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Academic Commons Search Resultsen-usOn the Power of Probabilistic Polynomial Time:
PNP[log] ⊆ PP
https://academiccommons.columbia.edu/catalog/ac:142778
Hemachandra, Lane A.; Wechsung, Gerd10.7916/D83X8FRJWed, 21 Dec 2011 14:11:31 +0000We show that every set in the ΘP2 level of the polynomial hierarchy -- that is, every set polynomial-time truth-table reducible to SAT -- is accepted by a probabilistic polynomialtime Turing machine: PNP[log] ⊆ PP.Computer scienceComputer ScienceReportsOn the Structure of Solutions of Computable Real Functions
https://academiccommons.columbia.edu/catalog/ac:142682
Hartmanis, Juris; Hemachandra, Lane A.10.7916/D8BK1MFNSat, 17 Dec 2011 05:14:37 +0000The relationship between the structure of a domain and the complexity of computing over that domain is a fundamental question of computer science. This paper studies how the structure of the real numbers constrains the behavior of computable real functions. In particular, we uncover a close correlation between the structure of the zero set of a computable real function, and the complexity of the zeros. We show that computable real functions with hard solutions perforce have many solutions. Furthermore, as the complexity of solutions increases, the number of solutions increases. We prove that computable real functions with nonrecursive, nonarithmetical, or random zeros have solution sets that are, respectively, infinite,“˜ uncountable, or of positive measure. In addition, we show that the computational complexity of the zero set of a computable real function is limited by its topological complexity. These results suggest an emerging paradigm-the inability of machines to name complex strings can serve as the basis of powerful proof techniques in computational complexity theory.Computer scienceComputer ScienceReportsStructure of Complexity Classes: Separations, Collapses, and Completeness
https://academiccommons.columbia.edu/catalog/ac:142679
Hemachandra, Lane A.10.7916/D86W9K5JSat, 17 Dec 2011 04:57:41 +0000During the last few years, unprecedented programs has been made in structural complexity theory; class inclusions and relativised separations were discovered, and hierarchies collapsed. We survey this progress, highlighting the central role of counting techniques. We also present a new result whose proof demonstrate the power of combinational arguments: there is a relativezed world in which UP has no Turing complete sets.Computer scienceComputer ScienceReports