1389:
3077:
2788:
2781:
3072:{\displaystyle b\cdot ((a\cdot b)^{-1}\cdot a)\mathrel {\overset {R17}{\rightsquigarrow }} b\cdot ((b^{-1}\cdot a^{-1})\cdot a)\mathrel {\overset {R3}{\rightsquigarrow }} b\cdot (b^{-1}\cdot (a^{-1}\cdot a))\mathrel {\overset {R2}{\rightsquigarrow }} b\cdot (b^{-1}\cdot 1)\mathrel {\overset {R11}{\rightsquigarrow }} b\cdot b^{-1}\mathrel {\overset {R13}{\rightsquigarrow }} 1}
2552:
438:, and devise a transformation system to rewrite those expressions to that form, in the process proving that all equivalent expressions will be rewritten to the same normal form. But not all solutions to the word problem use a normal form theorem - there are algebraic properties which indirectly imply the existence of an algorithm.
1605:
of finite arity, and a finite set of identities that these operations must satisfy. The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word
66:
one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an
2776:{\displaystyle ((a^{-1}\cdot a)\cdot (b\cdot b^{-1}))^{-1}\mathrel {\overset {R2}{\rightsquigarrow }} (1\cdot (b\cdot b^{-1}))^{-1}\mathrel {\overset {R13}{\rightsquigarrow }} (1\cdot 1)^{-1}\mathrel {\overset {R1}{\rightsquigarrow }} 1^{-1}\mathrel {\overset {R8}{\rightsquigarrow }} 1}
2546:. The rewrite rules are numbered incontiguous since some rules became redundant and were deleted during the algorithm run. The equality of two terms follows from the axioms if and only if both terms are transformed into literally the same normal form term. For example, the terms
4802:
Mostowski, Andrzej (September 1951). "A. Markov. Névožmoinost' nékotoryh algoritmov v téorii associativnyh sistém (Impossibility of certain algorithms in the theory of associative systems). Doklady Akadémii Nauk SSSR, vol. 77 (1951), pp. 19–20".
1254:, i.e. there is no general algorithm for solving this problem. This even holds if we limit the systems to have finite presentations, i.e. a finite set of symbols and a finite set of relations on those symbols. Even the word problem restricted to
4728:
259:
178:
996:
gives another proof that the word problem for groups is unsolvable, based on Turing's cancellation semigroups result and some of
Britton's earlier work. An early version of Britton's Lemma appears.
1309:
670:
591:
4200:
1429:
3673:
1564:
1519:
767:
poses a general problem of term rewriting on tree-like structures. He states "A solution of this problem in the most general case may perhaps be connected with unsurmountable difficulties".
1466:
4075:
3729:
938:
gives the first published proof that the word problem for groups is unsolvable, using Turing's cancellation semigroup result. The proof contains a "Principal Lemma" equivalent to
316:
4099:
4019:
3951:
3873:
3815:
3753:
3585:
3517:
3452:
1185:
3631:
3410:
3176:
3138:
4878:
4150:
1146:
727:
3995:
3927:
3561:
3339:
692:
616:
436:
396:
356:
537:
489:
3849:
3493:
3271:
3228:
3202:
2515:) denote the same value in every bounded lattice. The case of lattices that are not bounded is treated similarly, omitting rules 2 and 3 in the above construction of ≤
1071:
Britton presents a greatly simplified version of Boone's 1959 proof that the word problem for groups is unsolvable. It uses a group-theoretic approach, in particular
3791:
101:
3363:
3295:
3100:
1245:
1225:
1205:
5351:
3230:, respectively. Since the normal forms are literally different, the original terms cannot be equal in every group. In fact, they are usually different in
2491:; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words
1529:
solution for the word problem in the specific case where every object reduces to a unique normal form in a finite number of steps (i.e. the system is
4724:
1047:, connecting recursion theory with group theory in an unexpected way and giving a very different proof of the unsolvability of the word problem.
5444:
5327:
5290:
1247:? Note that the rewriting here is one-way. The word problem is the accessibility problem for symmetric rewrite relations, i.e. Thue systems.
919:
is unsolvable, by furthering Post's construction. The proof is difficult to follow but marks a turning point in the word problem for groups.
4526:
4466:
4317:
1365:, and the equivalence of two Turing machines is undecidable, it follows that the equivalence of two strings of combinators is undecidable.
191:
110:
4732:
5501:
4348:
1567:
1388:
2543:
264:
The most direct solution to a word problem takes the form of a normal form theorem and algorithm which maps every element in an
5636:
5626:
2532:
5605:
Apply rules in any order to a term, as long as possible; the result doesn't depend on the order; it is the term's normal form.
5267:
4287:
447:
4992:
1278:
1044:
966:
621:
542:
20:
4156:
1613:
is difficult. The only known results are that the free
Heyting algebra on one generator is infinite, and that the free
1395:
3637:
993:
1536:
1491:
2040:). This set of words contains many expressions that turn out to denote equal values in every lattice. For example, if
2528:
1376:: given two distinct lambda expressions, there is no algorithm which can discern whether they are equivalent or not;
892:
independently construct finitely presented semigroups with unsolvable word problem. Post's construction is built on
5631:
4665:
4613:
5621:
4218:
2536:
1477:
27:
1614:
1442:
1380:. For several typed variants of the lambda calculus, equivalence is decidable by comparison of normal forms.
4032:
3686:
1598:
1267:
1107:
787:
43:
4434:
2088:
916:
492:
442:
279:
19:
This article is about algorithmic word problems in mathematics and computer science. For other uses, see
4531:
4471:
4081:
4001:
3933:
3855:
3797:
3735:
3567:
3499:
3416:
2394:
1571:
1151:
746:
4425:
Steinby, Magnus; Thomas, Wolfgang (2000). "Trees and term rewriting in 1910: on a paper by Axel Thue".
3598:
3377:
3143:
3105:
2060:. The word problem for free bounded lattices is the problem of determining which of these elements of
5109:
5011:
4113:
2360:
866:
695:
1113:
818:
of genus greater than or equal to 2. Subsequent authors have greatly extended it to a wide range of
700:
4439:
3964:
3886:
3530:
3308:
1587:
1526:
1522:
1340:
1251:
969:
independently shows the word problem for groups is unsolvable, using Post's semigroup construction.
807:
742:
51:
4404:
Müller-Stach, Stefan (12 September 2021). "Max Dehn, Axel Thue, and the
Undecidable". p. 13.
675:
599:
401:
361:
321:
5572:
5537:
5160:
5125:
5082:
4855:
4820:
4777:
4769:
4704:
4685:
4643:
4564:
4504:
4405:
1273:
1075:. This proof has been used in a graduate course, although more modern and condensed proofs exist.
