Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Inductive Turing Machines

  • Mark Burgin
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_682-1

Glossary

Algorithm

is a compressed exact description of some activity (functioning, behavior, or computation), which allows reproducing this activity (functioning, behavior, or computation).

Complexity

is a measure of definite resources needed for activity (functioning, behavior, or computation).

Computation

is goal-oriented information processing, results of which can be accessed (registered) by an observer (user).

Information

for a system R is a capacity to change an infological system IF(R) of the system R.

Computability

is a possibility to compute values of a function or elements of a set.

Process

is a connected system, e.g., a sequence, of actions or events.

Introduction/History

Inductive Turing machine, as a rigorous mathematical model of algorithms and computation, was introduced by Mark Burgin in 1983 in the form of an abstract computational device more powerful than Turing machine (Burgin 1983). It was the first class of rigorously defined abstract automata with this...

Keywords

Turing Machine Control Device Kolmogorov Complexity Input Tape Basic Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Bibliography

  1. Adamatzky A (2010) Physarum machines: computers from slime mould. World Scientific Publishing Co., HackensackCrossRefGoogle Scholar
  2. Ades MJ, Burgin MS (2007) Evolutionary processes in modeling and simulation. In: Proceedings of the 2007 Spring Simulation MultiConference (SpringSim '07), March 25–29, 2007, Norfolk, pp 133–138Google Scholar
  3. Barzdin JM, Freivald RV (1972) On the prediction of general recursive functions. Soviet Math Dokl 13:1224–1228MATHGoogle Scholar
  4. Beros AA (2013) Learning theory in the arithmetical hierarchy, Preprint in mathematics, math.LO/1302.7069, 2013 (electronic edition: http://arXiv.org)
  5. Blum L, Blum M (1975) Toward a mathematical theory of inductive inference. Inf Control 28:125–155MathSciNetMATHCrossRefGoogle Scholar
  6. Boddy M, Dean T (1989) Solving time-dependent planning problems. Technical Report: CS-89-03, Brown UniversityGoogle Scholar
  7. Borodyanskiy YM, Burgin M (1994) Problems of artificial intelligence and trans-recursive operators, Visnik of the National Academy of Sciences of Ukraine, No. 11–12, pp 29–34 (in Ukrainian)Google Scholar
  8. Bradshaw J (ed) (1997) Software agents. AAAI Press/MIT Press, Menlo ParkGoogle Scholar
  9. Burgin M (1983) Inductive turing machines. Not Acad Sci USSR 270(6):1289–1293. translated from Russian, v. 27, No. 3MathSciNetMATHGoogle Scholar
  10. Burgin M (1984) Inductive Turing machines with multiple heads and Kolmogorov algorithms. Not Acad Sci USSR 275(2):280–284. (translated from Russian)MATHGoogle Scholar
  11. Burgin M (1987) The notion of algorithm and the Church-Turing thesis. VIII International Congress on logic, methodology and philosophy of science, Moscow, vol 5, pt. 1, pp 138–140Google Scholar
  12. Burgin M (1988) Arithmetic hierarchy and inductive Turing machines. Not Acad Sci USSR 299(3):390–393. (translated from Russian)MATHGoogle Scholar
  13. Burgin M (1992) Universal limit Turing machines. Not Russian Acad Sci 325(4):654–658. (translated from Russian)MATHGoogle Scholar
  14. Burgin M (1993) Procedures of sociological measurements. In: Catastrophe, chaos, and self-organization in social systems. University of Koblenz-Landau, Koblenz, pp 125–129Google Scholar
  15. Burgin M (1999) Super-recursive algorithms as a tool for high performance computing. In: Proceedings of the high performance computing symposium, San Diego, pp 224–228Google Scholar
