Abstract
It is a fundamental open problem in the randomized computation how to separate different randomized time or randomized small space classes (cf., e.g., [KV 87], [KV 88]). In this paper we study lower space bounds for randomized computation, and prove lower space bounds up to log n for the specific sets computed by the Monte Carlo Turing machines. This enables us for the first time, to separate randomized space classes below log n (cf. [KV 87], [KV 88]), allowing us to separate, say, the randomized space O (1) from the randomized space O (log* n). We prove also lower space bounds up to log log n and log n, respectively, for specific sets computed by probabilistic Turing machines, and one-way probabilistic Turing machines.
Research partially supported by Grant No. 93-599 from the Latvian Council of Science.
Research partially supported by the International Computer Science Institute, Berkeley, California, by the DFG Grant KA 673/4-1, and by the ESPRIT BR Grants 7079 and ECUS030.
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References
Allender, E., Beigel, R., Hertrampf, U., and Homer, S., Almost-Everywhere Complexity Hierarchies for Nondeterministic Time, Manuscript, 1992, A preliminary version has appeared in Proc. STACS'90, LNCS 415, Springer-Verlag, 1990, pp. 1–11
Borodin, A., Cook, S., and Pippenger, N., Parallel computation for wellendowed rings and space-bounded probabilistic machines, Information and Control 58, pp. 113–136.
Bukharaev, R. E., Probabilistic Automata, Kazan University Press, 1977 (Russian).
Dwork, C., and Stockmeyer, L., Interactive proof systems with finite state verifiers, Res. Rep. RJ6262. IBM Research Division, San Jose, Calif., May 1988.
Dwork, C., and Stockmeyer, L., Finite state verifiers I: The power of interaction, Journal of ACM, 39, 4 (Oct. 1992), pp. 800–828.
Freivalds, R., Fast computation by probabilistic Turing machines, Proceedings of Latvian State University, 233 (1975), pp. 201–205 (Russian).
Freivalds, R., Probabilistic machines can use less running time, In: Information Processing'77 (Proc. IFIP Congress'77), North Holland, 1977, pp. 839–842.
Freivalds, R., Speeding up recognition of some sets by usage of random number generators, Problemi kibernetiki, 36 (1979), pp. 209–224 (Russian).
Freivalds, R., Space and reversal complexity of probabilistic one-way Turing machines, Lecture Notes in Computer Science, 158 (1983), pp. 159–170.
Freivalds, R., Space and reversal complexity of probabilistic one-way Turing machines, Annals of Discrete Mathematics, 24 (1985), pp. 39–50.
Gill, J. T., Computational complexity of probabilistic Turing machines, SIAM J. Comput. 6 (1977), pp. 675–694.
Greenberg, A. G. and Weiss, A., A lower bound for probabilistic algorithms for finite state machines, Journal of Computer and System Science, 33 (1986), pp. 88–105.
Kaneps, J., Stochasticity of languages recognized by two-way finite probabilistic automata, Diskretnaya matematika, 1 (1989), pp. 63–77 (Russian).
Kaneps, J. and Freivalds, R., Minimal nontrivial space complexity of probabilistic one-way Turing machines, Lecture Notes in Computer Science, Springer, 452 (1990), pp. 355–361.
Karp, R. M., An Introduction to Randomized Algorithms, Technical Report TR-90-029, International Computer Science Institute, Berkeley, 1990.
Karpinski, M., and Verbeek, R., On the Monte Carlo space constructible functions and separation results for probabilistic complexity classes, Information and Computation, 75 (1987), pp. 178–189.
Karpinski, M., and Verbeek, R., Randomness, Probability, and the Separartion of Monte Carlo Time and Space, LNCS 270, Springer-Verlag, 1988, pp. 189–207.
Karpinski, M., and Verbeek, R., On Randomized versus Deterministic Computation, Proc. ICALP '93, LNCS 700 (1993), Springer-Verlag, pp. 227–240.
Kemeny, J. G., and Snell, J. L., Finite Markov Chains, Van Nostrand, 1960.
Leighton, F. T., and Rivest, R. L., The Markov Chain Tree Theorem, Rep. MIT/LCS, TM-249, Laboratory of Computer Science, MIT, Cambridge, Mass., 1983.
Leighton, F. T., and Rivest, R. L., Estimating a Probability Using Finite Memory, IEEE Trans. Inf. Theory IT-32 (1986), pp. 733–742.
Nasu, M. and Honda, N., A context-free language which is not acceptable by a probabilistic automaton, Information and Control, 18 (1971), pp. 233–236.
Paz, A., Introduction to Probabilistic Automata, Academic Press, 1971.
Rabin, M. O., Probabilistic automata, Information and Control, 6 (1963), pp. 230–245.
Seneta, E., Non-Negative Matrices and Markov Chains, 2nd Edition, Springer-Verlag, New York, 1981.
Trakhtenbrot, B. A., Notes on the complexity of computation by probabilistic machines, In: Theory of Algorithms and Mathematical Logics, VC AN SSSR, 1974, pp. 159–176 (Russian).
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Freivalds, R., Karpinski, M. (1994). Lower space bounds for randomized computation. In: Abiteboul, S., Shamir, E. (eds) Automata, Languages and Programming. ICALP 1994. Lecture Notes in Computer Science, vol 820. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58201-0_100
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DOI: https://doi.org/10.1007/3-540-58201-0_100
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