The complexity of small universal Turing machines: A survey

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 7147 LNCS, Issue , 2012, Pages 385-405

  Neary, Turlough ^a   Woods, Damien ^b  

^a UNIVERSITY COLLEGE DUBLIN   (Ireland)

^b California Institute of Technology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL DEVICES; MODELS OF COMPUTATION; MULTIPLE DIMENSIONS; POLYNOMIAL-TIME; TAG SYSTEMS; UNIVERSAL PROGRAMS; UNIVERSAL TURING MACHINE;

CELLULAR AUTOMATA; COMPUTER SCIENCE; POLYNOMIAL APPROXIMATION; SURVEYS; TURING MACHINES;

MACHINERY;

EID: 84856035046     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-27660-6_32     Document Type: Conference Paper

References (113)

1. Aanderaa, S., Fischer, P.C.: The solvability of the halting problem for 2-state post machines. Journal of the Association for Computing Machinery 14(4), 677-682 (1967)
2. Aaronson, S.: Book review: A new kind of science. Quantum Information and Computation 2(5), 410-423 (2002)
3. Baiocchi, C.: 3N+1, UTM e tag-system. Technical Report Pubblicazione 98/38, Dipartimento di Matematico, Università di Roma (1998) (in Italian)
4. Baiocchi, C.: Three Small Universal Turing Machines. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 1-10. Springer, Heidelberg (2001)
5. Bennett, C.H.: Logical reversibility of computation. IBM Journal of Research and Development 17(6), 525-532 (1973)
6. Brady, A.H.: The determination of the value of Rado's noncomputable function Σ(k) for four-state Turing machines. Mathematics of Computation 40(163), 647-665 (1983)
7. Brady, A.H.: The busy beaver game and the meaning of life. In: Herken, R. (ed.) The Universal Turing Machine: A Half-Century Survey, pp. 259-277. Oxford University Press (1988)
8. Cocke, J., Minsky, M.: Universality of tag systems with P = 2. Journal of the Association for Computing Machinery 11(1), 15-20 (1964)
9. Cook, M.: Universality in elementary cellular automata. Complex Systems 15(1), 1-40 (2004)
10. Cook, M.: A Concrete View of Rule 110 Computation. Electronic Proceedings in Theoretical Computer Science 1, 31-55 (2009)
11. De Mol, L.: Study of Limits of Solvability in Tag Systems. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 170-181. Springer, Heidelberg (2007)
12. Fischer, P.C.: On formalisms for Turing machines. Journal of the Association for Computing Machinery 12(4), 570-580 (1965)
13. Fischer, P.C., Meyer, A.R., Rosenberg, A.L.: Counter machines and counter languages. Mathematical Systems Theory 2(3), 265-283 (1968)
14. Friedman, J.: A decision procedure for computations of finite automata. Journal of the ACM 9(3), 315-323 (1962)
15. Gajardo, A., Moreira, A., Goles, E.: Complexity of Langton's ant. Discrete Applied Mathematics 117, 41-50 (2002)
16. Green, M.W.: A lower bound on Rado's sigma function for binary Turing machines. In: Proceedings of the 5th IEEE Annual Symposium on Switching Circuit Theory and Logical Design, pp. 91-94 (1964)
17. Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to parallel computation: P-completeness theory. Oxford university Press, Oxford (1995)
18. Harju, T., Margenstern, M.: Splicing Systems for Universal Turing Machines. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 149-158. Springer, Heidelberg (2005) (Pubitemid 41231392)
19. Hermann, G.T.: The uniform halting problem for generalized one state Turing machines. In: Proceedings of the Ninth Annual Symposium on Switching and Automata Theory (FOCS), pp. 368-372. IEEE Computer Society Press, Schenectady (1968)
20. Hermann, G.T.: The halting problem of one state Turing machines with n-dimensional tape. Mathematical Logic Quarterly 14(7-12), 185-191 (1968)
21. Hooper, P.: Some small, multitape universal Turing machines. Technical report, Computation Laboratory, Harvard University, Cambridge, Massachusetts (1963)
22. Hooper, P.: Some small, multitape universal Turing machines. Information Sciences 1(2), 205-215 (1969)
23. Ikeno, N.: A 6-symbol 10-state universal Turing machine. In: Proceedings, Institute of Electrical Communications, Tokyo (1958)
24. Kleine-Büning, H.: Über probleme bei homogener Parkettierung von ℤxℤ durch Mealy-automaten bei normierter verwendung. PhD thesis, Institut für Mathematische Logik, Münster (1977)
25. Kleine-Büning, H., Ottmann, T.: Kleine universelle mehrdimensionale Turingmaschinen. Elektronische Informationsverarbeitung und Kybernetik 13(4-5), 179-201 (1977) (in German)