502:
454:
273:
1072:
939:
5517:
5497:
5440:
5381:
5360:
5323:
5286:
5039:
4746:
4548:
4488:
4344:
4283:
4213:
3828:
3472:
3250:
3231:
3207:
3181:
2412:
2021:
1583:
1358:
1352:
889:
811:
265:
5317:
5564:
5529:
5473:
5432:
5278:
5246:
5212:
5152:
5117:
5074:
5047:
5029:
5019:
4974:
4945:
4914:
4887:
4847:
4812:
4761:
4677:
4633:
4625:
4591:
4540:
4480:
4384:
4336:
4328:
4258:
3766:
2017:
63:
35:
5179:
4560:
4500:
4448:
74:
5051:
4891:
4556:
4496:
4444:
2013:
1610:
1377:
1373:
1361:: when are two strings of combinators equivalent? Because combinators encode all possible
819:
2028:
that can be formulated using these operations on elements from a given set of generators
672:
is a unification problem in the same group; since the former terms happen to be equal in
5113:
5102:
Proceedings of the Royal
Society of London. Series A. Mathematical and Physical Sciences
5015:
3345:
3277:
3085:
2539:
2102:
1591:
1362:
1315:, the word problem is the algorithmic problem of deciding, given as input two words in
1230:
1210:
1190:
893:
269:
5180:"A Slick Proof of the Unsolvability of the Word Problem for Finitely Presented Groups"
5034:
4950:
4933:
4918:
5615:
5401:
5129:
4873:
4838:
Turing, A. M. (September 1950). "The Word
Problem in Semi-Groups With Cancellation".
4689:
4568:
4508:
1366:
1328:
1110:(semi-Thue systems or semigroups) can be stated as follows: Given a semi-Thue system
1036:
935:
5216:
4905:
Boone, William W. (1954). "Certain Simple, Unsolvable
Problems of Group Theory. I".
4781:
4647:
4389:
4372:
4263:
4246:
5372:
Matiyasevich, Yu. V. (1967). "Simple examples of undecidable associative calculi".
4609:
2397:
2025:
2009:
50:
of computational theory is that answering this question is in many important cases
1617:
on one generator exists (and has one more element than the free
Heyting algebra).
5282:
4996:
4729:
On
Formally Undecidable Propositions of Principia Mathematica and Related Systems
4332:
2016:
has a decidable solution. Bounded lattices are algebraic structures with the two
1606:
problems for groups and semigroups can be phrased as word problems for algebras.
5404:(1955). "On the algorithmic unsolvability of the word problem in group theory".
4876:(1955). "On the algorithmic unsolvability of the word problem in group theory".
4661:
1323:. The word problem is one of three algorithmic problems for groups proposed by
1255:
1092:
912:
104:
47:
5478:
5461:
5346:
5251:
5234:
4978:
3102:; therefore both terms are equal in every group. As another example, the term
5385:
5364:
4552:
4492:
2523:
Example: A term rewriting system to decide the word problem in the free group
737:
One of the most deeply studied cases of the word problem is in the theory of
5024:
885:
838:
764:
738:
39:
5121:
5043:
4595:
4582:
Greendlinger, Martin (June 1959). "Dehn's algorithm for the word problem".
4638:
4522:
4462:
2344:
1324:
1040:
815:
783:
5436:
5576:
5541:
5164:
5086:
4859:
4824:
4773:
4681:
4629:
4544:
4484:
4340:
594:
5100:
Higman, G. (8 August 1961). "Subgroups of finitely presented groups".
4703:
Power, James F. (27 August 2013). "Thue's 1914 paper: a translation".
1357:
One of the earliest proofs that a word problem is undecidable was for
272:- the word problem is then solved by comparing these normal forms via
107:- then a relevant solution to the word problem would, given the input
38:
whether two given expressions are equivalent with respect to a set of
4427:
Bulletin of the
European Association for Theoretical Computer Science
1091:
Gennady
Makanin proves that the existential theory of equations over
869:
emerges, defining formal notions of computability and undecidability.
5568:
5533:
5156:
5078:
4851:
4816:
4765:
4410:
254:{\displaystyle (x\cdot y)/z\mathrel {\overset {?}{=}} (x/x)\cdot y}
173:{\displaystyle (x\cdot y)/z\mathrel {\overset {?}{=}} (x/z)\cdot y}
4709:
1387:
5462:"Categorification, term rewriting and the Knuth–Bendix procedure"
1353:
Combinatory logic § Undecidability of combinatorial calculus
4280:
Computer algebra and symbolic computation: elementary algorithms
2469:) denote the same value in every bounded lattice if and only if
445:
are equal, a proper extension of the word problem known as the
5349:[Simple examples of undecidable associative calculi].
4965:
Britton, J. L. (October 1958). "The Word Problem for Groups".
1570:
can be used to transform a set of equations into a convergent
1347:
The word problem in combinatorial calculus and lambda calculus
5423:
Statman, Rick (2000). "On the Word Problem for Combinators".
1372:
Likewise, one has essentially the same problem in (untyped)
1293:
1258:
is not decidable for certain finitely presented semigroups.
5235:"Decision problems for semi-Thue systems with a few rules"
3460:
Term rewrite system obtained from Knuth–Bendix completion
4934:"Certain Simple, Unsolvable Problems of Group Theory. VI"
4373:"The word problem and the isomorphism problem for groups"
2075:
The word problem may be resolved as follows. A relation ≤
1020:
Boone publishes a simplified version of his construction.
841:
poses the word problem for finitely presented semigroups.
441:
While the word problem asks whether two terms containing
16:
Decision problem pertaining to equivalence of expressions
5233:
Matiyasevich, Yuri; Sénizergues, Géraud (January 2005).
5065:
Boone, William W. (September 1959). "The Word Problem".
1566:
if and only if they reduce to the same normal form. The
5347:"Простые примеры неразрешимых ассоциативных исчислений"
499:
that are equal, or in other words whether the equation
2068:) denote the same element in the free bounded lattice
5143:
Britton, John L. (January 1963). "The Word Problem".
4159:
4116:
4084:
4035:
4004:
3967:
3936:
3889:
3858:
3831:
3800:
3769:
3738:
3689:
3640:
3601:
3570:
3533:
3502:
3475:
3419:
3380:
3348:
3311:
3280:
3253:
3210:
3184:
3146:
3108:
3088:
2791:
2555:
1539:
1494:
1445:
1398:
1331:
in 1955 that there exists a finitely presented group
1281:
1233:
1213:
1193:
1154:
1116:
703:
678:
624:
602:
545:
505:
457:
404:
364:
324:
282:
194:
113:
77:
4527:"Transformation der Kurven auf zweiseitigen Flächen"
2531:
on an axiom set for groups. The algorithm yields a
1304:{\displaystyle \langle S\mid {\mathcal {R}}\rangle }
665:{\displaystyle 2+x\mathrel {\overset {?}{=}} 8+(-x)}
586:{\displaystyle 2+3\mathrel {\overset {?}{=}} 8+(-3)}
4879:
Proceedings of the Steklov Institute of Mathematics
4195:{\displaystyle \rightsquigarrow y^{-1}\cdot x^{-1}}
1424:{\displaystyle x{\stackrel {*}{\leftrightarrow }}y}
4194:
4144:
4093:
4069:
4013:
3989:
3945:
3921:
3867:
3843:
3809:
3785:
3747:
3723:
3668:{\displaystyle \rightsquigarrow x\cdot (y\cdot z)}
3667:
3625:
3579:
3555:
3511:
3487:
3446:
3404:
3357:
3333:
3289:
3265:
3222:
3196:
3170:
3132:
3094:
3071:
2775:
1558:
1513:
1460:
1423:
1303:
1239:
1219:
1199:
1179:
1140:
721:
686:
664:
610:
585:
531:
483:
430:
390:
350:
310:
253:
172:
95:
1559:{\displaystyle {\stackrel {*}{\leftrightarrow }}}
1514:{\displaystyle {\stackrel {*}{\leftrightarrow }}}
268:of expressions to a single encoding known as the
5496:, (1982) Cambridge University Press, Cambridge,
810:, and proves it solves the word problem for the
46:, but there are many other instances as well. A
5555:Whitman, Philip M. (1942). "Free Lattices II".
