\newpage \begin {thebibliography} {99} \bibitem{alg} T.H. Cormen, C.E. Leiserson, R.L. Rivest. Introduction to Algorithms. 1990, McGraw-Hill. \bibitem{Knuth} Donald E. Knuth. {\em The Art of Computer Programming.} Vol.~1-3. Addison-Wesley, 3d ed., 1997. Additions to v.2 can be found in \href {http://www-cs-faculty.stanford.edu/~knuth/err2-2e.ps.gz} {http://www-cs-faculty.stanford.edu/\~{}knuth/err2-2e.ps.gz} . \bibitem{feller} William Feller. {\em An Introduction to Probability Theory and Its Applications.} John Wiley \& Sons, 1968. \bibitem{Lang} S.~Lang. {\em Algebra.} 3rd ed. 1993, Addison-Wesley. \bibitem{rg} H. Rogers, Jr. {\em Theory of Recursive Functions and Effective Computability.} McGraw-Hill Book Co., 1967. \bibitem[]{sec1} References for section~\ref{models}: \bibitem{ba} Ja.M. Barzdin', Ja.Ja. Kalnin's. A Universal Automaton with Variable Structure. {\em Automatic Control and Computing Sciences.} 8(2):6-12, 1974. \bibitem{ww} E.R. Berlekamp, J.H. Conway, R.K. Guy. {\em Winning Ways.} Section 25. Academic Press, 1982. \bibitem{ku} A.N. Kolmogorov, V.A. Uspenskii. On the Definition of an Algorithm. {\em Uspekhi Mat. Nauk} 13:3-28, 1958; AMS Transl.\ 2nd ser.\ 29:217-245, 1963. \bibitem{pm} A. Schoenhage. Storage Modification Machines. {\em SIAM J. on Computing} 9(3):490-508, 1980. \bibitem{ofm} Yu. Ofman. A Universal Automaton. {\em Transactions of the Moscow Math. Society,} pp.200-215, 1965. \bibitem[]{sec2} Section~\ref{diagonal}: \bibitem{spd} M. Blum. A machine-independent theory of the complexity of recursive functions. {\em J. ACM} 14, 1967. \bibitem{und} M. Davis, ed. {\em The Undecidable.} Hewlett, N.Y. Raven Press, 1965.\\ (The reprints of the original papers of K.Goedel, A.Turing, A.Church, E.Post and others). \bibitem{Ikeno} Shinichi Ikeno. A 6-symbol 10-state Universal Turing Machine. {\em Proc.~Inst.~of Elec.~Comm.} Tokyo, 1958. \bibitem{sm} Joel I. Seiferas, Albert R. Meyer. Characterization of Realizable Space Complexities.\\ {\em Annals of Pure and Applied Logic} 73:171-190, 1995. \bibitem{rb} M.O. Rabin. Speed of computation of functions and classification of recursive sets. {\em Third Convention of Sci.Soc.} Israel, 1959, 1-2. Abst.: Bull. of the Research Council of Israel, 8F:69-70, 1959. \bibitem{ts} G.S. Tseitin. Talk: seminar on math. logic, Moscow university, 11/14, 11/21, 1956. Also pp.~44-45 in: S.A. Yanovskaya, Math.~Logic and Foundations of Math., {\em Math.~in the USSR for 40 Years,} 1:13-120, 1959, Moscow, Fizmatgiz, (in Russian). % 90 min TV movie about Alan Turing: "Breaking The Code," 1997, VHS. % http://www.amazon.com/exec/obidos/ASIN/6304361092/ \bibitem[]{sec3} Section~\ref{games}: \bibitem{1} J. v.Neumann, O. Morgenstern. {\em Theory of Games and Economic Behavior}. Princeton Univ. Press, 1944. \bibitem{2} A.K.Chandra and L.J.Stockmeyer, Alternation. {\em FOCS}-1976. \bibitem{4} Ashok K. Chandra, Dexter C. Kozen, Larry J. Stockmeyer. Alternation. {\em J. ACM}, 28(1):114-133, 1981. \bibitem{checkers} J.M. Robson. N by N checkers is EXPTIME-complete. {\em SIAM J. Comput} 13(2), 1984. \bibitem{go} J.M. Robson. The complexity of Go. {\em Proc. 1983 IFIP World Computer Congress}, p.~413-417. \bibitem{chess} A.S. Fraenkel and D. Lichtenstein. Computing a perfect strategy for $n\times n$ chess requires time exponential in $n$. {\em J. Combin. Theory\/} (Ser.~A) 31:199-214. ICALP-1981. \bibitem[]{sec4} Section~\ref{np}: \bibitem{7} W.J. Savitch. Relationships between nondeterministic and deterministic tape complexities. {\em J. Comput. Syst. Sci.} 4:177-190, 1970. \bibitem{YN} D.B. Yudin and A.S. Nemirovsky. Informational Complexity and Effective Methods for Solving Convex Extremum Problems. {\em Economica i Mat. Metody} 12(2):128-142; transl. {\em MatEcon} 13:3-25, 1976. \bibitem{8} E.M. Luks: Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time. {\em FOCS}-1980. \bibitem{GJ} M.R.Garey, D.S.Johnson. {\em Computers and Intractability.