26. Kudlek, M.: Small deterministic Turing machines. Theoretical Computer Science 168(2), 241-255 (1996)
27. Kudlek, M., Rogozhin, Y.: A Universal Turing Machine with 3 States and 9 Symbols. In: Kuich,W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 311-318. Springer, Heidelberg (2002)
28. Lafitte, G., Papazian, C.: The fabric of small Turing machines. In: Computation and Logic in the Real World, Third Conference on Computability in Europe, CiE 2007, Local Proceedings, pp. 219-227 (2007)
29. Langton, C.: Studying artificial life with cellular automata. Physica D 2(1-3), 120-149 (1986)
30. Lin, S., Rado, T.: Computer studies of Turing machine problems. Journal of the ACM 12(2), 196-212 (1965)
31. Lindgren, K., Nordahl, M.G.: Universal computation in simple one-dimensional cellular automata. Complex Systems 4(3), 299-318 (1990)
32. Margenstern, M.: Surprising areas in the quest for small universal devices. Electronic Notes in Theoretical computer Science 225, 201-220 (2009)
33. Margenstern, M.: Sur la frontière entre machines de Turing á arrêt décidable et machines de Turing universelles. Technical Report 92-83, LITP Institut Blaise Pascal (1992)
34. Margenstern, M.: Non-erasing Turing Machines: A Frontier Between a Decidable Halting Problem and Universality. In: Ésik, Z. (ed.) FCT 1993. LNCS, vol. 710, pp. 375-385. Springer, Heidelberg (1993)
35. Margenstern, M.: Une machine de Turing universelle sur {0,1}, non-effaçante et à trois instructions gauches. Technical Report 94-08, LITP Institut Blaise Pascal (1994)
36. Margenstern, M.: Non-Erasing Turing Machines: A New Frontier Between a Decidable Halting Problem and Universality. In: Baeza-Yates, R.A., Poblete, P.V., Goles, E. (eds.) LATIN 1995. LNCS, vol. 911, pp. 386-397. Springer, Heidelberg (1995)
37. Margenstern, M.: Une machine de Turing universelle non-effaçante à exactement trois instructions gauches. C. R. Acad. Sci. Paris, Série I 320, 101-106 (1995)
38. Margenstern, M.: Decidability and Undecidability of the Halting Problem on Turing Machines, a Survey. In: Adian, S., Nerode, A. (eds.) LFCS 1997. LNCS, vol. 1234, pp. 226-236. Springer, Heidelberg (1997)
39. Margenstern, M.: The laterality problem for non-erasing Turing machines on {0, 1} is completely solved. Theoretical Informatics and Applications 31(2), 159-204 (1997)
40. Margenstern, M.: Frontier between decidability and undecidability: a survey. In: Margenstern, M. (ed.) Machines, Computations and Universality (MCU), France, IUT, Metz., vol. 1, pp. 141-177 (1998)
41. Margenstern, M.: Frontier between decidability and undecidability: a survey. Theoretical Computer Science 231(2), 217-251 (2000)
42. Margenstern, M.: On quasi-unilateral universal Turing machines. Theoretical Computer Science 257(1-2), 153-166 (2001)
43. Margenstern, M.: An algorithm for building intrinsically universal cellular automata in hyperbolic spaces. In: Proceedings of the 2006 International Conference on Foundations of Computer Science (FCS), Las Vegas, NV, pp. 3-9. CSREA Press (2006)
44. Margenstern, M., Pavlotskaya, L.: Deux machines de Turing universelles á au plus deux instructions gauches. C. R. Acad. Sci. Paris, Série I 320, 1395-1400 (1995)
45. Margenstern, M., Pavlotskaya, L.: Vers ue nouvelle approche de l'universalitéconcernant les machines de Turing. Technical Report 95-58, LITP Institut Blaise Pascal (1995)
46. Margenstern, M., Pavlotskaya, L.: On the optimal number of instructions for universality of Turing machines connected with a finite automaton. International Journal of Algebra and Computation 13(2), 133-202 (2003)
47. Marxen, H., Buntrock, J.: Attacking the Busy Beaver 5. Bulletin of the EATCS 40, 247-251 (1990)
48. Michel, P.: Busy beaver competition and Collatz-like problems. Archive Mathematical Logic 32(5), 351-367 (1993)
49. Michel, P.: Small Turing machines and generalized busy beaver competition. Theoretical Computer Science 326, 45-56 (2004)
50. Michel, P.: The busy beaver competition: a historical survey, arXiv:0906.3749v2 (2010), http://www.logique.jussieu.fr/~michel/ha.html
51. Minsky, M.: A 6-symbol 7-state universal Turing machines. Technical Report 54-G-027, MIT (1960)
52. Minsky, M.: Recursive unsolvability of Post's tag problem. Technical Report 54-G-023, Massachusetts Institute of Technology (1960)
