can be understood as an
2870:
2833:
2813:
2793:
2770:
2744:
2724:
2700:
2680:
2654:
2612:
2592:
2572:
2502:
2482:
2453:
2433:
2413:
2393:
2370:
2344:
2312:
2292:
2272:
2252:
2232:
2204:
2184:
2164:
2144:
2124:
2104:
2057:
2023:
1975:
1946:
1913:
1880:
1845:
1808:
1775:
1742:
1709:
1682:
1626:
1601:{\displaystyle i(e,n)\in A\Leftrightarrow (e,n)\in B}
1549:
1514:
1471:
1428:
1396:
1346:
1313:
1281:
1218:
1167:
1136:
1104:
1076:
1028:
1004:
974:
944:
911:
887:
855:
810:
774:
754:
708:
675:
641:
621:
597:
492:
469:
445:
411:
350:
330:
310:
290:
270:
250:
230:
206:
186:
159:
135:
111:
91:
67:
47:
3217:
3171:
1094:of recursively enumerable sets. Thus, a necessary
2876:
2852:
2819:
2799:
2776:
2750:
2730:
2706:
2686:
2660:
2636:when a Turing reduction for the same sets exists.
2618:
2598:
2578:
2517:
2488:
2468:
2439:
2419:
2399:
2376:
2350:
2318:
2298:
2278:
2258:
2238:
2210:
2190:
2170:
2150:
2130:
2110:
2070:
2036:
1993:
1958:
1932:
1899:
1858:
1827:
1794:
1761:
1728:
1695:
1645:
1600:
1535:
1492:
1449:
1414:
1382:
1332:
1299:
1267:
1204:
1149:
1114:
1086:
1038:
1010:
990:
960:
930:
897:
869:
841:
793:
760:
730:
694:
661:
627:
603:
511:
475:
451:
431:
356:
336:
316:
296:
276:
256:
236:
212:
192:
165:
141:
117:
97:
73:
53:
1666:Every set is Turing equivalent to its complement.
284:at each place where the oracle machine computing
2138:has to determine whether a single element is in
1383:{\displaystyle e\in A\Leftrightarrow (e,e)\in B}
2178:. When the amount of information about the set
3018:Proceedings of the London Mathematical Society
2626:that runs in polynomial time. The concept of
3106:
2988:Bulletin of the American Mathematical Society
8:
3050:Notices of the American Mathematical Society
1262:
1225:
1199:
1174:
173:. The concept can be analogously applied to
563:, computes a partial function with domain
3113:
3099:
3091:
2218:is discussed, this is made precise by the
1268:{\displaystyle B=\{(e,n)\mid n\in W_{e}\}}
389:defined an equivalent concept in terms of
3000:
2947:, Raven, New York. Reprint, Dover, 2004.
2869:
2838:
2832:
2812:
2792:
2769:
2743:
2723:
2699:
2679:
2653:
2611:
2591:
2571:
2501:
2481:
2452:
2432:
2412:
2392:
2369:
2343:
2311:
2291:
2271:
2251:
2231:
2203:
2183:
2163:
2143:
2123:
2103:
2085:Every set is Turing reducible to its own
2062:
2056:
2028:
2022:
1974:
1945:
1921:
1912:
1888:
1879:
1850:
1844:
1816:
1807:
1783:
1774:
1750:
1741:
1717:
1708:
1687:
1681:
1634:
1625:
1548:
1513:
1508:. In particular, the machine with index
1470:
1427:
1395:
1345:
1321:
1312:
1280:
1256:
1217:
1193:
1166:
1141:
1135:
1106:
1105:
1103:
1078:
1077:
1075:
1030:
1029:
1027:
1003:
982:
981:
973:
952:
951:
943:
919:
910:
889:
888:
886:
863:
862:
854:
832:
831:
822:
821:
812:
811:
809:
776:
775:
773:
753:
716:
707:
683:
674:
658:
649:
640:
620:
596:
500:
491:
468:
444:
432:{\displaystyle A,B\subseteq \mathbb {N} }
425:
424:
410:
349:
329:
309:
289:
269:
249:
229:
205:
185:
158:
134:
110:
90:
66:
46:
244:can be used to produce an algorithm for
2915:
2890:is an important reducibility notion in
1340:can be constructed using the fact that
870:{\displaystyle A\subseteq \mathbb {N} }
1205:{\displaystyle A=\{e\mid e\in W_{e}\}}
742:of Turing equivalent sets are called
7:
3201:Computing Machinery and Intelligence
3208:The Chemical Basis of Morphogenesis
991:{\displaystyle A\in {\mathcal {X}}}
961:{\displaystyle X\in {\mathcal {X}}}
777:
3187:Systems of Logic Based on Ordinals
2586:if there is a Turing reduction of
794:{\displaystyle {\textbf {deg}}(X)}
25:
2098:Since every reduction from a set
264:, by inserting the algorithm for
3042:"What is...Turing Reducibility?"
