292:
331:
3062:
228:
Given any decision problem in NP, construct a non-deterministic machine that solves it in polynomial time. Then for each input to that machine, build a
Boolean expression that computes whether when that specific input is passed to the machine, the machine runs correctly, and the machine halts and
2789:
229:
answers "yes". Then the expression can be satisfied if and only if there is a way for the machine to run correctly and answer "yes", so the satisfiability of the constructed expression is equivalent to asking whether or not the machine will answer "yes".
599:
4255:, the question of whether there is a deterministic polynomial-time algorithm for SAT (and consequently all other NP-complete problems) is still a famous unsolved problem, despite decades of intense effort by complexity theorists,
246:
There are two parts to proving that the
Boolean satisfiability problem (SAT) is NP-complete. One is to show that SAT is an NP problem. The other is to show that every NP problem can be reduced to an instance of a SAT problem by a
3270:
2187:
4816:
First modify the proof of the Cook–Levin theorem, so that the resulting formula is in conjunctive normal form, then introduce new variables to split clauses with more than 3 atoms. For example, the clause
3057:{\displaystyle {\begin{array}{l}(H_{i,k}\land Q_{q,k}\land T_{i,\sigma ,k})\to \\\bigvee _{((q,\sigma ),(q',\sigma ',d))\in \delta }(H_{i+d,\ k+1}\land Q_{q',\ k+1}\land T_{i,\ \sigma ',\ k+1})\end{array}}}
4924:
1900:
2730:
2570:
2334:
4241:(CNF) with exactly three variables or negations of variables per clause) to be NP-complete by showing how to reduce (in polynomial time) any instance of SAT to an equivalent instance of 3SAT.
2039:
415:
2460:
144:
such that, for all subexponential deterministic-time complexity classes T, the relativized complexity class NP is not a subset of T. In particular, for this oracle, P ≠ NP.
4859:
4100:
3763:
1453:
857:
5086:
4233:
Karp showed each of his problems to be NP-complete by reducing another problem (already shown to be NP-complete) to that problem. For example, he showed the problem 3SAT (the
4162:
1494:
924:
5014:
512:
2602:
2366:
2219:
1932:
3629:
883:
507:
81:
problem can be solved by a deterministic polynomial-time algorithm. The question of whether such an algorithm for
Boolean satisfiability exists is thus equivalent to the
3674:
3585:
3187:
2781:
2651:
2268:
2090:
1981:
1233:
1088:
4694:
4976:
3805:
3411:
3098:
1552:
979:
1628:
4541:
3477:
3444:
2504:
2408:
1814:
1750:
1714:
1371:
1269:
1124:
455:
162:. Additionally he found for each of these problems an algorithm that solves it in optimal time (in particular, these algorithms run in polynomial time if and only if
1654:
804:
481:
4003:
3946:
3914:
3303:
648:
4944:
4503:
4023:
3970:
3885:
3865:
3845:
3825:
3694:
3537:
3517:
3497:
3372:
3352:
3332:
3140:
3120:
1774:
1674:
1594:
1574:
1396:
1331:
1311:
1291:
1186:
1166:
1146:
1041:
1021:
1001:
775:
755:
731:
711:
688:
668:
619:
435:
352:
316:
4211:
suffices) to an instance of the
Boolean satisfiability problem. This means that if the Boolean satisfiability problem could be solved in polynomial time by a
4768:
4390:
77:
An important consequence of this theorem is that if there exists a deterministic polynomial-time algorithm for solving
Boolean satisfiability, then every
4167:
The use of SAT to prove the existence of an NP-complete problem can be extended to other computational problems in logic, and to completeness for other
151:'s paper, "Universal search problems", was published in 1973, although it was mentioned in talks and submitted for publication a few years earlier.
4946:
is a new variable that will not be used anywhere else in the expression. Clauses with fewer than three atoms can be padded; for example,
3195:
4437:
248:
122:
5048:
4337:
4297:
2098:
4227:
118:
254:
SAT is in NP because any assignment of
Boolean values to Boolean variables that is claimed to satisfy the given expression can be
158:, which require finding solutions rather than simply determining existence. He provided six such NP-complete search problems, or
110:
4864:
359:
20:
4252:
4234:
32:
4388:
Dekhtiar, M. (1969). "On the impossibility of eliminating exhaustive search in computing a function relative to its graph".
2659:
117:'s subsequent paper, "Reducibility among combinatorial problems", generated renewed interest in Cook's paper by providing a
5035:
4212:
1830:
86:
52:
2512:
2276:
1989:
109:
published his paper "The complexity of theorem proving procedures" in conference proceedings of the newly founded ACM
3976:
the membership of the given problem in NP, the transformation cannot be effectively computed, unless an upper bound
4172:
5040:
364:
132:
The theoretical interest in NP-completeness was also enhanced by the work of
Theodore P. Baker, John Gill, and
2416:
121:. Karp also introduced the notion of completeness used in the current definition of NP-completeness (i.e., by
4596:
4238:
4820:
4317:
4036:
3699:
147:
In the USSR, a result equivalent to Baker, Gill, and
Solovay's was published in 1969 by M. Dekhtiar. Later
4191:. Analogously, dependency quantified boolean formulas encode computation with a Turing machine limited to
4180:
1405:
809:
44:
4256:
3973:
4554:
4105:
594:{\displaystyle \delta \subseteq ((Q\setminus F)\times \Sigma )\times (Q\times \Sigma \times \{-1,+1\})}
4183:
for its variables. The QBF problem can be used to encode computation with a Turing machine limited to
1458:
888:
4981:
4288:(1972). "Reducibility Among Combinatorial Problems". In Raymond E. Miller; James W. Thatcher (eds.).
4176:
4187:, proving that there exists a problem (the recognition of true quantified Boolean formulas) that is
101:
was developed in the late 1960s and early 1970s in parallel by researchers in North
America and the
4260:
1399:
295:
163:
82:
3590:
862:
486:
4728:
4619:
4577:
4454:
4428:
4343:
3949:
3634:
3545:
3147:
2741:
2611:
2228:
2050:
1941:
1193:
1048:
209:
201:
4661:
4949:
3771:
3377:
3068:
1518:
945:
5062:
5044:
4333:
4293:
4208:
2576:
2340:
2193:
1906:
1601:
4508:
3449:
3416:
2471:
2375:
1786:
1722:
1681:
1338:
1241:
1096:
440:
5030:
4797:
4743:
4707:
4611:
4569:
4446:
4370:
4325:
4216:
4168:
1633:
783:
460:
205:
175:
5058:
5054:
4192:
4188:
3979:
3922:
3890:
3765:. Thus the transformation is certainly a polynomial-time many-one reduction, as required.
3279:
624:
285:
190:
181:
98:
78:
48:
40:
36:
4226:'s landmark paper, "Reducibility among combinatorial problems", in which he showed that
4929:
4488:
4473:
4164:. The quasilinear result first appeared seven years after Cook's original publication.
4008:
3955:
3870:
3850:
3830:
3810:
3679:
3522:
3502:
3482:
3357:
3337:
3317:
3125:
3105:
1759:
1659:
1579:
1559:
1381:
1316:
1296:
1276:
1171:
1151:
1131:
1026:
1006:
986:
760:
740:
734:
716:
696:
673:
653:
604:
420:
337:
301:
186:
155:
137:
133:
4802:
4785:
291:
5080:
5026:
4747:
4712:
4657:
4207:
The proof shows that every problem in NP can be reduced in polynomial time (in fact,
154:
Levin's approach was slightly different from Cook's and Karp's in that he considered
4623:
4581:
4409:
4361:
T. P. Baker; J. Gill; R. Solovay (1975). "Relativizations of the P = NP question".
4347:
4313:
4285:
4223:
280:
are equivalent, and the proof can be found in many textbooks, for example Sipser's
148:
126:
114:
106:
71:
67:
63:
59:
16:
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
4458:
4248:, and new problems are still being discovered to be within that complexity class.