53. Minsky, M.: Recursive unsolvability of Post's problem of "tag" and other topics in theory of Turing machines. Annals of Mathematics 74(3), 437-455 (1961)
54. Minsky, M.: Size and structure of universal Turing machines using tag systems. In: Recursive Function Theory: Proceedings, Symposium in Pure Mathematics, Provelence, vol. 5, pp. 229-238 (1962)
55. Minsky, M.: Universality of (p=2) tag systems and a 4 symbol 7 state universal Turing machine. In: AIM-33, A.I. memo 33, Computer Science and Artificial Intelligence Laboratory. MIT, Cambridge (1962)
56. Minsky, M.: Computation, finite and infinite machines. Prentice-Hall, Englewood Cliffs (1967)
57. Moore, C., Mertens, S.: The Nature of Computation. Oxford University Press (2011)
58. Moore, C.: Quasi-linear cellular automata. Physica D 103, 100-132 (1997)
59. Moore, C.: Predicting non-linear cellular automata quickly by decomposing them into linear ones. Physica D 111, 27-41 (1998)
60. Moore, E.F.: A simplified universal Turing machine. In: ACM National Meeting, Toronto, Canada, pp. 50-54. ACM Press (1952)
61. Morita, K., Shirasaki, A., Gono, Y.: A 1-tape 2-symbol reversible Turing machine. The Transactions of the IEICE Japan E72(3), 223-228 (1989)
62. Morita, K., Yamaguchi, Y.: A Universal Reversible Turing Machine. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 90-98. Springer, Heidelberg (2007)
63. Neary, T.: Small universal Turing machines. PhD thesis, National University of Ireland, Maynooth (2008)
64. Neary, T., Woods, D.: P-Completeness of Cellular Automaton Rule 110. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006, Part I. LNCS, vol. 4051, pp. 132-143. Springer, Heidelberg (2006) (Pubitemid 44113295)
65. Neary, T., Woods, D.: Small fast universal Turing machines. Theoretical Computer Science 362(1-3), 171-195 (2006)
66. Neary, T., Woods, D.: Four Small Universal Turing Machines. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 242-254. Springer, Heidelberg (2007)
67. Neary, T., Woods, D.: Small weakly universal Turing machines, arXiv:0707.4489v1 [cs.CC] (2007)
68. Neary, T., Woods, D.: Small Weakly Universal Turing Machines. In: Kutyłowski, M., Charatonik, W., Gȩbala, M. (eds.) FCT 2009. LNCS, vol. 5699, pp. 262-273. Springer, Heidelberg (2009)
69. Neary, T., Woods, D.: Four small universal Turing machines. Fundamenta Informaticae 91(1), 123-144 (2009)
70. Nozaki, A.: On the notion of universality of Turing machine. Kybernetika Academia Praha 5(1), 29-43 (1969) (English translated version)
71. Ollinger, N.: The Quest for Small Universal Cellular Automata. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 318-329. Springer, Heidelberg (2002)
72. Ollinger, N., Richard, G.: Four states are enough! Theoretical Computer Science 412(1-2), 22-32 (2011)
73. Ottmann, T.: Eine universelle Turingmaschine mit zweidimensionalem band. Elektronische Informationsverarbeitung und Kybernetik 11(1-2), 27-38 (1975) (in German)
74. Ottmann, T.: Einfache universelle mehrdimensionale Turingmaschinen. Habilitationsschrift, Karlsruhe (1975)
75. Pavlotskaya, L.: Solvability of the halting problem for certain classes of Turing machines. Mathematical Notes 13(6), 537-541;
76. Translated from Matematicheskie Zametki, 13(6), 899-909 (1973)
77. Pavlotskaya, L.: O minimal'nom chisle razlichnykh kodov vershin v grafe universal'noj mashiny T'juringa. Disketnyj Analiz, Sbornik Trudov Instituta Matematiki SO AN SSSR 27, 52-60 (1975); On the minimal number of distinct codes for the vertices of the graph of a universal Turing machine (in Russian)
78. Pavlotskaya, L.: Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T'juring. Avtomaty i Mashiny, 91-118 (1978); Sufficient conditions for the halting problem decidability of Turing machines (in Russian)
79. Pavlotskaya, L.: On machines, universal by extensions. Theoretical Computer Science 168(2), 257-266 (1996)
80. Post, E.L.: Formal reductions of the general combinatorial decision problem. American Journal of Mathmatics 65(2), 197-215 (1943)
81. Post, E.L.: Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic 12(1), 1-11 (1947)
82. Post, E.L.: Absolutely unsolvable problems and relatively undecidable propositions - account of an anticipation. In: Davis, M. (ed.) The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, pp. 340-406. Raven Press, New York (1965); Corrected republication. Dover publications, New York (2004)
83. Priese, L.: Towards a precise characterization of the complexity of universal and nonuniversal Turing machines. SIAM J. Comput. 8(4), 508-523 (1979)
84. Rado, T.: On non-computable functions. Bell System Technical Journal 41(3), 877-884 (1962)
85. Richard, G.: A particular universal cellular automaton, HAL research report (oai:hal.archives-ouvertes.fr:hal-00095821 v1) (2006)
86. Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae 12(3), 177-209 (1971)
87. Robinson, R.M.: Minsky's small universal Turing machine. International Journal of Mathematics 2(5), 551-562 (1991)
88. Rogozhin, Y.: Sem' universal'nykh mashin T'juringa. In: Fifth All Union Conference on Mathematical Logic, Akad. Naul SSSR. Otdel. Inst. Mat., Novosibirsk, p. 27 (1979); Seven universal Turing machines (in Russian)
89. Rogozhin, Y.: Sem' universal'nykh mashin T'juringa. Systems and Theoretical Programming, Mat. Issled 69, 76-90 (1982); Seven universal Turing machines (in Russian)
90. Rogozhin, Y.: Universal'naja mashina T'juringa s 10 sostojanijami i 3 simvolami. Izvestiya Akademii Nauk Respubliki Moldova, Matematika 4(10), 80-82 (1992); A universal Turing machine with 10 states and 3 symbols (in Russian)
91. Rogozhin, Y.: About Shannon's problem for Turing machines. Computer Science Journal of Moldova 1(3), 108-111 (1993)
92. Rogozhin, Y.: Small universal Turing machines. Theoretical Computer Science 168(2), 215-240 (1996)
93. Rogozhin, Y.: A universal Turing machine with 22 states and 2 symbols. Romanian Journal of Information Science and Technology 1(3), 259-265 (1998) (in Russian)
94. Rogozhin, Y., Verlan, S.: On the Rule Complexity of Universal Tissue P Systems. In: Freund, R., Pǎ;un, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2005. LNCS, vol. 3850, pp. 356-362. Springer, Heidelberg (2006) (Pubitemid 43973484)
95. Rothemund, P.W.K.: A DNA and restriction enzyme implementation of Turing Machines. In: Lipton, R.J., Baum, E.B. (eds.) DNA Based Computers: Proceeding of a DIMACS Workshop. DIMACS, vol. 2055, pp. 75-119. Princeton University, AMS (1996)
96. Shannon, C.E.: A universal Turing machine with two internal states. Automata Studies, Annals of Mathematics Studies 34, 157-165 (1956)
97. Siegelmann, H.T., Margenstern, M.: Nine switch-affine neurons suffice for Turing universality. Neural Networks 12(4-5), 593-600 (1999)
98. Schroeppel, R.: A two counter machine cannot calculate 2n. Technical Report AIM-257, A.I. memo 257, Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA (1972)
99. Takahashi, H.: Keisankikai II. Iwanami, Tokyo (1958);
100. Computing machine II (in Japanese)
101. Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 2(42), 230-265 (1936)
102. Wagner, K.: Universelle Turingmaschinen mit n-dimensionale band. Elektronische Informationsverarbeitung und Kybernetik 9(7-8), 423-431 (1973); Universal Turing machines with n-dimensional tapes (in German)
103. Wang, H.: A variant to Turing's theory of computing machines. Journal of the Association for Computing Machinery 4(1), 63-92 (1957)
104. Wang, H.: Tag systems and lag systems. Mathematical Annals 152(4), 65-74 (1963)
105. Watanabe, S.: On a minimal universal Turing machine. Technical report, MCB Report, Tokyo (1960)
106. Watanabe, S.: 5-symbol 8-state and 5-symbol 6-state universal Turing machines. Journal of the ACM 8(4), 476-483 (1961)
107. Watanabe, S.: 4-symbol 5-state universal Turing machine. Information Processing Society of Japan Magazine 13(9), 588-592 (1972)
108. Wolfram, S.: Statistical mechanics of cellular automata. Reviews of Modern Physics 55(3), 601-644 (1983)
109. Wolfram, S.: A new kind of science. Wolfram Media, Inc. (2002)
110. Woods, D., Neary, T.: On the time complexity of 2-tag systems and small universal Turing machines. In: 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 439-446. IEEE, Berkeley (2006)
111. Woods, D., Neary, T.: Small Semi-Weakly Universal Turing Machines. In: Durand- Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 303-315. Springer, Heidelberg (2007)
112. Woods, D., Neary, T.: The complexity of small universal Turing machines: A survey. Theoretical Computer Science 410(4-5), 443-450 (2009)
113. Woods, D., Neary, T.: Small semi-weakly universal Turing machines. Fundamenta Informaticae 91(1), 179-195 (2009)