# Compressibility and resource bounded measure

## Abstract

We give a new definition of resource bounded measure based on compressibility of infinite binary strings. We prove that the new definition is equivalent to the one commonly used. This new characterization offers us a different way to look at resource bounded measure, shedding more light on the meaning of measure zero results and providing one more tool to prove such results.

The main contribution of the paper is the new definition and the proofs leading to the equivalence result. We then show how this new characterization can be used to prove that the class of linear auto-reducible sets has *p*-measure 0. We also prove that the class of sets that are truth-table reducible to a *p*-selective set has *p*-measure 0 and that the class of sets that Turing reduce to a sub-polynomial dense set has *p*-measure 0. This strengthens various results.

## Keywords

Turing Machine Measure Zero Random Oracle Kolmogorov Complexity Lebesgue Measure Zero## Preview

Unable to display preview. Download preview PDF.

## References

- [AS94a]E. Allender and M. Strauss. Measure on small complexity classes, with applications for BPP. In
*Proc. 35th IEEE Symposium on Foundations of Computer Science*, pages 807–818. IEEE Computer Society Press, 1994.Google Scholar - [AS94b]K. Ambos-Spies. p-mitotic sets. In
*Logic and Machines, Lecture Notes in Computer Science*, volume 177, pages 1–23. Springer-Verlag, 1994.Google Scholar - [ASNT94]K. Ambos-Spies, C. Neis, and S. A. Terwijn. Genericity and measure for exponential time. In
*Proceedings of the 19th Symposium on Mathematical Foundations of Computer Science*, pages 221–232. Springer-Verlag, 1994. To appear in Theoretical Computer Science.Google Scholar - [ASTZ94]K. Ambos-Spies, S.A. Terwijn, and X. Zheng. Genericity and measure for exponential time. In
*Proc. ISAAC94, Lecture Notes in Computer Science*, volume 834, pages 369–377. Springer-Verlag, 1994. To appear in Theoretical Computer Science.Google Scholar - [BG81]C. Bennett and J. Gill. Relative to a random oracle
*A*, P^{A}≠NP^{A}≠ Co-NP^{A}with probability 1.*SIAM J. Comput.*, 10(1):96–113, February 1981.CrossRefGoogle Scholar - [BH95]H. Buhrman and M. Hermo. On the sparse set conjecture for sets with low density. In Ernst W. Mayr and Claude Puech, editors,
*STACS 95*, volume 900 of*Lecture Notes in Computer Science*, pages 609–618, Berlin, 1995. Springer-Verlag.Google Scholar - [BvHT93]H. Buhrman, P. van Helden, and L. Torenvliet. P-selective self-reducible sets: A new characterization of P. In
*Proc. Structure in Complexity Theory eighth annual conference*, pages 44–51. IEEE Computer Society Press, 1993.Google Scholar - [JL95a]D. W. Juedes and J. H. Lutz. The complexity and distribution of hard problems.
*SIAM Journal on Computing*, 24:279–295, 1995.CrossRefGoogle Scholar - [JL95b]D. W. Juedes and J. H. Lutz. Weak compleness in
*E*and*E*_{2}.*Theoretical Computer Science*, 143:149–158, 1995.CrossRefGoogle Scholar - [Lev73]L. Levin. On the notion of a random sequence.
*Soviet Math. Dokl.*, 14:1413–1416, 1973.Google Scholar - [LM94a]J. Lutz and E. Mayordomo. Cook versus Karp-Levin: Separating completeness notions if NP is not small. In
*STACS 1994, Lectures Notes in Computer Science*, pages 415–426. Springer-Verlag, 1994.Google Scholar - [LM94b]J. Lutz and E. Mayordomo. Measure, stochasticity, and the density of hard languages.
*SIAM Journal on Computing*, 23:762–779, 1994.CrossRefGoogle Scholar - [Lut90]J. Lutz. Category and measure in complexity classes.
*Siam J. Computing*, 19(6):1100–1131, December 1990.CrossRefGoogle Scholar - [Lut92]J. Lutz. Almost everywhere high nonuniform complexity.
*J. Computer and System Sciences*, 44:220–258, 1992.CrossRefGoogle Scholar - [Lut94]J. Lutz. Weakly hard problems. In
*Proc. Structure in Complexity Theory ninth annual conference*, pages 146–161. IEEE Computer Society Press, 1994. To appear in SIAM J. on Computing.Google Scholar - [LV93]M. Li and Paul Vitányi.
*An Introduction to Kolmogorov Complexity and its Applications*. Texts and Monographs in Computer Science. Springer-Verlag, 1993.Google Scholar - [May94a]E. Mayordomo. Almost every set in exponential time is
*p*-bi-immune.*Theoretical Computer Science*, 136:487–506, 1994.CrossRefGoogle Scholar - [May94b]E. Mayordomo.
*Contributions to the Study of Resource-Bounded Measure*. PhD thesis, Universitat Politècnica de Catalunya, 1994.Google Scholar - [ML66]P. Martin-Löf. The definition of random sequences.
*Information and Control*, 9:602–619, 1966.Google Scholar - [Sch73]C. Schnorr. Process complexity and effective random tests.
*J. Comput. System Sci.*, 7:376–388, 1973.Google Scholar - [Sel79]A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP.
*Math. Systems Theory*, 13:55–65, 1979.CrossRefGoogle Scholar