4222:
The significance of NP-completeness was made clear by the publication in 1972 of
4196:
4033:
While the above method encodes a non-deterministic Turing machine in complexity
780:
The
Boolean expression uses the variables set out in the following table. Here,
217:
4762:
5066:
4786:"Lower bounds for multiplayer noncooperative games of incomplete information"
4244:
Garey and Johnson presented more than 300 NP-complete problems in their book
4450:
330:
4615:
4573:
4329:
85:, which is still widely considered the most important unsolved problem in
70:, based on an earlier proof (using a different notion of reducibility) by
5039:. Series of Books in the Mathematical Sciences (1st ed.). New York:
4215:, then all problems in NP could be solved in polynomial time, and so the
2794:
4102:, the literature describes more sophisticated approaches in complexity
5036:
Computers and Intractability: A Guide to the Theory of NP-Completeness
4729:"Short propositional formulas represent nondeterministic computations"
4246:
Computers and Intractability: A Guide to the Theory of NP-Completeness
3539:
that follows the steps indicated by the assignments to the variables.
258:
in polynomial time by a deterministic Turing machine. (The statements
4184:
4413:
4374:
4322:
Proceedings of the Third Annual ACM Symposium on Theory of Computing
4175:
problem (QBF) involves Boolean formulas extended to include nested
3265:{\displaystyle \bigvee _{0\leq k\leq p(n)}\bigvee _{f\in F}Q_{f,k}}
329:
290:
140:
models requires exponential time. That is, there exists an oracle
102:
4505:, except for the last table row, which leads to a clause with
2182:{\displaystyle T_{i,j,k}\land T_{i,j',k+1}\rightarrow H_{i,k}}
4643:
Proceedings of the 2nd Australian Computer Science Conference
621:
accepts or rejects an instance of the problem after at most
358:
Now suppose that a given problem in NP can be solved by the
3499:
is satisfiable, then there is an accepting computation for
4919:{\displaystyle (A\lor B\lor Z)\land (\lnot Z\lor C\lor D)}
3276:
Must finish in an accepting state, not later than in step
1676:, the initial symbol is the special default/blank symbol.
4485:
The number of literals in each clause does not depend on
4230:, each infamous for its intractability, are NP-complete.
3919:
The transformation makes extensive use of the polynomial
136:
who showed, in 1975, that solving NP-problems in certain
220:
to the variables that makes the entire expression true.
4228:
21 diverse combinatorial and graph theoretical problems
1402:
of the sub-expressions in the following table, for all
318:
to SAT. Data sizes and program runtimes are colored in
4595:
Nicholas Pippenger and Michael J. Fischer (Apr 1979).
3827:. The remaining lines depend only on the input length
3479:
their intended interpretations. On the other hand, if
2725:{\displaystyle \bigvee _{-p(n)\leq i\leq p(n)}H_{i,k}}
4984:
4952:
4932:
4867:
4823:
4664:
4511:
4491:
4108:
4039:
4011:
3982:
3958:
3925:
3893:
3873:
3853:
3833:
3813:
3774:
3702:
3682:
3637:
3593:
3548:
3525:
3505:
3485:
3452:
3419:
3380:
3360:
3340:
3320:
3282:
3198:
3150:
3128:
3108:
3071:
2792:
2744:
2662:
2614:
2579:
2515:
2474:
2419:
2378:
2343:
2279:
2231:
2196:
2101:
2053:
1992:
1944:
1909:
1833:
1789:
1762:
1725:
1684:
1662:
1636:
1604:
1582:
1562:
1521:
1461:
1408:
1384:
1341:
1319:
1299:
1279:
1244:
1196:
1174:
1154:
1134:
1099:
1051:
1029:
1009:
989:
948:
891:
865:
812:
786:
763:
743:
719:
699:
676:
656:
627:
607:
515:
489:
463:
443:
423:
367:
340:
304:
4639:
A new proof of the NP completeness of satisfiability
4784:Gary Peterson; John Reif; Salman Azhar (Apr 2001).
5008:
4970:
4938:
4918:
4853:
4696:reduction from RAM computations to satisfiability"
4688:
4535:
4497:
4156:
4094:
4017:
3997:
3964:
3940:
3908:
3879:
3859:
3839:
3819:
3799:
3757:
3688:
3668:
3623:
3579:
3531:
3511:
3491:
3471:
3438:
3405:
3366:
3346:
3326:
3297:
3264:
3181:
3134:
3114:
3092:
3056:
2775:
2724:
2645:
2596:
2564:
2498:
2454:
2402:
2360:
2328:
2262:
2213:
2181:
2084:
2033:
1975:
1926:
1895:{\displaystyle \neg T_{i,j,k}\lor \neg T_{i,j',k}}
1894:
1808:
1768:
1744:
1708:
1668:
1648:
1622:
1588:
1568:
1546:
1488:
1447:
1390:
1365:
1325:
1305:
1285:
1263:
1227:
1180:
1160:
1140:
1118:
1082:
1035:
1015:
995:
973:
918:
877:
851:
798:
769:
749:
725:
705:
682:
662:
642:
613:
593:
501:
475:
449:
429:
409:
346:
310:
4259:, and others. For more details, see the article
601:is the transition relation. Suppose further that
334:Schematized accepting computation by the machine
4251:Although many practical instances of SAT can be
2565:{\displaystyle \lnot H_{i,k}\lor \lnot H_{i',k}}
2329:{\displaystyle \lnot Q_{q,k}\lor \lnot Q_{q',k}}
2034:{\displaystyle \bigvee _{j\in \Sigma }T_{i,j,k}}
4555:"Satisfiability is quasilinear complete in NQL"
2223:Tape remains unchanged unless written by head.
4861:can be replaced by the conjunction of clauses
4318:"The complexity of theorem proving procedures"
4195:, proving that there exists a problem that is
198:instance of the Boolean satisfiability problem
4790:Computers & Mathematics with Applications
4219:NP would be equal to the complexity class P.
8:
4769:Symposium on Foundations of Computer Science
585:
567:
239:
5087:Theorems in computational complexity theory
4431:(1984). "A survey of Russian approaches to
4391:Proceedings of the USSR Academy of Sciences
3948:. As a consequence, the above proof is not
3587:Boolean variables, each encodable in space
3867:; they formalize a generic computation of
4983:
4951:
4931:
4866:
4822:
4801:
4711:
4663:
4510:
4490:
4107:
4083:
4038:
4010:
3981:
3957:
3924:
3892:
3872:
3852:
3832:
3812:
3779:
3773:
3746:
3701:
3681:
3657:
3636:
3592:
3568:
3547:
3524:
3504:
3484:
3457:
3451:
3424:
3418:
3385:
3379:
3359:
3339:
3319:
3314:If there is an accepting computation for
3281:
3250:
3234:
3203:
3197:
3170:
3149:
3127:
3107:
3102:Possible transitions at computation step
3070:
3012:
2979:
2945:
2874:
2842:
2823:
2804:
2793:
2791:
2764:
2743:
2710:
2667:
2661:
2634:
2613:
2578:
2545:
2523:
2514:
2473:
2440:
2424:
2418:
2377:
2342:
2309:
2287:
2278:
2251:
2230:
2195:
2167:
2131:
2106:
2100:
2073:
2052:
2013:
1997:
1991:
1964:
1943:
1908:
1869:
1841:
1832:
1794:
1788:
1761:
1730:
1724:
1683:
1661:
1635:
1603:
1581:
1561:
1526:
1520:
1460:
1407:
1383:
1340:
1318:
1298:
1278:
1249:
1243:
1216:
1195:
1173:
1153:
1133:
1104:
1098:
1071:
1050:
1028:
1008:
988:
953:
947:
890:
864:
811:
785:
762:
742:
718:
698:
675:
655:
626:
606:
514:
488:
462:
442:
422:
410:{\displaystyle M=(Q,\Sigma ,s,F,\delta )}
366:
339:
303:
282:Introduction to the Theory of Computation
4765:. In Ronald V. Book; Paul Young (eds.).