5316:Baader, Franz; Nipkow, Tobias (5 August 1999).
5228:
5226:
5004:Proceedings of the National Academy of Sciences
4666:"On Dehn's algorithm and the conjugacy problem"
1384:The word problem for abstract rewriting systems
5590:K. H. Bläsius and H.-J. Bürckert, ed. (1992).
5322:. Cambridge University Press. pp. 59–60.
4967:Proceedings of the London Mathematical Society
4747:"Recursive Unsolvability of a problem of Thue"
4584:Communications on Pure and Applied Mathematics
4316:Miller, Charles F. (2014). Downey, Rod (ed.).
4377:Bulletin of the American Mathematical Society
4282:. Natick, Mass.: A K Peters. pp. 90–92.
4251:Bulletin of the American Mathematical Society
3238:Group axioms used in Knuth–Bendix completion
1319:, whether they represent the same element of
8:
4467:"Über unendliche diskontinuierliche Gruppen"
1298:
1282:
716:
704:
5203:"Subgroups of finitely presented groups".
2117:(this can be restricted to the case where
896:while Markov's uses Post's normal systems.
42:identities. A prototypical example is the
5477:
5311:
5309:
5250:
5033:
5023:
4949:
4708:
4637:
4438:
4409:
4388:
4262:
4183:
4167:
4158:
4133:
4115:
4083:
4049:
4034:
4003:
3978:
3966:
3935:
3910:
3897:
3888:
3857:
3830:
3799:
3774:
3768:
3737:
3694:
3688:
3639:
3600:
3569:
3538:
3532:
3501:
3474:
3418:
3379:
3347:
3316:
3310:
3279:
3252:
3209:
3183:
3145:
3107:
3087:
3051:
3042:
3017:
2999:
2971:
2950:
2931:
2903:
2882:
2866:
2835:
2817:
2790:
2755:
2746:
2727:
2718:
2687:
2678:
2662:
2625:
2616:
2600:
2566:
2554:
2542:that transforms every term into a unique
1548:
1543:
1541:
1540:
1538:
1503:
1498:
1496:
1495:
1493:
1461:{\displaystyle x\downarrow =y\downarrow }
1444:
1410:
1405:
1403:
1402:
1397:
1292:
1291:
1280:
1232:
1212:
1192:
1171:
1153:
1115:
702:
680:
679:
677:
634:
623:
604:
603:
601:
555:
544:
523:
510:
504:
475:
462:
456:
411:
403:
371:
363:
340:
323:
299:
281:
234:
218:
210:
193:
153:
137:
129:
112:
76:
5275:Mathematics Today Twelve Informal Essays
4311:
4309:
4307:
4305:
4303:
4301:
4299:
1250:The accessibility and word problems are
539:has any solutions. As a common example,
4938:Indagationes Mathematicae (Proceedings)
4907:Indagationes Mathematicae (Proceedings)
4240:
4238:
4236:
4234:
4230:
1480:(ARS) is quite succinct: given objects
745:. A timeline of papers relevant to the
4070:{\displaystyle x\cdot (x^{-1}\cdot y)}
3724:{\displaystyle x^{-1}\cdot (x\cdot y)}
2527:Bläsius and Bürckert demonstrate the
2072:, and hence in every bounded lattice.
2024:) 0 and 1. The set of all well-formed
1392:Solving the word problem: deciding if
1102:The word problem for semi-Thue systems
5427:. Lecture Notes in Computer Science.
5425:Rewriting Techniques and Applications
1578:The word problem in universal algebra
814:of closed orientable two-dimensional
7:
1533:): two objects are equivalent under
311:{\displaystyle x\cdot y\cdot z^{-1}}
276:. For example one might decide that
5466:Journal of Pure and Applied Algebra
5178:Simpson, Stephen G. (18 May 2005).
4725:History of the Church–Turing thesis
1431:usually requires heuristic search (
5211:(145): 147–236. 13 February 1977.
4733:Systems of Logic Based on Ordinals
4318:"Turing machines to word problems"
4094:{\displaystyle \rightsquigarrow y}
4014:{\displaystyle \rightsquigarrow 1}
3946:{\displaystyle \rightsquigarrow x}
3868:{\displaystyle \rightsquigarrow x}
3810:{\displaystyle \rightsquigarrow 1}
3748:{\displaystyle \rightsquigarrow y}
3580:{\displaystyle \rightsquigarrow 1}
3512:{\displaystyle \rightsquigarrow x}
3447:{\displaystyle =x\cdot (y\cdot z)}
1621:The word problem for free lattices
1180:{\displaystyle u,v\in \Sigma ^{*}}
1168:
1126:
1043:of finitely presented groups with
14:
5520:(January 1941). "Free Lattices".
3626:{\displaystyle (x\cdot y)\cdot z}
3405:{\displaystyle (x\cdot y)\cdot z}
3171:{\displaystyle b\cdot (1\cdot a)}
3133:{\displaystyle 1\cdot (a\cdot b)}
3082:share the same normal form, viz.
1568:Knuth-Bendix completion algorithm
1521:? The word problem for an ARS is
2407:)/~ is the free bounded lattice
1525:in general. However, there is a
5506:(See chapter 1, paragraph 4.11)
5217:10.1070/SM1977v032n02ABEH002376
5205:Mathematics of the USSR-Sbornik
4390:10.1090/S0273-0979-1982-14963-1
4264:10.1090/S0002-9904-1978-14516-9
4145:{\displaystyle (x\cdot y)^{-1}}
2020:∨ and ∧ and the two constants (
1335:such that the word problem for
4160:
4130:
4117:
4085:
4064:
4042:
4005:
3937:
3907:
3890:
3859:
3801:
3739:
3718:
3706:
3662:
3650:
3641:
3614:
3602:
3571:
3503:
3441:
3429:
3393:
3381:
3165:
3153:
3127:
3115:
3053:
3019:
3014:
2992:
2973:
2968:
2965:
2943:
2924:
2905:
2900:
2891:
2859:
2856:
2837:
2832:
2814:
2801:
2798:
2757:
2729:
2715:
2702:
2689:
2675:
2671:
2649:
2640:
2627:
2613:
2609:
2587:
2581:
2559:
2556:
2423:)/~ are the sets of all words
1544:
1499:
1455:
1449:
1406:
1141:{\displaystyle T:=(\Sigma ,R)}
1135:
1123:
1106:The accessibility problem for
722:{\displaystyle \{x\mapsto 3\}}
710:
659:
650:
580:
571:
419:
405:
379:
365:
337:
325:
242:
228:
207:
195:
161:
147:
126:
114:
1:
5345:Matiyasevich, Yu. V. (1967).