} W.H.Freeman \& Co. 1979. \bibitem{trakh} B.A.Trakhtenbrot. A survey of Russian approaches to {\em Perebor} (brute-force search) algorithms. {\em Annals of the History of Computing,} 6(4):384-400, 1984. \bibitem[]{sec5} Section~\ref{rand}: \bibitem{14} M.O.Rabin. Probabilistic Algorithms for Testing Primality. {\em J. Number Theory}, 12: 128-138, 1980. \bibitem{15} G.L.Miller. Riemann's Hypothesis and tests for Primality. {\em J.~Comp.~Sys.~Sci.} 13(3):300-317, 1976. \bibitem{16} R. Solovay, V. Strassen. A fast Monte-Carlo test for primality. {\em SIComp} 6:84-85, 1977. \bibitem{12} R. Karp. Combinatorics, Complexity and Randomness. (Turing Award Lecture) {\em Communication of the ACM}, 29(2):98-109, 1986. \bibitem{17} David S. Johnson. The NP-Completeness Column. {\em J. of Algorithms} 5, (1984) 284-299. \bibitem{karp-pr} R. Karp. The probabilistic analysis of some combinatorial search algorithms. {\em Algorithms and Complexity.} (J.F.Traub, ed.) Academic Press, NY 1976, pp. 1-19. \bibitem{18} Y. Gurevich, Average Case Complexity, Internat. Symp.\ on Information Theory, IEEE, Proc.\ 1985. \bibitem{ku87} A.N.Kolmogorov, V.A.Uspenskii. Algorithms and Randomness. {\em Theoria Veroyatnostey i ee Primeneniya = Theory of Probability and its Applications}, 3(32):389-412, 1987. \bibitem{vitan} M. Li, P.M.B. Vitanyi. Introduction to Kolmogorov Complexity and its Applications. Springer Verlag, New York, 1993. \bibitem{22} M. Blum, S. Micali. How to generate Cryptographically Strong Sequences. {\em SIAM J. Comp.}, 13, 1984. \bibitem{24} A. C. Yao. Theory and Applications of Trapdoor Functions. {\em FOCS}-1982. \bibitem{25} O.Goldreich, L.Levin. A Hard-Core Predicate for all One-Way Functions. {\em STOC}-1989, pp. 25-32. \bibitem{26} R.Rivest, A.Shamir, L.Adleman. A Method for Obtaining Digital Signature and Public-Key Cryptosystems. {\em Comm. ACM}, 21:120-126, 1978. \bibitem{27} M. Blum, S. Goldwasser. An Efficient Probabilistic Encryption Scheme Hiding All Partial Information. {\em Crypto}-1982. \bibitem{29} M. Rabin. {\em Digitalized Signatures as Intractable as Factorization.} MIT/LCS/TR-212, 1979. \bibitem{color} R.Venkatesan, L.Levin. Random Instances of a Graph Coloring Problem are Hard. {\em STOC}-1988. % \bibitem{causal} Peter Gacs, Leonid A. Levin. Causal Nets or What Is % a Deterministic Computation? {\em Information and Control} 51(1), 1981. % \bibitem{min} M.L. Minsky. {\em Computation: Finite and Infinite Machines.} % Prentice-Hall, Englewood Cliffs, N.J., 1967. % \bibitem{Shannon} C.E.Shannon: Automata Studies. 157-165, 1956. %\bibitem{tur} A.M.Turing : Proc. London Math. Soc. Ser. 2, 42, 230-265, 1936. % \bibitem{fct} L.~Levin. On Storage Capacity for Algorithms. % {\em Soviet Math.\ Dokl.} 14(5):1464-1466, 1973. % \bibitem{9} S. Cook. The complexity of theorem proving procedures. % {\em STOC}, 1971. % \bibitem{11} R. Karp. Reducibility among combinatorial problems. % {\em Complexity of Computer Computations.} % R. Miller, J. Thatcher eds. Plenum, NY, 1972, pp. 85-103. % \bibitem{search} L.Levin, Universal Sequential Search Problems, {\em Probl.\ % Pered.\ Inf.\ = Probl.\ Inform.\ Transm.} 9(3), 1973. % \bibitem{19} A.N. Kolmogorov. Three Approaches to the Concept of the Amount % of Information. {\em Probl. Inf. Transm.} 1(1), 1965. % \bibitem{20} P. Martin-Lof. The Definition of Random Sequences. % {\em Inf. and Control} 9:602-619, 1966. % \bibitem{21} R.J. Solomonoff. A Formal Theory of Inductive Inference. % {\em Inf. and Control} 7(1):l-22, 1964. % \bibitem{23} O. Goldreich, S. Goldwasser, S. Micali. % How to Construct Random Functions. {\em J. ACM}, 33(4), 1986. \end {thebibliography} \end {document}