Abstract
We show the following results regarding complete sets.
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NP-complete sets and PSPACE-complete sets are many-one autoreducible.
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Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are many-one autoreducible.
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EXP-complete sets are many-one mitotic.
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NEXP-complete sets are weakly many-one mitotic.
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PSPACE-complete sets are weakly Turing-mitotic.
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If one-way permutations and quick pseudo-random generators exist, then NP-complete languages are m-mitotic.
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If there is a tally language in NP ∩ coNP - P, then, for every ε > 0, NP-complete sets are not 2n(1 + ε)-immune.
These results solve several of the open questions raised by Buhrman and Torenvliet in their 1994 survey paper on the structure of complete sets.
A full version of this paper is available as ECCC Technical Report TR05-011.
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References
Agrawal, M.: Pseudo-random generators and structure of complete degrees. In: 17th Annual IEEE Conference on Computational Complexity, pp. 139–145 (2002)
Ambos-Spies, K.: P-mitotic sets. In: Börger, E., Hasenjäger, G., Roding, D. (eds.) Logic and Machines. LNCS, vol. 177, pp. 1–23. Springer, Heidelberg (1984)
Ambos-Spies, K., Fleischhack, H., Huwig, H.: Diagonalizations over polynomial time computable sets. Theoretical Computer Science 51, 177–204 (1987)
Ambos-Spies, K., Neis, H., Terwijn, A.: Genericity and measure for exponential time. Theoretical Computer Science 168(1), 3–19 (1996)
Balcazar, J., Mayordomo, E.: A note on genericty and bi-immunity. In: Proceedings of the Tenth Annual IEEE Conference on Computational Complexity, pp. 193–196 (1995)
Balcázar, J., Schöning, U.: Bi-immune sets for complexity classes. Mathematical Systems Theory 18(1), 1–18 (1985)
Beigel, R., Feigenbaum, J.: On being incoherent without being very hard. Computational Complexity 2, 1–17 (1992)
Beigel, R., Gill, J.: Counting classes: Thresholds, parity, mods, and fewness. Theoretical Computer Science 103, 3–23 (1992)
Berman, L.: Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University, Ithaca, NY (1977)
Buhrman, H., Fortnow, L., van Melkebeek, D., Torenvliet, L.: Using autoreducibility to separate complexity classes. SIAM Journal on Computing 29(5), 1497–1520 (2000)
Buhrman, H., Hoene, A., Torenvliet, L.: Splittings, robustness, and structure of complete sets. SIAM Journal on Computing 27, 637–653 (1998)
Buhrman, H., Torenvliet, L.: On the structure of complete sets. In: Proceedings 9th Structure in Complexity Theory, pp. 118–133 (1994)
Buhrman, H., Torenvliet, L.: Complete sets and structure in subrecursive classes. In: Proceedings of Logic Colloquium 1996, pp. 45–78. Springer, Heidelberg (1998)
Buhrman, H., Torenvliet, L.: Separating complexity classes using structural properties. In: Proceedings of the 19th IEEE Conference on Computational Complexity, pp. 130–138 (2004)
Buhrman, H., Torenvliet, L.: A Post’s program for complexity theory. Bulleting of the EATCS 85, 41–51 (2005)
Ganesan, K., Homer, S.: Complete problems and strong polynomial reducibilities. SIAM Journal on Computing 21, 733–742 (1992)
Glaßer, C., Pavan, A., Selman, A., Sengupta, S.: Properties of NP-complete sets. In: Proceedings of the 19th Annual IEEE Conference on Computational Complexity, pp. 184–197 (2004)
Hertrampf, U., Lautemann, C., Schwentick, T., Vollmer, H., Wagner, K.W.: On the power of polynomial time bit-reductions. In: Proceedings 8th Structure in Complexity Theory, pp. 200–207 (1993)
Homer, S., Selman, A.: Computability and Complexity Theory. In: Texts in Computer Science, Springer, New York (2001)
Juedes, D.W., Lutz, J.H.: The complexity and distribution of hard problems. SIAM Joutnal on Computing 24, 279–295 (1995)
Ladner, R.: Mitotic recursively enumerable sets. Journal of Symbolic Logic 38(2), 199–211 (1973)
Ogiwara, M., Watanabe, O.: On polynomial-time bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal of Computing 20(3), 471–483 (1991)
Pavan, A., Selman, A.: Separation of NP-completeness notions. SIAM Journal on Computing 31(3), 906–918 (2002)
Schöning, U.: Probabilistic complexity classes and lowness. Journal of Computer and System Sciences 39, 84–100 (1989)
Stockmeyer, L.: The polynomial-time hierarchy. Theoretical Computer Science 3, 1–22 (1977)
Trahtenbrot, B.: On autoreducibility. Dokl. Akad. Nauk SSSR, 192 (1970); Translation in Soviet Math. Dokl. 11, 814–817 (1970)
Tran, N.: On P-immunity of nondeterministic complete sets. In: Proceedings of the 10th Annual Conference on Structure in Complexity Theory, pp. 262–263. IEEE Computer Society Press, Los Alamitos (1995)
Wagner, K.W.: Leaf language classes. In: Margenstern, M. (ed.) MCU 2004. LNCS, vol. 3354, pp. 60–81. Springer, Heidelberg (2004)
Wrathall, C.: Complete sets and the polynomial-time hierarchy. Theoretical Computer Science 3, 23–33 (1977)
Yao, A.: Coherent functions and program checkers. In: Proceedings of the 22nd Annual Symposium on Theory of Computing, pp. 89–94 (1990)
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Glaßer, C., Ogihara, M., Pavan, A., Selman, A.L., Zhang, L. (2005). Autoreducibility, Mitoticity, and Immunity. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_34
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DOI: https://doi.org/10.1007/11549345_34
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