2198:used to compute a single bit of
344:or the oracle machine computing
3002:10.1090/s0002-9904-1944-08111-1
1465:such that the program coded by
662:{\displaystyle A\equiv _{T}B\,}
27:Concept in computability theory
2930:for which no algorithm exists.
2845:
2839:
2512:
2506:
2463:
2457:
2246:to the largest natural number
1988:
1976:
1589:
1577:
1574:
1565:
1553:
1530:
1518:
1487:
1475:
1444:
1432:
1409:
1397:
1371:
1359:
1356:
1294:
1282:
1240:
1228:
1115:{\displaystyle {\mathcal {X}}}
1087:{\displaystyle {\mathcal {X}}}
1039:{\displaystyle {\mathcal {X}}}
898:{\displaystyle {\mathcal {X}}}
836:
828:
788:
782:
1:
3086:Prof. Jean Gallier’s Homepage
2853:{\displaystyle B^{(\alpha )}}
2714:is definable by a formula of
1457:can be constructed using the
748:. The Turing degree of a set
538:. In this case, we also say
2266:whose membership in the set
2078:. Thus this relation is not
1275:are Turing equivalent (here
731:{\displaystyle B\leq _{T}A.}
149:if it had available to it a
129:that could be used to solve
2013:is not Turing reducible to
2005:is not Turing reducible to
1940:does not necessarily imply
1933:{\displaystyle B\leq _{T}A}
1900:{\displaystyle A\leq _{T}B}
1828:{\displaystyle A\leq _{T}A}
1795:{\displaystyle A\leq _{T}C}
1762:{\displaystyle B\leq _{T}C}
1729:{\displaystyle A\leq _{T}B}
1646:{\displaystyle B\leq _{T}A}
1333:{\displaystyle A\leq _{T}B}
931:{\displaystyle X\leq _{T}A}
695:{\displaystyle A\leq _{T}B}
522:if and only if there is an
512:{\displaystyle A\leq _{T}B}
439:of natural numbers, we say
180:If a Turing reduction from
3274:
2537:weak truth table reduction
1620:is computable, this shows
3128:
2887:relative constructibility
2864:-iterated Turing jump of
2738:as a parameter. The set
2563:polynomial-time reducible
2386:total computable function
2071:{\displaystyle \leq _{T}}
2037:{\displaystyle \leq _{T}}
1859:{\displaystyle \leq _{T}}
1696:{\displaystyle \leq _{T}}
385:. Later in 1943 and 1952
2550:computational complexity
2222:function. Formally, the
1969:There are pairs of sets
1839:, and thus the relation
1616:. Because the function
1060:universal Turing machine
2800:{\displaystyle \alpha }
528:characteristic function
304:queries the oracle for
3253:Reduction (complexity)
2943:M. Davis, ed., 1965.