2455:{\displaystyle \bigvee _{q\in Q}Q_{q,k}}
1498:
928:
238:This proof is based on the one given by
4761:Gary L. Peterson; John H. Reif (1979).
4272:
3807:) actually depends on the input string
531:
55:to the Boolean satisfiability problem.
4280:
4278:
4276:
2736:At least one head position at a time.
4854:{\displaystyle (A\lor B\lor C\lor D)}
4597:"Relations among complexity measures"
4416:[Universal search problems].
4292:. New York: Plenum. pp. 85–103.
4095:{\displaystyle O(\log(p(n))p(n)^{3})}
3758:{\displaystyle O(\log(p(n))p(n)^{3})}
2606:At most one head position at a time.
1820:Initial position of read/write head.
926:is the number of a computation step.
7:
4463:Translation see appendix, p.399-400.
4435:(brute-force searches) algorithms".
4418:Problems of Information Transmission
1448:{\displaystyle -p(n)\leq i\leq p(n)}
852:{\displaystyle -p(n)\leq i\leq p(n)}
509:is the set of accepting states, and
286:in the Knowledge (XXG) article on NP
4290:Complexity of Computer Computations
2045:At least one symbol per tape cell.
1598:Initial contents of the tape. For
1148:'s read/write head is at tape cell
4895:
4438:Annals of the History of Computing
4025:'s time complexity is also known.
2538:
2516:
2302:
2280:
2004:
1936:At most one symbol per tape cell.
1862:
1834:
872:
561:
543:
444:
383:
249:polynomial-time many-one reduction
123:polynomial-time many-one reduction
14:
4157:{\displaystyle O(p(n)\log(p(n)))}
457:is the alphabet of tape symbols,
125:). Cook and Karp each received a
4656:John Michael Robson (May 1991).
4637:John Michael Robson (Feb 1979).
4553:Claus-Peter Schnorr (Jan 1978).
1489:{\displaystyle 0\leq k\leq p(n)}
919:{\displaystyle 0\leq k\leq p(n)}
670:is the size of the instance and
187:non-deterministic Turing machine
111:Symposium on Theory of Computing
5009:{\displaystyle (A\lor B\lor B)}
4414:"Универсальные задачи перебора"
713:, specify a Boolean expression
360:nondeterministic Turing machine
216:if there is some assignment of
119:list of 21 NP-complete problems
43:, and any problem in NP can be
21:computational complexity theory
5003:
4985:
4965:
4953:
4913:
4892:
4886:
4868:
4848:
4824:
4736:Information Processing Letters
4683:
4668:
4530:
4527:
4521:
4515:
4235:Boolean satisfiability problem
4151:
4148:
4145:
4139:
4133:
4124:
4118:
4112:
4089:
4080:
4073:
4067:
4064:
4058:
4052:
4043:
3992:
3986:
3935:
3929:
3903:
3897:
3752:
3743:
3736:
3730:
3727:
3721:
3715:
3706:
3663:
3654:
3647:
3641:
3618:
3615:
3609:
3597:
3574:
3565:
3558:
3552:
3292:
3286:
3225:
3219:
3176:
3167:
3160:
3154:
3087:
3081:
3047:
2938:
2927:
2924:
2896:
2890:
2878:
2875:
2863:
2860:
2797:
2770:
2761:
2754:
2748:
2701:
2695:
2680:
2674:
2640:
2631:
2624:
2618:
2493:
2490:
2484:
2478:
2466:At least one state at a time.
2397:
2394:
2388:
2382:
2257:
2248:
2241:
2235:
2160:
2079:
2070:
2063:
2057:
1970:
1961:
1954:
1948:
1703:
1700:
1694:
1688:
1656:, outside of the actual input
1483:
1477:
1442:
1436:
1421:
1415:
1378:Define the Boolean expression
1360:
1357:
1351:
1345:
1222:
1213:
1206:
1200:
1077:
1068:
1061:
1055:
913:
907:
846:
840:
825:
819:
637:
631:
588:
552:
546:
537:
525:
522:
404:
374:
242:, pp. 38–44, Section 2.6.
33:Boolean satisfiability problem
1:
4803:10.1016/S0898-1221(00)00333-3
4763:"Multiple-person alternation"
2370:At most one state at a time.
4748:10.1016/0020-0190(88)90152-4
4727:Stephen A. Cook (Jan 1988).
4713:10.1016/0304-3975(91)90177-4
4700:Theoretical Computer Science
4213:deterministic Turing machine
4193:logarithmic space complexity
3624:{\displaystyle O(\log p(n))}
3374:is satisfiable by assigning
878:{\displaystyle j\in \Sigma }
502:{\displaystyle F\subseteq Q}
298:showing Cook's reduction of
87:theoretical computer science
53:deterministic Turing machine
4426:Translated into English by
4253:solved by heuristic methods
4185:polynomial space complexity
3669:{\displaystyle O(p(n)^{3})}
3631:. The number of clauses is
3580:{\displaystyle O(p(n)^{2})}
3182:{\displaystyle O(p(n)^{2})}
2776:{\displaystyle O(p(n)^{2})}
2646:{\displaystyle O(p(n)^{3})}
2263:{\displaystyle O(p(n)^{2})}
2085:{\displaystyle O(p(n)^{2})}
1976:{\displaystyle O(p(n)^{2})}
1228:{\displaystyle O(p(n)^{2})}
1083:{\displaystyle O(p(n)^{2})}
284:, section 7.3., as well as
58:The theorem is named after
5103:
4689:{\displaystyle O(T\log T)}
4173:quantified Boolean formula
3768:Only the first table row (
1576:initially contains symbol
690:is a polynomial function.
185:if it can be decided by a
5041:W. H. Freeman and Company
4971:{\displaystyle (A\lor B)}
4773:. IEEE. pp. 348–363.