4951:10.1016/S1385-7258(57)50030-9
4919:10.1016/S1385-7258(54)50033-8
3990:{\displaystyle x\cdot x^{-1}}
3922:{\displaystyle (x^{-1})^{-1}}
3556:{\displaystyle x^{-1}\cdot x}
3334:{\displaystyle x^{-1}\cdot x}
2393:. One may then show that the
694:, the latter problem has the
5283:10.1007/978-1-4613-9435-8_10
5239:Theoretical Computer Science
4745:Post, Emil L. (March 1947).
4333:10.1017/CBO9781107338579.010
2105:one of the following holds:
687:{\displaystyle \mathbb {Z} }
611:{\displaystyle \mathbb {Z} }
431:{\displaystyle (y/z)\cdot x}
391:{\displaystyle (x/z)\cdot y}
351:{\displaystyle (x\cdot y)/z}
71:. For example, imagine that
69:solution to the word problem
5319:Term Rewriting and All That
1987:
1951:
1911:
1882:
1850:
1819:
1778:
1735:
1703:
1662:
1262:The word problem for groups
915:shows the word problem for
786:poses the word problem for
532:{\displaystyle t_{1}=t_{2}}
484:{\displaystyle t_{1},t_{2}}
5653:
5594:. Oldenbourg. p. 291.
5479:10.1016/j.jpaa.2010.06.019
5352:Doklady Akademii Nauk SSSR
4932:Boone, William W. (1957).
1488:are they equivalent under
1378:equivalence is undecidable
1350:
1327:in 1911. It was shown by
1265:
1045:Higman's embedding theorem
18:
5522:The Annals of Mathematics
5252:10.1016/j.tcs.2004.09.016
5145:The Annals of Mathematics
5067:The Annals of Mathematics
4840:The Annals of Mathematics
4805:Journal of Symbolic Logic
4754:Journal of Symbolic Logic
4727:. The dates are based on
4219:Group isomorphism problem
2052: ∨ 1 = 1 and
1609:The word problem on free
1478:abstract rewriting system
788:finitely presented groups
593:is a word problem in the
103:are symbols representing
58:Background and motivation
28:computational mathematics
5460:Beke, Tibor (May 2011).
5406:Trudy Mat. Inst. Steklov
5268:"What is a Computation?"
4371:Stillwell, John (1982).
3844:{\displaystyle x\cdot 1}
3488:{\displaystyle 1\cdot x}
3266:{\displaystyle 1\cdot x}
3223:{\displaystyle b\cdot a}
3197:{\displaystyle a\cdot b}
2453:. Two well-formed words
2012:and more generally free
1615:complete Heyting algebra
1476:The word problem for an
1148:and two words (strings)
1108:string rewriting systems
184:, and similarly produce
5025:10.1073/pnas.44.10.1061
4979:10.1112/plms/s3-8.4.493
4278:Cohen, Joel S. (2002).
1626:Example computation of
1369:observed this in 1936.
1268:Word problem for groups
1227:by applying rules from
917:cancellation semigroups
451:asks whether two terms
44:word problem for groups
5637:Computational problems
5627:Combinatorics on words
5266:Davis, Martin (1978).
5122:10.1098/rspa.1961.0132
4596:10.1002/cpa.3160130108
4245:Evans, Trevor (1978).
4196:
4146:
4095:
4071:
4015:
3991:
3947:
3923:
3869:
3845:
3811:
3787:
3786:{\displaystyle 1^{-1}}
3749:
3725:
3669:
3627:
3581:
3557:
3513:
3489:
3448:
3406:
3359:
3335:
3291:
3267:
3224:
3198:
3172:
3134:
3096:
3073:
2777:
2529:Knuth–Bendix algorithm
1560:
1515:
1473:
1462:
1425:
1305:
1241:
1221:
1201:
1181:
1142:
723:
688:
666:
612:
587:
533:
485:
432:
392:
352:
318:is the normal form of
312:
255:
174:
97:
67:algorithm is called a
5557:Annals of Mathematics
4670:Mathematische Annalen
4618:Mathematische Annalen
4614:"On Dehn's algorithm"
4532:Mathematische Annalen
4472:Mathematische Annalen
4197:
4147:
4096:
4072:
4016:
3992:
3948:
3924:
3870:
3846:
3812:
3788:
3750:
3726:
3670:
3628:
3582:
3558:
3514:
3490:
3449:
3407:
3360:
3336:
3292:
3268:
3225:
3199:
3173:
3135:
3097:
3074:
2778:
1572:term rewriting system
1561:
1516:
1463:
1426:
1391:
1306:
1242:
1222:
1202:
1182:
1143:
747:Novikov-Boone theorem
724:
689:
667:
613:
588:
534:
486:
433:
393:
353:
313:
256:
180:, produce the output
175:
98:
96:{\displaystyle x,y,z}
5492:Peter T. Johnstone,
4157:
4114:
4082:
4033:
4002:
3965:
3934:
3887:
3856:
3829:
3798:
3767:
3736:
3687:
3638:
3599:
3568:
3531:
3500:
3473:
3417:
3378:
3346:
3309:
3278:
3251:
3208:
3182:
3178:has the normal form
3144:
3106:
3086:
2789:
2553:
2361:equivalence relation
2008:The word problem on
1588:algebraic structures
1537:
1492:
1468:is straightforward (
1443:
1396:
1279:
1231:
1211:
1207:be transformed into
1191:
1152:
1114:
867:Church-Turing thesis
701:
676:
622:
600:
543:
503:
455:
402:
362:
322:
280:
192:
111:
75:
5437:10.1007/10721975_14
5114:1961RSPSA.262..455H
5016:1958PNAS...44.1061B
3461:
3239:
2540:term rewrite system
2413:equivalence classes
2056: ∧ 1 =
2044:is some element of
1651:
448:unification problem
36:problem of deciding
5596:; here: p.126, 134
5592:Deduktionsssysteme
5518:Whitman, Philip M.
5374:Soviet Mathematics
4997:"The word problem"
4682:10.1007/BF01350654
4630:10.1007/BF01361168
4612:(September 1966).
4545:10.1007/BF01456725
4485:10.1007/BF01456932
4192:
4142:
4091:
4067:
4011:
3987:
3943:
3919:
3865:
3841:
3807:
3783:
3745:
3721:
3665:
3623:
3577:
3553:
3509:
3485:
3459:
3444:
3402:
3358:{\displaystyle =1}
3355:
3331:
3290:{\displaystyle =x}
3287:
3263:
3237:
3232:non-abelian groups
3220:
3194:
3168:
3130:
3092:
3069:
2773:
2363:can be defined by
2022:nullary operations
1625:
1597:, a collection of
1556:
1511:
1474:
1458:
1439:), while deciding
1421:
1301:
1237:
1217:
1197:
1177:
1138:
1039:characterises the
822:decision problems.
812:fundamental groups
719:
684:
662:
608:
583:
529:
481:
428:
388:
348:
308:
274:syntactic equality
251:
170:
93:
5632:Rewriting systems
5446:978-3-540-67778-9
5329:978-0-521-77920-3
5292:978-1-4613-9437-2
5108:(1311): 455–475.
5010:(10): 1061–1065.
4993:Boone, William W.
4214:Conjugacy problem
4205:
4204:
3457:
3456:
3095:{\displaystyle 1}
3064:
3030:
2984:
2916:
2848:
2768:
2740:
2700:
2638:
2395:partially ordered
2087:) may be defined
2018:binary operations
2006:
2005:
2002:
2001:
1768:
1767:
1584:universal algebra
1553:
1508:
1415:
1359:combinatory logic
1240:{\displaystyle R}
1220:{\displaystyle v}
1200:{\displaystyle u}
890:Andrey Markov Jr.