2878:
2854:
2821:
2801:
2778:
2752:
2732:
2708:
2688:
2662:
2620:
2600:
2580:
2519:
2490:
2470:
2441:
2421:
2401:
2378:
2352:
2320:
2300:
2280:
2260:
2240:
2212:
2192:
2172:
2152:
2132:
2112:
2094:The use of a reduction
2072:
2038:
1995:
1960:
1934:
1901:
1860:
1829:
1796:
1763:
1730:
1697:
1647:
1602:
1537:
1536:{\displaystyle i(e,n)}
1494:
1493:{\displaystyle i(e,n)}
1451:
1450:{\displaystyle i(e,n)}
1416:
1384:
1334:
1301:
1269:
1206:
1161:halts. Then the sets
1151:
1116:
1088:
1040:
1012:
992:
962:
932:
899:
871:
843:
795:
762:
732:
696:
663:
629:
605:
586:-computably enumerable
578:recursively enumerable
513:
477:
453:
433:
358:
338:
318:
298:
278:
258:
238:
214:
194:
167:
143:
119:
99:
75:
61:to a decision problem
55:
3225:Legacy of Alan Turing
3194:Intelligent Machinery
3180:On Computable Numbers
2964:Annals of Mathematics
2879:
2855:
2822:
2802:
2779:
2753:
2733:
2709:
2689:
2663:
2621:
2601:
2581:
2531:truth table reduction
2520:
2491:
2471:
2442:
2422:
2407:such that an element
2402:
2379:
2353:
2321:
2301:
2281:
2261:
2241:
2213:
2193:
2173:
2153:
2133:
2113:
2073:
2039:
1996:
1994:{\displaystyle (A,B)}
1961:
1935:
1902:
1861:
1830:
1797:
1764:
1731:
1698:
1648:
1603:
1538:
1495:
1452:
1417:
1415:{\displaystyle (e,n)}
1385:
1335:
1302:
1300:{\displaystyle (-,-)}
1270:
1207:
1152:
1150:{\displaystyle W_{e}}
1117:
1089:
1041:
1013:
993:
963:
933:
900:
872:
844:
796:
763:
733:
697:
664:
630:
606:
534:when run with oracle
514:
478:
454:
434:
359:
339:
319:
299:
279:
259:
239:
215:
195:
168:
144:
120:
100:
85:that decides problem
76:
56:
2967:v. 2 n. 59, 379–407.
2922:It is possible that
2868:
2831:
2811:
2791:
2768:
2742:
2722:
2698:
2678:
2652:
2646:Church–Turing thesis
2610:
2590:
2570:
2518:{\displaystyle f(n)}
2500:
2480:
2469:{\displaystyle f(n)}
2451:
2431:
2411:
2391:
2368:
2342:
2310:
2290:
2270:
2250:
2230:
2202:
2182:
2162:
2142:
2122:
2102:
2055:
2021:
1973:
1944:
1911:
1878:
1843:
1835:holds for every set
1806:
1773:
1740:
1707:
1680:
1624:
1547:
1512:
1469:
1426:
1394:
1344:
1311:
1279:
1216:
1165:
1134:
1102:
1074:
1026:
1002:
972:
942:
909:
885:
853:
808:
772:
752:
706:
673:
639:
619:
595:
490:
467:
443:
409:
381:in 1939 in terms of
348:
328:
308:
288:
268:
248:
228:
204:
184:
157:
133:
109:
105:given an oracle for
89:
65:
45:
32:computability theory
3133:Turing completeness
2928:undecidable problem
2827:is computable from
2629:log-space reduction
2330:Stronger reductions
1959:{\displaystyle A=B}
1657:, discussed below.
1655:many-one reductions
1056:Turing completeness
740:equivalence classes
391:recursive functions
220:exists, then every
18:Turing complete set
2874:
2850:
2817:
2797:
2774:
2748:
2728:
2704:
2684:
2658:
2616:
2596:
2576:
2515:
2486:
2466:
2437:
2417:
2397:
2374:
2361:many-one reducible
2348:
2316:
2296:
2276:
2256:
2236:
2208:
2188:
2168:
2148:
2128:
2108:
2068:
2034:
1991:
1956:
1930:
1897:
1856:
1825:
1792:
1759:
1726:
1703:is transitive: if
1693:
1643:
1598:
1533:
1490:
1447:
1412:
1380:
1330:
1297:
1265:
1202:
1147:
1112:
1084:
1036:
1008:
988:
968:. If additionally
958:
928:
895:
867:
839:
791:
758:
728:
692:
659:
625:
601:
526:that computes the
509:
473:
449:
429:
354:
334:
314:
294:
274:
254:
234:
210:
190:
163:
139:
115:
95:
71:
51:
3240:
3239:
3040:(November 2006).