  16. Burgin M (2000) Theory of super-recursive algorithms as a source of a new paradigm for computer simulation. In: Proceedings of the business and industry simulation symposium, Washington, DC, pp 70–75Google Scholar
  17. Burgin M (2001a) Mathematical models for computer simulation. In: Proceedings of the business and industry simulation symposium, SCS, Seattle, pp 111–118Google Scholar
  18. Burgin M (2001b) How we know what technology can do. Commun ACM 44(11):82–88CrossRefGoogle Scholar
  19. Burgin M (2001c) Topological algorithms. In: Proceedings of the ISCA 16th international conference “Computers and their applications”, ISCA, Seattle, pp 61–64Google Scholar
  20. Burgin M (2003a) Nonlinear phenomena in spaces of algorithms. Int J Comput Math 80(12):1449–1476MathSciNetMATHCrossRefGoogle Scholar
  21. Burgin M (2003b) Cluster computers and grid automata. In: Proceedings of the ISCA 17th international conference “Computers and their applications”, International Society for Computers and their Applications, Honolulu, pp 106–109Google Scholar
  22. Burgin M (2004) Algorithmic complexity of recursive and inductive algorithms. Theor Comput Sci 317(1/3):31–60MathSciNetMATHCrossRefGoogle Scholar
  23. Burgin M (2005a) Super-recursive algorithms. Springer, New YorkMATHGoogle Scholar
  24. Burgin M (2005b) Superrecursive hierarchies of algorithmic problems. In: Proceedings of the 2005 international conference on foundations of computer science, CSREA Press, Las Vegas, pp 31–37Google Scholar
  25. Burgin M (2006a) Algorithmic control in concurrent computations. In: Proceedings of the 2006 international conference on foundations of computer science, CSREA Press, Las Vegas, pp 17–23Google Scholar
  26. Burgin M (2006b) Book Review: Juraj Hromkovic “Design and analysis of randomized algorithms: introduction to design paradigms”, Series: Texts in theoretical computer science, EATCS Series, Springer, (2005), Comput J 49(2): 249–250Google Scholar
  27. Burgin M (2007) Algorithmic complexity as a criterion of unsolvability. Theor Comput Sci 383(2/3):244–259MathSciNetMATHCrossRefGoogle Scholar
  28. Burgin M (2010a) Theory of information: Fundamentality, diversity and unification, World Scientific, New York/London/SingaporeGoogle Scholar
  29. Burgin M (2010b) Algorithmic complexity of computational problems. Int J Comput Inform Technol 2(1):149–187Google Scholar
  30. Burgin M (2011) Super-recursive Kolmogorov Komplexity, Glossarium – BITri, (electronic edition: http://glossarium.bitrum.unileon.es)
  31. Burgin M (2013) Evolutionary information theory, information, vol 4, No.2, pp. 224–268Google Scholar
  32. Burgin M (2014a) Functioning of inductive Turing machines. Int J Unconv Comput (IJUC) 10(1–2):19–35Google Scholar
  33. Burgin M (2014b) Periodic Turing machines. J Comput Technol Appl (JoCTA) 5(3):6–18Google Scholar
  34. Burgin M (2015a) Inductive cellular automata. Int J Data Structures Algorithms 1(1):1–9MathSciNetGoogle Scholar
  35. Burgin M (2015b) Properties of stabilizing computations. Theory Appl Math Comput Sci 5(1):71–93MathSciNetGoogle Scholar
  36. Burgin M (2015c) Super-recursive algorithms and modes of computation. In: Proceedings of the 2015 European conference on software architecture workshops, Dubrovnik/Cavtat, 7–11 Sept 2015, ACM, pp 10:1–10:5Google Scholar
  37. Burgin M (2016a) Inductively computable sets and functions. J Comput Technol Appl (JoCTA) 7(1):12–22MathSciNetGoogle Scholar
  38. Burgin M (2016b) Inductively computable Hierarchies and inductive algorithmic complexity. Global J Comp Sci Technol,, Ser. H: Inf Technol 16(1): 35–45Google Scholar
