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Complexity of algorithms and computations

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Abstract

A survey is given of some results on the complexity of algorithms and computations published up to 1973.

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Literature cited

  1. V. N. Agafonov, “Normal sequences and finite automata,” in: Probl. Kibern., No. 20, Nauka, Moscow (1968), pp. 123–129.

    Google Scholar 

  2. V. N. Agafonov, The Complexity of Algorithms and Computations. 2, Novosibirsk State Univ. (1975).

  3. V. N. Agafonov, “The complexity of computing pseudorandom sequences,” Algebra Logika, Seminar,7, No. 2, 4–19 (1968).

    Google Scholar 

  4. Ya. M. Barzdin', “Complexity and accuracy of solving initial fragments of the problem of belonging to a recursively enumerable set,” Dokl. Akad. Nauk SSSR,199, No. 2, 262–264 (1971).

    Google Scholar 

  5. Ya. M. Barzdin', “The complexity of programs recognizing whether quantities not exceeding natural numbers belong to a recursively enumerable set,” Dokl. Akad. Nauk SSSR,182, No`. 6, 1249–1252 (1968).

    Google Scholar 

  6. Ya. M. Barzdin', “The complexity of recognizing symmetry on Turing machines,” in: Probl. Kibern., No. 15, Nauka, Moscow (1965), pp. 245–248.

    Google Scholar 

  7. Yu. Ya. Breitbart, “On automatic and ‘zone’ complexity of the predicate ‘be the power of a number’,” Dokl. Akad. Nauk SSSR,196, No. 1, 16–19 (1971).

    Google Scholar 

  8. Yu. A. Bukhshtab, “The realizability of functions on one-dimensional iterative circuits in real time,” in: Probl. Kibern., No. 22, Nauka, Moscow (1970), pp. 85–94.

    Google Scholar 

  9. M. K. Valiev, “Some estimates of computation time for Turing machines with input,” Kibernetika, No. 6, 26–32 (1970).

    Google Scholar 

  10. A. V. Gladkii and A. Ya. Dikovskii, “The theory of formal grammars,” Teor. Veroyatn. Mat. Statistika. Teor. Kibern., Vol. 7 (Itogi Nauki i Tekh.) (1972), pp. 107–142.

    Google Scholar 

  11. V. G. Zharov, “On estimating the complexity of terms of constructive sequences of normal algorithms,” Dokl. Akad. Nauk SSSR,203, No. 4, 746–748 (1972).

    Google Scholar 

  12. I. D. Zaslavskii, “On the pseudofunctions of Shannon,” Zap. Nauchn. Sem. Leningr. Otd. Mat.Inst. Akad. Nauk SSSR,16, 65–76 (1969).

    Google Scholar 

  13. A. K. Zvonkin and L. A. Levin, “The complexity of finite objects and the foundation of the concepts of information and randomness by means of the theory of algorithms,” Usp. Mat. Nauk,25, No. 6, 86–127 (1970).

    Google Scholar 

  14. Ya. B. Kazanovich, “Classification of primitively recursive functions by means of Turing machines,” in: Probl. Kibern., No. 22, Nauka, Moscow (1970), pp. 95–106.

    Google Scholar 

  15. M. I. Kanovich, “On the domains of optimal algorithms,” Dokl. Akad. Nauk SSSR,198, No. 2, 283–285 (1971).

    Google Scholar 

  16. M. I. Kanovich, “On the universality of strongly nonsolvable sets,” Dokl. Akad. Nauk SSSR,204, No. 3, 533–535 (1972).

    Google Scholar 

  17. M. I. Kanovich, “On the complexity of minimizing Boolean functions,” Dokl. Akad. Nauk SSSR,198, No. 1, 35–38 (1971).

    Google Scholar 

  18. M. I. Kanovich, “On the complexity of the resolution of algorithms,” Dokl. Akad. Nauk SSSR,186, No. 5, 1008–1009 (1969).

    Google Scholar 

  19. M. I. Kanovich, “On the complexity of enumerating and resolving predicates,” Dokl. Akad. Nauk SSSR,190, No. 1, 23–26 (1970).

    Google Scholar 

  20. M. I. Kanovich, “The complexity of bounded resolution of semienumerable sets,” Dokl. Akad. Nauk SSSR,203, No. 6, 1246–1248 (1972).

    Google Scholar 

  21. M. I. Kanovich, “The complexity of resolving an enumerable set as a criterion of its universality,” Dokl. Akad. Nauk SSSR,194, No. 3, 500–503 (1970).

    Google Scholar 

  22. M. I. Kanovich and B. A. Kushner, “On an estimate of the complexity of some mass problems of analysis,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,16, 81–90 (1969).

    Google Scholar 

  23. M. I. Kanovich and N. V. Petri, “Some theorems on the complexity of normal algorithms and computations,” Dokl. Akad. Nauk SSSR,184, No. 6, 1275–1276 (1969).

    Google Scholar 

  24. A. A. Karatsuba and Yu. P. Ofman, “Multiplication of multidigit numbers on automata,” Dokl. Akad. Nauk SSSR,145, No. 2, 293–294 (1962).

