Structure in Complexity Theory pp 184-195 | Cite as

# Resource-bounded Kolmogorov complexity of hard languages

## Abstract

This paper explores further the relationship between randomness and complexity. The concept of time-/space-bounded Kolmogorov complexity of languages is introduced. Among others we show that there exists a language L in DTIME(2^{2lin}) such that the 2^{poly}-time-bounded Kolmogorov complexity of L is exponential almost everywhere. We study time-/space- bounded Kolmogorov complexity of languages that are DTIME(2^{2lin})-(SPACE (2^{lin})-) hard under polynomial-time Turing reductions. The connection between Kolmogorov-random languages and almost everywhere hard languages is investigated. We also show that Kolmogorov randomness implies Church randomness. Finally, we work out the relationship between time-/space- bounded Kolmogorov complexity and time-/space- bounded descriptional complexity of Boolean circuits and formulas for hard languages. This result provides a classification of exponential-size circuits and formulas in terms of the amount of information contained in them.

## Keywords

Binary String Boolean Formula Kolmogorov Complexity Input Instance Boolean Circuit## Preview

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## References

- [Adlx]Adleman, L.: "Two Theorems on Random Polynomial Time", Proc. 19th FOCS, pp. 75–83, 1978.Google Scholar
- [B-S]Balcázar, J.L. & Schöning, U.: "Bi-immune Sets for Complexity Classes", to appear in Math. Syst. Theor.Google Scholar
- [Ber]Berman, L.: "On the Structure of Complete Sets: Almost Everywhere Complexity And Infinitely Often Speedup", Proc. 17th FOCS, pp. 76–80, 1976.Google Scholar
- [B-H]Berman, L. & Hartmanis, J.: "On Isomorphisms and Density of NP and Other Complete Sets", SIAM J. Comput. 6, pp. 305–322, 1977.CrossRefGoogle Scholar
- [Cha]Chaitin, G.J.: "On the Length of Programs for Computing Finite Binary Sequences", J.ACM 13, pp. 547–569, 1966.Google Scholar
- [Chu]Church, A.: "On the Concept of Random Sequence", Bull. AMS 46, pp. 130–135, 1940.Google Scholar
- [CoS]Cook, S.A.: "The Classification of Problems Which Have Fast Parallel Algorithms", Proc. FCT'83, LNCS 158, pp. 78–93, 1983.Google Scholar
- [DiP]Di Paola, R.A.: "Random Sets in Subrecursive Hierarchies", J. ACM 16, pp. 621–630, 1969.Google Scholar
- [G-J]Garey, M.R. & Johnson, D.S.: "Computers and Intractability", Freeman & Co., 1979.Google Scholar
- [Harx]Hartmanis, J.: "Generalized Kolmogorov Complexity and the Structure of Feasible Computations", Proc. 24th FOCS, pp. 439–445, 1983.Google Scholar
- [H-U]Hopcroft, J.E. & Ullman, J.D.: "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, 1979.Google Scholar
- [Huy]Huynh, D.T.: "Some Observations about the Randomness of Hard Problems", Manuscript, 1985.Google Scholar
- [Kan]Kannan, R.: "Circuit-Size Lower Bounds and Non-reducibility to Sparse Sets", Infor. & Contr. 55, pp. 40–56, 1982.Google Scholar
- [K-L]Karp, R.M. & Lipton, R.J.: "Some Connections between Nonuniform and Uniform Complexity Classes", Proc. 12th STOC, pp. 302–309, 1980.Google Scholar
- [KoK]Ko, K.: "Resource-Bounded Program-Size Complexity and Pseudo-Random Sequences", to appear.Google Scholar
- [Kol]Kolmogorov, A.N.: "Three Approaches for Defining the Concept of Information Quantity", Probl. Inform. Trans. 1, pp. 1–7, 1965.Google Scholar
- [Lad]Ladner, R.E.: "The Circuit Value Problem Is Log-space Complete for P", SIGACT News 7, pp. 18–20, 1975.CrossRefGoogle Scholar
- [Lov]Loveland, D.W.: "A Variant of the Kolmogorov Concept of Complexity", Infor. & Contr. 15, pp. 510–526, 1969.Google Scholar
- [Lyn]Lynch, N.: ‘Log Space Recognition and Translation of Parenthesis Languages", J. ACM 24, pp. 583–590, 1977.CrossRefGoogle Scholar
- [ML1]Martin-Löf, P.: "On the Definition of Random Sequences", Infor. & Contr. 9, pp. 602–619, 1966.Google Scholar
- [ML2]Martin-Löf, P.: "Complexity Oscillations in Infinite Binary Sequences", Z. Wahrsch.-theor. verw. Geb. 19, pp. 225–230, 1971.Google Scholar
- [Meh]Mehlhorn, K.: "Bracket Languages Are Recognizable in Logarithmic Space", TR Univ. Saarlandes, Saarbrücken, 1975.Google Scholar
- [M-M]Meyer, A.R. & McCreight, E.M.: "Computationally Complex and Pseudo-random Zero-one Valued Functions", in "Theory of Machines and Computations", Kohavi and Paz, eds., Academic Press, pp. 19–42, 1971.Google Scholar
- [PSS]Paul, W., Seiferas, J. & Simon, J.: "An Information-theoretic Approach to Time Bounds for On-line Computation", Proc. 12th STOC. pp. 357–367, 1980.Google Scholar
- [Rog]Rogers, H.: "Theory of Recursive Functions and Effective Computability", McGraw-Hill, 1967.Google Scholar
- [Ruz]Ruzzo, W.L.: "On Uniform Circuit Complexity", J. Comput. Syst. Sci. 22, pp. 365–383, 1981.Google Scholar
- [Savx]Savage, J.E.: "The Complexity of Computing", Wiley & Sons, New York, 1976.Google Scholar
- [Sch]Schnorr, C.P.: "Zulfalligkeit und Wahrscheinlichkeit", LNM 218, Springer Verlag, 1971.Google Scholar
- [Sip]Sipser, M.: "A Complexity Theoretic Approach to Randomness", Proc. 15th STOC, pp. 330–335, 1983.Google Scholar
- [Sto]Stockmeyer, L.J.: "The Complexity of Decision Problems in Automata Theory and Logic", MAC TR-133, MIT, 1974.Google Scholar
- [Val]Valiant, L.G.: "Completeness Classes in Algebra", Proc. 11th STOC, pp. 249–261, 1979.Google Scholar
- [Wlb]Wilber, R.E.: "Randomness and the Density of Hard Problems", Proc. 24th FOCS, pp. 335–342, 1983.Google Scholar
- [Wls]Wilson, C.B.: "Relativized Circuit Complexity", Proc. 24th FOCS, pp. 329–334, 1983.Google Scholar
- [Yao]Yao, A.C.: "Theory and Applications of Trapdoor Functions", Proc. 23rd FOCS, pp. 80–91, 1982.Google Scholar
- [Z-L]Zvonkin, A.K. & Levin, L.A.: The Complexity of Finite Objects and the Development of the Concepts of Information and Randomness by Means of the Theory of Algorithms", Russian Math. Survey 25, pp. 83–124, 1970.Google Scholar