4363:SIAM Journal on Computing
3800:{\displaystyle T_{i,j,0}}
3406:{\displaystyle T_{i,j,k}}
3122:when head is at position
3093:{\displaystyle k<p(n)}
1547:{\displaystyle T_{i,j,0}}
974:{\displaystyle T_{i,j,k}}
650:computation steps, where
2597:{\displaystyle i\neq i'}
2361:{\displaystyle q\neq q'}
2214:{\displaystyle j\neq j'}
1927:{\displaystyle j\neq j'}
1623:{\displaystyle i>n-1}
935:Intended interpretation
274:in polynomial time by a
263:in polynomial time by a
240:Garey & Johnson 1979
212:. Such an expression is
4536:{\displaystyle O(p(n))}
4451:10.1109/MAHC.1984.10036
4239:conjunctive normal form
4181:existential quantifiers
3472:{\displaystyle Q_{i,k}}
3439:{\displaystyle H_{i,k}}
2499:{\displaystyle O(p(n))}
2403:{\displaystyle O(p(n))}
1809:{\displaystyle H_{0,0}}
1745:{\displaystyle Q_{s,0}}
1709:{\displaystyle O(p(n))}
1366:{\displaystyle O(p(n))}
1264:{\displaystyle Q_{q,k}}
1119:{\displaystyle H_{i,k}}
450:{\displaystyle \Sigma }
5010:
4972:
4940:
4920:
4855:
4690:
4537:
4499:
4257:mathematical logicians
4158:
4096:
4019:
3999:
3966:
3942:
3910:
3881:
3861:
3841:
3821:
3801:
3759:
3690:
3670:
3625:
3581:
3533:
3513:
3493:
3473:
3440:
3407:
3368:
3348:
3328:
3299:
3266:
3183:
3136:
3116:
3094:
3058:
2777:
2726:
2647:
2598:
2566:
2500:
2456:
2404:
2362:
2330:
2264:
2215:
2183:
2086:
2035:
1977:
1928:
1896:
1810:
1770:
1746:
1710:
1670:
1650:
1649:{\displaystyle i<0}
1624:
1590:
1570:
1548:
1490:
1449:
1392:
1367:
1327:
1307:
1287:
1265:
1229:
1182:
1162:
1142:
1120:
1084:
1037:
1017:
997:
975:
920:
885:is a tape symbol, and
879:
853:
800:
799:{\displaystyle q\in Q}
771:
751:
727:
707:
684:
664:
644:
615:
595:
503:
483:is the initial state,
477:
476:{\displaystyle s\in Q}
451:
437:is the set of states,
431:
411:
355:
348:
327:
312:
66:. The proof is due to
5011:
4973:
4941:
4921:
4856:
4691:
4616:10.1145/322123.322138
4574:10.1145/322047.322060
4538:
4500:
4472:This column uses the
4330:10.1145/800157.805047
4177:universal quantifiers
4159:
4097:
4020:
4000:
3967:
3943:
3911:
3882:
3862:
3842:
3822:
3802:
3760:
3691:
3671:
3626:
3582:
3534:
3514:
3494:
3474:
3441:
3408:
3369:
3349:
3329:
3300:
3267:
3184:
3137:
3117:
3095:
3059:
2778:
2727:
2648:
2599:
2567:
2501:
2457:
2405:
2363:
2331:
2265:
2216:
2184:
2087:
2036:
1978:
1929:
1897:
1811:
1771:
1747:
1711:
1671:
1651:
1625:
1591:
1571:
1549:
1491:
1450:
1393:
1368:
1328:
1308:
1288:
1266:
1230:
1183:
1163:
1143:
1121:
1085:
1038:
1018:
998:
976:
921:
880:
854:
801:
772:
752:
728:
708:
685:
665:
645:
616:
596:
504:
478:
452:
432:
412:
349:
333:
313:
294:
39:. That is, it is in
4982:
4950:
4930:
4865:
4821:
4662:
4509:
4489:
4324:. pp. 151–158.
4106:
4037:
4009:
3998:{\displaystyle p(n)}
3980:
3956:
3941:{\displaystyle p(n)}
3923:
3909:{\displaystyle p(n)}
3891:
3871:
3851:
3831:
3811:
3772:
3700:
3680:
3635:
3591:
3546:
3523:
3503:
3483:
3450:
3417:
3378:
3358:
3338:
3318:
3298:{\displaystyle p(n)}
3280:
3196:
3148:
3126:
3106:
3069:
2790:
2742:
2660:
2612:
2577:
2513:
2472:
2417:
2376:
2341:
2277:
2229:
2194:
2099:
2051:
1990:
1942:
1907:
1831:
1787:
1760:
1723:
1682:
1660:
1634:
1602:
1580:
1560:
1519:
1459:
1406:
1382:
1339:
1333:of the computation.
1317:
1297:
1277:
1242:
1194:
1188:of the computation.
1172:
1152:
1132:
1097:
1049:
1043:of the computation.
1027:
1007:
987:
946:
889:
863:
859:is a tape position,
810:
806:is a machine state,
784:
761:
741:
733:that is satisfiable
717:
697:
674:
654:
643:{\displaystyle p(n)}
625:
605:
513:
487:
461:
441:
421:
365:
338:
302:
4978:can be replaced by
4429:Trakhtenbrot, B. A.
4261:P versus NP problem
4237:for expressions in
3847:and on the machine
296:Commutative diagram
83:P versus NP problem
5006:
4968:
4936:
4916:
4851:
4767:Proc. 20th Annual
4686:
4604:Journal of the ACM
4562:Journal of the ACM
4533:
4495:
4169:complexity classes
4154:
4092:
4015:
3995:
3962:
3938:
3906:
3877:
3857:
3837:
3817:
3797:
3755:
3686:
3666:
3621:
3577:
3529:
3509:
3489:
3469:
3436:
3403:
3364:
3344:
3324:
3295:
3262:
3245:
3229:
3179:
3132:
3112:
3090:
3054:
3052:
2937:
2773:
2722:
2705:
2643:
2594:
2562:
2496:
2452:
2435:
2400:
2358:
2326:
2260:
2211:
2179:
2082:
2031:
2008:
1973:
1924:
1892:
1806:
1766:
1742:
1706:
1666:
1646:
1620:
1586:
1566:
1544:
1486:
1445:
1388:
1363:
1323:
1303:
1283:
1261:
1225:
1178:
1158:
1138:
1116:
1080:
1033:
1013:
993:
983:True if tape cell
971:
916:
875:
849:
796:
767:
747:
723:
703:
680:
660:
640:
611:
591:
499:
473:
447:
427:
407:
356:
344:
328:
308:
202:Boolean expression
160:universal problems
31:, states that the
25:Cook–Levin theorem
5031:Johnson, David S.
5027:Garey, Michael R.
4939:{\displaystyle Z}
4645:. pp. 62–70.
4498:{\displaystyle n}
4209:logarithmic space
4018:{\displaystyle M}
3965:{\displaystyle M}
3880:{\displaystyle M}
3860:{\displaystyle M}
3840:{\displaystyle n}
3820:{\displaystyle I}
3689:{\displaystyle B}
3532:{\displaystyle I}
3512:{\displaystyle M}
3492:{\displaystyle B}
3367:{\displaystyle B}
3347:{\displaystyle I}
3327:{\displaystyle M}
3312:
3311:
3230:
3199:
3135:{\displaystyle i}
3115:{\displaystyle k}
3035:
3021:
2993:
2960:
2870:
2663:
2420:
1993:
1769:{\displaystyle M}
1756:Initial state of
1669:{\displaystyle I}
1589:{\displaystyle j}
1569:{\displaystyle i}
1391:{\displaystyle B}
1376:
1375:
1326:{\displaystyle k}
1306:{\displaystyle q}
1286:{\displaystyle M}
1181:{\displaystyle k}
1161:{\displaystyle i}
1141:{\displaystyle M}
1036:{\displaystyle k}
1016:{\displaystyle j}
996:{\displaystyle i}
770:{\displaystyle I}
750:{\displaystyle M}
726:{\displaystyle B}
706:{\displaystyle I}
683:{\displaystyle p}
663:{\displaystyle n}
614:{\displaystyle M}
430:{\displaystyle Q}
347:{\displaystyle M}
311:{\displaystyle M}
276:non-deterministic
210:Boolean operators
206:Boolean variables
5094:
5071:
5070:
5023:
5017:
5015:
5013:
5012:
5007:
4977:
4975:
4974:
4969:
4945:
4943:
4942:
4937:
4925:
4923:
4922:
4917:
4860:
4858:
4857:
4852:
4814:
4808:
4807:
4805:
4796:(7–8): 957–992.