    Google Scholar 

  25. B. M. Kloss, “On the definition of the complexity of algorithms,” Dokl. Akad. Nauk SSSR,157, No. 1, 38–40 (1964).

    Google Scholar 

  26. V. A. Kozmidiadi, “On a generalization of finite automata which forms a hierarchy analogous to the classification of A. Grzegorczyk of primitively recursive functions,” in: Probl. Kibern., No. 23, Nauka, Moscow (1970), pp. 127–170.

    Google Scholar 

  27. V. A. Kozmidiadi and S. S. Marchenkov, “On multihead automata,” in: Probl. Kibern., No. 21, Nauka, Moscow (1969), pp. 127–158.

    Google Scholar 

  28. A. N. Kolmogorov, “Three approaches to the definition of the concept of the ‘quantity of information’,” Probl. Peredachi Inf.,1, No. 1, 3–11 (1965).

    Google Scholar 

  29. Yu. A. Kryukov, “Turing machines with two symbols and two states,” Algebra Logika. Seminar,6, No. 3, 51–60 (1967).

    Google Scholar 

  30. Yu. A. Kryukov, “Turing machines with three states and two symbols and one state and symbols,” Kibernetika, No. 1, 12–13 (1971).

    Google Scholar 

  31. A. V. Kuznetsov, “On a theorem on the canonical form for ordinal recursive functions,” Appendix to the book of R. L. Goodstein, Mathematical Logic [Russian translation], IL, Moscow (1961), pp. 149–154.

    Google Scholar 

  32. V. A. Kuz'min, “Realization of functions of a logic algebra by automata, normal algorithms, and Turing machines,” in: Probl. Kibern., No. 13, Nauka, Moscow (1955), pp. 75–96.

    Google Scholar 

  33. G. B. Marandzhyan, “The hierarchy of recursive functions and asymptotic optimality,” Aikakan SSR Gitutyunneri Akademia, Zekuitsner. Dokl. Akad. Nauk ArmSSR,48, No. 4, 193–197 (1969).

    Google Scholar 

  34. G. B. Marandzhyan, “On some properties of asymptotical optimal recursive functions,” Aikakan SSR Gitutyunneri Akademiai Tegekagir. Matematika, Izv. Akad. Nauk ArmSSR, Mat.,4, No. 1, 3–22 (1969).

    Google Scholar 

  35. G. B. Marandzhyan, “On strictly effective immunity of the stems of additively optimal recursive functions,” Aikakan SSR Gitutyunneri Akademia Tegekagir. Matematika, Izv. Akad. Nauk ArmSSR,7, No. 6, 391–398 (1972).

    Google Scholar 

  36. A. A. Markov, “On normal algorithms computing Boolean functions,” Dokl. Akad. Nauk SSSR,157, No. 2, 262–264 (1964).

    Google Scholar 

  37. A. A. Markov, “On normal algorithms connected with the computation of Boolean functions,” Izv. Akad. Nauk SSSR, Ser. Mat.,31, No. 1, 161–208 (1967).

    Google Scholar 

  38. P. Martin-Löf, “On the concept of a random sequence,” Teor. Veroyatn. Ee Primen.,11, No, 2, 198–202 (1966).

    Google Scholar 

  39. S. S. Marchenkov, “On bounded recursions,” Math. Balkan.,2, 124–142 (1972).

    Google Scholar 

  40. S. G. Matveeva, “On the theorem of Rabin on the complexity of computable functions,” Sib. Mat. Zh.,4, No. 3, 546–555 (1965).

    Google Scholar 

  41. Yu. V. Matiyasevich, “On the recognizing in real time of the relation of containment,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,20, 104–114 (1971).

    Google Scholar 

  42. V. A. Moshchenskii, “On the question of the complexity of Turing computations,” Dokl. Akad. Nauk BSSR,13, No. 10, 871–878 (1969).

    Google Scholar 

  43. V. A. Moshchenskii, “On the estimate of certain functions characterizing the work of Turing machines,” Kibernetika, No. 1, 34–40 (1971).

    Google Scholar 

  44. V. A. Nepomnyashchii, “On a basis for recursively enumerable sets,” Dokl. Akad. Nauk SSSR,170, No. 6, 1262–1264 (1966).

    Google Scholar 

  45. V. A. Nepomnyashchii, “A rudimentary interpretation of two-tape Turing computations,” Kibernetika, No. 2, 29–35 (1970).

    Google Scholar 

  46. V. A. Nepomnyashchii, “Rudimentary predicates and Turing computations,” Dokl. Akad. Nauk SSSR,195, No. 2, 282–284 (1970).

    Google Scholar 

  47. E. S. Orlovskii, “Some questions of the theory of algorithms,” Tr. Mat. Inst. Akad. Nauk SSSR,52, 140–171 (1958).

    Google Scholar 

  48. D. A. Ostroukhov, “On the coding of natural numbers by schemes of normal algorithms,” Z. Math. Log. Grundl. Math.,16, No. 4, 347–352 (1970).

    Google Scholar 

  49. D. A. Ostroukhov, “On an estimate of the complexity of normal algorithms,” Dokl. Akad. Nauk SSSR,184, No. 6, 1292–1293 (1969).

    Google Scholar 

  50. N. V. Petri, “On algorithms connected with predicates and Boolean functions,” Dokl. Akad. Nauk SSSR,185, No. 1, 37–39 (1969).