2884:. The notion of
2877:{\displaystyle B}
2820:{\displaystyle A}
2786:recursive ordinal
2777:{\displaystyle B}
2761:hyperarithmetical
2751:{\displaystyle A}
2731:{\displaystyle B}
2707:{\displaystyle A}
2687:{\displaystyle B}
2661:{\displaystyle A}
2644:According to the
2640:Weaker reductions
2619:{\displaystyle B}
2599:{\displaystyle A}
2579:{\displaystyle B}
2489:{\displaystyle B}
2440:{\displaystyle A}
2420:{\displaystyle n}
2400:{\displaystyle f}
2377:{\displaystyle B}
2351:{\displaystyle A}
2319:{\displaystyle A}
2299:{\displaystyle n}
2279:{\displaystyle B}
2259:{\displaystyle m}
2239:{\displaystyle n}
2211:{\displaystyle A}
2191:{\displaystyle B}
2171:{\displaystyle B}
2151:{\displaystyle A}
2131:{\displaystyle B}
2111:{\displaystyle A}
1068:many-one complete
1011:{\displaystyle A}
779:
761:{\displaystyle X}
628:{\displaystyle B}
613:Turing equivalent
604:{\displaystyle A}
476:{\displaystyle B}
452:{\displaystyle A}
357:{\displaystyle A}
337:{\displaystyle B}
317:{\displaystyle B}
297:{\displaystyle A}
277:{\displaystyle B}
257:{\displaystyle A}
237:{\displaystyle B}
213:{\displaystyle B}
193:{\displaystyle A}
175:function problems
166:{\displaystyle B}
142:{\displaystyle A}
118:{\displaystyle B}
98:{\displaystyle A}
74:{\displaystyle B}
54:{\displaystyle A}
16:(Redirected from
3265:
3158:Turing reduction
3115:
3108:
3101:
3092:
3065:
3063:
3062:
3046:
3012:
3010:
3009:
3004:
2984:
2931:
2920:
2883:
2881:
2880:
2875:
2859:
2857:
2856:
2851:
2849:
2848:
2826:
2824:
2823:
2818:
2806:
2804:
2803:
2798:
2783:
2781:
2780:
2775:
2757:
2755:
2754:
2749:
2737:
2735:
2734:
2729:
2716:Peano arithmetic
2713:
2711:
2710:
2705:
2693:
2691:
2690:
2685:
2667:
2665:
2664:
2659:
2625:
2623:
2622:
2617:
2605:
2603:
2602:
2597:
2585:
2583:
2582:
2577:
2524:
2522:
2521:
2516:
2495:
2493:
2492:
2487:
2475:
2473:
2472:
2467:
2446:
2444:
2443:
2438:
2426:
2424:
2423:
2418:
2406:
2404:
2403:
2398:
2383:
2381:
2380:
2375:
2357:
2355:
2354:
2349:
2325:
2323:
2322:
2317:
2305:
2303:
2302:
2297:
2285:
2283:
2282:
2277:
2265:
2263:
2262:
2257:
2245:
2243:
2242:
2237:
2217:
2215:
2214:
2209:
2197:
2195:
2194:
2189:
2177:
2175:
2174:
2169:
2157:
2155:
2154:
2149:
2137:
2135:
2134:
2129:
2117:
2115:
2114:
2109:
2077:
2075:
2074:
2069:
2067:
2066:
2043:
2041:
2040:
2035:
2033:
2032:
2000:
1998:
1997:
1992:
1965:
1963:
1962:
1957:
1939:
1937:
1936:
1931:
1926:
1925:
1906:
1904:
1903:
1898:
1893:
1892:
1865:
1863:
1862:
1857:
1855:
1854:
1834:
1832:
1831:
1826:
1821:
1820:
1801:
1799:
1798:
1793:
1788:
1787:
1768:
1766:
1765:
1760:
1755:
1754:
1735:
1733:
1732:
1727:
1722:
1721:
1702:
1700:
1699:
1694:
1692:
1691:
1652:
1650:
1649:
1644:
1639:
1638:
1607:
1605:
1604:
1599:
1542:
1540:
1539:
1534:
1499:
1497:
1496:
1491:
1456:
1454:
1453:
1448:
1421:
1419:
1418:
1413:
1390:. Given a pair
1389:
1387:
1386:
1381:
1339:
1337:
1336:
1331:
1326:
1325:
1306:
1304:
1303:
1298:
1274:
1272:
1271:
1266:
1261:
1260:
1211:
1209:
1208:
1203:
1198:
1197:
1156:
1154:
1153:
1148:
1146:
1145:
1121:
1119:
1118:
1113:
1111:
1110:
1096:but insufficient
1093:
1091:
1090:
1085:
1083:
1082:
1045:
1043:
1042:
1037:
1035:
1034:
1017:
1015:
1014:
1009:
997:
995:
994:
989:
987:
986:
967:
965:
964:
959:
957:
956:
937:
935:
934:
929:
924:
923:
904:
902:
901:
896:
894:
893:
876:
874:
873:
868:
866:
848:
846:
845:
840:
835:
827:
826:
817:
816:
800:
798:
797:
792:
781:
780:
767:
765:
764:
759:
737:
735:
734:
729:
721:
720:
701:
699:
698:
693:
688:
687:
668:
666:
665:
660:
654:
653:
634:
632:
631:
626:
610:
608:
607:
602:
518:
516:
515:
510:
505:
504:
482:
480:
479:
474:
461:Turing reducible
458:
456:
455:
450:
438:
436:
435:
430:
428:
363:
361:
360:
355:
343:
341:
340:
335:
323:
321:
320:
315:
303:
301:
300:
295:
283:
281:
280:
275:
263:
261:
260:
255:
243:
241:
240:
235:
219:
217:
216:
211:
199:
197:
196:
191:
172:
170:
169:
164:
148:
146:
145:
140:
124:
122:
121:
116:
104:
102:
101:
96:
80:
78:
77:
72:
60:
58:
57:
52:
40:decision problem
36:Turing reduction
21:
3273:
3272:
3268:
3267:
3266:
3264:
3263:
3262:
3243:
3242:
3241:
3236:
3213:
3167:
3124:
3119:
3072:
3060:
3058:
3057:(10): 1218–1219
3044:
3036:
3007:
3005:
2978:
2970:
2940:
2935:
2934:
2921:
2917:
2912:
2900:
2866:
2865:
2834:
2829:
2828:
2809:
2808:
2789:
2788:
2766:
2765:
2740:
2739:
2720:
2719:
2696:
2695:
2676:
2675:
2650:
2649:
2642:
2608:
2607:
2588:
2587:
2568:
2567:
2498:
2497:
2478:
2477:
2449:
2448:
2447:if and only if
2429:
2428:
2409:
2408:
2389:
2388:
2366:
2365:
2340:
2339:
2332:
2308:
2307:
2288:
2287:
2268:
2267:
2248:
2247:
2228:
2227:
2200:
2199:
2180:
2179:
2160:
2159:
2140:
2139:
2120:
2119:
2100:
2099:
2096:
2058:
2053:
2052:
2024:
2019:
2018:
1971:
1970:
1942:
1941:
1917:
1909:
1908:
1884:
1876:
1875:
1846:
1841:
1840:
1812:
1804:
1803:
1779:
1771:
1770:
1746:
1738:
1737:
1713:
1705:
1704:
1683:
1678:
1677:
1663:
1630:
1622:
1621:
1545:
1544:
1510:
1509:
1467:
1466:
1462:
1424:
1423:
1392:
1391:
1342:
1341:
1317:
1309:
1308:
1277:
1276:
1252:
1214:
1213:
1189:
1163:
1162:
1137:
1132:
1131:
1128:
1100:
1099:
1072:
1071:
1064:halting problem
1052:
1024:
1023:
1020:Turing complete
1000:
999:
970:
969:
940:
939:
915:
907:
906:
883:
882:
851:
850:
806:
805:
770:
769:
750:
749:
712:
704:
703:
679:
671:
670:
645:
637:
636:
617:
616:
593:
592:
496:
488:
487:
465:
464:
441:
440:
407:
406:
405:Given two sets
403:
383:oracle machines
366:polynomial time
346:
345:
326:
325:
306:
305:
286:
285:
266:
265:
246:
245:
226:
225:
202:
201:
182:
181:
155:
154:
131:
130:
107:
106:
87:
86:
63:
62:
43:
42:
28:
23:
22:
15:
12:
11:
5:
3271:
3269:
3261:
3260:
3255:
3245:
3244:
3238:
3237:
3235:
3234:
3233:
3232:
3221:
3219:
3215:
3214:
3212:
3211:
3204:
3197:
3190:
3183:
3175:
3173:
3169:
3168:
3166:
3165:
3160:
3155:
3153:Turing's proof
3150:
3148:Turing pattern
3145:
3143:Turing machine
3140:
3135:
3129:
3126:
3125:
3120:
3118:
3117:
3110:
3103:
3095:
3089:
3088:
3083:
3078:
3071:
3070:External links
3068:
3067:
3066:
3034:
3028:
3022:
3013:
2995:(5): 284–316.