4781:
4775:
4774:
4758:
4752:
4751:
4733:
4724:
4718:
4717:
4715:
4695:
4693:
4692:
4687:
4653:
4647:
4646:
4634:
4628:
4627:
4601:
4592:
4586:
4585:
4559:
4550:
4544:
4542:
4540:
4539:
4534:
4504:
4502:
4501:
4496:
4483:
4477:
4470:
4464:
4462:
4425:
4406:
4400:
4399:
4385:
4379:
4378:
4358:
4352:
4351:
4310:
4304:
4303:
4286:Karp, Richard M.
4282:
4217:complexity class
4163:
4161:
4160:
4155:
4101:
4099:
4098:
4093:
4088:
4087:
4024:
4022:
4021:
4016:
4004:
4002:
4001:
3996:
3971:
3969:
3968:
3963:
3947:
3945:
3944:
3939:
3915:
3913:
3912:
3907:
3886:
3884:
3883:
3878:
3866:
3864:
3863:
3858:
3846:
3844:
3843:
3838:
3826:
3824:
3823:
3818:
3806:
3804:
3803:
3798:
3796:
3795:
3764:
3762:
3761:
3756:
3751:
3750:
3695:
3693:
3692:
3687:
3675:
3673:
3672:
3667:
3662:
3661:
3630:
3628:
3627:
3622:
3586:
3584:
3583:
3578:
3573:
3572:
3538:
3536:
3535:
3530:
3518:
3516:
3515:
3510:
3498:
3496:
3495:
3490:
3478:
3476:
3475:
3470:
3468:
3467:
3445:
3443:
3442:
3437:
3435:
3434:
3412:
3410:
3409:
3404:
3402:
3401:
3373:
3371:
3370:
3365:
3353:
3351:
3350:
3345:
3333:
3331:
3330:
3325:
3304:
3302:
3301:
3296:
3271:
3269:
3268:
3263:
3261:
3260:
3244:
3228:
3188:
3186:
3185:
3180:
3175:
3174:
3141:
3139:
3138:
3133:
3121:
3119:
3118:
3113:
3099:
3097:
3096:
3091:
3063:
3061:
3060:
3055:
3053:
3046:
3045:
3033:
3029:
3019:
3004:
3003:
2991:
2987:
2971:
2970:
2958:
2936:
2917:
2906:
2859:
2858:
2834:
2833:
2815:
2814:
2782:
2780:
2779:
2774:
2769:
2768:
2731:
2729:
2728:
2723:
2721:
2720:
2704:
2652:
2650:
2649:
2644:
2639:
2638:
2603:
2601:
2600:
2595:
2593:
2571:
2569:
2568:
2563:
2561:
2560:
2553:
2534:
2533:
2505:
2503:
2502:
2497:
2461:
2459:
2458:
2453:
2451:
2450:
2434:
2409:
2407:
2406:
2401:
2367:
2365:
2364:
2359:
2357:
2335:
2333:
2332:
2327:
2325:
2324:
2317:
2298:
2297:
2269:
2267:
2266:
2261:
2256:
2255:
2220:
2218:
2217:
2212:
2210:
2188:
2186:
2185:
2180:
2178:
2177:
2159:
2158:
2145:
2123:
2122:
2091:
2089:
2088:
2083:
2078:
2077:
2040:
2038:
2037:
2032:
2030:
2029:
2007:
1982:
1980:
1979:
1974:
1969:
1968:
1933:
1931:
1930:
1925:
1923:
1901:
1899:
1898:
1893:
1891:
1890:
1883:
1858:
1857:
1815:
1813:
1812:
1807:
1805:
1804:
1775:
1773:
1772:
1767:
1751:
1749:
1748:
1743:
1741:
1740:
1715:
1713:
1712:
1707:
1675:
1673:
1672:
1667:
1655:
1653:
1652:
1647:
1629:
1627:
1626:
1621:
1595:
1593:
1592:
1587:
1575:
1573:
1572:
1567:
1553:
1551:
1550:
1545:
1543:
1542:
1499:
1495:
1493:
1492:
1487:
1454:
1452:
1451:
1446:
1397:
1395:
1394:
1389:
1372:
1370:
1369:
1364:
1332:
1330:
1329:
1324:
1312:
1310:
1309:
1304:
1292:
1290:
1289:
1284:
1270:
1268:
1267:
1262:
1260:
1259:
1234:
1232:
1231:
1226:
1221:
1220:
1187:
1185:
1184:
1179:
1167:
1165:
1164:
1159:
1147:
1145:
1144:
1139:
1125:
1123:
1122:
1117:
1115:
1114:
1089:
1087:
1086:
1081:
1076:
1075:
1042:
1040:
1039:
1034:
1022:
1020:
1019:
1014:
1003:contains symbol
1002:
1000:
999:
994:
980:
978:
977:
972:
970:
969:
929:
925:
923:
922:
917:
884:
882:
881:
876:
858:
856:
855:
850:
805:
803:
802:
797:
776:
774:
773:
768:
756:
754:
753:
748:
732:
730:
729:
724:
712:
710:
709:
704:
693:For each input,
689:
687:
686:
681:
669:
667:
666:
661:
649:
647:
646:
641:
620:
618:
617:
612:
600:
598:
597:
592:
508:
506:
505:
500:
482:
480:
479:
474:
456:
454:
453:
448:
436:
434:
433:
428:
416:
414:
413:
408:
353:
351:
350:
345:
325:
321:
317:
315:
314:
309:
176:decision problem
27:, also known as
5102:
5101:
5097:
5096:
5095:
5093:
5092:
5091:
5077:
5076:
5075:
5074:
5051:
5025:
5024:
5020:
4980:
4979:
4948:
4947:
4928:
4927:
4863:
4862:
4819:
4818:
4815:
4811:
4783:
4782:
4778:
4760:
4759:
4755:
4731:
4726:
4725:
4721:
4660:
4659:
4655:
4654:
4650:
4636:
4635:
4631:
4599:
4594:
4593:
4589:
4557:
4552:
4551:
4547:
4507:
4506:
4487:
4486:
4484:
4480:
4471:
4467:
4427:
4408:
4407:
4403:
4387:
4386:
4382:
4375:10.1137/0204037
4360:
4359:
4355:
4340:
4312:
4311:
4307:
4300:
4284:
4283:
4274:
4269:
4205:
4189:PSPACE-complete
4104:
4103:
4079:
4035:
4034:
4031:
4007:
4006:
3978:
3977:
3954:
3953:
3921:
3920:
3889:
3888:
3869:
3868:
3849:
3848:
3829:
3828:
3809:
3808:
3775:
3770:
3769:
3742:
3698:
3697:
3678:
3677:
3676:so the size of
3653:
3633:
3632:
3589:
3588:
3564:
3544:
3543:
3521:
3520:
3501:
3500:
3481:
3480:
3453:
3448:
3447:
3420:
3415:
3414:
3381:
3376:
3375:
3356:
3355:
3336:
3335:
3316:
3315:
3278:
3277:
3246:
3194:
3193:
3166:
3146:
3145:
3124:
3123:
3104:
3103:
3067:
3066:
3051:
3050:
3022:
3008:
2980:
2975:
2941:
2910:
2899:
2867:
2866:
2838:
2819:
2800:
2788:
2787:
2760:
2740:
2739:
2706:
2658:
2657:
2630:
2610:
2609:
2586:
2575:
2574:
2546:
2541:
2519:
2511:
2510:
2470:
2469:
2436:
2415:
2414:
2374:
2373:
2350:
2339:
2338:
2310:
2305:
2283:
2275:
2274:
2247:
2227:
2226:
2203:
2192:
2191:
2163:
2138:
2127:
2102:
2097:
2096:
2069:
2049:
2048:
2009:
1988:
1987:
1960:
1940:
1939:
1916:
1905:
1904:
1876:
1865:
1837:
1829:
1828:
1790:
1785:
1784:
1758:
1757:
1726:
1721:
1720:
1680:
1679:
1658:
1657:
1632:
1631:
1600:
1599:
1578:
1577:
1558:
1557:
1522:
1517:
1516:
1508:Interpretation
1457:
1456:
1404:
1403:
1380:
1379:
1337:
1336:
1315:
1314:
1295:
1294:
1275:
1274:
1245:
1240:
1239:
1212:
1192:
1191:
1170:
1169:
1150:
1149:
1130:
1129:
1100:
1095:
1094:
1067:
1047:
1046:
1025:
1024:
1005:
1004:
985:
984:
949:
944:
943:
887:
886:
861:
860:
808:
807:
782:
781:
759:
758:
739:
738:
715:
714:
695:
694:
672:
671:
652:
651:
623:
622:
603:
602:
511:
510:
485:
484:
459:
458:
439:
438:
419:
418:
363:
362:
336:
335:
326:, respectively.