642:
563:
266:equivalence class
226:
145:
5644:
5622:Abstract algebra
5606:
5603:
5597:
5595:
5587:
5581:
5580:
5552:
5546:
5545:
5514:
5508:
5490:
5484:
5483:
5481:
5457:
5451:
5450:
5420:
5414:
5413:
5398:
5392:
5389:
5368:
5359:(6): 1264–1266.
5340:
5334:
5333:
5313:
5304:
5303:
5301:
5299:
5272:
5263:
5257:
5256:
5254:
5230:
5221:
5220:
5200:
5194:
5193:
5191:
5189:
5184:
5175:
5169:
5168:
5140:
5134:
5133:
5097:
5091:
5090:
5062:
5056:
5055:
5037:
5027:
5001:
4989:
4983:
4982:
4962:
4956:
4955:
4953:
4929:
4923:
4922:
4902:
4896:
4895:
4870:
4864:
4863:
4835:
4829:
4828:
4799:
4793:
4792:
4790:
4788:
4751:
4742:
4736:
4721:
4715:
4714:
4712:
4700:
4694:
4693:
4658:
4652:
4651:
4641:
4610:Lyndon, Roger C.
4606:
4600:
4599:
4579:
4573:
4572:
4519:
4513:
4512:
4459:
4453:
4452:
4442:
4422:
4416:
4415:
4413:
4401:
4395:
4394:
4392:
4368:
4362:
4361:
4359:
4357:
4322:
4313:
4294:
4293:
4275:
4269:
4268:
4266:
4242:
4201:
4199:
4198:
4193:
4191:
4190:
4175:
4174:
4151:
4149:
4148:
4143:
4141:
4140:
4100:
4098:
4097:
4092:
4076:
4074:
4073:
4068:
4057:
4056:
4020:
4018:
4017:
4012:
3996:
3994:
3993:
3988:
3986:
3985:
3952:
3950:
3949:
3944:
3928:
3926:
3925:
3920:
3918:
3917:
3905:
3904:
3874:
3872:
3871:
3866:
3850:
3848:
3847:
3842:
3816:
3814:
3813:
3808:
3792:
3790:
3789:
3784:
3782:
3781:
3754:
3752:
3751:
3746:
3730:
3728:
3727:
3722:
3702:
3701:
3674:
3672:
3671:
3666:
3632:
3630:
3629:
3624:
3586:
3584:
3583:
3578:
3562:
3560:
3559:
3554:
3546:
3545:
3518:
3516:
3515:
3510:
3494:
3492:
3491:
3486:
3462:
3458:
3453:
3451:
3450:
3445:
3411:
3409:
3408:
3403:
3364:
3362:
3361:
3356:
3340:
3338:
3337:
3332:
3324:
3323:
3296:
3294:
3293:
3288:
3272:
3270:
3269:
3264:
3240:
3236:
3229:
3227:
3226:
3221:
3203:
3201:
3200:
3195:
3177:
3175:
3174:
3169:
3139:
3137:
3136:
3131:
3101:
3099:
3098:
3093:
3078:
3076:
3075:
3070:
3065:
3063:
3052:
3050:
3049:
3031:
3029:
3018:
3007:
3006:
2985:
2983:
2972:
2958:
2957:
2939:
2938:
2917:
2915:
2904:
2890:
2889:
2874:
2873:
2849:
2847:
2836:
2825:
2824:
2782:
2780:
2779:
2774:
2769:
2767:
2756:
2754:
2753:
2741:
2739:
2728:
2726:
2725:
2701:
2699:
2688:
2686:
2685:
2670:
2669:
2639:
2637:
2626:
2624:
2623:
2608:
2607:
2574:
2573:
2125:are elements of
2014:bounded lattices
1772:
1771:
1656:
1655:
1652:
1624:
1611:Heyting algebras
1590:consisting of a
1565:
1563:
1562:
1557:
1555:
1554:
1552:
1547:
1542:
1520:
1518:
1517:
1512:
1510:
1509:
1507:
1502:
1497:
1471:
1467:
1465:
1464:
1459:
1438:
1434:
1430:
1428:
1427:
1422:
1417:
1416:
1414:
1409:
1404:
1310:
1308:
1307:
1302:
1297:
1296:
1246:
1244:
1243:
1238:
1226:
1224:
1223:
1218:
1206:
1204:
1203:
1198:
1186:
1184:
1183:
1178:
1176:
1175:
1147:
1145:
1144:
1139:
1097:
1096:
1088:
1086:
1077:
1076:
1068:
1066:
1061: – 1963
1060:
1058:
1049:
1048:
1033:
1031:
1022:
1021:
1017:
1015:
1010: – 1959
1009:
1007:
998:
997:
990:
988:
983: – 1958
982:
980:
971:
970:
963:
961:
956: – 1957
955:
953:
944:
943:
932:
930:
921:
920:
909:
907:
898:
897:
882:
880:
871:
870:
862:
860:
855: – 1938
854:
852:
843:
842:
835:
833:
824:
823:
808:Dehn's algorithm
803:
801:
792:
791:
780:
778:
769:
768:
761:
759:
728:
726:
725:
720:
693:
691:
690:
685:
683:
671:
669:
668:
663:
643:
635:
617:
615:
614:
609:
607:
592:
590:
589:
584:
564:
556:
538:
536:
535:
530:
528:
527:
515:
514:
490:
488:
487:
482:
480:
479:
467:
466:
437:
435:
434:
429:
415:
397:
395:
394:
389:
375:
357:
355:
354:
349:
344:
317:
315:
314:
309:
307:
306:
260:
258:
257:
252:
238:
227:
219:
214:
187:
183:
179:
177:
176:
171:
157:
146:
138:
133:
102:
100:
99:
94:
64:computer algebra
5652:
5651:
5647:
5646:
5645:
5643:
5642:
5641:
5612:
5611:
5610:
5609:
5604:
5600:
5589:
5588:
5584:
5569:10.2307/1968883
5554:
5553:
5549:
5534:10.2307/1969001
5516:
5515:
5511:
5491:
5487:
5459:
5458:
5454:
5447:
5422:
5421:
5417:
5400:
5399:
5395:
5371:
5344:
5341:
5337:
5330:
5315:
5314:
5307:
5297:
5295:
5293:
5270:
5265:
5264:
5260:
5232:
5231:
5224:
5202:
5201:
5197:
5187:
5185:
5182:
5177:
5176:
5172:
5157:10.2307/1970200
5142:
5141:
5137:
5099:
5098:
5094:
5079:10.2307/1970103
5064:
5063:
5059:
4999:
4991:
4990:
4986:
4964:
4963:
4959:
4931:
4930:
4926:
4904:
4903:
4899:
4872:
4871:
4867:
4852:10.2307/1969481
4837:
4836:
4832:
4817:10.2307/2266407
4801:
4800:
4796:
4786:
4784:
4766:10.2307/2267170
4749:
4744:
4743:
4739:
4722:
4718:
4702:
4701:
4697:
4662:Schupp, Paul E.
