TY - JOUR
T1 - On characterizations of randomized computation using plain Kolmogorov complexity
AU - Hirahara, Shuichi
AU - Kawamura, Akitoshi
N1 - Funding Information:
The authors thank Eric Allender for introducing the second author to the relevant line of research and thank anonymous reviewers of Computability for helpful comments. They are also grateful for the advice and suggestions provided by Hiroshi Imai and the members of his group at the University of Tokyo. This work was supported in part by the Kayamori Foundation of Informational Science Advancement and the KAKENHI projects 24106002 and 26700001.
PY - 2018
Y1 - 2018
N2 - Allender, Friedman, and Gasarch recently proved an upper bound of PSPACE for the class DTTR K of decidable languages that are polynomial-time truth-table reducible to the set of prefix-free Kolmogorov-random strings regardless of the universal machine used in the definition of Kolmogorov complexity. It is conjectured that DTTRK in fact lies closer to BPP, a lower bound established earlier by Buhrman, Fortnow, Koucký, and Loff. It is also conjectured that we have similar bounds for the analogous class DTTRC defined by plain Kolmogorov randomness. In this paper, we provide further evidence for these conjectures. First, we show that the time-bounded analogue of DTTR C sits between BPP and PSPACE ∩ P/poly. Next, we show that the class DTTRC,α obtained from DTTRC by imposing a super-constant minimum query length restriction on the reduction lies between BPP and PSPACE. Finally, we show that the class P/RCt=log obtained by further restricting the reduction to ask queries of logarithmic length lies between BPP and ∑2p ∩ P/poly.
AB - Allender, Friedman, and Gasarch recently proved an upper bound of PSPACE for the class DTTR K of decidable languages that are polynomial-time truth-table reducible to the set of prefix-free Kolmogorov-random strings regardless of the universal machine used in the definition of Kolmogorov complexity. It is conjectured that DTTRK in fact lies closer to BPP, a lower bound established earlier by Buhrman, Fortnow, Koucký, and Loff. It is also conjectured that we have similar bounds for the analogous class DTTRC defined by plain Kolmogorov randomness. In this paper, we provide further evidence for these conjectures. First, we show that the time-bounded analogue of DTTR C sits between BPP and PSPACE ∩ P/poly. Next, we show that the class DTTRC,α obtained from DTTRC by imposing a super-constant minimum query length restriction on the reduction lies between BPP and PSPACE. Finally, we show that the class P/RCt=log obtained by further restricting the reduction to ask queries of logarithmic length lies between BPP and ∑2p ∩ P/poly.
UR - http://www.scopus.com/inward/record.url?scp=85048092979&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85048092979&partnerID=8YFLogxK
U2 - 10.3233/COM-170075
DO - 10.3233/COM-170075
M3 - Article
AN - SCOPUS:85048092979
SN - 2211-3568
VL - 7
SP - 45
EP - 56
JO - Computability
JF - Computability
IS - 1
ER -