323:
319:
300:
299:
235:
226:
191:polynomial time
172:
156:search problems
129:for this work.
99:NP-completeness
97:The concept of
95:
49:polynomial time
17:
12:
11:
5:
5100:
5098:
5090:
5089:
5079:
5078:
5073:
5072:
5049:
5018:
5005:
5002:
4999:
4996:
4993:
4990:
4987:
4967:
4964:
4961:
4958:
4955:
4935:
4915:
4912:
4909:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4873:
4870:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4809:
4776:
4753:
4742:(5): 269–270.
4719:
4706:(1): 141–149.
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4648:
4629:
4610:(2): 361–381.
4587:
4568:(1): 136–145.
4545:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4494:
4478:
4474:big O notation
4465:
4445:(4): 384–400.
4420:(in Russian).
4401:
4394:(in Russian).
4380:
4369:(4): 431–442.
4353:
4338:
4305:
4298:
4271:
4270:
4268:
4265:
4204:
4201:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4091:
4086:
4082:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4030:
4027:
4014:
3994:
3991:
3988:
3985:
3961:
3937:
3934:
3931:
3928:
3905:
3902:
3899:
3896:
3876:
3856:
3836:
3816:
3794:
3791:
3788:
3785:
3782:
3778:
3754:
3749:
3745:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3685:
3665:
3660:
3656:
3652:
3649:
3646:
3643:
3640:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3576:
3571:
3567:
3563:
3560:
3557:
3554:
3551:
3528:
3508:
3488:
3466:
3463:
3460:
3456:
3433:
3430:
3427:
3423:
3400:
3397:
3394:
3391:
3388:
3384:
3363:
3343:
3323:
3310:
3309:
3306:
3294:
3291:
3288:
3285:
3274:
3272:
3259:
3256:
3253:
3249:
3243:
3240:
3237:
3233:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3202:
3190:
3189:
3178:
3173:
3169:
3165:
3162:
3159:
3156:
3153:
3143:
3131:
3111:
3100:
3089:
3086:
3083:
3080:
3077:
3074:
3064:
3049:
3044:
3041:
3038:
3032:
3028:
3025:
3018:
3015:
3011:
3007:
3002:
2999:
2996:
2990:
2986:
2983:
2978:
2974:
2969:
2966:
2963:
2957:
2954:
2951:
2948:
2944:
2940:
2935:
2932:
2929:
2926:
2923:
2920:
2916:
2913:
2909:
2905:
2902:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2873:
2869:
2868:
2865:
2862:
2857:
2854:
2851:
2848:
2845:
2841:
2837:
2832:
2829:
2826:
2822:
2818:
2813:
2810:
2807:
2803:
2799:
2796:
2795:
2784:
2783:
2772:
2767:
2763:
2759:
2756:
2753:
2750:
2747:
2737:
2734:
2732:
2719:
2716:
2713:
2709:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2666:
2654:
2653:
2642:
2637:
2633:
2629:
2626:
2623:
2620:
2617:
2607:
2604:
2592:
2589:
2585:
2582:
2572:
2559:
2556:
2552:
2549:
2544:
2540:
2537:
2532:
2529:
2526:
2522:
2518:
2507:
2506:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2467:
2464:
2462:
2449:
2446:
2443:
2439:
2433:
2430:
2427:
2423:
2411:
2410:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2371:
2368:
2356:
2353:
2349:
2346:
2336:
2323:
2320:
2316:
2313:
2308:
2304:
2301:
2296:
2293:
2290:
2286:
2282:
2271:
2270:
2259:
2254:
2250:
2246:
2243:
2240:
2237:
2234:
2224:
2221:
2209:
2206:
2202:
2199:
2189:
2176:
2173:
2170:
2166:
2162:
2157:
2154:
2151:
2148:
2144:
2141:
2137:
2134:
2130:
2126:
2121:
2118:
2115:
2112:
2109:
2105:
2093:
2092:
2081:
2076:
2072:
2068:
2065:
2062:
2059:
2056:
2046:
2043:
2041:
2028:
2025:
2022:
2019:
2016:
2012:
2006:
2003:
2000:
1996:
1984:
1983:
1972:
1967:
1963:
1959:
1956:
1953:
1950:
1947:
1937:
1934:
1922:
1919:
1915:
1912:
1902:
1889:
1886:
1882:
1879:
1875:
1872:
1868:
1864:
1861:
1856:
1853:
1850:
1847:
1844:
1840:
1836:
1825:
1824:
1821:
1818:
1816:
1803:
1800:
1797:
1793:
1781:
1780:
1777:
1765:
1754:
1752:
1739:
1736:
1733:
1729:
1717:
1716:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1677:
1665:
1645:
1642:
1639:
1619:
1616:
1613:
1610:
1607:
1596:
1585:
1565:
1554:
1541:
1538:
1535:
1532:
1529:
1525:
1513:
1512:
1509:
1506:
1503:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1387:
1374:
1373:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1334:
1322:
1302:
1282:
1271:
1258:
1255:
1252:
1248:
1236:
1235:
1224:
1219:
1215:
1211:
1208:
1205:
1202:
1199:
1189:
1177:
1157:
1137:
1126:
1113:
1110:
1107:
1103:
1091:
1090:
1079:
1074:
1070:
1066:
1063:
1060:
1057:
1054:
1044:
1032:
1012:
992:
981:
968:
965:
962:
959:
956:
952:
940:
939:
936:
933:
915:
912:
909:
906:
903:
900:
897:
894:
874:
871:
868:
848:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
795:
792:
789:
766:
746:
735:if and only if
722:
702:
679:
659:
639:
636:
633:
630:
610:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
498:
495:
492:
472:
469:
466:
446:
426:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
343:
307:
278:Turing machine
267:Turing machine
234:
231:
225:
222:
204:that combines
171:
168:
138:oracle machine
134:Robert Solovay
94:
91:
29:Cook's theorem
15:
13:
10:
9:
6:
4:
3:
2:
5099:
5088:
5085:
5084:
5082:
5068:
5064:
5060:
5056:
5052:
5050:9780716710455
5046:
5042:
5038:
5037:
5032:
5028:
5022:
5019:
5000:
4997:
4994:
4991:
4988:
4962:
4959:
4956:
4933:
4910:
4907:
4904:
4901:
4898:
4889:
4883:
4880:
4877:
4874:
4871:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4813:
4810:
4804:
4799:
4795:
4791:
4787:
4780:
4777:
4772:
4770:
4764:
4757:
4754:
4749:
4745:
4741:
4737:
4730:
4723:
4720:
4714:
4709:
4705:
4701:
4697:
4680:
4677:
4674:
4671:
4665:
4652:
4649:
4644:
4640:
4633:
4630:
4625:
4621:
4617:
4613:
4609:
4605:
4598:
4591:
4588:
4583:
4579:
4575:
4571:
4567:
4563:
4556:
4549:
4546:
4524:
4518:
4512:
4492:
4482:
4479:
4475:
4469:
4466:
4460:
4456:
4452:
4448:
4444:
4440:
4439:
4434:
4430:
4424:(3): 115–116.
4423:
4419:
4415:
4411:
4410:Levin, Leonid
4405:
4402:
4397:
4393:
4392:
4384:
4381:
4376:
4372:
4368:
4364:
4357:
4354:
4349:
4345:
4341:
4339:9781450374644
4335:
4331:
4327:
4323:
4319:
4315:
4314:Cook, Stephen
4309:
4306:
4301:
4299:0-306-30707-3
4295:
4291:
4287:
4281:
4279:
4277:
4273:
4266:
4264:
4262:
4258:
4254:
4249:
4247:
4242:
4240:
4236:
4231:
4229:
4225:
4220:
4218:
4214:
4210:
4202:
4200:
4198:
4194:
4190:
4186:
4182:
4178:
4174:
4170:
4165:
4142:
4136:
4130:
4127:
4121:
4115:
4109:
4084:
4076:
4070:
4061:
4055:
4049:
4046:
4040:
4028:
4026:
4012:
3989:
3983:
3975:
3959:
3951:
3932:
3926:
3917:
3900:
3894:
3874:
3854:
3834:
3814:
3792:
3789:
3786:
3783:
3780:
3776:
3766:
3747:
3739:
3733:
3724:
3718:
3712:
3709:
3703:
3683:
3658:
3650:
3644:
3638:
3612:
3606:
3603:
3600:
3594:
3569:
3561:
3555:
3549:
3540:
3526:
3506:
3486:
3464:
3461:
3458:
3454:
3431:
3428:
3425:
3421:
3398:
3395:
3392:
3389:
3386:
3382:
3361:
3341:
3321:
3307:
3289:
3283:
3275:
3273:
3257:
3254:
3251:
3247:
3241:
3238:
3235:
3231:
3222:
3216:
3213:
3210:
3207:
3204:
3200:
3192:
3191:
3171:
3163:
3157:
3151:
3144:
3129:
3109:
3101:
3084:
3078:
3075:
3072:
3065:
3042:
3039:
3036:
3030:
3026:
3023:
3016:
3013:
3009:
3005:
3000:
2997:
2994:
2988:
2984:
2981:
2976:
2972:
2967:
2964:
2961:
2955:
2952:
2949:
2946:
2942:
2933:
2930:
2921:
2918:
2914:
2911:
2907:
2903:
2900:
2893:
2887:
2884:
2881:
2871:
2855:
2852:
2849:
2846:
2843:
2839:
2835:
2830:
2827:
2824:
2820:
2816:
2811:
2808:
2805:
2801:
2786:
2785:
2765:
2757:
2751:
2745:
2738:
2735:
2733:
2717:
2714:
2711:
2707:
2698:
2692:
2689:
2686:
2683:
2677:
2671:
2668:
2664:
2656:
2655:
2635:
2627:
2621:
2615:
2608:
2605:
2590:
2587:
2583:
2580:
2573:
2557:
2554:
2550:
2547:
2542:
2535:
2530:
2527:
2524:
2520:
2509:
2508:
2487:
2481:
2475:
2468:
2465:
2463:
2447:
2444:
2441:
2437:
2431:
2428:
2425:
2421:
2413:
2412:
2391:
2385:
2379:
2372:
2369:
2354:
2351:
2347:
2344:
2337:
2321:
2318:
2314:
2311:
2306:
2299:
2294:
2291:
2288:
2284:
2273:
2272:
2252:
2244:
2238:
2232:
2225:
2222:
2207:
2204:
2200:
2197:
2190:
2174:
2171:
2168:
2164:
2155:
2152:
2149:
2146:
2142:
2139:
2135:
2132:
2128:
2124:
2119:
2116:
2113:
2110:
2107:
2103:
2095:
2094:
2074:
2066:
2060:
2054:
2047:
2044:
2042:
2026:
2023:
2020:
2017:
2014:
2010:
2001:
1998:
1994:
1986:
1985:
1965:
1957:
1951:
1945:
1938:
1935:
1920:
1917:
1913:
1910:
1903:
1887:
1884:
1880:
1877:
1873:
1870:
1866:
1859:
1854:
1851:
1848:
1845:
1842:
1838:
1827:
1826:
1822:
1819:
1817:
1801:
1798:
1795:
1791:
1783:
1782:
1778:
1763:
1755:
1753:
1737:
1734:
1731:
1727:
1719:
1718:
1697:
1691:
1685:
1678:
1663:
1643:
1640:
1637:
1617:
1614:
1611:
1608:
1605:
1597:
1583:
1563:
1555:
1539:
1536:
1533:
1530:
1527:
1523:
1515:
1514:
1510:
1507:
1504:
1501:
1500:
1497:
1480:
1474:
1471:
1468:
1465:
1462:
1439:
1433:
1430:
1427:
1424:
1418:
1412:
1409:
1401:
1385:
1354:
1348:
1342:
1335:
1320:
1300:
1280:
1272:
1256:
1253:
1250:
1246:
1238:
1237:
1217:
1209:
1203:
1197:
1190:
1175:
1155:
1135:
1127:
1111:
1108:
1105:
1101:
1093:
1092:
1072:
1064:
1058:
1052:
1045:
1030:
1010:
990:
982:
966:
963:
960:
957:
954:
950:
942:
941:
937:
934:
931:
930:
927:
910:
904:
901:
898:
895:
892:
869:
866:
843:
837:
834:
831:
828:
822:
816:
813:
793:
790:
787:
778:
764:
744:
736:
720:
700:
691:
677:
657:
634:
628:
608:
582:
579:
576:
573:
570:
564:
558:
555:
549:
540:
534:
528:
519:
516:
496:
493:
490:
470:
467:
464:
424:
401:
398:
395:
392:
389:
386:
380:
377:
371:
368:
361:
341:
332:
305:
297:
293:
289:
287:
283:
279:
277:
273:
268:
266:
265:deterministic
262:
257:
252:
250:
244:
243:
241:
232:
230:
223:
221:
219:
215:
211:
207:
203:
199:
194:
192:
188:
184:
183:
177:
169:
167:
165:
161:
157:
152:
150:
145:
143:
139:
135:
130:
128:
124:
120:
116:
112:
108:
104:
100:
93:Contributions
92:
90:
88:
84:
80:
75:
73:
69:
65:
61:
56:
54:
50:
46:
42:
38:
34:
30:
26:
22:
5034:
5021:
4812:
4793:
4789:
4779:
4766:
4756:
4739:
4735:
4722:
4703:
4699:
4651:
4642:
4638:
4632:
4607:
4603:
4590:
4565:
4561:
4548:
4481:
4468:
4442:
4436:
4432:
4421:
4417:
4404:
4398:: 1146–1148.
4395:
4389:
4383:
4366:
4362:
4356:
4321:
4308:
4289:
4250:
4245:
4243:
4232:
4224:Richard Karp
4221:
4206:
4203:Consequences
4166:
4032:
3950:constructive
3918:
3767:
3541:
3313:
1377:
1293:is in state
779:
737:the machine
692:
357:
281:
275:
271:
270:
264:
260:
259:
255:
253:
245:
237:
236:
227:
218:truth values
213:
197:
195:
179:
173:
159:
153:
149:Leonid Levin
146:
141:
131:
127:Turing Award
115:Richard Karp
107:Stephen Cook
96:
76:
68:Richard Karp
64:Leonid Levin
60:Stephen Cook
57:
28:
24:
18:
4197:NL-complete
1505:Conditions
1502:Expression
1400:conjunction
214:satisfiable
170:Definitions
105:. In 1971,
37:NP-complete
4267:References
4029:Complexity
3974:witnessing
3972:is known,
3952:: even if
3887:for up to
3542:There are
1556:Tape cell
1511:How many?
1398:to be the
938:How many?
932:Variables
261:verifiable
5067:247570676
4998:∨
4992:∨
4960:∨
4908:∨
4902:∨
4896:¬
4890:∧
4881:∨
4875:∨
4843:∨
4837:∨
4831:∨
4678:
4543:literals.
4131:
4050:
3713:
3604:
3519:on input
3334:on input
3239:∈
3232:⋁
3214:≤
3208:≤
3201:⋁
3024:σ
3006:∧
2973:∧
2934:δ
2931:∈
2912:σ
2888:σ
2872:⋁
2864:→
2850:σ
2836:∧
2817:∧
2690:≤
2684:≤
2669:−
2665:⋁
2584:≠
2539:¬
2536:∨
2517:¬
2429:∈
2422:⋁
2348:≠
2303:¬
2300:∨
2281:¬
2201:≠
2161:→
2125:∧
2005:Σ
2002:∈
1995:⋁
1914:≠
1863:¬
1860:∨
1835:¬
1615:−
1472:≤
1466:≤
1431:≤
1425:≤
1410:−
902:≤
896:≤
873:Σ
870:∈
835:≤
829:≤
814:−
791:∈
571:−
565:×
562:Σ
559:×
550:×
544:Σ
541:×
532:∖
520:⊆
517:δ
494:⊆
468:∈
445:Σ
402:δ
384:Σ
5081:Category
5033:(1979).
4926:, where
4412:(1973).
4316:(1971).
3027:′
2985:′
2915:′
2904:′
2591:′
2551:′
2355:′
2315:′
2208:′
2143:′
1921:′
1881:′
1313:at step
1273:True if
1168:at step
1128:True if
1023:at step
757:accepts
417:, where
272:solvable
256:verified
5059:0519066
4624:2432526
4582:1929802
4433:perebor
4348:7573663
3916:steps.
3354:, then
45:reduced
5065:
5057:
5047:
4771:(SFCS)
4622:
4580:
4459:950581
4457:
4346:
4336:
4296:
4171:. The
3034:
3020:
2992:
2959:
320:orange
208:using
164:P = NP
23:, the
4732:(PDF)
4620:S2CID
4600:(PDF)
4578:S2CID
4558:(PDF)
4455:S2CID
4344:S2CID
324:green
233:Proof
200:is a
51:by a
5063:OCLC
5045:ISBN
4658:"An
4334:ISBN
4294:ISBN
4179:and
3446:and
3076:<
1641:<
1630:and
1609:>
1455:and
322:and
269:and
224:Idea
103:USSR
72:Cook
62:and
4798:doi
4744:doi
4708:doi
4675:log
4612:doi
4570:doi
4447:doi
4371:doi
4326:doi
4128:log
4047:log
4005:of
3710:log
3696:is
3601:log
288:).
196:An
189:in
180:in
178:is
166:).
47:in
35:is
19:In
5083::
5061:.
5055:MR
5053:.
5043:.
5029:;
4794:41
4792:.
4788:.
4740:26
4738:.
4734:.
4704:82
4702:.
4698:.
4641:.
4618:.
4608:26
4606:.
4602:.
4576:.
4566:25
4564:.
4560:.
4453:.
4441:.
4396:14
4365:.
4342:.
4332:.
4320:.
4275:^
4263:.
4199:.
3413:,
3308:1
3305:.
3142:.
1823:1
1779:1
1776:.
1496::
777:.
251:.
193:.
182:NP
174:A
113:.
89:.
79:NP
74:.
41:NP
5069:.
5016:.
5004:)
5001:B
4995:B
4989:A
4986:(
4966:)
4963:B
4957:A
4954:(
4934:Z
4914:)
4911:D
4905:C
4899:Z
4893:(
4887:)
4884:Z
4878:B
4872:A
4869:(
4849:)
4846:D
4840:C
4834:B
4828:A
4825:(
4806:.
4800::
4750:.
4746::
4716:.
4710::
4684:)
4681:T
4672:T
4669:(
4666:O
4626:.
4614::
4584:.
4572::
4531:)
4528:)
4525:n
4522:(
4519:p
4516:(
4513:O
4493:n
4476:.
4461:.
4449::
4443:6
4422:9
4377:.
4373::
4367:4
4350:.
4328::
4302:.
4152:)
4149:)
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4140:(
4137:p
4134:(
4125:)
4122:n
4119:(
4116:p
4113:(
4110:O
4090:)
4085:3
4081:)
4077:n
4074:(
4071:p
4068:)
4065:)
4062:n
4059:(
4056:p
4053:(
4044:(
4041:O
4013:M
3993:)
3990:n
3987:(
3984:p
3960:M
3936:)
3933:n
3930:(
3927:p
3904:)
3901:n
3898:(
3895:p
3875:M
3855:M
3835:n
3815:I
3793:0
3790:,
3787:j
3784:,
3781:i
3777:T
3753:)
3748:3
3744:)
3740:n
3737:(
3734:p
3731:)
3728:)
3725:n
3722:(
3719:p
3716:(
3707:(
3704:O
3684:B
3664:)
3659:3
3655:)
3651:n
3648:(
3645:p
3642:(
3639:O
3619:)
3616:)
3613:n
3610:(
3607:p
3598:(
3595:O
3575:)
3570:2
3566:)
3562:n
3559:(
3556:p
3553:(
3550:O
3527:I
3507:M
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3465:k
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3459:i
3455:Q
3432:k
3429:,
3426:i
3422:H
3399:k
3396:,
3393:j
3390:,
3387:i
3383:T
3362:B
3342:I
3322:M
3293:)
3290:n
3287:(
3284:p
3258:k
3255:,
3252:f
3248:Q
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3236:f
3226:)
3223:n
3220:(
3217:p
3211:k
3205:0
3177:)
3172:2
3168:)
3164:n
3161:(
3158:p
3155:(
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3130:i
3110:k
3088:)
3085:n
3082:(
3079:p
3073:k
3048:)
3043:1
3040:+
3037:k
3031:,
3017:,
3014:i
3010:T
3001:1
2998:+
2995:k
2989:,
2982:q
2977:Q
2968:1
2965:+
2962:k
2956:,
2953:d
2950:+
2947:i
2943:H
2939:(
2928:)
2925:)
2922:d
2919:,
2908:,
2901:q
2897:(
2894:,
2891:)
2885:,
2882:q
2879:(
2876:(
2861:)
2856:k
2853:,
2847:,
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2821:Q
2812:k
2809:,
2806:i
2802:H
2798:(
2771:)
2766:2
2762:)
2758:n
2755:(
2752:p
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2718:k
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2641:)
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2172:,
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2018:,
2015:i
2011:T
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1966:2
1962:)
1958:n
1955:(
1952:p
1949:(
1946:O
1918:j
1911:j
1888:k
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1802:0
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1422:)
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375:(
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