    Google Scholar 

  51. N. V. Petri, “Complexity of algorithms and their working time,” Dokl. Akad. Nauk SSSR,186, No, 1, 30–31 (1969).

    Google Scholar 

  52. D. Skordev, “Some simple examples of universal functions,” Dokl. Akad. Nauk SSSR,190, No. 1, 45–46 (1970).

    Google Scholar 

  53. N. P. Ter-Zakharyan, “On some quantitative characteristics of algorithmic languages,” Dokl. Akad. Nauk SSSR,190, No. 3, 538–540 (1970).

    Google Scholar 

  54. A. L. Toom, “On the complexity of a scheme of functional elements realizing the multiplication of whole numbers,” Dokl. Akad. Nauk SSSR,150, No. 3, 496–498 (1963).

    Google Scholar 

  55. B. A. Trakhtenbrot, “On normalized signalings for Turing computations,” Algebra Logika. Seminar,5, No. 6, 61–70 (1966).

    Google Scholar 

  56. B. A. Trakhtenbrot, “Optimal computations and the frequency phenomenon of Yablonskii,” Algebra Logika. Seminar,4, No. 5, 79–93 (1965).

    Google Scholar 

  57. B. A. Trakhtenbrot, Complexity of Algorithms and Computations [in Russian], Novosibirsk State Univ. (1967).

  58. B. A. Trakhtenbrot, “Turing computations with logarithmic delay,” Algebra Logika. Seminar,3, No. 4, 33–48 (1964).

    Google Scholar 

  59. R. V. Freivald, “On the order of growth of precise time signalings for Turing computations,” Algebra Logika. Seminar,5, No. 5, 85–93 (1966).

    Google Scholar 

  60. R. V. Freivald, “The complexity of symmetry recognition on Turing machines with input,” Algebra Logika. Seminar,4, No. 1, 47–58 (1965).

    Google Scholar 

  61. V. I. Khomich, “On the complexity of algorithms connected with the realization of logical-arithmetic and propositional formulas,” Dokl. Akad. Nauk SSSR,191, No. 5, 1004–1006 (1970).

    Google Scholar 

  62. G. S. Tseitin, “A lower bound for the number of steps for an inverting normal algorithm and two analogous algorithms,” Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,20, 243–262 (1971).

    Google Scholar 

  63. G. S. Tseitin, “An estimate of the number of steps in applying a normal algorithm,” Mathematics in the USSR after Forty Years, Moscow,1, 44–45 (1959).

    Google Scholar 

  64. S. Aanderaa and P. C. Fischer, “The solvability of halting problem for 2-state Post machines,” J. Assoc. Comput. Mach.,14, No. 4, 677–682 (1967).

    Google Scholar 

  65. M. A. Arbib, Speed-Up Theorems and Incompleteness Theorems in Automata Theory, Academic Press, New York-London (1966), pp. 6–24.

    Google Scholar 

  66. M. A. Arbib and M. Blum, “Machine dependence of degrees of difficulty,” Proc. Am. Math. Soc.,16, No. 3, 442–447 (1965).

    Google Scholar 

  67. A. J. Atrubin, “A one-dimensional real-time iterative multiplier,” IEEE Trans. Electron. Comput.,14, No. 3, 394–399 (1965).

    Google Scholar 

  68. G. Ausiello, “Abstract computational complexity and cyclic function,” Conf. Rec. of Second Annual ACM Symposium on Theory of Computing,4, 1–47 (1970).

    Google Scholar 

  69. P. Axt, “Enumeration and Grzegorczyk hierarchy,” Z. Math. Logik Grundl. Math.,9, No. 1, 53–65 (1963).

    Google Scholar 

  70. P. Axt, “Iteration of primitive recursion,” Z. Math. Logik Grundl. Math.,11, No. 3, 253–255 (1965).

    Google Scholar 

  71. P. Axt, “Note on the 3-recursive functions,” Z. Math. Logik Grundl. Math.,7, No. 2, 97–98 (1961).

    Google Scholar 

  72. J. Becvar, “Probleme der Komplexität in der Theorie der Algorithmen und Automaten,” Int. Ser. Number. Math. 6, Basel-Stuttgart, 142–157 (1967).

    Google Scholar 

  73. J. Becvar, “Programmkomplexität von borechenbaren Funktionen,” Z. Angew. Math. Mech.,50, Sonderh, 1–4, 82 (1970).

    Google Scholar 

  74. J. Becvar, “Programmkomplexität von Funktionen und Mengen,” Ber. Math. Forschunginst. Oberwolfach, No. 3, 317–326 (1970).

    Google Scholar 

  75. J. Becvar, “Real-time and complexity problems in automata theory,” Kybernetika,1, No. 6, 475–498 (1965).

    Google Scholar 

  76. M. Blum, “A machine-independent theory of the complexity of recursive functions,” J. Assoc. Comput. Mach.,14, No. 2, 322–326 (1967).

    Google Scholar 

  77. M. Blum, “Measures on the computational speed of partial recursive functions,” Q. Progr. Report, 72, Res. Lab. Electronics, MIT, 237–253 (1964).

  78. M. Blum, “On effective procedures for speeding up algorithms,” J. Assoc. Comput. Mach.,18, No. 2, 290–305 (1971).

    Google Scholar 

  79. M. Blum, “On the size of machines,” Inf. Control,11, No. 3, 257–265 (1968).

    Google Scholar 

  80. M. Blum, “Recursive function theory and speed of computation,” Can. Math. Bull.,9, No. 6, 745–750 (1966).

    Google Scholar 

  81. R. V. Book and S. A. Greibach, “Quasi-realtime languages,” Math. Syst. Theory,4, No. 2, 97–111 (1970).

    Google Scholar 

  82. C. Böhm, “A three-tape, three-state, three-symbol universal Turing machine,” Pubbl. Inst. Applic. Calcolo, No. 698 (1968).

  83. A. Borodin, “Computational complexity and the existence of complexity gaps,” J. Assoc. Comput. Mach.,19, 158–174 (1972).

    Google Scholar 

  84. A. Borodin, “Computational complexity 2 A survey,” Proc. Fourth Ann. Princeton Conf. on Information Sci. and Systems, 257–262 (1970).

  85. A. Borodin, R. L. Constable, and J. E. Hopcroft, “Dense and nondense families of complexity classes,” IEEE Conf. Rec. 10th Annual Sympos. Switch. and Automata Theory, Waterloo, 1969, New York, 7–19 (1969).

  86. J. Bulnes, “On the speed of addition and multiplication on one-tape, off-line Turing machines,” Inf. Control,20, No. 5, 415–431 (1972).

    Google Scholar 

  87. W. A. Burkhard, “Complexity problems in real time computation,” Conf. Rec. of Second Ann. ACM Symposium on Theory of Computing, 62–69 (1970).

  88. W. A. Burkhard and P. P. Varaiya, “Complexity problems in real time languages,” Inf. Sci.,3, No. 1, 87–100 (1971).

    Google Scholar 

  89. J. R. Buchi, “Weak second-order arithmetic and finite automata,” Z. Math. Logik Grundl. Math.,6, No. 1, 66–92 (1960).

    Google Scholar 

  90. G. J. Chaitin, “On the length of programs for computing finite binary sequences,” J. Assoc. Comput. Mach.,13, 547–569 (1966).

    Google Scholar 

  91. T. S. Chow, “On the structure of Blum measure,” AFIPS Conf. Proc, Vol. 40, Spring Joint Comput. Conf., Atlantic City, N.J., 1972, Montvale, New Jersey, 503–506 (1972).

  92. J. P. Cleave, “A hierarchy of primitive recursive functions,” Z. Math. Logik Grundl. Math.,9, No. 4, 331–346 (1963).

    Google Scholar 

  93. A. Cobham, “The intrinsic computational difficulty of functions,” Logic, Methodol. Philos. Sci., Amsterdam, 24–30 (1965).

  94. A. Cobham, “Time and memory capacity bounds for machines which recognize squares of palindroms,” IBM Res. Report RC 1621 (1966).

  95. “Complexity of computer computations,” Panel Discussion, Complexity Comput. Computat., Proc. Symp., Yorktown Heights, New York, 1972, New York-London, 169–185 (1972).

  96. R. L. Constable, “On the size of programs in subrecursive formalisms,” Ann. ACM. Symp. Theory Comput., 1–9 (1970).

  97. R. L. Constable, “Two types of hierarchy theorems for axiomatic complexity classes,” Courant. Comput. Sci. Symp. J. Comput., Complex., 1971, New York, New York, 37–63 (1973).

  98. R. L. Constable, “Upward and downward diagonalisation over axiomatic complexity classes,” Technical Report 69-32, Dept. of Computer Science, Cornell University (1969).

  99. R. L. Constable and A. Borodin, “On the efficiency of programs in subrecursive formalisms.” Incomplete version, Extend. Abstr. IEEE Conf. Rec. 11th Annual Symp. Switch. and Automata Theory, Santa Monica, Calif., 1970, New York, New York (1970).

  100. R. L. Constable and J. Hartmanis, “Complexity of formal translations and speed-up results,” Conf. Rec. 3rd Annu. ACM Symp. Theory Comput., Shaker Heights, Ohio, 1971, New York, New York (1971).

  101. S. A. Cook, “Characterizations of pushdown machines in terms of time-bounded computers,” J. Assoc. Comput. Mach.,18, No. 1, 4–18 (1971).

    Google Scholar 

  102. S. A. Cook, “The complexity of theorem-proving procedures,” Conf. Rec. 3rd Annu. ACM Symp. Theory Comput., Shaker Heights, Ohio, 1971, New York, New York (1971).

  103. S. A. Cook and S. O. Aanderaa, “On the minimum computation time of functions,” Trans. Am. Math. Soc.,142, 291–314 (1969).

    Google Scholar 

  104. H. B. Enderton, “Degrees of computational complexity,” J. Comput. Syst. Sci.,6, No. 5, 389–396 (1972).

    Google Scholar 

  105. S. Feferman, “Classifications of recursive functions by means of hierarchies,” Trans. Am. Math. Soc.,104, No. 1, 101–122 (1962).

    Google Scholar 

  106. P. C. Fischer, “The reduction of tape reversals for off-line one-tape Turing machines,” J. Comput. Syst. Sci.,2, No. 2, 136–147 (1968).

    Google Scholar 

  107. P. C. Fischer, “Turing machines with a schedule to keep,” Inf. Control,11, Nos. 1–2, 138–146 (1967).

    Google Scholar 

  108. P. C. Fischer, J. Hartmanis, and M. Blum, “Tape reversal complexity hierarchies,” IEEE Conf. Rec. of 1968 Ninth Annual Symp. on Switching and Automata Theory, 373–382 (1968).

  109. P. C. Fischer, A. R. Meyer, and A. L. Rosenberg, “Counter machines and counter languages,” Math. Syst. Theory,2, No. 3, 265–283 (1968).

    Google Scholar 

  110. P. C. Fischer, A. R. Meyer, and A. L. Rosenberg, “Real time simulation of multihead tape units,” J. Assoc. Comput. Mach.,19, No. 4, 590–607 (1972).

    Google Scholar 

  111. P. C. Fischer, A. R. Meyer, and A. L. Rosenberg, “Time-restricted sequence generation,” J. Comput. Syst. Sci.,4, No. 1, 50–73 (1970).

    Google Scholar 

  112. E. P. Glinert, “On restricted Turing computability,” Math. Syst. Theory,5, No. 4, 331–343 (1971).

    Google Scholar 

  113. A. Grzegorczyk, “Some classes of recursive functions,” Instytut Matematyczny Polskiej Akademii Nauk. Rozprawy Matematyczne, 4 (1953).

  114. J. Hartmanis, “Computational complexity of one tape Turing machine computations,” J. Assoc. Comput. Mach.,15, No. 2, 325–339 (1968).

    Google Scholar 

  115. J. Hartmanis, “On the complexity of undecidable problems on automata theory,” J. Assoc. Comput. Mach.,16, No. 1, 160–167 (1969).

    Google Scholar 

  116. J. Hartmanis, “Size argument in the study of computation speeds,” Proc. Symp. Comput. and Automata, New York, New York, 1971, Brooklyn, New York (1971).

  117. J. Hartmanis, “Tape reversal bounded Turing machine computations,” J. Comput. Syst. Sci.,2, No. 2, 117–135 (1968).

    Google Scholar 

  118. J. Hartmanis and J. E. Hopcroft, “An overview of the theory of computational complexity,” J. Assoc. Comput. Mach.,18, No. 3, 444–475 (1971).

    Google Scholar 

  119. J. Hartmanis, P. M. Lewis II, and R. E. Stearns, “Classifications of computations by time and memory requirements,” Proc. of IFIP Congress I, Washington, D. C. (1965).

  120. J. Hartmanis and H. Shank, “Two memory bounds for the recognition of primes by automata,” Math. Syst. Theory,3, No. 2, 125–129 (1969).

    Google Scholar 

  121. J. Hartmanis and R. E. Stearns, “Automata-based computational complexity,” Inf. Sci.,1, No. 2, 173–184 (1969).

    Google Scholar 

  122. J. Hartmanis and R. E. Stearns, “On the computational complexity of algorithms,” Trans. Am. Math. Soc.,117, No. 5, 285–306 (1965).

    Google Scholar 

  123. I. M. Havel, “Weak complexity measures,” SIGACT News,8, 21–30 (1971).

    Google Scholar 

  124. J. P. Helm and P. R. Young, “On size vs efficiency for programs admitting speed-ups,” J. Symbol. Log.,36, No. 1, 21–27 (1971).

    Google Scholar 

  125. F. C. Hennie, “Crossing sequences and off-line Turing machine computations,” IEEE Conf. Rec. Switch. Circuit. Theory and Logic Design, Ann Arbor, Mich., 1965, New York, New York, Inst. Electr. and Electron Engrs. Inc. (1965).

    Google Scholar 

  126. F. C. Hennie, “One-tape, off-line Turing machine computations,” Inf. Control,8, 553–578 (1965).

    Google Scholar 

  127. F. C. Hennie, “On-line Turing machine computations,” IEEE Trans. Electron. Comput.,15, No. 1, 35–44 (1966).

    Google Scholar 

  128. F. C. Hennie and R. E. Stearns, “Two-tape simulation of multitape Turing machines,” J. Assoc. Comput. Mach.,13, No. 4, 533–546 (1976).

    Google Scholar 

  129. G. T. Herman, “A new hierarchy of elementary functions,” Proc. Am. Mach. Soc.,20, No. 2, 557–562 (1969).

    Google Scholar 

  130. G. T. Herman, “The equivalence of different hierarchies of elementary functions,” Z. Math. Log. Grundl.,17, No. 3, 219–224 (1971).

    Google Scholar 

  131. G. T. Herman, “The halting problem of one state Turing machines with n-dimensional tape,” Z. Math. Logik Grundl. Math.,14, No. 2, 185–191 (1968).

    Google Scholar 

  132. P. K. Hooper, “Some small, multitape universal Turing machines,” Inf. Sci.,1, No. 2, 209–215 (1969).

    Google Scholar 

  133. J. E. Hopcroft and J. D. Ullman, “Relation between time and tape complexities,” J. Assoc. Comput. Mach.,15, No. 3, 414–427 (1968).

    Google Scholar 

  134. J. E. Hopcroft and J. D. Ullman, “Some results on tape-bounded Turing machines,” J. Assoc. Comput. Mach.,16, No. 1, 168–177 (1969).

    Google Scholar 

  135. O. H. Ibarra, “A note concerning nondeterministic tape complexities,” J. Assoc. Comput. Mach.,18, No. 4, 608–612 (1972).

    Google Scholar 

  136. L. Kalmar, “Eguszerü pelda eldönthetetlen aritmetikai problemara” (Ein einfaches Beispiel für ein unentscheibares arithmetisches Problem), Matematikai es Fizikai Lapok,50, 1–23 (1943).

    Google Scholar 

  137. T. Kameda, “Constant-tape-reversal bounded nondeterministic Turing machine computations,” Comput. Symp., 1970, Bonn, Proc., Frankfurt (1973).

  138. T. Kameda and K. Vollmar, “Note on tape reversal complexity of languages,” Inf. Control,17, No. 2, 203–215 (1970).

    Google Scholar 

  139. T. Kameda and R. Vollmar, “Zur Umkehrkomplexität von Sprachen,” Ber. Math. Forschungsinst. Oberwolfach,3, 327–339 (1970).

    Google Scholar 

  140. R. M. Karp, “Reducibility among combinatorial problems,” Complexity Comput. Computat., Proc. Symp., Yorktown Heights, New York, 1972, New York-London (1972).

  141. R. M. Karp, “Some bounds on the storage requirements of sequential machines and Turing machines,” J. Assoc. Comput. Mach.,14, No. 3, 478–489 (1967).

    Google Scholar 

  142. S. C. Kleene, “Extension of an effectively generated class of functions by enumeration,” Colloq. Math.,6, 67–78 (1958).

    Google Scholar 

  143. D. E. Knouth, The Art of Computer Programming, 2, Addison-Wesley (1969).

  144. L. H. Landweber and E. L. Robertson, “Recursive properties of abstract complexity classes,” J. Assoc. Comput. Mach.,19, No. 2, 296–308 (1972).

    Google Scholar 

  145. E. L. Lawler, “The complexity of combinatorial computations, a survey,” Proc. Symp. Comput. and Automata, New York, New York, 305–311 (1971).

  146. F. D. Lewis, “The enumerability and invariance of complexity classes,” J. Comput. Syst. Sci.,5, No. 3, 286–303 (1971).

    Google Scholar 

  147. G. Longo, “Towards an abstract analysis of time progression of consumption of resources during computation,” Int. Comput. Symp., Venice (1972).

  148. L. Longo, “Axioms for time dependence of resource consumption in computing recursive functions,” SIGACT News,12, 14–24 (1971).

    Google Scholar 

  149. D. W. Loveland, “A variant of the Kolomogorov concept of complexity,” Inf. Control,15, No. 6, 510–526 (1969).

    Google Scholar 

  150. M. H. Löb and S. S. Wainer, “Hierarchies of number-theoretic functions. I,” Arch.Mach. Log. Grundl.,13, Nos. 1–2, 39–51 (1970).

    Google Scholar 

  151. M. H. Löb and S. S. Wainer, “Hierarchies of number-theoretic functions. II,” Arch. Math. Log. Grundl.,13, Nos. 3–4, 97–113 (1970).

    Google Scholar 

  152. N. A. Lynch, “Relativization of the theory of computational complexity,” Proj. MAC Tech. Rept., No. 99 (1972).

  153. P. Martin-Löf, “Complexity oscillations in infinite binary sequences,” Z. Wahrscheinlichkeitstheory Verw. Geb.,19, No. 3, 225–237 (1971).

    Google Scholar 

  154. P. Martin-Löf, “On the notion of randomness,” Intuitionism and proof theory, New York, 73–78 (1968).

  155. P. Martin-Löf, “The definition of random sequences,” Inf. Control,9, No. 6, 602–619 (1966).

    Google Scholar 

  156. E. M. McCreight and A. R. Meyer. “Classes of computable functions defined by bounds on computation: preliminary report,” Conf. Rec. of ACM Symp. on Theory of Computing, 79–88 (1969).

  157. A. R. Meyer, “Program size in restricted programming languages,” Inf. Control,21, No. 4, 382–394 (1972).

    Google Scholar 

  158. A. R. Meyer, “Theories of computational complexity,” Comput. Sci. Res. Review, Carnegie -Mellon Univ., 17–22 (1968).

  159. A. R. Meyer, “Weak monadic second-order theory of successor is not elementary recursive,” Preliminary Report, Cambridge, Mass., 1–24 (1972).

  160. A. R. Meyer and A. Bagchi, “Program size and economy of descriptions,” Preliminary Report, 4th Annual ACM Symp. Theory Comput., Denver, Colo., 183–186 (1972).

  161. A. R. Meyer and P. Fischer, “Economy of descriptions by automata, grammars and formal systems,” Conf. Rec. 12th Annual Symp. Switch. and Automata Theory, East Lansing, Mich., 1971, New York, New York, 188–191 (1971).

  162. A. R. Meyer and P. Fischer, “Computational speed-up by effective operators,” J. Symbol. Log.,37, No. 1, 55–68 (1972).

    Google Scholar 

  163. A. R. Meyer and E. M. McCreight, “Computationally complex and pseudorandom zero-one valued functions,” Theory Mach. Comput., New York-London, 19–42 (1971).

  164. A. R. Meyer and R. Moll, “Honest bounds for complexity classes of recursive functions,” 13th Annual Symp. Switch. and Automata Theory, 1972, New York, New York, 61–66 (1972).

  165. A. R. Meyer, A. L. Rosenberg, and P. C. Fischer, “Turing machines with several read-write heads,” Preliminary Report, IEEE Conf. Rec. 8th Annual Symp. Switch. and Automata Theory, Austin, Texas, 1967, New York, New York, 117–127 (1967).

  166. A. R. Meyer and D. M. Ritchie, “A classification of the recursive functions,” Z. Math. Log. Grundl. Math.,18, No. 1, 71–81 (1972).

    Google Scholar 

  167. M. L. Minsky, Computation: Finite and Infinite Machines, Prentice-Hall, Englewood Cliffs, New Jersey (1967).

    Google Scholar 

  168. M. L. Minsky, “Size and structure of universal Turing machines using Tag systems,” Recurs. Funct. Theory, Am. Math. Soc., 229–238 (1962).

  169. J. Myhill, “A stumblingblock in constructive mathematics,” Abstract, J. Symbol. Logic,18, 190–191 (1953).

    Google Scholar 

  170. A. Nozaki, “On the notion of universality of Turing machine,” Kybernetika,5, No. 1 (1969).

  171. D. Pager, “On the efficiency of algorithms,” J. Assoc. Comput. Mach.,17, No. 4, 708–714 (1970).

    Google Scholar 

  172. R. A. Di Paola, “Random sets in subrecursive hierarchies,” J. Assoc. Comput. Mach.,16, No. 4, 621–630 (1969).

    Google Scholar 

  173. Ch. Parsons, “Hierarchies of primitive recursive functions,” Z. Math. Logik Grundl. Math.,14, No. 4, 357–376 (1968).

    Google Scholar 

  174. M. S. Paterson, “Tape bounds for time bounded Turing machines,” J. Comput. Syst. Sci.,6, No. 2, 116–124 (1972).

    Google Scholar 

  175. R. Peter, “Mehrfache und transfinite Rekursionen,” J. Symbol. Logic,15, 248–272 (1950).

    Google Scholar 

  176. R. Peter, Rekursive Funktione, Budapest (1951).

  177. R. Peter, “Über die mehrfache rekursion,” Math. Annalen,113, 489–527 (1936).

    Google Scholar 

  178. M. O. Rabin, “Real time computation,” Israel J. Math.,1, No. 4, 203–211 (1963).

    Google Scholar 

  179. M. O. Rabin, “Speed of computation of functions and classification of recursive sets,” Bull. Res. Council Israel,8, No. 1, 69–70 (1959).

    Google Scholar 

  180. R. W. Ritchie, “Classes of predictably computable functions,” Trans. Am. Math. Soc.,106, 139–173 (1963).

    Google Scholar 

  181. R. W. Ritchie, “Classes of recursive functions based on Ackermann's function,” Pac. J. Math.,15, No. 3, 1027–1044 (1965).

    Google Scholar 

  182. E. L. Robertson, “A corrected definition of ‘local speedup’,” SIGACT News,6, 15–16 (1970).

    Google Scholar 

  183. E. L. Robertson, “Complexity classes of partial recursive functions (preliminary version),” Conf. Rec. 3rd Annual ACM Symp. Theory Comput., Shaker Heights, Ohio, 1971, New York, New York, 258–266 (1971).

  184. A. L. Rosenberg, “Real-time definable languages,” J. Assoc. Comput. Mach.,14, No. 4, 645–662 (1967).

    Google Scholar 

  185. N. A. Routledge, “Ordinal recursion,” Proc. Cambridge Philos. Soc.,49, Part 2, 175–182 (1953).

    Google Scholar 

  186. S. S. Ruby and P. C. Fischer, “Translational methods and computational complexity,” IEEE Conf. Rec. Switch. Circuit Theory and Log. Design, Ann Arbor, Mich., 1965, New York, New York, Inst. Electr. and Electron. Eng. Inc., 173–178 (1965).

    Google Scholar 

  187. W. J. Savitch, “Relationships between nondeterministic and deterministic tape complexities,” J. Comput. Syst. Sci.,4, No. 2, 177–192 (1970).

    Google Scholar 

  188. A. Schmitt, “Die Zustands-Komplexitats-Klassen von Turing-machinen,” Ber. Math. Forschungsinst. Oberwolfach, No. 3, 343–350 (1970).

    Google Scholar 

  189. A. Schmitt, “The state complexity of Turing machines,” Inf. Control,17, No. 3, 217–225 (1970).

    Google Scholar 

  190. C. P. Schnorr, “A unified approach to the definition of random sequences,” Math. Syst. Theory,5, No. 3, 246–258 (1971).

    Google Scholar 

  191. C. P. Schnorr, “Eine Bemerkung zum Begriff der zufalligen Folge,” Z. Wahrscheinlichkeitstheor. Verw. Geb.,14, No. 1, 27–35 (1969).