4660:
4659:
4655:
4608:
4607:
4603:
4581:
4580:
4576:
4521:
4520:
4516:
4461:
4460:
4456:
4424:
4423:
4419:
4403:
4402:
4398:
4370:
4369:
4365:
4355:
4353:
4351:
4325:Turing's Legacy
4320:
4315:
4314:
4297:
4290:
4277:
4276:
4272:
4247:"Word problems"
4244:
4243:
4232:
4227:
4210:
4179:
4163:
4155:
4154:
4129:
4112:
4111:
4080:
4079:
4045:
4031:
4030:
4000:
3999:
3974:
3963:
3962:
3932:
3931:
3906:
3893:
3885:
3884:
3854:
3853:
3827:
3826:
3796:
3795:
3770:
3765:
3764:
3734:
3733:
3690:
3685:
3684:
3636:
3635:
3597:
3596:
3566:
3565:
3534:
3529:
3528:
3498:
3497:
3471:
3470:
3415:
3414:
3376:
3375:
3344:
3343:
3312:
3307:
3306:
3276:
3275:
3249:
3248:
3206:
3205:
3180:
3179:
3142:
3141:
3104:
3103:
3084:
3083:
3056:
3038:
3022:
2995:
2976:
2946:
2927:
2908:
2878:
2862:
2840:
2813:
2787:
2786:
2760:
2742:
2732:
2714:
2692:
2674:
2658:
2630:
2612:
2596:
2562:
2551:
2550:
2525:
2518:
2487:
2476:
2449:
2438:
2389:
2378:
2350:
2343:This defines a
2338:
2332:
2324:
2318:
2310:
2303:
2289:
2283:
2275:
2269:
2261:
2254:
2237:
2233:
2223:
2219:
2212:
2205:
2188:
2184:
2174:
2170:
2163:
2156:
2098:
2078:
2032:will be called
1960:
1924:
1863:
1832:
1791:
1716:
1684:
1623:
1580:
1535:
1534:
1490:
1489:
1469:
1441:
1440:
1436:
1432:
1394:
1393:
1386:
1374:lambda calculus
1363:Turing machines
1355:
1349:
1277:
1276:
1270:
1264:
1229:
1228:
1209:
1208:
1189:
1188:
1167:
1150:
1149:
1112:
1111:
1104:
1090:
1084:
1082:
1080:
1073:Britton's Lemma
1070:
1064:
1062:
1056:
1054:
1052:
1035:
1029:
1027:
1025:
1019:
1013:
1011:
1005:
1003:
1001:
992:
986:
984:
978:
976:
974:
965:
959:
957:
951:
949:
947:
940:Britton's Lemma
934:
928:
926:
924:
911:
905:
903:
901:
894:Turing machines
884:
878:
876:
874:
864:
858:
856:
850:
848:
846:
837:
831:
829:
827:
820:group-theoretic
805:
799:
797:
795:
782:
776:
774:
772:
763:
757:
755:
753:
749:is as follows:
735:
729:as a solution.
699:
698:
674:
673:
620:
619:
598:
597:
541:
540:
519:
506:
501:
500:
471:
458:
453:
452:
400:
399:
360:
359:
320:
319:
295:
278:
277:
190:
189:
185:
181:
109:
108:
73:
72:
60:
24:
17:
12:
11:
5:
5650:
5648:
5640:
5639:
5634:
5629:
5624:
5614:
5613:
5608:
5607:
5598:
5582:
5563:(1): 104–115.
5547:
5528:(1): 325–329.
5509:
5485:
5452:
5445:
5415:
5408:(in Russian).
5402:Novikov, P. S.
5393:
5391:
5390:
5380:(2): 555–557.
5369:
5355:(in Russian).
5335:
5328:
5305:
5291:
5258:
5245:(1): 145–169.
5222:
5195:
5170:
5135:
5092:
5073:(2): 207–265.
5057:
4984:
4973:(4): 493–506.
4957:
4924:
4897:
4882:(in Russian).
4874:Novikov, P. S.
4865:
4846:(2): 491–505.
4830:
4794:
4737:
4716:
4695:
4676:(2): 119–130.
4653:
4624:(3): 208–228.
4601:
4574:
4539:(3): 413–421.
4514:
4479:(1): 116–144.
4454:
4440:10.1.1.32.8993
4417:
4396:
4363:
4349:
4295:
4288:
4270:
4229:
4228:
4226:
4223:
4222:
4221:
4216:
4209:
4206:
4203:
4202:
4189:
4186:
4182:
4178:
4173:
4170:
4166:
4162:
4152:
4139:
4136:
4132:
4128:
4125:
4122:
4119:
4109:
4102:
4101:
4090:
4087:
4077:
4066:
4063:
4060:
4055:
4052:
4048:
4044:
4041:
4038:
4028:
4022:
4021:
4010:
4007:
3997:
3984:
3981:
3977:
3973:
3970:
3960:
3954:
3953:
3942:
3939:
3929:
3916:
3913:
3909:
3903:
3900:
3896:
3892:
3882:
3876:
3875:
3864:
3861:
3851:
3840:
3837:
3834:
3824:
3818:
3817:
3806:
3803:
3793:
3780:
3777:
3773:
3762:
3756:
3755:
3744:
3741:
3731:
3720:
3717:
3714:
3711:
3708:
3705:
3700:
3697:
3693:
3682:
3676:
3675:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3633:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3594:
3588:
3587:
3576:
3573:
3563:
3552:
3549:
3544:
3541:
3537:
3526:
3520:
3519:
3508:
3505:
3495:
3484:
3481:
3478:
3468:
3455:
3454:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3412:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3373:
3366:
3365:
3354:
3351:
3341:
3330:
3327:
3322:
3319:
3315:
3304:
3298:
3297:
3286:
3283:
3273:
3262:
3259:
3256:
3246:
3219:
3216:
3213:
3193:
3190:
3187:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3091:
3080:
3079:
3068:
3062:
3059:
3055:
3048:
3045:
3041:
3037:
3034:
3028:
3025:
3021:
3016:
3013:
3010:
3005:
3002:
2998:
2994:
2991:
2988:
2982:
2979:
2975:
2970:
2967:
2964:
2961:
2956:
2953:
2949:
2945:
2942:
2937:
2934:
2930:
2926:
2923:
2920:
2914:
2911:
2907:
2902:
2899:
2896:
2893:
2888:
2885:
2881:
2877:
2872:
2869:
2865:
2861:
2858:
2855:
2852:
2846:
2843:
2839:
2834:
2831:
2828:
2823:
2820:
2816:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2784:
2772:
2766:
2763:
2759:
2752:
2749:
2745:
2738:
2735:
2731:
2724:
2721:
2717:
2713:
2710:
2707:
2704:
2698:
2695:
2691:
2684:
2681:
2677:
2673:
2668:
2665:
2661:
2657:
2654:
2651:
2648:
2645:
2642:
2636:
2633:
2629:
2622:
2619:
2615:
2611:
2606:
2603:
2599:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2572:
2569:
2565:
2561:
2558:
2524:
2521:
2516:
2485:
2474:
2447:
2436:
2387:
2376:
2348:
2341:
2340:
2336:
2330:
2322:
2316:
2308:
2301:
2291:
2287:
2281:
2273:
2267:
2259:
2252:
2242:
2235:
2231:
2221:
2217:
2210:
2203:
2193:
2186:
2182:
2172:
2168:
2161:
2154:
2144:
2137:
2130:
2103:if and only if
2096:
2076:
2004:
2003:
2000:
1999:
1994:
1991:
1986:
1983:
1980:
1978:
1976:
1974:
1972:
1970:
1967:
1966:
1961:
1958:
1955:
1950:
1947:
1944:
1942:
1940:
1938:
1936:
1934:
1931:
1930:
1925:
1922:
1919:
1910:
1907:
1904:
1902:
1893:
1890:
1881:
1878:
1874:
1873:
1864:
1861:
1858:
1849:
1847:
1845:
1842:
1833:
1830:
1827:
1818:
1815:
1811:
1810:
1792:
1789:
1786:
1777:
1775:
1769:
1766:
1765:
1761:
1760:
1756:
1755:
1746:
1743:
1734:
1731:
1727:
1726:
1717:
1714:
1711:
1702:
1699:
1695:
1694:
1685:
1682:
1679:
1661:
1659:
1622:
1619:
1592:generating set
1579:
1576:
1551:
1546:
1506:
1501:
1457:
1454:
1451:
1448:
1420:
1413:
1408:
1401:
1385:
1382:
1351:Main article:
1348:
1345:
1300:
1295:
1290:
1287:
1284:
1266:Main article:
1263:
1260:
1236:
1216:
1196:
1174:
1170:
1166:
1163:
1160:
1157:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1103:
1100:
1099:
1098:
1078:
1050:
1023:
999:
972:
945:
922:
899:
872:
844:
825:
806:Dehn presents
793:
770:
734:
731:
718:
715:
712:
709:
706:
682:
661:
658:
655:
652:
649:
646:
641:
638:
633:
630:
627:
606:
582:
579:
576:
573:
570:
567:
562:
559:
554:
551:
548:
526:
522:
518:
513:
509:
478:
474:
470:
465:
461:
427:
424:
421:
418:
414:
410:
407:
387:
384:
381:
378:
374:
370:
367:
347:
343:
339:
336:
333:
330:
327:
305:
302:
298:
294:
291:
288:
285:
250:
247:
244:
241:
237:
233:
230:
225:
222:
217:
213:
209:
206:
203:
200:
197:
169:
166:
163:
160:
156:
152:
149:
144:
141:
136:
132:
128:
125:
122:
119:
116:
92:
89:
86:
83:
80:
59:
56:
15:
13:
10:
9:
6:
4:
3:
2:
5649:
5638:
5635:
5633:
5630:
5628:
5625:
5623:
5620:
5619:
5617:
5602:
5599:
5593:
5586:
5583:
5578:
5574:
5570:
5566:
5562:
5558:
5551:
5548:
5543:
5539:
5535:
5531:
5527:
5523:
5519:
5513:
5510:
5507:
5503:
5502:0-521-23893-5
5499:
5495:
5489:
5486:
5480:
5475:
5471:
5467:
5463:
5456:
5453:
5448:
5442:
5438:
5434:
5430:
5426:
5419:
5416:
5411:
5407:
5403:
5397:
5394:
5387:
5383:
5379:
5375:
5370:
5366:
5362:
5358:
5354:
5353:
5348:
5343:
5342:
5339:
5336:
5331:
5325:
5321:
5320:
5312:
5310:
5306:
5294:
5288:
5284:
5280:
5276:
5269:
5262:
5259:
5253:
5248:
5244:
5240:
5236:
5229:
5227:
5223:
5218:
5214:
5210:
5206:
5199:
5196:
5181:
5174:
5171:
5166:
5162:
5158:
5154:
5150:
5146:
5139:
5136:
5131:
5127:
5123:
5119:
5115:
5111:
5107:
5103:
5096:
5093:
5088:
5084:
5080:
5076:
5072:
5068:
5061:
5058:
5053:
5049:
5045:
5041:
5036:
5031:
5026:
5021:
5017:
5013:
5009:
5005:
4998:
4994:
4988:
4985:
4980:
4976:
4972:
4968:
4961:
4958:
4952:
4947:
4943:
4939:
4935:
4928:
4925:
4920:
4916:
4912:
4908:
4901:
4898:
4893:
4889:
4885:
4881:
4880:
4875:
4869:
4866:
4861:
4857:
4853:
4849:
4845:
4841:
4834:
4831:
4826:
4822:
4818:
4814:
4810:
4806:
4798:
4795:
4783:
4779:
4775:
4771:
4767:
4763:
4759:
4755:
4748:
4741:
4738:
4734:
4730:
4726:
4720:
4717:
4711:
4706:
4699:
4696:
4691:
4687:
4683:
4679:
4675:
4671:
4667:
4664:(June 1968).
4663:
4657:
4654:
4649:
4645:
4640:
4639:2027.42/46211
4635:
4631:
4627:
4623:
4619:
4615:
4611:
4605:
4602:
4597:
4593:
4589:
4585:
4578:
4575:
4570:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4538:
4534:
4533:
4528:
4524:
4518:
4515:
4510:
4506:
4502:
4498:
4494:
4490:
4486:
4482:
4478:
4474:
4473:
4468:
4464:
4458:
4455:
4450:
4446:
4441:
4436:
4432:
4428:
4421:
4418:
4412:
4407:
4400:
4397:
4391:
4386:
4382:
4378:
4374:
4367:
4364:
4352:
4350:9781107338579
4346:
4342:
4338:
4334:
4330:
4326:
4319:
4312:
4310:
4308:
4306:
4304:
4302:
4300:
4296:
4291:
4285:
4281:
4274:
4271:
4265:
4260:
4256:
4252:
4248:
4241:
4239:
4237:
4235:
4231:
4224:
4220:
4217:
4215:
4212:
4211:
4207:
4187:
4184:
4180:
4176:
4171:
4168:
4164:
4153:
4137:
4134:
4126:
4123:
4120:
4110:
4108:
4107:
4104:
4103:
4088:
4078:
4061:
4058:
4053:
4050:
4046:
4039:
4036:
4029:
4027:
4024:
4023:
4008:
3998:
3982:
3979:
3975:
3971:
3968:
3961:
3959:
3956:
3955:
3940:
3930:
3914:
3911:
3901:
3898:
3894:
3883:
3881:
3878:
3877:
3862:
3852:
3838:
3835:
3832:
3825:
3823:
3820:
3819:
3804:
3794:
3778:
3775:
3771:
3763:
3761:
3758:
3757:
3742:
3732:
3715:
3712:
3709:
3703:
3698:
3695:
3691:
3683:
3681:
3678:
3677:
3659:
3656:
3653:
3647:
3644:
3634:
3620:
3617:
3611:
3608:
3605:
3595:
3593:
3590:
3589:
3574:
3564:
3550:
3547:
3542:
3539:
3535:
3527:
3525:
3522:
3521:
3506:
3496:
3482:
3479:
3476:
3469:
3467:
3464:
3463:
3438:
3435:
3432:
3426:
3423:
3420:
3413:
3399:
3396:
3390:
3387:
3384:
3374:
3372:
3371:
3368:
3367:
3352:
3349:
3342:
3328:
3325:
3320:
3317:
3313:
3305:
3303:
3300:
3299:
3284:
3281:
3274:
3260:
3257:
3254:
3247:
3245:
3242:
3241:
3235:
3233:
3217:
3214:
3211:
3191:
3188:
3185:
3162:
3159:
3156:
3150:
3147:
3124:
3121:
3118:
3112:
3109:
3089:
3066:
3060:
3057:
3046:
3043:
3039:
3035:
3032:
3026:
3023:
3011:
3008:
3003:
3000:
2996:
2989:
2986:
2980:
2977:
2962:
2959:
2954:
2951:
2947:
2940:
2935:
2932:
2928:
2921:
2918:
2912:
2909:
2897:
2894:
2886:
2883:
2879:
2875:
2870:
2867:
2863:
2853:
2850:
2844:
2841:
2829:
2826:
2821:
2818:
2810:
2807:
2804:
2795:
2792:
2785:
2770:
2764:
2761:
2750:
2747:
2743:
2736:
2733:
2722:
2719:
2711:
2708:
2705:
2696:
2693:
2682:
2679:
2666:
2663:
2659:
2655:
2652:
2646:
2643:
2634:
2631:
2620:
2617:
2604:
2601:
2597:
2593:
2590:
2584:
2578:
2575:
2570:
2567:
2563:
2549:
2548:
2547:
2545:
2541:
2538:
2534:
2530:
2522:
2520:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2483:
2479:
2472:
2468:
2464:
2460:
2456:
2452:
2445:
2441:
2434:
2430:
2426:
2422:
2418:
2414:
2410:
2406:
2402:
2399:
2396:
2392:
2385:
2381:
2374:
2370:
2366:
2362:
2358:
2354:
2346:
2335:
2328:
2321:
2314:
2307:
2300:
2296:
2292:
2286:
2279:
2272:
2265:
2258:
2251:
2247:
2243:
2240:
2230:
2226:
2216:
2209:
2202:
2198:
2194:
2191:
2181:
2177:
2167:
2160:
2153:
2149:
2145:
2142:
2138:
2135:
2131:
2128:
2124:
2120:
2116:
2112:
2108:
2107:
2106:
2104:
2101:
2094:
2090:
2086:
2082:
2073:
2071:
2067:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2019:
2015:
2011:
2010:free lattices
1998:
1995:
1992:
1990:
1984:
1981:
1979:
1977:
1975:
1973:
1971:
1969:
1968:
1965:
1962:
1956:
1954:
1948:
1945:
1943:
1941:
1939:
1937:
1935:
1933:
1932:
1929:
1926:
1920:
1918:
1914:
1908:
1905:
1903:
1901:
1897:
1894:
1891:
1889:
1885:
1879:
1876:
1875:
1872:
1868:
1865:
1859:
1857:
1853:
1848:
1846:
1843:
1841:
1837:
1834:
1828:
1826:
1822:
1816:
1813:
1812:
1808:
1804:
1800:
1796:
1793:
1787:
1785:
1781:
1776:
1774:
1773:
1770:
1763:
1762:
1758:
1757:
1754:
1750:
1747:
1744:
1742:
1738:
1732:
1729:
1728:
1725:
1721:
1718:
1712:
1710:
1706:
1700:
1697:
1696:
1693:
1689:
1686:
1680:
1677:
1673:
1669:
1665:
1660:
1658:
1657:
1654:
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1620:
1618:
1616:
1612:
1607:
1604:
1600:
1596:
1593:
1589:
1585:
1577:
1575:
1573:
1569:
1549:
1532:
1528:
1524:
1504:
1487:
1483:
1479:
1452:
1446:
1418:
1411:
1399:
1390:
1383:
1381:
1379:
1375:
1370:
1368:
1367:Alonzo Church
1364:
1360:
1354:
1346:
1344:
1342:
1338:
1334:
1330:
1329:Pyotr Novikov
1326:
1322:
1318:
1314:
1288:
1285:
1275:
1269:
1261:
1259:
1257:
1253:
1248:
1234:
1214:
1194:
1172:
1164:
1161:
1158:
1155:
1132:
1129:
1120:
1117:
1109:
1101:
1094:
1079:
1074:
1051:
1046:
1042:
1038:
1037:Graham Higman
1024:
1000:
995:
973:
968:
967:William Boone
946:
941:
937:
936:Pyotr Novikov
923:
918:
914:
900:
895:
891:
887:
873:
868:
845:
840:
826:
821:
817:
813:
809:
794:
789:
785:
771:
766:
752:
751:
750:
748:
744:
740:
732:
730:
713:
707:
697:
656:
653:
647:
644:
639:
636:
631:
628:
625:
596:
595:integer group
577:
574:
568:
565:
560:
557:
552:
549:
546:
524:
520:
516:
511:
507:
498:
494:
476:
472:
468:
463:
459:
450:
449:
444:
439:
425:
422:
416:
412:
408:
385:
382:
376:
372:
368:
345:
341:
334:
331:
328:
303:
300:
296:
292:
289:
286:
283:
275:
271:
267:
262:
248:
245:
239:
235:
231:
223:
220:
215:
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4757:
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4324:
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2398:quotient set
2390:
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1274:presentation
1271:
1256:ground terms
1249:
1105:
1095:is solvable.
1093:free monoids
994:John Britton
736:
696:substitution
496:
446:
440:
263:
105:real numbers
68:
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32:word problem
31:
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21:Word problem
5431:: 203–213.
5277:: 257–259.
4944:: 227–232.
4913:: 231–237.
4760:(1): 1–11.
4433:: 256–269.
4341:11343/51723
2544:normal form
2262:and either
2213:and either
2091:by setting
2089:inductively
2026:expressions
1523:undecidable
1341:undecidable
1252:undecidable
913:Alan Turing
491:containing
270:normal form
52:undecidable
48:deep result
5616:Categories
5472:(5): 730.
5298:5 December
5188:6 December
5052:0086.24701
4892:0068.01301
4811:(3): 215.
4787:6 December
4411:1703.09750
4356:6 December
4289:1568811586
4257:(5): 790.
4225:References
2537:noetherian
1599:operations
1531:convergent
1527:computable
1450:↓ =
739:semigroups
5386:0197-6788
5365:0869-5652
5130:120100270
4886:: 1–143.
4710:1308.5858
4690:120429853
4569:122988176
4553:0025-5831
4523:Dehn, Max
4509:123478582
4493:0025-5831
4463:Dehn, Max
4435:CiteSeerX
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2164:and both
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1500:↔
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1127:Σ
1041:subgroups
886:Emil Post
839:Axel Thue
816:manifolds
765:Axel Thue
711:↦
654:−
575:−
497:instances
493:variables
443:constants
423:⋅
383:⋅
332:⋅
301:−
293:⋅
287:⋅
246:⋅
202:⋅
186:NOT_EQUAL
165:⋅
121:⋅
40:rewriting
5412:: 1–143.
5044:16590307
4995:(1958).
4782:30320278
4648:36469569
4525:(1912).
4465:(1911).
4208:See also
2345:preorder
1325:Max Dehn
1272:Given a
784:Max Dehn
618:, while
5577:1968883
5542:1969001
5165:1970200
5110:Bibcode
5087:1970103
5012:Bibcode
4860:1969481
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4774:2267170
4561:1511705
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4705:arXiv
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2192:hold,
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