  39. Burgin M (2016c) Decreasing complexity in inductive computations. In: Advances in Unconventional computing, series emergence, complexity and computation, vol 22. Springer, Switzerland, pp 183–203Google Scholar
  40. Burgin M (2016d) On the power of oracles in the context of hierarchical intelligence. J Artif Intell Res Adv 3(2):6–17Google Scholar
  41. Burgin M (2016e) Inductive complexity and Shannon entropy. In: Information and complexity. World Scientific, New York/London/Singapore, pp 16–32Google Scholar
  42. Burgin M, Ades M (2009) Monte Carlo methods and superrecursive algorithms. In: Proceedings of the spring simulation multiconference (ADS, BIS, MSE, and MSEng), Society for Modeling and Simulation International, San Diego, pp 289–294Google Scholar
  43. Burgin M, Borodyanskiy YM (1991) Infinite processes and super-recursive algorithms. Not Acad Sci USSR 321(5):876–879. translated from Russian: 1992, v.44, No. 1Google Scholar
  44. Burgin MS, Borodyanskiy YM (1993a) Social processes and limit computations. In: Catastrophe, chaos, and self-organization in social systems. University of Koblenz-Landau, Koblenz, pp 117–123Google Scholar
  45. Burgin M, Borodyanskiy YM (1993b) Alphabetic operators and algorithms. Cybern Syst Anal (3): 42–57Google Scholar
  46. Burgin M, Debnath, N (2004) Measuring software maintenance. In: Proceedings of the ISCA 19th international conference “Computers and their applications”, ISCA, Seattle, pp 118–121Google Scholar
  47. Burgin M, Debnath N (2005) Complexity Measures for software engineering. J Comput Methods Sci Eng 5(1):127–143MATHGoogle Scholar
  48. Burgin M, Debnath N (2009) Super-recursive algorithms in testing distributed systems. In: Proceedings of the ISCA 24th international conference “Computers and their applications” (CATA-2009), ISCA, New Orleans, pp 209–214Google Scholar
  49. Burgin M, Dodig-Crnkovic G (2013) From the closed classical algorithmic universe to an open World of algorithmic constellations. In: Computing nature, studies in applied philosophy, epistemology and rational ethics, vol 7. Springer, Berlin/Heidelberg, pp 241–254Google Scholar
  50. Burgin M, Eberbach E (2009a) Universality for Turing machines, inductive Turing machines and evolutionary algorithms. Fundam Inf 91(1):53–77MathSciNetMATHGoogle Scholar
  51. Burgin M, Eberbach E (2009b) On Foundations of evolutionary computation: an evolutionary automata approach. In: Mo H (ed) Handbook of research on artificial immune systems and natural computing: applying complex adaptive technologies. IGI Global, Hershey, pp 342–360CrossRefGoogle Scholar
  52. Burgin M, Eberbach E (2012) Evolutionary automata: expressiveness and convergence of evolutionary computation. Comput J 55(9):1023–1029CrossRefGoogle Scholar
  53. Burgin M, Gupta B (2012) Second-level algorithms, Superrecursivity, and recovery problem in distributed systems. Theory Comput Syst 50(4):694–705MathSciNetMATHCrossRefGoogle Scholar
  54. Burgin M, Klinger A (2004a) Three aspects of super-recursive algorithms and hypercomputation or finding black swans. Theor Comput Sci 317(1/3):1–11MATHCrossRefGoogle Scholar
  55. Burgin M, Klinger A (2004b) Experience, generations, and limits in machine learning. Theor Comput Sci 317(1/3):71–91MathSciNetMATHCrossRefGoogle Scholar
  56. Burgin M, Mikkilineni R (2014) Semantic network organization based on distributed intelligent managed elements. In: Proceeding of the 6th international conference on advances in future Internet, Lisbon, pp 16–20Google Scholar