    Google Scholar 

  192. C. P. Schnorr, “Klassifikation der Zufallsgesetze nach Komplexität und Ordnung,” Z. Wahrschienlichkeitstheor. Verw. Geb.,16, No. 1, 1–21 (1971).

    Google Scholar 

  193. C. P. Schnorr, “The process complexity and effective random tests,” 4th Annual ACM Symp. Theory of Comput., Denver, Colo., 1972, S. 1, 168–176 (1972).

  194. C. P. Schnorr, “Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begrundung der Wahrscheinlichkeitstheorie,” Lect. Notes Math.,218, IV (1971).

    Google Scholar 

  195. A. Schönhage, “Multiplikation grosser Zahlen,” Computing (Arch. Elektron. Rechnen),1, 182–196 (1966).

    Google Scholar 

  196. A. Schönhage and V. Strassen, “Schnelle Multiplikation grosser Zahlen,” Computing,7, Nos. 3–4, 281–292 (1971).

    Google Scholar 

  197. H. Schwichtenberg, “Rekursionszahlen und die Grzegorczyk-Hierarchie,” Arch. Math. Logik Grundl.,12, Nos. 1–2, 85–97 (1969).

    Google Scholar 

  198. C. E. Shannon, “A universal Turing machine with two internal states,” Ann. Math. Stud.,34, 157–165 (1959).

    Google Scholar 

  199. T. Skolem, “A theorem on recursively enumerable sets,” Abstr. of Short Comm. Int. Congress Math., Stockholm, 11 (1962).

  200. A. R. Smith, “Real-time language recognition by one-dimensional cellular automata,” J. Comput. Syst. Sci.,6, No. 3, 233–253 (1972).

    Google Scholar 

  201. R. M. Smullyan, Theory of Formal Systems (Ann. Math. Studies, No. 47), Univ. Press, Princeton, New Jersey, XTV (1961).

    Google Scholar 

  202. R. J. Solomonoff, “A formal theory of inductive inference. Part I,” Inf. Control,7, No. 1, 1–22 (1964).

    Google Scholar 

  203. R. E. Stearns, J. Hartmanis, and P. M. Lewis II, “Hierarchies of memory limited computations,” IEEE Conf. Rec. Switch. Circuit Theory and Logic Design, Ann Arbor, Mich, 1965, New York, New York, Inst. Electr. and Electron. Eng., Inc., 179–190 (1965).

    Google Scholar 

  204. L. J. Stockmeyer and A. R. Meyer, “Word problems requiring exponential time,” Preliminary Report, Cambridge, Mass., 1–17 (1972).

  205. H. J. Stoss, “A two-tape simulation of Turing machines,” Computing,7, 222–235 (1971).

    Google Scholar 

  206. H. J. Stoss, “k-band simulation von k-kops Turing machinen,” Computing,6, Nos. 3–4, 309–317 (1970).

    Google Scholar 

  207. P. Strnad, “On-line Turing machine recognition,” Inf. Control,12, 442–452 (1968).

    Google Scholar 

  208. P. Strnad, “O reprezentovatelnosti jiste mnozine slov automatem v realnem case,” Sb. Vedeck. Praci Vysoke Skoly Strojni a Textil, Liberei, 1966, Prague, 23–27 (1966).

  209. D. B. Thompson, “Subrecursiveness: machine-independent notions of computability in restricted time and storage,” Math. Syst. Theory,6, No. 1, 3–15 (1972).

    Google Scholar 

  210. S. S. Wainer, “A classification of the ordinal recursive functions,” Arch. Math. Log. Grundl.,13, Nos. 3–4, 136–153 (1970).

    Google Scholar 

  211. S. S. Wainer, “Ordinal recursion and refinement of the extended Grzegorczyk hierarchy,” J. Symbol. Logic,37, No. 2, 281–292 (1972).

    Google Scholar 

  212. Shigeru Watanabe, “5-symbol 8-state and 5-symbol 6-state universal Turing machines,” J. Assoc. Comput. Machinery,8, No. 4, 476–483 (1961).

    Google Scholar 

  213. S. Winograd, “On the time required to perform addition,” J. Assoc. Comput. Mach.,12, No. 2, 277–285 (1965).

    Google Scholar 

  214. S. Winograd, “On the time required to perform multiplication,” J. Assoc. Comput. Mach.,14, No. 4, 793–802 (1967).

    Google Scholar 

  215. H. Yamada, “Real-time computation and recursive functions not real-time computable,” IRE Trans. Electron. Comput.,11, No. 6, 753–760 (1962).

    Google Scholar 

  216. P. R. Young, “A note on dense and nondense families of complexity classes,” Math. Syst. Theory,5, No. 1, 66–70 (1971).

    Google Scholar 

  217. P. R. Young, “Speed-ups by changing the order in which sets are enumerated,” Math. Syst. Theory,5, 145–156 (1971).

    Google Scholar 

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Translated from Itogi Nauki i Tekhniki. Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 16, pp. 103–149, 1979.

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Marchenkov, S.S., Matrosov, V.L. Complexity of algorithms and computations. J Math Sci 15, 140–165 (1981). https://doi.org/10.1007/BF01084283

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