2968:
2959:
2956:
2939:
2936:
2933:
2932:
2914:
2913:
2911:
2908:
2907:
2906:
2904:Karp reduction
2899:
2896:
2873:
2847:
2844:
2841:
2837:
2816:
2796:
2784:if there is a
2773:
2747:
2727:
2703:
2683:
2668:is said to be
2657:
2641:
2638:
2615:
2595:
2575:
2546:
2545:
2526:
2514:
2511:
2508:
2505:
2485:
2465:
2462:
2459:
2456:
2436:
2416:
2396:
2384:if there is a
2373:
2347:
2331:
2328:
2315:
2295:
2275:
2255:
2235:
2207:
2187:
2167:
2147:
2127:
2107:
2095:
2092:
2091:
2090:
2083:
2065:
2061:
2049:
2031:
2027:
1990:
1987:
1984:
1981:
1978:
1967:
1955:
1952:
1949:
1929:
1924:
1920:
1916:
1896:
1891:
1887:
1883:
1853:
1849:
1824:
1819:
1815:
1811:
1791:
1786:
1782:
1778:
1758:
1753:
1749:
1745:
1725:
1720:
1716:
1712:
1690:
1686:
1674:
1671:computable set
1667:
1662:
1659:
1642:
1637:
1633:
1629:
1608:holds for all
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1532:
1529:
1526:
1523:
1520:
1517:
1489:
1486:
1483:
1480:
1477:
1474:
1460:
1446:
1443:
1440:
1437:
1434:
1431:
1422:, a new index
1411:
1408:
1405:
1402:
1399:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1329:
1324:
1320:
1316:
1296:
1293:
1290:
1287:
1284:
1264:
1259:
1255:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1201:
1196:
1192:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1144:
1140:
1127:
1124:
1109:
1081:
1051:
1048:
1033:
1007:
985:
980:
977:
955:
950:
947:
927:
922:
918:
914:
892:
865:
861:
858:
838:
834:
830:
825:
820:
815:
790:
787:
784:
757:
745:Turing degrees
727:
724:
719:
715:
711:
691:
686:
682:
678:
657:
652:
648:
644:
624:
600:
571:is said to be
524:oracle machine
520:
519:
508:
503:
499:
495:
472:
448:
427:
423:
420:
417:
414:
402:
399:
387:Stephen Kleene
371:Cook reduction
368:is known as a
353:
333:
313:
293:
273:
253:
233:
209:
189:
162:
138:
114:
94:
83:oracle machine
70:
50:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3270:
3259:
3256:
3254:
3251:
3250:
3248:
3231:
3228:
3227:
3226:
3223:
3222:
3220:
3216:
3209:
3205:
3202:
3198:
3195:
3191:
3188:
3184:
3181:
3177:
3176:
3174:
3170:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3138:Turing degree
3136:
3134:
3131:
3130:
3127:
3123:
3116:
3111:
3109:
3104:
3102:
3097:
3096:
3093:
3087:
3084:
3082:
3079:
3077:
3074:
3073:
3069:
3056:
3052:
3051:
3043:
3039:
3038:Davis, Martin
3035:
3032:
3029:
3026:
3023:
3020:
3019:
3014:
3003:
2998:
2994:
2990:
2989:
2982:
2977:
2973:
2969:
2966:
2965:
2960:
2957:
2954:
2953:0-486-43228-9
2950:
2946:
2942:
2941:
2937:
2929:
2925:
2919:
2916:
2909:
2905:
2902:
2901:
2897:
2895:
2893:
2889:
2888:
2871:
2863:
2842:
2835:
2814:
2794:
2787:
2771:
2764:
2762:
2745:
2725:
2717:
2701:
2681:
2674:
2672:
2655:
2647:
2639:
2637:
2633:
2631:
2630:
2613:
2593:
2573:
2565:
2564:
2559:
2555:
2551:
2543:
2539:
2538:
2533:
2532:
2527:
2509:
2503:
2483:
2460:
2454:
2434:
2414:
2394:
2387:
2371:
2363:
2362:
2345:
2337:
2336:
2335:
2329:
2327:
2313:
2293:
2273:
2253:
2233:
2225:
2221:
2205:
2185:
2165:
2145:
2125:
2105:
2093:
2088:
2084:
2081:
2063:
2059:
2050:
2047:
2029:
2025:
2016:
2012:
2008:
2004:
1985:
1982:
1979:
1968:
1953:
1950:
1947:
1927:
1922:
1918:
1914:
1894:
1889:
1885:
1881:
1873:
1872:partial order
1870:(it is not a
1869:
1851:
1847:
1838:
1822:
1817:
1813:
1809:
1802:. Moreover,
1789:
1784:
1780:
1776:
1756:
1751:
1747:
1743:
1723:
1718:
1714:
1710:
1688:
1684:
1676:The relation
1675:
1672:
1668:
1665:
1664:
1660:
1658:
1656:
1640:
1635:
1631:
1627:
1619:
1615:
1611:
1595:
1592:
1586:
1583:
1580:
1571:
1568:
1562:
1559:
1556:
1550:
1527:
1524:
1521:
1515:
1507:
1503:
1484:
1481:
1478:
1472:
1464:
1441:
1438:
1435:
1429:
1406:
1403:
1400:
1377:
1374:
1368:
1365:
1362:
1353:
1350:
1347:
1327:
1322:
1318:
1314:
1291:
1288:
1285:
1257:
1253:
1249:
1246:
1243:
1237:
1234:
1231:
1222:
1219:
1194:
1190:
1186:
1183:
1180:
1177:
1171:
1168:
1160:
1142:
1138:
1125:
1123:
1097:
1069:
1065:
1061:
1057:
1049:
1047:
1021:
1005:
978:
975:
948:
945:
925:
920:
916:
912:
880:
859:
856:
818:
802:
785:
755:
747:
746:
741:
725:
722:
717:
713:
709:
689:
684:
680:
676:
655:
650:
646:
642:
622:
614:
598:
589:
587:
585:
580:
579:
575:
570:
566:
562:
557:
555:
553:
548:
546:
541:
537:
533:
529:
525:
506:
501:
497:
493:
486:
485:
484:
470:
462:
446:
421:
418:
415:
412:
400:
398:
396:
392:
388:
384:
380:
375:
373:
372:
367:
351:
331:
311:
291:
271:
251:
231:
223:
207:
187:
178:
176:
160:
152:
136:
128:
112:
92:
84:
68:
48:
41:
37:
33:
19:
3172:Publications
3157:
3059:. Retrieved
3054:
3048:
3016:
3006:. Retrieved
2992:
2986:
2962:
2944:
2923:
2918:
2885:
2861:
2759:
2671:arithmetical
2669:
2643:
2634:
2632:is similar.
2627:
2561:
2557:
2547:
2541:
2535:
2529:
2359:
2333:
2223:
2219:
2097:
2080:well-founded
2014:
2010:
2006:
2002:
1836:
1654:
1617:
1613:
1609:
1505:
1501:
1158:
1129:
1095:
1070:for the set
1053:
1019:
878:
804:Given a set
803:
743:
612:
590:
583:
582:
573:
572:
568:
564:
560:
558:
551:
550:
544:
543:
539:
535:
531:
521:
460:
404:
376:
369:
179:
153:for solving
35:
29:
3258:Alan Turing
3163:Turing test
3122:Alan Turing
2972:Post, E. L.
2542:truth table
2087:Turing jump
2046:total order
879:Turing hard
768:is written
554:-computable
379:Alan Turing
3247:Categories
3061:2008-01-16
3008:2015-12-17
2938:References
2892:set theory
2807:such that
2001:such that
1661:Properties
1018:is called
877:is called
635:and write
547:-recursive
483:and write
401:Definition
393:. In 1944
151:subroutine
3230:namesakes
3025:H. Rogers
2843:α
2795:α
2566:to a set
2556:. A set
2118:to a set
2060:≤
2044:is not a
2026:≤
1919:≤
1886:≤
1848:≤
1814:≤
1781:≤
1748:≤
1715:≤
1685:≤
1632:≤
1593:∈
1575:⇔
1569:∈
1504:on input
1375:∈
1357:⇔
1351:∈
1319:≤
1292:−
1286:−
1250:∈
1244:∣
1187:∈
1181:∣
979:∈
949:∈
917:≤
860:⊆
819:⊆
714:≤
681:≤
647:≡
498:≤
422:⊆
395:Emil Post
222:algorithm
127:algorithm
3210:" (1952)
3203:" (1950)
3196:" (1948)
3189:" (1939)
3182:" (1936)
3031:R. Soare
2974:(1944).
2898:See also
2017:. Thus
1874:because
1868:preorder
938:for all
849:, a set
669:if both
3218:Related
1463:theorem
1126:Example
1062:if its
591:We say
567:, then
38:from a
2951:
2926:is an
2860:, the
2476:is in
2427:is in
1669:Every
81:is an
3045:(PDF)
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