  57. Burgin M, Mikkilineni R (2016) Agent technology, superrecursive algorithms and DNA as a tool for distributed clouds and grids. In: Proceedings of the 25th IEEE international conference on enabling technologies: infrastructure for collaborative enterprises (WETICE 2016), Paris, 12–15 June, pp 89–94Google Scholar
  58. Burgin M, Shmidski Y (1996) Is it possible to compute non-computable or why programmers need the theory of algorithms. Comput Softw 5:4–8Google Scholar
  59. Burgin M, Debnath N, Lee HK (2009) Measuring testing as a distributed component of the software life cycle. J Comput Methods Sci Eng 9(1/2)(Suppl 2):211–223MathSciNetMATHGoogle Scholar
  60. Burgin M, Calude CS, Calude E (2011) Inductive complexity measures for mathematical problems, Int J Found Comput Sci 24(4): 487–500, 2013Google Scholar
  61. Burgin M, Mikkilineni R, Morana G (2016) Intelligent organization of semantic networks, DIME network architecture and grid automata. Int J Embed Syst (IJES) 8(4) pp. 352–366Google Scholar
  62. Calude CS, Calude E (2012) Algorithmic complexity of mathematical problems: an overview of results and open problems, CDMTCS Research Report 410Google Scholar
  63. Calude CS, Calude E, Queen MS (2012) Inductive complexity of P versus NP problem. In: Unconventional computation and natural computation. Lecture notes in computer science, vol 7445. Springer, New York, pp 2–9Google Scholar
  64. Chaitin GJ (1977) Algorithmic information theory. IBM J Res Dev 21(4):350–359MathSciNetMATHCrossRefGoogle Scholar
  65. Church A (1932/33) A set of postulates for the foundations of logic. Ann Math 33: 346–366; 34: 839–864Google Scholar
  66. Cockshott P, Michaelson G (2007) Are there new models of computation? Reply to Wegner and Eberbach. Comput J 50(2):232–247CrossRefGoogle Scholar
  67. Criscuolo G, Minicozzi E, Tratteur G (1975) Limiting recursion and the arithmetic hierarchy. Rev Franfaise Informat Recherche Opdrationnelle (Dec. 1975) 9:5–12Google Scholar
  68. Dinverno M, Luck M (eds) (2001) Understanding Agent Systems. Springer, New YorkGoogle Scholar
  69. Doyle J (1983) What is rational psychology? Toward a modern mental philosophy. AI Mag 4(3):50–53Google Scholar
  70. Dyson FJ (1972) Missed opportunities. Bull Amer Math Soc 78:635–652MathSciNetMATHCrossRefGoogle Scholar
  71. Eberbach E, Burgin M (2009) Evolutionary automata as foundations of evolutionary computation: Larry Fogel was right. In: Proceedings of 2009 congress on evolutionary computation (CEC’2009), Trondheim, pp 2149–2156Google Scholar
  72. Feferman S (1992) Turing’s ‘Oracle’: From absolute to relative computability - and back, in The Space of Mathematics, Walter de Gruyter, Berlin, pp. 314–348Google Scholar
  73. Gödel K (1934) On undecidable propositions of Formal mathematical Systems. Lectures given at the Institute for Advanced Studies, Princeton, in The Undecidable, Raven Press, 1965, pp 39–71Google Scholar
  74. Gödel K (1990) Collected works, Vol. II, Publications 1938–1974. Oxford University Press, New YorkMATHGoogle Scholar
  75. Gold EM (1965) Limiting recursion. J Symb Log 30(1):28–46MathSciNetMATHCrossRefGoogle Scholar
  76. Gold EM (1967) Language identification in the limit. Inf Control 10:447–474MathSciNetMATHCrossRefGoogle Scholar
  77. Hertel J (2012) Inductive complexity of Goodstein’s theorem. Unconventional computation and natural computation. Lecture notes in computer science, vol 7445. Springer, New York, pp 141–151Google Scholar
  78. Hintikka Ja, Mutanen A (1998) An alternative concept of computability. In: Language, truth, and logic in mathematics. Springer, Dordrecht, pp 174–188Google Scholar
