Abstract
Glaßer et al. (SIAMJCOMP 2008 and TCS 2009 (The two papers have slightly different sets of authors)) proved existence of two sparse sets A and B in EXP, where A is 3-tt (truth-table) polynomial-time autoreducible but not weakly polynomial-time Turing mitotic and B is polynomial-time 2-tt autoreducible but not weakly polynomial-time 2-tt mitotic. We unify and strengthen both of those results by showing that there is a sparse set in EXP that is polynomial-time 2-tt autoreducible but not even weakly polynomial-time Turing mitotic. All these results indicate that polynomial-time autoreducibilities in general do not imply polynomial-time mitoticity at all with the only exceptions of the many-one and 1-tt reductions. On the other hand, however, we proved that every autoreducible set for the polynomial-time bounded disjunctive or conjunctive tt reductions is weakly mitotic for the polynomial-time tt reduction that makes logarithmically many queries only. This shows that autoreducible sets for reductions making more than one query could still be mitotic in some way if they possess certain special properties.
L. Zhang—Research supported in part by NSF CCF grant 1218093.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ambos-Spies, K.: On the structure of the polynomial time degrees of recursive sets. Habilitationsschrift, Zur Erlangung der Venia Legendi Für das Fach Informatik an der Abteilung Informatik der Universität Dortmund, September 1984
Buhrman, H., Fortnow, L., van Melkebeek, D., Torenvliet, L.: Using autoreducibility to separate complexity classes. SIAM J. Comput. 29(5), 1497–1520 (2000)
Buhrman, H., Torenvliet, L.: A Post’s program for complexity theory. Bull. EATCS 85, 41–51 (2005)
Glaßer, C., Selman, A., Travers, S., Zhang, L.: Non-mitotic sets. Theoret. Comput. Sci. 410(21–23), 2011–2033 (2009)
Glaßer, C., Nguyen, D.T., Reitwießner, C., Selman, A.L., Witek, M.: Autoreducibility of complete sets for log-space and polynomial-time reductions. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7965, pp. 473–484. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39206-1_40
Glaßer, C., Nguyen, D.T., Selman, A.L., Witek, M.: Introduction to autoreducibility and mitoticity. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds.) Computability and Complexity. LNCS, vol. 10010, pp. 56–78. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-50062-1_5
Glaßer, C., Ogihara, M., Pavan, A., Selman, A., Zhang, L.: Autoreducibility and mitoticity. ACM SIGACT News 40(3), 60–76 (2009)
Glaßer, C., Ogihara, M., Pavan, A., Selman, A.L., Zhang, L.: Autoreducibility, mitoticity, and immunity. J. Comput. Syst. Sci. 73, 735–754 (2007)
Glaßer, C., Pavan, A., Selman, A., Zhang, L.: Splitting NP-complete sets. SIAM J. Comput. 37(5), 1517–1535 (2008)
Hemaspaandra, L., Ogihara, M.: The Complexity Theory Companion. Springer, Heidelberg (2002). https://doi.org/10.1007/978-3-662-04880-1
Homer, S., Selman, A.: Computability and Complexity Theory. Texts in Computer Science, 2nd edn. Springer, New York (2011). https://doi.org/10.1007/978-1-4614-0682-2
Trakhtenbrot, B.: On autoreducibility. Dokl. Akad. Nauk SSSR 192(6), 1224–1227 (1970). Transl. Soviet Math. Dokl. 11(3), 814–817 (1790)
Yao, A.: Coherent functions and program checkers. In: Proceedings of the 22nd Annual Symposium on Theory of Computing, pp. 89–94 (1990)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Zhang, L., Quweider, M., Lei, H., Khan, F. (2018). Weak Mitoticity of Bounded Disjunctive and Conjunctive Truth-Table Autoreducible Sets. In: Wang, L., Zhu, D. (eds) Computing and Combinatorics. COCOON 2018. Lecture Notes in Computer Science(), vol 10976. Springer, Cham. https://doi.org/10.1007/978-3-319-94776-1_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-94776-1_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94775-4
Online ISBN: 978-3-319-94776-1
eBook Packages: Computer ScienceComputer Science (R0)