  79. Hromkovic J (2005) Design and analysis of randomized algorithms. Springer, New YorkMATHCrossRefGoogle Scholar
  80. Kleene S (1936) General recursive functions of natural numbers. Math Ann 112(5):727–729MathSciNetMATHCrossRefGoogle Scholar
  81. Kolmogorov AN (1953) On the concept of algorithm. Uspehi Mat Nauk 8(4):175–176Google Scholar
  82. Kolmogorov AN (1965) Three approaches to the definition of the quantity of information. Probl Inf Transm 1(1):3–11MathSciNetMATHGoogle Scholar
  83. Kugel P (2004) Toward a theory of intelligence. Theor Comput Sci 317(1/3):13–30MathSciNetMATHCrossRefGoogle Scholar
  84. Kugel P (2005) It’s time to think outside the computational box. Commun ACM 48(11):32–37CrossRefGoogle Scholar
  85. Li M, Vitanyi P (1997) An introduction to Kolmogorov complexity and its applications. Springer, New YorkMATHCrossRefGoogle Scholar
  86. Manin Y (2012) Renormalisation and computation II: time cut-off and the halting problem. Comput Sci 22:729–751MathSciNetMATHGoogle Scholar
  87. McCarthy T, Shapiro S (1987) Turing projectability. Notre Dame J Formal Logic 28:520–535Google Scholar
  88. Mikkilineni R, Comparini A, Morana G (2012) The Turing o-machine and the DIME network architecture: injecting the architectural resiliency into distributed computing. In: Turing-100, The Alan Turing Centenary, EasyChair Proceedings in Computing, www.easychair.org/publications/?page=877986046
  89. Minsky M (1986) The Society of Mind. Simon and Schuster, New YorkGoogle Scholar
  90. Mutanen A (2004) From computation to truth via learning. Dissertation, University of HelsinkiGoogle Scholar
  91. Putnam H (1965) Trial and error predicates and the solution to a problem of Mostowski. J Symbolic Logic 30(1):49–57MathSciNetMATHCrossRefGoogle Scholar
  92. Rogers H (1987) Theory of recursive functions and effective computability. MIT Press, Cambridge, MAMATHGoogle Scholar
  93. Roglic D (2007) The universal evolutionary computer based on super-recursive algorithms of evolvability. Preprint in Computer Science, http://arxiv.org/ftp/arxiv/papers/0708/0708.2686.pdf
  94. Roglic D (2011) Super-recursive features of evolutionary processes and the models for computational evolution. In: Information and computation. World Scientific, Singapore, pp 331–379CrossRefGoogle Scholar
  95. Schmidhuber J (2002) Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit. Int J Found Comput Sci 3(4):587–612MathSciNetMATHCrossRefGoogle Scholar
  96. Schubert LK (1974) Iterated limiting recursion and the program minimization problem. J Assoc Comput Mach 21:436–445MathSciNetMATHCrossRefGoogle Scholar
  97. Sloman A (2002) The irrelevance of Turing machines to AI (http://www.cs.bham.ac.uk/~axs/)
  98. Soare RI (2015) Turing Oracle machines, online computing, and three displacements in computability theory. Ann Pure Appl Logic 160:368–399MathSciNetMATHCrossRefGoogle Scholar
  99. Syropoulos A (2008) Hypercomputation: computing beyond the Church-Turing barrier. Springer, New YorkMATHCrossRefGoogle Scholar
  100. Thomas WJ (1979) A simple generalization of Turing computability. Notre Dame J Formal Logic 20:95–102MathSciNetMATHCrossRefGoogle Scholar
  101. Turing A (1936) On computable numbers with an application to the Entscheidungs-problem. Proc Lond Math Soc, Ser 2 42: 230–265Google Scholar
  102. Turing A (1950) Computing machinery and intelligence, mind 59:433–460Google Scholar
  103. Zilberstein S (1996) Using anytime algorithms in intelligent systems. AI Mag 17(3):73–83Google Scholar

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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA